Computational Analysis of Biochemical Networks For
Drug Target Identification and Therapeutic
Intervention Design
by
MASSCHUSETTS INSW1 E
OF TECHNOLOGY
JUN 18 2014
Nirmala Paudel
MEng, University of Oxford (2009)
LIBRARIES
Submitted to the Department of Biological Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
© Massachusetts Institute of Technology 2014. All rights reserved.
A uthor ........................
Signature redacted
Department of Biological Engineering
May 22, 2014
Certified by ..................
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'I
Bruce Tidor
Professor of Biological Engineering and Computer Sciences
,rhesis Supervisor
Accepted by....................Signature
redacted..........
Forest M. White
Associate Professor of Biological Engineering
Chairman, Department Committee on Graduate Theses
Thesis Committee
Accepted
bySignature
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K. Dane Wittrup
C.P. Dubbs Professor of Chemical and Biological Engineering
Chairman, Thesis Committee
Accepted by
.................
....
Bruce Tidor
Professor of Biological Engineering and Computer Science
Signature redacted
A ccepted by ...
Thesis Supervisor
.......................
Domitilla Del Vecchio
W.M. Keck Career Development Professor in Biomedical Engineering
Member, Thesis Committee
Computational Analysis of Biochemical Networks For Drug
Target Identification and Therapeutic Intervention Design
by
Nirmala Paudel
Submitted to the Department of Biological Engineering
on May 22, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Identification of effective drug targets to intervene, either as single agent therapy or in combination, is a critical question in drug development. As complexity of disease like cancer
is revealed, it has become clear that a holistic network approach is needed to identify drug
targets that are specially positioned to provide desired leverage on disease phenotypes. In
this thesis we develop a computational framework to exhaustively evaluate target behaviors
in biochemical network, either as single agent or combination therapies. We present our
single target therapy work as a problem of identifying good places to intervene in a network.
We quantify a relationship between how interventions at different places in network affect an
output of interest. We use this quantitative relationship between target inhibited and output
of interest as a metric to compare targets. In network analyzed here, most targets show a
sub-linear behavior where a large percentage of targeted molecule needs to be inhibited to
see a small change on output. The other key observation is that targets at the top of the network exerted relatively small control compared to the targets at the bottom of the network.
In the combination therapy work we study how combination of drug concentrations affect
network output of interest compared to when one of the drugs was given alone at equivalent
concentrations. By adapting the definitions of additive, synergistic, and antagonistic combination behaviors developed by Ting Chao-Chou (Chou TC, Talalay P (1984), Advances
in enzyme regulation 22: 27-55) for our system and systematically perturbing biochemical
pathway, we explore the range of combination behaviors for all plausible combination targets. This holistic approach reveals that most target combinations show additive behaviors.
Synergistic, and antagonistic behaviors are rare. Even when combinations are classified as
synergistic or antagonistic, they show this behavior only in a small range of the inhibitor
concentrations. This work is developed in a particular variant of the epidermal growth factor (EGF) receptor pathway for which a detailed mathematical model was first proposed
by Schoeberl et al. Computational framework developed in this work is applicable to any
biochemical network.
Thesis Supervisor: Bruce Tidor
Title: Professor of Biological Engineering and Computer Sciences
Acknowledgments
I would like to thank my advisor, Bruce Tidor, for all his support and guidance during
my time here at MIT. I particularly appreciate his role in helping me develop an insight
into rigorous, hypothesis driven research with a strong emphasis on principled execution of
scientific methods and effective communication of scientific findings.
I am equally grateful to all the members of the Tidor lab that I have had the privilege of
interacting with during my time here. A sincere thanks to David Hagen, Ishan Patel, Andrew
Horning, David Flowers, Raja Srinivas, Brian Bonk, Kevin Shi, Nate Silver, Gil Kwak,
Pradeep Ravindranath, Devanathan Raghunathan, Sudipta Samanta, and Sarah Guthrie
for helpful scientific (and others) discussions and feedbacks. David Hagen deserves a special
mention for maintaining the KorneckerBio toolbox and helping me understand mathematical
formulations behind it in my early days in the lab. I would particularly like to thank Tina
Toni, Yang Shen, Yuanyuan Cui, and Filipe Gracio for their friendship, encouragement, and
support. I would also like to thank Nira Manokharan, Tidor lab administrator, for friendly
chats, and stocked up stationery cupboard and tea counter.
I appreciate the contributions of my thesis committee members, Dane Wittrup and Domitilla Del Vecchio, for their guidance and support. They have played an important role in
encouraging me to ask important research questions and have provided helpful feedback
during committee meetings and beyond.
I would like to take this opportunity to specially mention two organizations, without
whose support my academic journey would not have come this far. PestalozziWorld and
Pestalozzi International Village Trust changed the course of my life by financially supporting
my education from primary school right up to undergraduate level. I will forever be indebted
to them for this unparalleled opportunity.
Finally, I would like to thank my family for their love, support, and encouragement. The
trust they have bestowed on me and the freedom they have provided me from an early age
have been crucial in my exploration of opportunities across continents. I would also like to
thank Suresh Sitaula for his support and encouragement.
Contents
1
. . . . . . . . . . . . . . . . . . . . . . . . . . .
10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Epidermal Growth Factor (EGF) Receptor Pathway . . . . . . . . . .
14
Computational Modeling of Biochemical Pathways . . . . . . . . . . . . . . .
15
. . . . . . . . . . . . . . .
17
. . . . . . . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.1
Background and Motivation
1.2
Biochemical Pathways
1.2.1
1.3
1.4
2
10
Introduction
1.3.1
Computational Models of EGFR Pathway
1.3.2
Variant Models of EGFR Pathway
Structure of This Thesis
A Framework for Evaluating Efficacies of Single Agent Therapy
2.1
2.2
2.3
24
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.1.1
The Biochemical Model
. . . . . . . . . . . . . . . . . . . . . . . . .
26
2.1.2
M odel Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.3
Format of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.4
Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
M ethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
The Normal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.2
Cancer Variant Models . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.2.3
Drug Intervention Models
. . . . . . . . . . . . . . . . . . . . . . . .
29
2.2.4
Target and Output Effect Metrics . . . . . . . . . . . . . . . . . . . .
32
2.2.5
Signal Transduction between MEK and ERK . . . . . . . . . . . . . .
32
2.2.6
Parameter Variability Study . . . . . . . . . . . . . . . . . . . . . . .
33
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5
2.4
3
2.3.1
Intervention-free Models
. . . . . . . . . . . . . . . . . . . . . . . . .
34
2.3.2
Intervention Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.3.3
Normal Model -
. . . . . . . . . . . . . . . . .
39
2.3.4
Cancer Variant Models . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.3.5
Signal Transduction Between MEK and ERK
. . . . . . . . . . . . .
42
2.3.6
Parameter Variability Analysis . . . . . . . . . . . . . . . . . . . . . .
44
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Target Comparisons
Computational Approach to Analyze Drug Combination for Synergy and
51
Antagonism
3.1
3.2
3.3
3.4
4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.1.1
Biochemical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.1.2
Format of Study
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.1.3
Summary of Results
. . . . . . . . . . . . . . . . . . . . . . . . . . .
56
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.2.1
Combination Behavior Definitions . . . . . . . . . . . . . . . . . . . .
57
3.2.2
Drug Intervention Models
. . . . . . . . . . . . . . . . . . . . . . . .
61
3.2.3
Target and Output Effect Metrics . . . . . . . . . . . . . . . . . . . .
63
3.2.4
Combination Summary Metric . . . . . . . . . . . . . . . . . . . . . .
64
3.2.5
Parameter Variability Analysis . . . . . . . . . . . . . . . . . . . . . .
65
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.3.1
General Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.3.2
Additive Targets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.3.3
Synergistic Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.3.4
Antagonistic Targets . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.3.5
Parameter Variability Analysis . . . . . . . . . . . . . . . . . . . . . .
75
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
M ethod
Therapeutic Design Strategies for Safety and Efficacy
80
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.2
Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
6
5
4.2.1
Model Details and Setup . . . . . . . . . . . . . . . . . . . . . . . .
82
4.2.2
O bjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.2.3
Design of Intervention Strategies and Optimization Framework . . .
84
4.2.4
Targets D esign
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.3
Analysis of the Optimized Designs . . . . . . . . . . . . . . . . . . . . . . .
88
4.4
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . .. .
. . .. .
92
Summary and Future Directions
105
A Single Target Intervention
A.1
Target Effect Metric
91
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
A.1.1
Single Substrate Enzymatic Reactions
. . . . . . . . . . . . . . . .
105
A.1.2
Two or More Substrate Enzymatic Reactions . . . . . . . . . . . . .
107
A.2
Biochemical Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
A.3
Molecular Identities of Targets Inhibited
. . . . . . . . . . . . . . . . . . .
122
129
B Combination Target Intervention
B.1
Molecular Targets Inhibited
. . . . . . . . . . . . . . . . . . . . . . . . . .
135
C Effects Exerted by Interventions
C.1
129
Mathematical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
C.1.1
Kinetically-Tuned Inhibitors . . . . . . . . . . . . . . . . . . . . . .
135
C.1.2
Feedback and Feed-Forward Loops
. . . . . . . . . . . . . . . . . .
136
7
List of Figures
1-1
Schematic of epidermal growth factor receptor (EGFR) pathway . . . . . . .
1-2
Definitions of normal and cancer phenotypes in terms of ERK-pp signal dy-
19
n am ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
. . . . . . . . . . . . . . . . . .
31
2-1
Overview of single target evaluation method
2-2
Schematic summary of three overstimulated (cancer) variants of EGFR pathway 35
2-3
Overview of signal propagation dynamics in EGFR models .
37
2-4
Experimental comparison of target inhibition behavior
. . .
39
2-5
Representative range of target behaviors in the normal and overstimulated
. . . . . . . . . . . . . . . . . . . . . . . .
43
2-6
Signal transduction dynamics between MEK-pp and ERK-pp
44
2-7
Effects of parameter variability on target behaviors
. . . . .
46
3-1
Overview of target combination evaluation method
. . . . .
59
3-2
Definitions of combination target behaviors . . . . . . . . . .
60
3-3
Schematic representation of combination metric
. . . . . . .
66
3-4
Summary of all combination behaviors in EGFR pathway . .
69
3-5
Representative combinations showing additive and synergistic targets in EGFR
EGFR pathways
p athw ay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3-6
Representative combinations showing antagonistic behaviors in EGFR pathway 74
3-7
Model ensemble behaviors of combination targets
. . . . . . . . . . . . . . .
75
3-8
Combination behaviors in over-stimulated models . . . . . . . . . . . . . . .
76
4-1
Objectives for therapeutic designs . . . . . . . . . . . . . . . . . . . . . . . .
84
8
4-2
Schematics of therapeutic design strategies . . . . . . . . . . . . . . . . . . .
85
4-3
Optimized designs for normal trajectory objective . . . . . . . . . . . . . . .
89
4-4
Optimized designs for signal block objective
. . . . . . . . . . . . . . . . . .
90
9
Chapter 1
Introduction
1.1
Background and Motivation
Biology is increasingly turning into a quantitative science with advancements in experimental
measurement techniques. The precision and the breadth with which cellular and molecular
events in cells and tissues can be measured mean a detailed picture of the inner working
principles of a cell, either in isolation or as a member of tissue, is emerging rapidly. Central to
this endeavor is a quest to elucidate and understand the networks (be it at genetic, protein, or
metabolic levels) that control cellular decisions and phenotypes both in normal physiological
conditions and in pathology.
In parallel, mathematical and computational methods are
being developed to calibrate these networks based on the experimental measurements both
at single-cell and population levels. One of the major goals of this endeavor is to use these
quantitative models to make useful clinical predictions that can aid in drug design and
discovery processes.
The goal of this thesis research is to develop methods to aid in drug design and discovery
based on knowledge of biochemical networks as it becomes available. There are two questions
that we address in this work. The first is "What makes a good target?" We develop and
use holistic network-level strategies to find drug targets, either for single agent therapy or
in combination, that are especially well positioned to exert suitably large effects on the
very pathway the drug is trying to alter. This approach ties in with the popular idea of
'druggability of the target' that is well established in the drug design world and literature
10
[61].
The question of 'druggability of a target' has, so far, been looked at mainly from
a molecular design perspective [22].
The question has been primarily interpreted as ease
or difficulty with which a small molecule inhibitor can be designed to block a target of
interest.
Here, we take a step back and look at the problem from a network level, which
adds an extra dimension to the problem and consequently its solution. The aim here is not
to replace the idea of 'druggability' as it exists but to complement it with a holistic view
of signal propagation in network of interest. The question we ask is, given the network of
interactions required for a particular cellular phenotype, what are the most suitable targets
in the pathway that are likely to produce desirable change at the output or output signature
of interest.
This framing differentiates the target at which drug acts from the output of
interest.
The motivation for this question comes from the fact that a 'druggable' target in the
traditional sense has little meaning if intervening at this target does not affect the phenotype
of interest. On the other hand, an intervention molecule that is difficult to design precisely
at molecular level and hence only affects the target minimally, can still have a desired effect
on the output of interest given the non-obvious way in which this effect propagates down the
signaling cascade. This question would have been redundant if all biological networks had a
single path for signal propagation with linear dynamics. In such a case, it would not really
matter where one decides to intervene. A proportional effect would be observed at output.
However, given the highly non-linear and branched nature of biological networks [44, 40, 102],
we believe that this question can complement well the idea of 'druggability of the target' in
designing suitable intervention molecules for suitable targets. It is important to point out
that 'target identification' is a well established concept within drug development [19, 92].
But the question is framed differently. Most of the efforts so far has been to find mutated
protein (resulting from mutated genes) [27, 80, 99, 29] that leads to the deregulation of the
pathway activity and propose it as a drug target. Though this approach has been successful
to an extent, it is our belief that a holistic network-based method, as we propose here, leads
to an unbiased and potentially more fruitful approach to target selection. Mutated proteins
of a particular pathway might not always be the suitable weak points that we might want
to exploit in the design endeavor.
However, if they truly are important nodes for drug
11
interventions, our analysis has a potential to identify those without any bias.
The second question that this thesis explores is intervention strategies that actively consider the multi-factorial, and often conflicting, objectives that drugs would ideally meet.
The most common objective is selectivity, not just in terms of the molecules they target
but also in terms of targeting only a fraction of the cells or tissues that are deregulated
without affecting the ones in the vicinity that are functioning normally. An efficient way
of doing this would be to target molecules specific to the pathology that are not present
in normal cells.
Despite immense effort in finding disease-specific molecules that can be
targeted for interventions [21, 115], the picture that is emerging is that, in complex diseases
like cancer, normal and diseased cells contain the same or very similar molecules, albeit in
different quantities and functioning slightly differently [46, 106].
Hence, attempts to find
specific disease markers for targeting have met with limited success. To complement this
process, we take a systems approach and probe this question by considering classic, simple
engineering design principles to explore the capabilities they provide in differentially regulating the normal and disease phenotypes. The major motivation for this comes from the need
to include adverse effects of drug as an active part of the design process. Drug side effects,
such as those arising from chemotherapies in the case of cancer, not only limit the amount
of drug that can be given to the patients but also affect greatly the quality of life of patients.
An ability to address this aspect right at the design level has the potential to help identify
intervention strategies with fewer side effects. While this kind of designs are not quite as
common in drug development industry yet, emergence of field of synthetic biology with a
goal of forward engineering biological components from known part is heading towards that
directions. In this light, we computationally explore design principles that are little more
complex than simple inhibitors that may also provide higher level of flexibility in terms of
their design specifications.
In summary, we explore capabilities of three kinds of intervention strategies to evaluate
their potentials and limitations in achieving two multi-factorial design objectives. The first
objective is formulated to make both disease (cancer) and normal cells signal as normal cells.
The second objective is formulated as a little more challenging problem of allowing normal
cells to function as normal, but blocking signaling in disease (cancer) cells thus halting
12
their growth and proliferation. An important question that this part of thesis tries to raise
is that, given the challenges posed by the design goals, can simple inhibitor molecules that
regulate biology in almost boolean-logic like fashion (i.e., "on" and "off') meet multi-factorial
objectives of minimizing the side effects and maximizing the efficacy of drugs, or are there
inherent limitations in the designs and hence the need to think beyond inhibitors to more
modular intervention strategies with advanced logical functionality?
Combined together, research work that is described in this thesis uses network level information of biochemical pathways to study where the better or worse drug targets are, where
beneficial combination targets are, and what kind of design strategies need to be adopted
and explored in order to design drugs that maximize the therapeutic window by actively
maximizing for efficacy and minimizing for toxicity. Both computational and experimental
approaches can be taken to evaluate these questions. In this thesis we use a computational
framework. This allows us to look at the trends much more exhaustively and evaluate how
the results depend on uncertainty about biochemical reactions in signaling pathways.
1.2
Biochemical Pathways
Cells, either in isolation or as a member of a tissue organization, can carry out large array of
computations and functions. Most of these functions are achieved by interaction networks of
proteins. These interaction networks can exist at genetic, proteomic, or metabolic levels. In
genetic level networks, proteins or protein complexes are used to control when one or more
genes are activated or inactivated. Most of the protein-protein interaction networks are used
to transmit or integrate information from one part of the cell to another. At the metabolic
level, proteins are used either to break down molecules from food, like sugar or fat, to energy
units (adenosine triphosphates - ATPs) that cells can use or to utilize the resources available
to synthesize macromolecules and other biomolecules that are needed for the cells to grow or
divide properly. In this viewpoint almost all the cellular processes that range from sensing
the environment, integrating large array of information, transmitting information from one
part or compartment to another, making a cellular phenotypic decision and executing it is,
dependent on the interaction networks that exist within cell.
13
Understanding the network based interactions of proteins in normal cells should give a
complete picture of how a cell functions. Study of these interactions in pathology should
provide a guide to understand how these interactions are deregulated, paving ways to rationally counteract these deregulations. However, given the complexity of studying all the
interactions in a cell as whole, the task is made more tractable by studying interactions of
small pathways present within larger networks in isolation. The hope is that detailed understanding of the working principles of each pathway can be put together to understand the
system as a whole. More importantly studying and understanding the currently tractable
subsets of biochemical networks can give actionable insights that can be used to prolong life,
or reduce the discomfort that result from disease process. For these actionable goals, we have
to make decisions in the face of uncertainty. Understanding of small subset of interactions
does reduce this uncertainty though cannot completely eliminate it.
1.2.1
Epidermal Growth Factor (EGF) Receptor Pathway
The epidermal growth factor (EGF) induced EGF receptor (EGFR) pathway is one such
example of a cellular pathway that has been extensively studied to understand how cells
sense the external environment and then transmit and integrate this information to make
a phenotypic decisions [112, 83, 48, 26]. EGFR belongs to a family of receptors that have
kinase activity leading to tyrosine phosphorylation at specific sites [59]. This receptor family
is collectively called the receptor tyrosine kinase (RTK) family. EGFR is arguably the most
well studied and understood system within the RTK family. This detailed understanding of
EGFR receptor system has been crucial in elucidating functions, behaviors, and regulations
of other members of the RTK family [112, 48]. This receptor system has been used in understanding basic processes such as receptor-mediated endocytosis [96], oncogenesis [117, 52],
mitogen-activated-protein-kinase (MAPK) signaling pathways, and receptor trans-activation
[18]. It is also the first system to emphasize the role of mechanistic mathematical modeling
to understand complex, integrated biological systems [65, 7].
Most of our current under-
standing of receptor binding, internalization, and degradation is derived from quantitative
models of these processes in EGFR system that were aided by related experimental measurements [37, 111]. Further, discoveries of mutations in some of the signaling proteins of this
14
cascade in a number of human epithelial cancers have provided us a basis for understanding
processes like oncogenesis. Detailed understanding of the signal transduction in this pathway and deregulation of this signaling pathway in various types of cancer has meant that
proteins of this pathway have been successful candidates of some targeted cancer therapeutic
strategies [91, 34].
In normal physiology, EGFR pathway is initiated by binding of EGF ligand to form EGFEGFR monomer. The monomers come together to form dimers that are activated by autophosphorylation at number of tyrosine residues [75]. There are at least 20 phosphorylation
sites on the receptor [114], although exact detail of how many and what combinations of
phosphorylation are needed for the activation or what the exact roles of these multiple
phosphorylations sites are is unclear. The activated (phosphorylated) receptor dimer can
signal and thus activate a number of downstream adaptor proteins that eventually lead to
activation of important transcription factors that trans-locate into the nucleus to activate
genes associated with growth, differentiation, or proliferation [83]. This pathway, which is
normally associated with cellular phenotypes like cell growth and proliferation, [71, 66, 45, 93]
has been found to be over-stimulated (signaling above the normal values) in large number of
solid human cancers [116, 81]. A number of proteins that single in this pathway are mutated
in cancers and are the sources of pathway over-stimulations [64, 105, 103].
Identification
of this direct clinical relevance of this pathway has meant that it has been studied both
in academic settings to discover the signaling principles and in pharmaceutical industries
to understand the pathway such that suitable drugs can be designed to reverse the disease
(cancer) progression.
1.3
Computational Modeling of Biochemical Pathways
Towards the end of 20th century, molecular biology, which is concerned with study of network of interacting molecules that define cellular behaviors (and subsequently higher level
interactions that define tissue, organ, and organism behaviors), started to see changes in
the ways it was being studied and analyzed. The field slowly started moving away from
reductionist way of studying molecular network one reaction at a time to a systems level
15
study which focuses on looking at molecular network as a whole by measuring and analyzing systemic level changes. This shift lead to an emergence of new field of biology called
systems biology [54, 109, 110] that started to draw a lot parallels with studying and analyzing man-made engineered systems. Unlike engineered systems, where there is a detailed
knowledge of how parts are assembled together, components of these biological systems are
largely unknown and have to be established through perturbation and inference [4, 47, 15].
This can be thought of as reverse engineering the cellular processes to understand how cells
achieve their functions. In other words, it is framed as a problem of trying to understand a
systems that was already 'engineered' (in this case evolved) but its blueprint was missing.
As the molecular networks started to be studied as a system rather than its component
parts, the qualitative nature of analysis and interpretation of experimental data collected
became cumbersome and non-intuitive paving a way for quantitative modeling, analysis,
interpretation of the results in process giving rise to a sub -field of computational systems
biology [63, 104, 54, 112] that emphasized on mathematical modeling of biochemical interactions. Mathematical models had been used to understand, interpret, and model biological
processes much earlier [67, 42, 72]. However, their use in study of molecular network biology really took off towards the beginning of the 21st century. A crucial contribution in the
emergence of this field comes from technological advances that lead to high-throughput data
collection possible.
Despite the role of computations and bioinformatics in completing the human genome
project, computational biology is still a young field.
The most popularly known form of
computational biology refers to a subfield of bioinformatics. This mostly comprises of using
computer science algorithms to shift through, align, and stitch together vast amount of
genomics data. There is a second field of computational biology, sometimes referred to as
computational systems biology, that is involved with modeling and understanding molecular
interaction networks (be it at genetic, proteomic, or metabolic levels) as dynamic systems.
The structure of these models are derived from the knowledge of the underlying biology
which they aim to quantify. Broadly speaking, one can think of bioinformatics as a process
of assembling a static picture of the cellular make up, and dynamic modeling as way of
understanding how these systems respond to changes or perturbations in their environments.
16
The second class of computational biology that aims to build dynamic molecular networks of
biological systems (sometimes called computational systems biology) forms a basis for this
thesis.
The work described here builds up on the mathematical dynamic models of molecular
networks to get insights on how they behave in response to stimuli.
How this dynamic
network level understanding of biology can be exploited to find suitable drug targets for single
and combination therapies is one of the goals of the work presented here. We also exploit how
these network biology models can be utilized in designing suitable interventions strategies to
maximize efficacy and minimize toxicity. There are number of mathematical techniques that
are used to model molecular network dynamics. Some of the more prevalent techniques are
deterministic kinetic models described by systems of ordinary differential equations [5, 86,
102], stochastic kinetic models [100, 101], fuzzy logics [3, 77], and agent-based models [85,
1071. These modeling techniques have played a crucial role in understanding the underlying
mechanism of biological and cellular functions.
1.3.1
Computational Models of EGFR Pathway
For the purpose of this thesis project that is concerned with development of system level
methods for target identification and therapeutic intervention design, EGFR (section 1.2)
pathway provides a natural place to start. Given a detailed biological understanding of
the system, well developed mathematical models, and its deregulation in number of human
epithelial cancers, it lays the right background for us to ask questions that go beyond model
calibration. Further, key topological features like non-linearity arising from the bimolecular
nature of interactions, the branched nature of signal transduction, and the signaling events
that expand across multiple compartments encompass the key aspects of signaling pathways
that we want to train my methods on.
Among various available mathematical models of this pathway [50, 62, 20, 90] we choose
to start with the model proposed by Schoeberl et al. [94] which incorporates signaling events
associated with this pathway from binding of EGF ligand to EGFR eventually leading to
the activation mitogen activated protein kinase (MAPK) called extracellular-regulated kinase
(ERK). The choice is mainly influenced by the level of details incorporated in the model,
17
actual size of the model, and key topological features like the branched nature of the signal
transduction and signaling events that expands across multiple compartments, namely extracellular matrix, cytosol, and endosome. Easy access to the model in our group and
familiarity with it in the context of other related projects were also factors that contributed
to this model choice. It is important to point out that while this model will be used as basis
for developing our methods, the methods themselves are generalizable to other variants of
the pathway or models of other biochemical processes.
Biochemical model of EGFR pathway proposed by Schoeberl et al. [94] is a deterministic
mass-action model represented by a system of ordinary differential equations (ODEs). The
exact variant of the model that we use is the one that was modified by Apgar et al. [6]. The
modification accounts for the synthesis of the adaptor proteins that are degraded along with
the receptor complexes. In the original model, when the receptor complexes are degraded,
the receptors are synthesized to return the cell to the pre-stimulus state so that it can respond
to future stimuli, but the adaptor proteins that were degraded along with receptors were not
synthesized. So, although an attempt was made to return the cells to pre-stimulus state, this
was not achieved as some of the adaptor proteins that were degraded were not synthesized.
Apgar et al. [6] updated this aspect of the model by including synthesis and degradation for
adaptor proteins such that the model does return to the pre-stimulus state ready to respond
to the next set of input signals. A schematic of this biochemical pathway is shown in Figure
1-1. Only key reaction stages are shown in the schematic for clarity. The detailed model
contains 101 species (protein or protein complexes), that interact in 148 different chemical
reactions and are characterized by 107 zeroth, first, or second order rate constants.
For both the drug target identification and intervention strategy design parts of the
project, the model system that we are using is Epidermal Growth Factor (EGF) induced
EGF Receptor (EGFR) pathway proposed by Schoeberl [94], Hornberg [50] and modified by
Apgar et al [6]. This pathway is activated in the presence of extracellular ligand EGF which
can bind to the EGF Receptor (EGFR) on the cell membrane resulting in its dimerization
and cross phosphorylation. The ligand bound phosphorylated receptor dimer recruits and
activates a number of Adaptor Proteins in the cell eventually leading to the activation of
Mitogen Activated Protein Kinase (MAPK) called Extracellular Regulated Kinase (ERK)
18
gExtracellular
EGV
Cytoplasm
Endosorm
Figure 1-1: Schematic biochemical pathway of EGF induced EGFR system [6].
by phosphorylations at two different residues. This pathway captures most of the essential
features of a biological signal transduction pathway. The pathway is highly non-linear with
bimolecular nature of interactions. The signal can flow in a number of parallel branches down
the cascade and the signaling can take place both in the cytosol and in endosome before the
endocytosed molecules are either degraded or recycled back to the membrane. This is an
important pathway that has been well studied experimentally and has been extensively
calibrated making it a suitable model choice for us to ask some of the questions that go
beyond model calibration and are the focus of this thesis project. Further, besides having
the suitable topological features to develop our methods, the pathway is of huge interest
in the academic and pharmaceutical community as it is deregulated in a large fraction of
human cancers of epithelial origin [82, 97]. The common mutations that are widely observed
in cancers that are associated with deregulations of this pathway are: (1) EGFR mutation
or over-expression [57, 87, 1] (2) mutation of Ras protein [12, 13] and (3) mutation of the
Raf protein [13, 30, 33, 74]. EGF is usually taken as the input to this pathway and activated
(doubly phosphorylated) ERK is taken as its output. We adopt the same convention in this
project. The schematic representation of the pathway is shown in figure 1-1. Only a part of
the biochemical interactions are shown in the figure for clarity reasons. The detailed model
contains 148 chemical reactions between 101 reacting species and are modeled by 107 kinetic
rate constants which are either zeroth, first or second order.
19
This biochemical network is modeled using systems of Ordinary Differential Equations
(ODEs).
A toy model is shown here as an example of setting up the ODE model from
biochemical network of reactions. In this model, species A and B react with second order
(bimolecular) rate constant k1 to produce C, C can dissociate back into A and B with first
order (unimolecular) rate constant k2 or can go on to form species D with a first order
rate (unimolecular) constant k3. Equation (1.1) here represents the biochemical process and
equations (1.2),(1.3),(1.4) and (1.5) show how the species A,B, C, and D in the system evolve
over time.
A+B
k1
C-
k3
(1.1)
>D
k2
d[ A]
dt dt -kl [ A] [B] + k2[C]
d[B]
d
-kl[ A][B] + k2[C]
dt
d[] - k2[C]
dt
d[D]
=tk3[C]
(1.2)
(1.3)
(1.4)
(1.5)
A generic rate law for all the species in a system can be represented as shown below in
equation (1.6):
dt
S= Aix + A2x Ox + Blu+ B2U o x + B3U &u + k
(1.6)
Here, x is the vector of all the species in the biochemical system, u represents the input
vectors and 9 represents the Kronecker product. Kronecker product between vectors a and
b is a vector with all the possible product combinations of elements in a and elements in b.
A 1 , A 2 , B 1 , B 2, B 3 contain the suitable coefficients for the reactions which are a combination
of stoichiometric matrix, unimolecular, and bimolecular rate constants. k is a vector with
constants which is used to capture the fixed production rate of one or more of the species in
x.
The assumptions that are needed to model a system with Ordinary Differential equations
20
(ODEs) like the well mixed compartment (mainly resulting from large number of protein
involved) is taken to be valid for the system within a given reaction compartment. There are
at least 10,000 copies (~
20nM) of each protein present in this pathway in a cell. The model
includes three compartments: (1) the extracellular environment where the ligand is present
(2) cytosol where the majority of the signaling events take place (3) endosome where some
signaling can still take place before the proteins/molecules are either degraded or recycled
back to the plasma membrane. The relative differences in the volumes in these compartments
is introduced implicitly through the rate constants of the reactions and hence no explicit
correction volume ratio is present in the reaction equations. The ODEs are integrated using
odel5s function in matlab 2009a (Mathworks) to evaluate the temporal dynamics of all the
proteins and the complexes in the system. The stiff integration provided by the ode15s is
necessary for the simulation of the model because the rate constants associated with this
model vary across several orders of magnitude. Design, simulation, and optimization of the
model parameters as appropriate is core to both the questions that we are trying to explore
in this project and these aspects will be discussed in greater details in the sections to follow.
1.3.2
Variant Models of EGFR Pathway
Three variants of the model, besides the 'Original' model (sometimes also referred to as
'normal') proposed by schoeberl [94], Hornberg [50] and modified by Apgar et al [6], are
to be used in both parts of the project. These variants are modeled by introducing one of
the three common mutations associated with this pathway to obtain an aberrant signaling
of the output protein ERKpp. This aberrant ERKpp dynamics will be considered to be
the 'cancerous' phenotype of the model. The three mutations considered are: (1)
EGFR
overexpression and Endocytosis defect, sometimes referred to as 'Cancer 1' (2) Defective
form of Ras GTP that has impaired GTPase activity and stabilizes in the GTP bound state,
sometimes referred to as 'Cancer 2' and (3) Defective form of Raf protein that cannot be
deactivated (dephosphorylated) once it has been activated (through phosphorylation) by up
stream Ras-GTP, sometimes referred to as 'Cancer 3'. It is important to note here that RasGTP is not a kinase as such, but most of the modeling work of this network is implemented as
if this were the case because the direct kinase that phosphorylates Raf protein is not known
21
yet. To the best of our knowledge, we are not aware of experimental data that supports
whether each one of these mutations alone is able to transform a normal cell to a cancer cell.
Further, we do not know what temporal dynamics of the ERKpp signal can be classified as
a cancer phenotype. So, for the purpose of modeling these mutations here, we will consider
the transient response (Figure 1-2) as a normal phenotype and sustained high signal as a
cancer phenotype.
(A)
6
X 106
(B)
6
X 106
Normal Model
Cancer Model
U
U
10
38
8
0
E 4
E
6
4
0- 2
2
1cL-
0
"'
1000 2000 3000 4000 5000
Time (s)
W
0
1000 2000 3000 4000 5000
Time (s)
Figure 1-2: Definitions of normal and cancer phenotypes in terms of ERK-pp signaling dynamics
1.4
Structure of This Thesis
This thesis, over a span of next three chapters (Chapter 2, Chapter 3, Chapter 4), explores
two key questions of early stage of drug discovery. The first question is concerned with
identifying the best places for intervention, either as single agent therapies or combination
therapies. Chapter 2 explores the effectiveness of single agent therapies at different places in
the network to evaluate where the best places for intervention are. Chapter 3 evaluates the
same question for combination therapies. Specifically, all plausible target combinations are
evaluated to quantify an effect they exert on an output of interest in combination compared
to when one of the drug was given alone at equivalent concentrations. Chapter 4 explores the
capabilities and limitations of inhibitor therapies that mostly work as "on" or "off switches"
for safety and efficacy when they affect both normal and deregulated signaling networks
22
as is commonly the case in disease site like cancer.
We then explore some protein based
intervention strategies that may be better suited for multi-factorial objectives that drugs
should ideally meet.
23
Chapter 2
A Framework for Evaluating Efficacies
of Single Agent Therapy
Abstract
Mechanistic systems biology models describe normal and diseased processes of cellular events
and serve to represent our current state of knowledge of the relationship between biology and
disease. A key goal of this endeavor is to inform clinical decision-making and drug discovery
to improve therapeutic approaches using a systems-level view. In this work we focus on the
important challenge of selecting effective drug targets. We develop a computational approach
that uses network-level information and simulation methodology to probe for the optimal
places for intervention. Our method evaluates the amount of control provided by each
potential target over network output, thus identifying proteins best poised for intervention.
We apply the method to signal transduction in the epidermal growth factor receptor pathway,
in which aberrant behavior has been linked to many cancer processes. The results exhibit
a wide range in the level of control exerted by different potential targets. Targets near
the top of the pathway exert relatively weak control, consistent with known experimental
results; some targets near the bottom of the pathway exert much stronger control due to
network properties that are analyzed. These behaviors observed are robust to details of the
parameterization of the model, suggesting that the specific results obtained here will not be
strongly affected by model uncertainty. Taken together, the results of this study provide
strong evidence that effects of network structure and dynamics can have a strong influence
on drug target effectiveness.
24
2.1
Introduction
Significant work in cell biology is focused on elucidating the networks of protein interactions
responsible for key cellular processes and that lead to individual phenotypes [53, 10, 8]. An
emerging picture is that these interactions are deregulated to some extent in certain diseases
such as cancer [46, 55], leading to studies undertaken to understand networks in the contexts
of both normal and abnormal physiology [41, 56, 51, 68]. Furthermore, computational and
mathematical approaches are being applied to quantitatively represent these biochemical
networks with different levels of mathematical abstraction [83, 63, 39, 53, 77]. In order to
understand the mechanistic details of signal transduction in these pathways, a class of models
represented by systems of ordinary differential equations (ODEs) is being widely developed
and calibrated using experimental data [94, 14, 2, 20]. An additional benefit from this type
of endeavor could be to develop therapeutic intervention strategies able to address network
deregulation problems from a holistic viewpoint.
A key challenge within this framework is to find nodes (protein or protein complexes) in
a network that are most suited to alter the deregulated network behavior in a desired way
[38, 49, 84, 19]. This challenge, popularly referred to as the 'Target Identification' problem,
has been dominated by molecular-level perspectives. The question of what makes a good drug
target has typically been addressed by identifying proteins whose active sites are especially
amenable to tight-binding by molecules with the size, shape, relative hydrophilicity, and other
properties matching those of current drugs [61, 22]. This focus on "druggability" has led to
targets for which it may be relatively straightforward to develop a tight-binding inhibitor
without assessing the effectiveness with which the node can be used to control biological
functions and disease processes. Here we report the development of a network engineering
method to identify suitable drug targets based on their relative control over disease processes.
This approach is not only new, but it has the potential to lead to especially effective drugs,
rather than just tight-binding inhibitors.
25
2.1.1
The Biochemical Model
We develop and apply the method in the context of the epidermal growth factor (EGF)
receptor signaling pathway.
It is one of the most thoroughly studied biochemical signal
transduction pathways with a wealth of experimental data supporting well established models of network behavior 18, 83, 62, 90. The relatively detailed and well-calibrated nature of
the model makes it a suitable candidate for our study, which is concerned with using detailed
network-level understanding of biochemical processes to identify suitable places for intervention. The particular pathway version of the model that we used for this work was initially
developed by Schoeberl et al [94], later updated by Hornberg et al [50], and further modified
in our group by Apgar et al [6]. The pathway is modeled by a system of ordinary differential
equations (ODEs) in which each ODE describes how a particular protein or protein complex
in the pathway evolves over time in the presence of an EGF stimulus. The integration of the
system of ODEs then gives the temporal dynamics for the concentration of all the proteins
and protein complexes in the pathway as a function of time.
The pathway is shown schematically in Figure 2-1A. It is induced by the binding of
EGF ligand to the trans-membrane EGF receptor (EGFR) [11, 89].
For the purposes of
this work, we consider the EGF ligand to be the input of the system. Upon ligand binding
the receptor can dimerize and autophosphorylate [11, 89, 31].
This autophosphorylated
and activated EGF-EGFR dimer can recruit and activate a number of adaptor proteins by
providing suitable binding sites. The sequential activation of the adaptor proteins leads
to the activation of Ras [16] and of a canonical mitogen-activated protein kinase (MAPK)
cascade composed of the proteins Raf, MEK, and ERK [78, 74]. ERK protein is the final
protein as modeled, with ERK-pp representing a doubly phosphorylated and activated form,
which is treated as the output of the pathway here. ERK-pp is itself an important signaling
molecule; it is a kinase with a large number of substrates in both the cytoplasm and the
nucleus, and can also act as a transcription factor to activate a number of growth and
proliferation related genes [74, 17].
26
2.1.2
Model Variants
The model proposed by Schoeberl et al [94] describes normal pathway dynamics in the
presence of an EGF stimulus.
Given that this pathway is deregulated (over-stimulated)
in a large number of human cancers of epithelial origin [91], we chose to study the ability
of inhibitors targeted to different nodes in the pathway to attenuate the over-stimulated
pathway response.
To this end, we modeled three variants of the pathway that we refer
to as 'cancer variants'. These variants were modeled by introducing one of three common
mutations that are associated with over-stimulation of this pathway in various types of
cancers.
These three mutations are: (1) EGFR over-expression together with a defect in
endocytosis [79, 31, 28, 9] (2) mutation of the Ras protein, which is a frequently mutated
oncogene [13, 9] and (3) mutation of Raf protein, which is again a common mutation in a
large number of cancers [13, 9]. Each of the three mutations leads to overstimulation of the
EGFR pathway represented by prolonged activation of ERK protein.
2.1.3
Format of Study
Here the question of target identification is formulated from a network perspective.
The
goal is to find protein targets whose inhibition reduces network output most effectively. For
each candidate target protein or protein complex in the pathway, we augmented the models
to introduce and simulate a competitive inhibitor at a range of different concentrations and
evaluated the effect on pathway output (ERK-pp). This approach is shown systematically
in Figure 2-1B.
The relationship between the amount of inhibitor introduced in the model, its direct effect
on the target inhibited, and the effect on the pathway output was evaluated for 14 candidate
targets in the pathway. This comparative approach provided a quantitative evaluation of
relative target effectiveness and helped identify species in the network predicted to be most
effective in network attenuation.
Metrics were used to quantify the relationship between target inhibition and output
27
attenuation.
The 'Target Effect' is a direct measure of the fraction of target inhibited
whereas the 'Output Effect' is quantified as the reduction in ERK-pp signal measured as
either the integral under the curve of the concentration trajectory or the peak height for the
trajectory [50, 8].
2.1.4
Summary of Findings
The results of the study show a very wide range of effectiveness across the panel of potential targets examined, with more effective targets found downstream, close to the output.
These observations are not strongly sensitive to which pathway model of EGFR signaling
signaling was used or the particular parameters used in simulating models. Furthermore,
we demonstrate that network dissection and detailed analysis of signaling dynamics of the
pathway can provide important insights that can be used to understand the basis for the
target behaviors observed.
2.2
2.2.1
Methods
The Normal Model
The ODE pathway model for signaling downstream from EGFR utilized in the current study
evolved from the original model by Schoeberl et al [94], as modified by Hornberg et al [50]
and further updated by Apgar et al [6].
Here our normal model is the Apgar et al [6]
version. The term "normal" refers to the published model of the pathway to distinguish
it from perturbed versions containing cancer-associated changes that lead to exaggerated
responsiveness. The model has 13 unique proteins that comprise 101 unique chemical species
through the formation of complexes and catalytic modification. Model dynamics are driven
by 163 elementary chemical reactions that are described using mass-action kinetics. A feature
of mass-action kinetic formulations is that they contain only zeroth-, first-, and secondorder reactions; all higher-order abstracted reactions are written as a series of these more
28
elementary ones. Parameters of the model include 107 distinct rate constants and 101 initial
concentrations; in addition, there is 1 input (EGF).
2.2.2
Cancer Variant Models
Three variants of the normal model were constructed as plausible mechanisms of deregulation
that might represent processes operating in cancer cells. Variant I was obtained by increasing
the rate of production of EGFR protein by 10 fold while also increasing its recycling rate
from endosomes to the plasma membrane by 10 fold (Figure 2-2A) [79, 1]. Whereas the
unstimulated normal model has a steady-state receptor number of 8.28 x 10' cell- 1 , for
the Variant I model this value was increased 53 fold to 4.38 x 10 5 cell-
fold. Variant II
was obtained by short-circuiting the activation-deactivation--reactivation process of Ras to
reflect compromised GTPase activity that arises from point mutations of the same class
as G12V, which we model as preventing GTP hydrolysis thus leading to prolonged RasGTP activity (Figure 2-2B). In the published model, upon activation of one molecule of Raf
protein, Ras-GTP is hydrolyzed back to Ras-GDP to start the next round of the activation
cycle. This hydrolysis step was removed in the Variant II model here, keeping Ras-GTP in
an activated form longer [95]. Variant III was obtained by decreasing the association rate
constant for Raf-p binding the phosphatase for the dephosphorylation step of activated Rafp protein to Raf by 1000 fold. This mimics the presence of a constitutively activated form
of the protein in the model (Figure 2-2C) and acts similarly to a common mutation V600E
[30]. While different investigators might have chosen different implementation details, the
processes represented here are directly drawn from common mutational alterations known
to affect tumor cells.
2.2.3
Drug Intervention Models
A series of modified versions of the normal model and of cancer variants I, II, and III
were constructed. Each modified version represented the effect of drug treatment with a
29
competitive inhibitor that specifically targeted one of 14 plausible chemical species in each
model; 14 modified versions of the normal and of each of the three variants were constructed
to represent targeting each plausible species in the system. In the work reported here each
target bound inhibitor in a second-order reaction to form a complex that was completely
inactive. This inhibitor-target complex was either allowed to dissociate back to the target
and the inhibitor or degrade at the rate of degradation of the target protein.
In other
work we treated the inhibitor-target complex as inhibiting only some of the activities of
the target or as being a non-competitive inhibitor of target. The models used here had the
inhibitor act in a non-depleting manner to simulate the effect of a large volume of drug
present in cell culture or in circulation that replenished drug that bound to target. Two
new parameters were introduced in each model variant for the kon and koff for the inhibitor
1
binding to target. Values of 1.66 x 10-6 cell molecule- s-1 and 1 x 10-
3
S-1 were used
for second-order association rate and first-order dissociation rate, respectively. These values
are equivalent to 1 x 106 M-
1
S-1 for the association and 1 x 10-3 S-1 for the dissociation
rate constants using typical dimensions of a mammalian cell (1 x 10-l
L) [94], giving a unit
nanomolar equilibrium dissociation constant.
In simulations the pathway was equilibrated in the presence of the inhibitor before stimulation with the EGF growth signal. For each target of interest, inhibitor was introduced at
100 different logarithmically spaced concentrations, between 6 and 6 x 108 molecules cell- 1 ,
which corresponded to a maximum concentration of 1 mM using typical dimensions for a
mammalian cell.
For each level of inhibitor concentration introduced, output signatures of interest were
measured and compared with the case in which the intervention was not present in the
pathway. A schematic of this process is shown in Figure 2-1B, where stage (i) represents
the model with no EGF stimulus and no inhibitor (intervention), stage (ii) represents the
pathway behavior when the model has been equilibrated in the presence of the inhibitor but
no EGF stimulus (input) is present, stage (iii) represents the model in the presence of EGF
30
stimulus without any intervention, and stage (iv) represents the system with intervention in
the presence of EGF stimulus. Simply stated, we equilibrated at (i) and used that as the
starting state for a type (iii) simulation; likewise, we equilibrated at (ii) as a starting point
for a type (iv) simulation.
(B)
(A)
00i
Wi
Extracellular
V ------
No EGF
No Inhib
Cytoplasm
M
m
-n
n
Inhibitor
C
Endosome
C
(iv
Cii
RAF
.RAF
6
x10
'L6(D)
10 6
Normal Model
8
.210
6
00.8
C0
0
Time (s)
0.8
U
aj
0.
2
0.2
0
(F)
110.8
4
LU0
1000 2000 3000 4000 5000
(E)
1.
Cancer Model
1000 2000 3000 4000 5000
0.2
0.C0
-
0.0 0.2
Time (s)
0.61 /
0.4 0.6 0.8
Target Effect
1.0
0. r
0
90 99 99.9 99.99
%Target Inhibition
Figure 2-1: Overview of target evaluation strategy. (A) Schematic representation of the EGFR signaling
pathway studied here. (B) The strategy compares network behavior in the presence and absence of candidate
inhibitors. (C) Dynamics of pathway output (ERK-pp) upon stimulation with step increase of 8 nM EGF at
time zero, for the normal model. (D) Idealized representation of overstimulated ERK-pp output dynamics in
the presence of activating mutations in the pathway. This is the phenotype typical of variants of the pathway
with activating mutations. (E) Illustrative behaviors expected for different types of targets, depending on the
relationship between the target and the output. The black line represents the case with a linear relationship
between the target and output. The non-linear nature of signal transduction means the actual trend can
deviate from this linear behavior either in a sub-linear (green line) or super-linear (red line) manner. (F)
Expected trends of (E) on a semi-log scale, as this is the scale used in presenting the simulation results.
31
2.2.4
Target and Output Effect Metrics
The focus of this study is to quantify the relationship between target binding and output
attenutation.
Thus, metrics were chosen to quantify each of these system perturbations.
In each case we chose a fraction (or percentage) metric. The Target Effect is defined as
the fraction of available target that is bound by inhibitor and thus inactivated; the Output
Effect is the fractional associated decrease in output signal. For each model (normal and the
variants) these fractional changes were calculated with respect to the signal strength in the
absence of any intervention in the model.
Target Effect
(2.1)
[1 ] + Ki
Where, [I] = inhibitor concentration, Ki = inhibitor binding affinity.
Output Effect
=
(unperturbed output - perturbed output)
unperturbed output
(2.2)
Where, "unperturbed" represents model without inhibitor and "perturbed" represents model
with inhibitor.
2.2.5
Signal Transduction between MEK and ERK
In order to understand why MEK went from being super-linear in normal model to sub-linear
in overstimulated variant models, the signal transduction dynamics between MEK and ERK
was analyzed further by perturbing the normal model and measuring the signals at MEK-pp
and ERK-pp levels. More specifically, we varied a parameter in the model ('k42'- one that
affects the binding of phosphorylated-Raf to its phosphotase) by three orders of magnitude
on either side of its nominal value in the normal model. This parameter span was sampled
at 100 different values in a log scale. The normal model was simulated for each of these 100
values for 'k42'. This resulted in 100 different signals at MEK-pp and ERK-pp levels (the
signal here refers to the area under the curve in agreement with the rest of our work). The
32
resulting ERK-pp values were plotted against MEK-pp values to evaluate the relationship
between these two species in the network. Further, we evaluated the MEK-pp and ERK-pp
signals for normal and the three variant models. These four data points were overlayed on
the curve describing the relationship between MEK-pp and ERK-pp values obtained from
'k42' variation.
2.2.6
Parameter Variability Study
Each parameter in the unperturbed normal model was sampled using Latin Hyper-cube
Sampling (LHS) method. We used a log-normal distribution with mean values of the normal
model and the standard deviation of 0.5.
This meant that 95% (2o-)
of the parameters
sampled were within 10 fold of the nominal parameter values (i.e. the 95% of the parameter
range was from 0.1 x nominal values to 10 x nominal values). 10,000 parameter sets and
hence 10,000 models were generated from this parameter space. Each model was first run to
steady state before applying an EGF stimulus of 8-nM.
Only the models with parameter sets that were able to stimulate the pathway were chosen
for further analysis. The criterion used in selecting model for further analysis was that the
model should produce at least half the output (ERK-pp area for 5000s) of the unperturbed
normal model. For each of the chosen models, the target behavior was evaluated by inhibiting
the model at each of the 14 nodes in the network at 100 different concentrations of inhibitor.
Resulting target behavior for each target was classified as sub-linear, super-linear, and inbetween (ambiguous) behaviors using a three point classifier.
The three point classifier
compared the output effect of the inhibition to that of linear effect at three different values
of target effect. If all the three point showed output effect less than that expected from a
linear response then that particular target behavior of the model under question was classified
as sub-linear. Similarly, if the output effect was greater than what would be expected from
linear response at all the three target effect levels the behavior was classified as super-linear.
If the output effect was greater than expected linear value in some target effects and less
33
than this value in other target effects that was classified as ambiguous (or partly-sub-partlysuper-linear).
2.3
2.3.1
Results
Intervention-free Models
We studied the effect of various cancer-associated mutations on the signal propagation dynamics resulting from an EGF stimulus, with a focus on the dynamics of ERK-pp, which we
treat as the pathway output. Figure 2-1C shows the dynamics of the output (ERK-pp) in
the normal model in response to an 8-nM EGF stimulus (step stimulus). There is a transient
peak followed by a return to the pre-stimulus level response. This type of adaptation is characteristic of normal cells that do not have significant deregulation. This transient response
has been observed in a large number of experimental studies [62, 90].
In the presence of
an activating mutation or pathway deregulation, the network can produce an excessive or
prolonged output growth signal (Figure 2-1D).
We constructed three variants based on the normal model, each incorporating a different
type of cancer-associated deregulation, and each demonstrated a similar but quantitatively
different over-activated ERK-pp phenotype. Figure 2-2 shows schematically the mutations
introduced into each of the three variant models and the resulting ERK-pp dynamics for
each of the three cancer variants when the models were stimulated with 8-nM EGF (see
Methods).
To establish a baseline from which to measure the effect of intervention, we studied
signal propagation dynamics for all four models (one normal and three cancer variants) in
the absence of any intervention. The models were stimulated with multiple levels of EGF,
and signal strength was measured as a function of time at species downstream from the
input.
Figure 2-3 shows a selection of the results; each panel represents a different model and
34
(C)
(B)
(A)
Receptor Complex-Sos (GEF)
RasGTP
0 EGFR
0
EGFRi
'1
RasGTP
GAP
0
{iox
t ox
RasGDP
Raf-P
Raf
Raf-RasGTP
0
ptasel
1000X
RasGTP*
(D) 12
EGFROE + EndoDefect Model
o6
(E)
6
(F) 1
RasGTPMut Model
6
RafMut Model
~
Cancer Variant III
10
10
Cancer Variant
_
I
Cancer Variant 11
0
E
4
4
4
2
r2
2
0
1000
2000 3000
4000
5000
6
00
Time (s)
1000
2000 3000
Time (s)
4000
5000
0
1000
2000
3000
4000
5000
Time (s)
Figure 2-2: Schematic summary of the three cancer variants of the model. Here the red text, lines,
and arrows represent changes to the original model to obtain each variant. (A) Changes made to obtain
the EGFR over-expression and endocytosis defect model variant (Cancer Variant I). (B) Changes made to
represent Ras mutation. Large red 'X' above the black lines represents elimination of these reactions from
the model (Cancer Variant II). (C) Changes made to represent Raf mutation (Cancer Variant III). Panels
(D), (E), and (F) show the dynamics of the ERK-pp protein when the three variant models were stimulated
with a step 8-nM EGF input signal. All three cancer variant models show over-stimulation in their ERK-pp
dynamics.
each subpanel shows the temporal dynamics of an individual protein species at a position
of the network. The four proteins shown in the figure are (1) phosphorylated EGF-EGFR
dimer (labeled EGFR*), (2) activated, phosphorylated Raf protein (Raf-p), (3) activated,
doubly phosphorylated MEK protein (MEK-pp), and (4) activated, doubly phosphorylated
ERK protein (ERK-pp).
These probe activation at the top of the network and for the
three levels of the MAP kinase cascade at the bottom, and were chosen because they are
representative of the full set of signals.
Two network features particularly stood out from this analysis. The first is amplification;
the strength of the signal (concentration of activated protein) increased progressively down
the MAP kinase cascade from Raf-p to MEK-pp to ERK-pp. This is in contrast with the
behavior upstream, in which the concentration of activated species remains in the range
35
102 - 10 4 molecules/cell, whereas it increased to 10 4 - 10 7 molecules/cell for MEK and ERK
activation. The second feature, saturation, can be seen in protein signals that are insensitive
to increases in the level of EGF stimulus beyond 0.8 nM. The input used in our analysis (8
nM) lies beyond this threshold and the results are thus relatively insensitive to EGF input
levels.. The amplification and saturation features were observed in the normal model (Figure
2-3A) and the three deregulated variants (Figures 2-3B-D).
Figure 2-3 shows increased and prolonged signaling in the cancer variant models as compared to the normal model. In the model with EGFR overexpression and the endocytosis
defect (Variant I), the Raf-p signal peak is about twice that of the normal model and the
width of the signal pulse is not changed (Figure 2-3B). In the model with the RasGTP
protein mutation (Variant II), the peak of the Raf-p signal is about four times that of the
normal model and the signal is present for a much longer time (Figure 2-3C). In the model
with the Raf mutation (Variant III), the Raf-p signal peak is three orders of magnitude
higher than the normal model and also lasts longer (Figure 2-3D). Based on these signaling
dynamics of the models used here, Variant I is the weakest form of over-stimulation, Variant II is intermediate, and Variant III is the strongest. Interestingly, Variants II and II I
exhibit relatively large differences in Raf-p dynamics that produce much smaller differences
in MEK-pp dynamics, and even smaller changes in ERK-pp dynamics.
2.3.2
Intervention Analysis
The observed non-linear nature of signal transduction could lead to a non-linear and nonobvious relationship between the effect of an intervention at its target and that observed
further downstream, such as at the output. One might expect three classes of targets (Figure
2-1E) - those with a roughly linear relationship between target inhibition and output effect
(as would be expected from a network with linear signal propagation); those with a strongly
sub-linear relationship, in which a large effect at the target is necessary to produce a smaller
effect on the output; and those with a strongly super-linear effect, in which less effort is
36
,03
(A)
8
-
Normal M( YU'I
80
8
60
EGFR*
-
6
40
=
EGF=8e-13M
EGF 8e- 12M
EGF =e-11M
EGF =8e-0M
EGF=8e-91M
-
EGF =e- M
EGF =8e-7M
4
20
2
3
_U
1000 2000 3000 4000 5000
0
x
104
MEK-pp
0
EGFR*
Cancer Variant c 10 2
p2.0
Raf-p
41.5
10
0.5
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0
1000 2000 3000 4000 5000
ERK-pp
x
10
SM
= -G
20
0
x 105
Raf-p
-
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x 105
3
MEK-pp
0.0 L
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x 106
15
ERK-pp
8
E
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E 2
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5
1
i
224
0
1000 2000 3000 4000 5000
00
00
1000 2000 3000 4000 5000
(C)
x 10
3
6I
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(D)
Raf-p
1000 2000 3000 4000 5000
x 106
EKp
15
Cancer Variant 1110
Ra-
6
3
4
2
2
1
kA
ERK-pp
00
0
1000 2000 3000 4000 5000
3 x10
7
MEK-pp
15
1000 2000 3000 4000 5000
x 10
6
ERK-pp
0
E2
E
10
1
00
0
1000 2000 3000 4000 5000
Q X107
3
EF*
2
\22
0
x13
4
4
'
1000 2000 3000 4000 5000
Time (s)
Cancer Variant I1
3 x 10
EGFR*
0
1000 2000 3000 4000 5000
Time (
1
5
1000 2000 3000 4000 5000
0
Time (s)
10
2
1000 2000 3000 4000 5000
S[5
0
1000
2000 3000 4000 5000
0
1000 2000 3000 4000 5000
Time (s)
Figure 2-3: Depiction of the two key features of signal propagation in this signaling cascade - amplification
and saturation. (A) The normal model, (B) Cancer Variant I, (C) Cancer Variant II, (D) Cancer Variant
III. Four panels within each subfigure show signal strengths at different points in network when the model is
stimulated with varying EGF concentrations. A general trend that holds across the models is that the signal
strengths increases as one progresses down the cascade - amplification - and the network response increase
in response to the increase in input stimulus only until a certain level, after which changes in stimulus value
are not reflected in the propagated strength -
saturation.
necessary at the target to create a greater effect at the output. These three scenarios are
shown schematically in Figure 2-1E, and other behaviors are also possible. The black line
represents the linear behavior, the green line sub-linear behavior, and the red line superlinear. Figure 2-iF shows the same behaviors replotted on a semi-logarithmic scale, which
37
is used throughout this work to show better distinction between targets.
Note that the
super-linear behavior might be especially useful for a therapeutic that is aimed at pathway
down-regulation, both because less target inhibition is required to achieve a given output
reduction and the output reduction is insensitive to inhibitor concentration for a larger range
of inhibitor concentration, but the details may depend on the goal at hand.
We evaluated the effect of unphosphorylated EGF-EGFR dimer inhibition on ERK-pp
output using our framework to measure target effectiveness in controlling network output.
The resulting quantification is shown in Figures 2-4A and 2-4B for the two metrics of effect
on network output, the peak height and the area-under-the-curve for activated ERK (ERKpp protein), respectively. The thin black line represents linear behavior for reference, and the
thick blue line shows the calculated relationship between the two metrics. The target shows
sub-linear behavior and while results from the two output measures used are not identical,
their trends are very similar.
The non-obvious nature of the signal transduction in this
cascade results in a scenario in which 50% target inhibition of EGFR has very little effect
on signaling dynamics of the ERK-pp protein, and greater than 98% receptor inhibition
is needed to affect ERK-pp.
We evaluated the validity of this prediction by comparing
the simulation results to cell-based measurements of the ERK-pp signal in response to 5 nM
EGF in A431 cells treated with varying doses of the EGFR inhibitor lapatinib by Chen et al.
(2009). The binding affinity of lapatinib to EGFR receptor has been measured to be ~ 3 nM
[113]. We simulated our normal model with the specified EGF stimulus level and inhibition
binding constant and the results (Figure 2-4C) agreed closely with the experimental ones
(Figure 2-4D). Both cases showed that the inhibitor exerts little effect on the output signal
until a 100-nM inhibitor concentration is administered; the output signal was essentially
completely attenuated at just over 1 pM concentration. This quantitative agreement lends
confidence to our use of the model to examine the potential of various clinical interventions
as well as the more specific observation of strongly sub-linear behavior for EGFR inhibition.
38
(A)
(B)
(EGF-EGFR)2 Inhibition
1
1
'I)
CL
0.8
0.8
0.6
0.6
UU
C
0.4
C
0.4
0
0
4-J
(EGF-EGFR)2 Inhibition
0.2
0.2
LU
00
(C)
0
99 99.9 99.99
%Target Inhibition
90
99 99.9 99.99
90
% Target Inhibition
0
(D)
Lapatinib Equivalent
.. Normalized Area
.Normaized
Peak
U
100
100
C
CL
75
50
U
LU
0-
75
50
25
25
-6
-7
Log[Inhibitor (M)]
U
-5
3
-7
-6
Log[lapatinib (M)]
-5
Figure 2-4: Comparison between the simulation results and the results reported in Chen et al [20] for
a cell-based assay with inhibition of equivalent targets. In the simulations unphosphorylated EGF-EGFR
dimer was inhibited. The clinically available EGFR. inhibitor lapatinib that was used is an ATP analogue
that competes in the phosphorylation step of the ligand-receptor dimer. (A), (B) show the simulated effect
on the ERK-pp peak and ERK-pp area, respectively, when the target was inhibited at different levels.
(c)
The linear reference line is black and the actual simulation results for the target are blue (sub-linear).
al
et
Chen
of
results
experimental
to
Representation of results from (A), (B) appropriate for comparison
[20]. (D) Experimental results of Chen et al [20].
2.3.3
Normal Model -
Target Comparisons
In the previous subsection we characterized a single target in terms of how its inhibition
affected pathway output. Here we extend that analysis to a comprehensive set of 14 plausible
targets in the pathway.
39
Simulation results for the normal model are shown for four representative targets selected
from various locations in the network (Figure 2-5).
Figure 2-5A shows the effect on the
output (area under the curve of ERK-pp) in response to inhibition of unphosphorylated
EGF-EGFR monomer. Inhibition of this protein complex showed dramatically sub-linear
behavior; almost 98% of this target had to be inhibited to cause a 50% reduction in the
output signal.
The dynamic range for this was from - 40% to 99.5% inhibition of the
target, in that less than 40% target inhibition produced negligible effect and 99.5% inhibition
produced nearly a complete effect. Similarly extreme sub-linear behavior was observed for all
the targets upstream of Raf (results not shown), although the precise degree of sub-linearity
varied among targets.
Figure 2-5B shows the effect on the output when Raf, the first protein of the MAP
kinase cascade, was targeted with different inhibitor concentrations.
Here ~ 92% of this
target protein had to be inhibited to see a 50% reduction in the output. The dynamic range
for this target was from - 20% to 99% inhibition of the target. Although both EGF-EGFR
monomer and Raf showed sub-linear response, their quantitative behavior was different as
reflected in their 50% output effect levels and the dynamic ranges. For example, if the goal
of an intervention in this cascade were to be to completely signal arrest at ERK-pp, the
amount of inhibitor (assuming the value of inhibitor equilibrium constant, K) to achieve
this by inhibiting Raf was almost 10 times less than that required by inhibiting EGF-EGFR.
Figure 2-5C shows the effect on the output resulting from inhibition of MEK protein, the
second enzyme of the MAP kinase cascade. Here ~ 10% inhibition of the target resulted in
50% reduction of the output, which represents a slightly super-linear behavior, and the full
dynamic range of output response was accessed with 0% to a 20% target inhibition. ERK
inhibition produced a profile that was both qualitatively and quantitatively very similar
to that for MEK inhibition (Figure 2-5D). The level of EGF stimulus used for all these
simulations was 8 nM, although variation in stimulus levels produced very similar effects
and results.
40
In summary, the normal model exhibited sub-linear behavior for all the targets examined
upstream of the MAP kinase cascade and for Raf. Only MEK and ERK showed greater
leverage over the pathway output and demonstrated slightly super-linear behavior. While it
is tempting to try to apply these results to target selection for cancer, it should be remarked
that the model is probably closer to simulating the behavior of normal cells than it is to cancer
cells. In fact, because of the large number of pathway modifications that may lead to cancer,
one may question whether there are targets broadly useful across large numbers of patients.
To address this question we studied potential targets in three cancer models created by
introducing perturbations associated with cancer into the normal model, representing three
mechanisms for overstimulating this pathway.
2.3.4
Cancer Variant Models
The same procedure to examine 14 potential targets was carried out on the three cancer
variant models, and the general target behavior along the signaling cascade was similar to
that observed in the normal model (Section 2.3.3).
Representative results for the normal
model are in Figure 2-2A-D and for the variants in Figure 2-5E -P. However, the sublinear behaviors were more pronounced, resulting in even higher levels of target inhibition
required to achieve 50% reduction in the output signal. The distance between the reference
black line and the simulation for targets was wider for cancer variants compared to the
normal model. This widening of the distance depended on the over-stimulation strength
of each model variant (see below). Furthermore, the transitions were sharper, resulting in
narrower dynamic ranges. For the first two targets shown, unphosphorylated EGF-EGFR
monomer and Raf protein, respectively, the general behavior observed was that increasing
overstimulation of the pathway led to greater sub-linearity of target behavior. For MEK
inhibition (the third target shown), a change from super-linear in the normal model to
partly sub- and partly super-linear was observed in the model with EGFR overexpression
(Variant I). With the stronger overstimulation resulting from mutation of RasGTP and Raf
41
proteins (Variants II and III, respectively) the target became fully sub-linear. The trend at
the level of ERK inhibition remained the same across all models considered. Thus, a similar
pattern of upstream sub-linearity was observed for the cancer models as had been seen for
the normal, except that the transition to slight super-linearity occurred later in the pathway
for more over-stimulated cancer models.
2.3.5
Signal Transduction Between MEK and ERK
In order to understand why MEK changed its target behavior from super-linear in normal
model to sub-linear in cancer variant models, signal transduction dynamics in this part of the
cascade was analyzed further quantifying the signals between MEK-pp and ERK-pp. The
results from this analysis is shown in Figure 2-6. This figure describes how ERK-pp signal
varies with varying levels of MEK-pp signal. There are 4 distinct regions on interest here.
The first is the linear region in a log-log plot thus describing a power law relationship between
MEK-pp and ERK-pp. The second is a region of even steeper signal where small increase in
MEK-pp signal leads to large increase in ERK-pp signal. This steep region slowly tapers off
to form the third region where the changes in MEK-pp levels have modest effects on ERK-pp
levels. With further increase in MEK-pp signal, we end up in a region of saturation where
the increase in MEK-pp levels have little of no effect on the ERK-pp signal levels. Figure 2-6
shows that normal (red asterisk) and variant I (green asterisk) models are operating on the
third region where the steep relationship between MEK-pp and ERK-pp is tapering off, but
model variant II (magenta asterisk) and III (black asterisk) are operating on the region of
the curve where ERK-pp signal has already saturated. This, we propose, is a contributor to
the variation of MEK target behavior in normal and the three variant models. A meaningful
way to interpret this graph is to notice that if you reduce the MEK-pp level by small amount,
as is the case for inhibition of MEK in the normal model, we reach a highly sensitive region
of the curve where small change in MEK-pp level leads to a large change in ERK-pp level.
This region is a bit further for the variant I model and hence a slightly more inhibition of
42
(A)
EGF-EGFR Inhibition
1
0.5
0
90
0
99 99.9 99.99
(E)
0.5
a--
(C)
1.0
1.0
(D)
1.0
0.5
0.5
0.5
Raf Inhibition
(B)
0.00
(F)
90
99
99.9 99.99
0
0
90
99
99.9 99.99
0
99
00
00
009
99090
0
S
(K)()
9o
no
0
an
999 999
99
99.9 99.99
9n
99
999 9999nno
90
99 99.9 99.99
1
0,5
0.5
.50.5
90
0.5
1
U i1
0
*
0.5
0
ERK Inhibition
E
0
(H)
99.9 99.99
1
(J
(C
90
(G)
0.5
0
MEK Inhibition
LUr
0
90
99
0
99.9 99.99
(M)
1
0
0
0
0
90
99
0
99.9 99.99
(N)
1
-1
90
99
0
99.9 99.99
(0)
1
(P)
C
-
0
0
0
90
99 99.9 99.99
0
90
99
99.9 99.99
C:
0.5
0.5
0.5
0
0
0
90
99 99.9 99.99
0
90
99
99.9 99.99
%Target Inhibition
Figure 2-5: Representative Target Behaviors: Depiction of the leverage provided by intervention at the
area in the 4 variants of
four different targets (EGF-EGFR, Raf, MEK, ERK) to the output signal ERK-pp
the simulation result
the pathway (one normal and 3 cancer) modeled in this work. The blue line represents
lie to the right of
that
lines
from the analysis of the target, and the black line is the linear reference. Blue
(D) The first row
the green line represent sub-linear behavior and to the left represent super-linear. (A) resulting
simulations
shows
shows simulations resulting from the normal model. (E) - (H) The second row
(Cancer Variant
from the variant model with overexpressed EGFR and deregulated endocytosis mechanism
in hydrolysis
mutation
with
I). (I) - (L) The third row shows simulations resulting from the variant model
resulting
mechanism of Ras-GTP protein (Cancer Variant II). (M) - (P) The fourth row show simulations
once
inactivate
to
ability
compromised
from the variant model with Raf mutation where the protein has
activated (Cancer Variant III).
a very large
MEK-pp is needed reach this sensitive region. For variant II and III, however,
to
change in MEK-pp level is needed to enter the regime where ERK-pp level is sensitive
these changes.
43
Signal Transduction Dynamics between MEK-pp and ERK-pp
1012
1010
J/
17
10 i8
10 6
Normal Model
Variant I Model
Variant 11 Model
Variant Ill Model
I
10
LU
100
10
10
.2
-40
2
0
-
-
10 4
106
10 8
1010
MEK-pp Area
Figure 2-6: Signal transduction dynamics between MEK-pp and ERK-pp: This figure evaluates relationship
between cumulative MEK-pp dynamics (MEK-pp Area) and cumulative ERK-pp dynamics (ERK-pp Area).
The plot was obtained my changing parameter 'k42' in the model over a span of six orders of magnitude, three
orders on either side of its nominal value. The red, green, magenta, and black asterisks show the MEK-pp area
and corresponding ERK-pp area values in normal, variant I, variant II, and variant III models respectively.
The figure shows that the relationship between these two proteins in the cascade is in responsive region of the
curve for normal model (red asterisk) and the variant I model with EGFR overexpression and endocytosis
defect (green asterisk). However, the variant II model with Ras protein mutation (magenta asterisk) and
Variant III model with Raf protein mutation (black asterisk) shift this relationship to the saturated region
in the curve. This explains with inhibiting MEK shows super-linear response in the normal model, partly
sub-linear, partly super-linear response in the variant I model and completely sub-linear behavior in the
variants II and III.
2.3.6
Parameter Variability Analysis
Effect of parameter variability on the target behaviors was studied using model ensemble
with varying kinetic parameters and initial protein concentrations (see Methods for details).
44
10,000 parameter sets were sampled and subjected to a signaling test (see Methods for
details). Only about one third of the parameter sets (2029 of 10,000) passed the signaling
test, resulting in a ensemble consisting of 2029 models. The signaling test criterion ensures
that the chosen models have some physiological relevance as there is little point in attempting
to block non-signaling pathway. For each of these models in the ensemble target behaviors of
14 targets were classified as sub-linear, super-linear, or ambiguous (somewhere in between)
using a three point classifier. The details of how this classifier was constructed is describes
in the methods section.
The results from this analysis are shown in Figure 2-7. Each column represents a target
and the height of the column represent the percentage of models in the ensemble showing
a particular target behavior. The three plausible target behaviors are represented by three
different colors and stacked on top of each other to make each of the column to add up to
100%. The first color represents the percentage of models that show a particular targetbehavior to be sub-linear. The second color represents the percentage of models that show
a particular target-behavior to be partly-sub-linear, partly-super-linear and the third color
represents the percentage of models that show a particular target-behavior to be super-linear.
An important result here is that, except for the last two targets, which correspond to
MEK and ERK, most of the targets are predicted to be sub-linear by most of the models.
This agrees with our results from analysis of these targets in normal and three cancer variant
models.
Of the remaining two targets, MEK is partly sub-linear, partly super-linear and
mostly in-between. ERK remains mostly super-linear but a significant fraction of of goes to
in-between state and some show sub-linear behavior. Overall, ERK is still the best target
followed by MEK. However, given the uncertainty in the parameters, Raf, Sos, GAP and
(EGF-EGFR) monomer have very similar target behaviors on whole. Cytosolic Ras shows
a more favorable overall behavior than these targets.
45
(B)
Target Behaviors Summary for Parameter and Initial Concentration Variability
120
Sub-Linear
Ambigious
Super-Linear
1"
100
80
"
0
Og
I
60
40
0
20
I"
0
'2
t'2
,0
'7
U~
0
C7
U4~(
'
0
'79
Targets
Parameter Variability and Target Behaviors: The parameters (kinetic rate constants and
initial protein concentrations) were sampled from log-normal distribution with the mean value set to the
value published in [94], and standard deviation of 0.5. This means that 95% of sampled parameters are within
10 fold of the published parameter values. The models used for the above analysis satisfied a criteria that
they should signal at least half the cumulative response at ERK-pp level. Here, x-axis is the names of the
Figure
2-7:
14 plausible targets that are analyzed in this paper. Each stack in each of the bar represents the percentage
of model with a given target behavior. Cyan here refers to the percentage of selected models with sub-linear
target behaviors, purple refers to percentage of selected models that show behavior in between sub-linear
and super-linear (labeled ambiguous here), and magenta refers to the percentage of selected models that
show a super-linear target behaviors.
46
2.4
Discussion
Other discussion of the problem of drug target identification have focused on the structural
and chemical properties of potential targets that suggest a tight-binding ligand can be discovered [73, 108, 69]. This so-called "druggability" approach provides a molecular imperative
from the perspective of potential binding affinity, which is certainly important. Here we
complement that view by considering the perspective of how inhibiting the target affects
relevant biological function. We address this problem by evaluating the leverage provided by
intervention at a target to a downstream output. This method can be paired with "druggability" approaches to select targets that are desirable for biological as well as physicochemical
reasons. Our analysis reveals that targets at different places in the network do exert different
effects at a downstream output. Some of the targets are dramatically sub-linear while others
show more potency and might make better interventions points by this criteria.
The IC 50 value gives the quantity of inhibition necessary to block half of the target
activity. Often quantities much larger than the in vitro IC 50 measured in a target enzyme
assay are required to reduce the activity by half in cell-based assays, animal models, or
patients. While in some cases poor bioavailability or reduced affinity in the biological context
may contribute to the apparent reduction in inhibition potency, the current work suggests
that non-linear signal propagation dynamics may also be at least partially responsible. More
detailed measurement and analysis of assay results may be useful to distinguish among effects.
We emphasize that the use of computational methods and quantitative mathematical
modes of the biochemical pathway is not critical for the key results that we observe in this
analysis. Our fundamental observation -
that proteins in different places in a biochemical
network exert different strength effects in the network context -
could equally be demon-
strated using experimental approaches, such as mass spectroscopy or quantitative antibody
assays. Because models are approximate and may omit unknown but important reactivity,
experiments benefit from an inherent correctness. However, the computational approach
may be more efficient and has the advantage of providing insight into why the relationships
47
between target inhibition and output are as observed. The current study illustrates the role
of signal saturation in its ability to attenuate a downstream output.
Saturation observed downstream in a biochemical network could be due to a single saturated step upstream; it is not necessary that every stage in the cascade is saturated. In fact,
in exploring the signal propagation dynamics between MEK-pp and ERK-pp to evaluate why
the target behavior for MEK intervention changes in the normal and the variant models (Figures 2-5C, 2-5G, 2-5K, and 2-50), we discovered that changes in MEK-pp level can change
the dynamics in ERK-pp level in the normal and to a lesser extent in the variant I model
but not in the variant II and the variant III models. Hence, the saturation between MEK-pp
and ERK-pp levels exists only in the over-stimulated cancer variant models. However, it is
not possible to change the MEK-pp dynamics and thus the ERK-pp dynamics in the normal
model by just changing the EGF signal. The distinction between these two saturations is
important in our analysis. Saturation of various stages of signal transduction to the original
EGF signal (Figure 2-3) tells us that our results are independent of the EGF stimulus as
long as it is in a broad physiological range. The second saturation between MEK-pp and
ERK-pp, which only exists in over-stimulated models, is a reason for why MEK target behavior varies between models. The insight here is that targeting after, rather than before, a
saturation step could be generally advantageous in shutting down signaling network.
Given the large scale of experimental data required to calibrate large biochemical models,
we appreciate that models of appropriate detail exist for only a small number of biochemical
and disease processes, and doing the exhaustive analysis like the one presented here, though
possible, may not be currently feasible in an experimental setting. One possibility is that the
study of a few well characterized networks will lead to a catalog of advantageous intervention
strategies that can be applied to new situations with less well characterized models and in
which all the mechanistic details might not have been worked out but the overall nature of
signal propagation is known.
Inhibition of EGF receptor is a clinical intervention used to attenuate over-stimulated
48
signals of this pathway [25].
A variety of drugs and drug candidates has been developed
that target proteins within the EGFR family and downstream of it. Most are competitive
analogs of ATP that aim to regulate the pathway by decreasing the amount of EGF-EGFR
dimer available for phosphorylation and thus reducing signal propagation downstream; some
are antibodies to the ligand binding domain. A non-exhaustive list of currently available
(or underdevelopment) drugs for this pathway is summarized in Table 2.1. EGFR family
targeting has been a generally successful mode of therapy for a number of cancers, and the
fact that they show sub-linear behavior may be less important as long these concentrations
are not toxic.
Alternative targets have been explored for the cases where EGFR family
targets have shown limited effectiveness. Recently (in 2013) MEK inhibitor trametinib was
approved by the Food and Drug Administration (FDA) for tumors with the BRaf V600E
mutation (akin to our third cancer variant). Our analysis here shows that despite both being
sub-linear, MEK is a better target than EGFR because it achieves its effect at a relatively
lower level of inhibitor concentration.
There are a number of considerations in applying these results in a translational setting.
Here the output of the system was a molecular entity within the pathway, which was used
as a proxy for disease process. A more direct phenotype of disease, like cell proliferation,
differentiation or cell death, may be desirable to evaluate the therapeutic effect of an intervention. Furthermore, in this study target inhibition was treated as competitive and the
inhibited complex was treated as completely inactive; the only reactions it participated in
were inhibitor dissociation and degradation. This was useful in the current study to probe
the range of target behaviors and to examine similarities and differences across related models. For more detailed target identification studies, non-competitive inhibition should also be
considered, and inhibition of individual functions of candidate targets should be studied. For
example, one candidate mode of inhibition could prevent EGFR dimerization and another
could interfere with catalysis. Such studies at the systems level could lead to very detailed
prescriptions for what should be achieved at the molecular level in order to reach desired
49
therapeutic goals, which is an important goal of systems medicine.
Target
EGFR
Drug Name
Cetuximab
EGFR
Panitumumab
EGFR
Erlotinib
Mechanism
Monoclonal anti-EGFR antibody:
compete for the ligand binding domain of the receptor.
Monoclonal anti-EGFR antibody:
compete for the ligand binding domain of the receptor.
Compete with ATP:
bind the catalytic kinase domain.
Gefitinib
Compete with ATP:
EGFR
Lapatinib
bind the catalytic kinase domain.
Compete with ATP:
bind the catalytic kinase domain.
MEK
BRaf
trametinib
Dabrafenib
MEK inhibitor
BRaf inhibitor
EGFR
Table 2.1: A sample of intervention strategies that are currently available (or under development) for down
regulation of EGFR pathway [25, 70].
50
Chapter 3
Computational Approach to Analyze
Drug Combination for Synergy and
Antagonism
Abstract
Given limited success of single agent targeted therapy in treatment of complex network disease like cancer, either because of emergence of drug resistance or due to a narrow therapeutic
windows, combination therapy has emerged as a potential way to circumvent some of these
shortcomings. An attractive idea within the field of combination therapy is that when two
or more potential targets are inhibited simultaneously, their desired effects can interact and
propagate in such a way that the resulting effect observed is much greater than that would
be observed by each drug alone after adjusting for differences in the total amount of drug in
the system. This idea is often referred to as drug synergism. Choosing target combinations
that can provide this kind of synergistic benefit is, however, a challenging task. In the work
presented in this chapter we use a computational framework to study combination effects of
every possible target combinations in a biochemical network that is modeled using a systems
of ordinary differential equations (ODEs).
In particular, we explore the range of combination target behaviors that exists within
epidermal growth factor receptor (EGFR) pathway by exhaustively inhibiting all the possible
sets of two target combinations systematically. We carefully evaluate what it means for two
targets to be additive, antagonistic, and synergistic based on the work of Ting Chao-Chou
(Chou TC, Talalay P (1984), Advances in enzyme regulation 22: 27-55) on inhibition of
enzymes by two or more drugs simultaneously. These careful definitions of combination behaviors reveal that most drug combinations are additive. Synergistic and antagonistic target
51
combinations are rare to find. Even in the cases where targets are classified as synergistic
or antagonistic, they show these behaviors only in a small range of inhibitor concentrations.
Analyzing synergistic targets more closely reveals that a lot (but not all) of synergistic targets
are binding partners of each other. Further there are a few targets (three in this particular
pathway that we have analyzed) that show synergistic behavior in combination a lot of the
other targets.
52
3.1
Introduction
Signaling protein networks within cells have emerged to be key factors in determining normal
and diseased cellular phenotypes. Deregulation of the parts of these networks cause most of
the clinically observed human cancers [46, 55] among other diseases. This insight has meant
that quantitative understanding of these networks is crucial in diagnosis and treatment of
these diseases. Computational and mathematical approaches are widely being used [83, 32,
39, 63, 98] in these efforts to quantitatively understand and evaluate the signaling networks.
A common mathematical tool that is employed to understand the signaling nature of
biochemical networks at mechanistic level of detail is the system of ordinary differential
equations (ODEs). ODE based modeling framework is capable of capturing how each protein
or protein complexes in the network evolve over time under a given input.
This means
available experimental data on the systems can be directly used to calibrate the model of
interest. This feature allows these models to be compared to experimental data while also
making them amenable to use the available experimental data to better define the models.
Well calibrated models of networks of interest can then be used in predictive realm to guide
therapeutic interventions strategies that are able to address and correct for the deregulated
network phenotypes holistically.
In this work we use an ODE based model to computationally evaluate the effect of
combinatorially inhibiting two plausible targets simultaneously. We quantify whether the
combinatorial inhibition of two targets is better than inhibition of one or the other target
alone at equivalent inhibitor concentrations. We define terms and metrics to quantitatively
evaluate whether the combinations are additive, synergistic, or antagonistic. We analyze the
general trends of combinations that exists in a epidermal growth factor receptor (EGFR)
network [94]. We develop approaches to derive insights into features of network properties
that make for additive, synergistic, or antagonistic target combination.
Furthermore, we
evaluate these properties and behaviors by encompassing the biological uncertainty of the
biochemical model used in this work. This is an important aspect in being able to make a
53
holistic decision about target behaviors. Decisions about target selection have to be made
in the face of, often, uncertain and conflicting information about biology. It studying the
combination behavior for a range of model parameter values, we encompass some of this
uncertainty in design and decision process.
It is important to point out that, although
the we present this work in the context of EGFR pathway, the strategy developed for the
analysis is generally applicable to any biochemical network that is modeled quantitatively
with a system of ODEs.
3.1.1
Biochemical Model
We develop and apply the combination target analysis in the context of the epidermal growth
factor (EGF) receptor signaling pathway. It is an extensively studied biochemical signal
transduction pathways with a number of well established mathematical models that are
calibrated and supported using experimental data [18, 83, 62, 90].
Further, it is also a
pathway that is of great clinical significance. Many proteins in this pathway are shown to be
mutated in cancer, leading to a overstimulated nature of signal transduction in the network.
The relatively well-calibrated nature of the model makes it a suitable candidate for our
study here that is concerned with using detailed network-level understanding of biochemical
processes to guide combination target selection strategies. The particular version of the
model that we use in this work was initially developed by Schoeberl et al [94], later updated
by Hornberg et al [50], and further modified in our group by Apgar et al [6]. The pathway
is modeled by a system of ordinary differential equations (ODEs). Each ODE in the model
describes how a particular protein or protein complex in the pathway evolves over time in
the presence of an EGF stimulus. The integration of the system of ODEs then gives the
temporal dynamics of concentration of all the proteins and protein complexes in the pathway.
The biochemical pathway is shown schematically in Figure 1-1. It is initiated by the binding of EGF ligand to the trans-membrane EGF receptor (EGFR) [11, 89]. For the purposes
of this work, we consider the EGF ligand to be an input to the system. Upon ligand binding
54
the receptor can dimerize and autophosphorylate [11, 89, 31].
This autophosphorylated,
hence activated, EGF-EGFR dimer can recruit and activate a number of adaptor proteins
by providing suitable docking sites. The sequential activation of the adaptor proteins eventually leads to the activation of Ras protein [16] and of a canonical mitogen-activated protein
kinase (MAPK) cascade composed Raf, MEK, and ERK [78, 74] proteins. ERK protein is
the final protein as modeled, with ERK-pp representing a doubly phosphorylated and activated form. This activated ERK protein (ERK-pp) is treated as the output of the pathway
here. ERK-pp is itself an important signaling molecule; it is a kinase with a large number of
substrates in both the cytoplasm and the nucleus and can also act as a transcription factor
to activate a number of growth and proliferation related genes [74, 17].
3.1.2
Format of Study
In this work, we formulate the question of target identification for combination therapy from
a network perspective by evaluating a combination metric between two targets. This combination metric forms the basis for classification of target behaviors into additive, synergistic,
or antagonistic category. To evaluate the combination metric, for each pair of targets inhibited, we evaluated the effect of this simultaneous inhibition on the model output (ERK-pp).
We then compared the concentration of combinations of the two inhibitors needed to produce
a particular output (ERK-PP) effect to the concentrations that would be needed if only one
the targets were inhibited instead. This comparison allowed us to quantitatively evaluate
if the combination was better, worse, or the same as inhibiting just one target, forming a
basis for classification of synergistic, antagonistic, or additive combination behavior. This
approach is shown schematically in Figure 3-1B. Figure 3-1E graphically shows the combination metrics that we use to classify a combination as additive, or synergistic, or antagonistic.
The green line in the figure represents an additive target, the blue line represents a synergistic target, and the other three lines represent (orange, magenta, and red) all fall under
antagonistic targets. These definitions are adapted from the work of Chou et al [23].
55
There are a total of 31 plausible single targets (detailed identities of these targets are
given in section B.1) in this pathway resulting in 465 combination pairs in this analysis. The
combination metrics provided a way to evaluate the overall trend in the nature of additive,
synergistic, or antagonistic targets in the pathway. We studied the nature of signal flow in
the network to dissect and eventually explain some of the trends. In particular our analysis
shows that there are three proteins or protein complexes that act as synergistic with almost
all of the other targets. We explore one of these cases (case where EGFR is synergistic with
a lot of other targets) in a greater detail.
3.1.3
Summary of Results
There are some key findings in this work that are worth highlighting here. In the published
model of the EGFR pathway [94, 50, 6] most of the target combinations showed an additive
behavior (Figure 3-4).
However, there are also non-negligible number of synergistic and
antagonistic targets. Most of the synergistic targets are binding partners of each other (but
not all the binding partners are synergistic). On the other hand, most of the antagonistic
targets are one of the reactants and the downstream product it forms.
of synergistic targets, the reverse does not hold for antagonistic targets.
Like in the case
That is to say
that not all the combinations where one of the reactant and the downstream product are
inhibited simultaneously show antagonistic behavior, but most of the combinations that show
antagonistic behavior adhere to this rule.
The overall trends of additive, synergistic, and antagonistic target behaviors deviated
fairly under the parameter variability. We studied the effect of parameter variability in the
model using two different methods (details in section 3.2). In the first method we created
three specific variants of the model with over-stimulated signaling dynamics in the network
and studied the combination target behaviors in these three models. We call these overstimulated or cancer variant models.
In the second method we sampled each parameter
in the model from a pre-defined distribution to model the uncertainty in each of the pa-
56
rameter in the model and selected an ensemble of models to study the combination target
behaviors on. These methods show that targets which were synergistic in the published
model remained synergistic, but the targets that were additive shifted to being synergistic,
and the targets that were antagonistic shifted to show additive trends in the models with
over-stimulated signaling in the pathway. We note here that the criteria used for selecting
model ensemble (second method that was used to study the effect of parameter variability in
pathway combination behavior) biased the selection of models with over-stimulated signaling
dynamics. This is justified choice because the cancer therapies in general (either single agent
or combination) are relevant to the over-stimulated variants of the EGFR pathway studied
here.
3.2
Method
The ODE pathway model for signaling downstream from EGFR utilized in the current study
evolved from the original model by Schoeberl et al. [94], which was modified by Hornberg
et al. [50] and further updated by Apgar et al. [6].
Here our model is the Apgar et al.
[6] version. The model has 13 unique proteins that comprise 101 unique chemical species
through the formation of complexes and catalytic modifications. Model dynamics are driven
by 163 elementary chemical reactions that are described using mass-action kinetics. A feature
of mass-action kinetic formulations is that they contain only zeroth-, first-, and secondorder reactions; all higher-order abstracted reactions are written as a series of these more
elementary ones. Parameters of the model include 107 distinct rate constants and 101 initial
concentrations; in addition, there is 1 input (EGF).
3.2.1
Combination Behavior Definitions
There is a generally accepted working guideline for vocabulary used in combination target
behaviors analysis [23].
Target combination behavior can be broadly categorized into 5
57
different classes - namely - additive, synergistic, antagonistic, independent, and suppression.
Additive refers to the case where a linear combination of the two drug concentration leads to
same overall effect in the system. Synergy refers to the case where the presence of one drug
in the system reduces the amount of second drug required to get the same overall effect at the
output compared to a linear combination. Said differently, the total amount of drug needed
to achieve a particular output effect is less when the two drugs are given in combination
than when one or the other is given alone. Antagonism is a scenario where the total amount
of drug needed to produce the same overall effect on the output in combination is more than
that would be needed if either of the drug was given alone. These are the three broad classes
of target behaviors. Independent and suppression are special cases of antagonistic behaviors.
Independent refers to the case where the presence of first drug in the system has no bearing
in the amount of second drug needed to produce the same overall output effect. Suppression
refers to a scenario where the first drug counteracts the effect of the second drug. Hence,
the presence of the first drug in the system means that the more of second drug is required
compared to the case where the second drug was given alone. These definitions are shown
schematically in Figure 3-1E. The green straight line refers to the additive behavior, the blue
line represents a synergistic behavior, the orange line refers to the antagonistic behavior, the
black line refers to the independent behavior, and the red line refers to suppression.
In the work presented here, we first categorized the combination behaviors with slight
modification to these general definitions such that we could get a sense of what was happening
in the system qualitatively.
For this scenario, instead of looking at the combinations of
drug concentrations needed to achieve a particular overall output effect, we enumerated the
overall effect on the output for all combinations of the two drugs.
The simulations were
designed such that each drug was administered at 100 different log-spaced concentrations
going from 6 molecules/cell to 6e8 molecules/cell. So, we calculated the overall output effects
at 1e4 possible concentration combinations. We then evaluated how this overall output effect
metric (or image) varies along with constant overall drug concentration in the system. In
58
(A)
E(G
00i
0i)
EGFEGF
Extracellular
No EGF
No Inhib
Cytoplasm
m
m
-n
2 Inhibitors ,
OF
Endosome
.
RAF
...
C
RAF
(E)
X 106
Normal Model
h.
Drug 1
8
Drug 2
6
:3
2
0
0
x
Combined
1000 2000 3000 4000 5000
Time (s)
0
) (Drug_1/Drug-x1)
Figure 3-1: Overview of target combination evaluation strategy. (A) Schematic representation of
the EGFR signaling pathway. (B) Schematic method used to study the effect of simultaneous inhibition of two
targets on the output. (C) Dynamics of the pathway output (ERK-pp) upon stimulation with step increase
of 8 nM at time zero in the presence of no inhibitor. (D) The target behavior is evaluated for the cases where
the total effective inhibitor (drug) concentration in the system is constant, but this total effective inhibitor
concentration can come from drug 1 (blue triangle) alone, drug 2 (red) alone, or some combinations of the
two drugs. The vertical green lines show a representative sampling strategy for maintaining constant total
effective inhibitors concentration. (E) Traditional approach to looking at the combination target behaviors
and the classes of possible combination behaviors. Each line represents the relative amount of drug 1 and
drug 2 required to achieve a fixed effect in the system output. The green line represents an additive target,
blue line represents a synergistic target, orange line represents an antagonistic target, magenta represents
independent targets, and red represents suppressive targets.
this setup, additive targets are the ones where the overall output effect remains the same
along the lines of constant total concentration (Figure 3-2A, D). A synergistic targets are
the ones where the overall effect at the output changes along the lines of the constant total
concentrations such that the higher overall output effects are achieved at some combinations
of the drugs concentrations (Figure 3-2B, E). An antagonistic target, on the other hand,
is the one where the overall effect at the output changes along the lines of constant total
59
concentrations such that the lower overall output effects are achieved at some combinations
of the drug concentrations (Figure 3-2C, F) compared to the effects observed at the edges
in the presence of one of the two drugs.
Combination Outcome - Linear Scale
12
-J
12
6
6
2
2
2
4
6 8 10
Drugi Levels
12
2
4
6 8 10
Drugi Levels
12
2
4
2
4
6 8 10
Drug 1 Levels
12
10
12
Combination Outcome - Log-log Scale
12
1
'A
0)
2
4
6
8
10
12
2
4
6 8 10
Drugl Levels
12
6
8
Figure 3-2: A general definition for combination target behaviors: This is an expansion of the
more commonly used definition of the drug target combination behaviors described in Figure 3-1E. Here, we
evaluate the output effect of inhibitor for every possible combination of two drug levels and then evaluate
the output effect along the lines of constant total effective inhibitor (over-layed black lines) in the system
to categorize the targets into additive (first column), synergistic (second column), or antagonistic (third
column). The data shown here is pure for illustration purpose. In each of the sub-figure x-axis shows the
drug 1 inhibitor levels, y-axis shows the drug 2 inhibitor levels. The color metric goes from 0 (blue) to 1
(red) linearly and represents an output effect.
Within these modified guidelines, we do not have a way of clearly identifying independent
and suppression behaviors. They fall in a more general category of antagonistic targets. The
choice of these modified definitions allows for drug combination to show more than one
60
combination outcome at different levels of the total effective inhibitor concentrations in the
model. Stated differently, the outcome of the two drug combination could be additive in
one part of the inhibitor regime, synergistic in the second regime, and/or antagonistic in the
third. This allowed for a more comprehensive evaluation of the combination behaviors arising
from inhibiting two plausible drug targets at varying levels of inhibitor concentrations. It is
worth noting that these modified definitions are nothing more but a different way of looking
at (or evaluating) the concepts of combination behavior by more traditional definitions [23].
We return to the traditional definitions when we want to develop a quantitative classifier to
identify additive, synergistic, or antagonistic targets (section 3.2.4).
3.2.2
Drug Intervention Models
This is an extension of our earlier work in developing a quantitative method for finding
better and worse places for single target inhibition in Chapter 2. In the project described
in chapter 2, we explored a more typical scenario, where each target referred to a particular
binding pocket in the protein and, hence, a target referred to a protein and the complexes
it formed with other proteins. In this work, we explored a more specific scenario where each
protein or protein complex can be treated as a separate target. This distinction accounts
for 14 plausible targets in our single target work (chapter 2) and 31 plausible targets in
the combination work described here. So, the question that we asked here is that if one
could selectively disable one of the many functions of a protein (designing inhibitors with
this level of specificity has been considered within drug design communities) what particular
functions of the proteins or the proteins should we choose to inhibit in combination for a
desired outcome.
We identified 31 unique protein and protein complexes that could be inhibited in this
pathway. This resulted in a total of 465 (31 choose 2) possible combination behaviors to
analyze and evaluate. 465 variants of the original model were created, each containing inhibitors for 2 of the 31 plausible targets in the model. As with our earlier work on single
61
target (chapter 2), in the work reported here each target bound an inhibitor in a secondorder reaction to form an inhibitor-target complex that was completely inactive and unable
to contribute to downstream signal. The inhibitor-target complex was either allowed to
dissociate back to the target and the inhibitor or degrade at the same kinetic rate as the
degradation rate of the target protein. The inhibitors and relevant reactions were augmented
in the model such that the inhibitor levels were present in the system at fixed concentrations
and did not vary with time. This aimed to simulate the effect of a large volume of drug
present in cell culture or in circulation that replenishes drug that binds to target maintaining
a constant concentration at the site of action within a cell. Two new parameters were introduced in each model variant for the k0 , and kff for the inhibitor binding to target. Values
of 1.66 x 10-6 cell molecule-- s-1 and 1 x 10-3 S-1 were used for second-order association
rate ( kon ) and first-order dissociation rate ( koff ), respectively. These values convert to
1 x 106 M-
1
s-1 for the association and 1 X 10-- 3 s1 for the dissociation rate constants using
typical dimensions of a mammalian cell (1 x 10-12 L) [94] giving a unit nanomolar equilibrium dissociation constant (kd = f).
Both the inhibitors were assumed to have the same
binding kinetics with their respective targets. The target-inhibitor complexes for both the
targets were allowed to degrade at the same rate as the degradation rate of their respective
targets in the model.
In simulations, the pathway was equilibrated in the presence of both the inhibitors before stimulation with the EGF growth signal, which is a stimulus input of the model. For
each target of of interest, inhibitor was introduced at 100 different logarithmically spaced
concentrations between 6 and 6 x 108 molecules cell- 1 , which corresponds to a maximum
concentration of 1 mM using typical dimensions for a mammalian cell [94]. This means that
each model was simulated for 100 x 100 = 1e4 different concentration combinations of the
two inhibitors introduced in the system simultaneously.
For each combination concentration of the two inhibitors introduced, output signatures of
interest were measured and compared with the case where the intervention was not present in
62
the pathway. A schematic of this process is shown in Figure 3-1 B, where stage (i) represents
the model with no EGF stimulus and no inhibitor, stage (ii) represents the pathway behavior
when the model has been equilibrated in the presence of both the inhibitors at predetermined
concentrations, but no EGF stimulus (input) is present, stage (iii) represents the model in
the presence of EGF stimulus without any intervention, and stage (iv) represents the system
with both the interventions (inhibitors) in the presence of EGF stimulus. Simply stated, we
equilibrated at (i) and used that as a starting state for a type (iii) stimulation; likewise, we
equilibrated at (ii) as a starting point for a type (iv) simulation. The effect of the combined
inhibition was evaluated by comparing ERK-PP signal (area under this signal curve for 5000
seconds) in state (iv) with that obtained in state (iii).
3.2.3
Target and Output Effect Metrics
The focus of this study is to quantify the potential differences between intervention free model
and models with interventions for two of the plausible targets at the same time. To evaluate
these effect, we chose two metrics - target effect and output effect. Target effect is evaluated
for both the targets inhibited in the model and the resulting effect of the two inhibitors on
the model output is evaluated as the output effect. These are the same metrics used in our
earlier work in understanding single target behavior (Chapter 2), in this work they have been
extended to two target inhibition simultaneously. These metrics are mathematically defined
as:
Target Effect
[I] + Ki
(3.1)
Where, [I] = inhibitor concentration, Ki = inhibitor binding affinity.
Output Effect = (unperturbed output - perturbed output)
unperturbed output
63
(3.2)
Where, "unperturbed" represents model without any inhibitor and "perturbed" represents
model with both the inhibitors present in within the model.
3.2.4
Combination Summary Metric
Overall pattern in combination behavior of the targets in this network was analyzed by
summarizing the combination behavior matrix resulting from introduction of each of the
inhibitor at 100 different levels by a single metric. The motivation for this metric came from
the need to summarize the rich behaviors observed by inhibiting two targets at different
inhibitor level on the output of the system (Figure 3-5, Figure 3-6) into a single number
such that the combination behaviors of all the targets in the pathway could be observed
holistically rather than looking a one surface plot like in Figure 3-5 or Figure 3-6 one at
a time. This summary metric was equally needed for practical reasons of evaluating large
number of combination target outcomes as it was not possible to do so by manual inspection.
Given a output effect of two target combination behaviors, seven contours of output
effect at effect levels of 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 were extracted. For each of these
contours, drug 1 and drug 2 levels that defined the contours were extracted from surface
plots like in figures 3-2 and 3-6 and normalized to the scale of 0-to-1. Each extracted point
contained two values - amount of the first inhibitor, and the amount of the second inhibitor
needed to produce an output effect of interest. These values of inhibitor 1 and inhibitor 2 at
each point was normalized by the amount of inhibitor 1 alone needed to produce the output
effect level of interest, and the amount of inhibitor 2 alone needed to produce the output
effect level of interest respectively. A straight line was drawn between the first and the last
pair of points. This straight line describes the contour of the extracted points if the targets
under analysis were in fact additive.
Now, to evaluate how the actual contour points deviated from this straight line, we
evaluated the area between the additive straight line and the actual contour line using
a trapezoidal method (matlab: trapz(straight additiveReference - actualicontour-points)).
64
This means that if the extracted actual contour points are smaller than the corresponding
points from additive reference line (synergistic case, Figure 3-1E) then this metric has a
positive value. Further, because the reference additive line traces a triangle with area of 0.5,
the maximum positive value that can be obtained using this approach is 0.5.
In the case
of antagonistic combination behavior, the contour points are larger than the corresponding
points from the reference additive line, and hence the trapz(straight-additiveReference actual-contour-points) results in a negative number. Because there is no bound on where
this contour could extend, there is no lower limit in the case. In ideal case, value between
0 and -0.5 would be simple antagonistic, value of -0.5 would be independent, and value less
than -0.5 would be suppression case. This combination metric is shown schematically in
figure 3-3. Here, the red area represents combination metric for each effect level of interest.
These strict theoretical cut off are not ideal in reality. Hence, we decided to differentiate
only negative and positive values allowing us to differentiate synergistic or partly synergistic
targets from additive and antagonistic targets. This process is repeated for all the 7 different
effect levels and an average values is taken to summarize the overall target behavior for every
possible target combinations (465 in this model).
3.2.5
Parameter Variability Analysis
We took two different approaches to analyze target combination behaviors in the presence of
variability in model parameters. The first method involved building ensemble of models of the
biochemical network with range of plausible values for each of the parameters in the model.
This ensemble of models is the same as the one used in the case of single target behavior
in section 2.2.6.
Three target combinations with additive, synergistic, and antagonistic
behaviors as evaluated in the originally published normal models were evaluated again for
their combination metric values in each of the model in the ensemble. Then a distribution
of each of these combination pairs was used to evaluate the distribution on target behaviors
that each pair may exert depending on the exact nature of the model which is hard to know
65
Area = Combination Metric
Area = 0: Additive
Area >0 : Synergistic
Area <0: Antagonistic
0
eN
0
0
D1/D_x1
The red area is the metric that is defined as combination metric to classify target behaviors.
producing
The x-axis and the y-axis represent the normalized amounts of inhibitor 1 and inhibitor 2 used in
line
curved
the
and
behavior,
additive
an output effect. The straight line with slope of -1 represents an
interest.
of
effect
the
represents the actual combination of inhibitors that produces
Figure 3-3:
exactly a priori. This work focused on analysis of this ensemble behavior only for three
combinations because of the computational constraint carry the same calculations on all
possible combinations.
The second approach evaluated the combination target behaviors for all the plausible
combination pairs for three variants of the normal model. These over-stimulated models
that we described in section 1.3.2 can be thought of as variants of the published normal [94]
with different parameter/concentration values.
3.3
Results
In this work we focused our efforts in exploring the range of combination behaviors that
exists between all plausible targets in EGFR pathway. These behavior range were quantified
by evaluating the effect of the combined interventions of two targets simultaneously on the
dynamics of ERK-pp protein, which we treat as an output of this pathway. In particular we
66
evaluated the integrated response of the ERK-pp protein over 5000 seconds as a metric for
drug effectiveness. We call this metric the output effect. Drug effectiveness (output effect)
values for different combinations of inhibitor concentrations are then collated together to
evaluate how the combination affects the output compared to inhibition of one or the other
target alone (section 3.2.4). Further, we have made an attempt to dissect and understand
some of these trends in detail to learn underlying network features or signal propagation
dynamics that may describe the combination outcome of the two targets.
3.3.1
General Trends
The range of target behaviors that we observed in analyzing the combination behaviors of this
biochemical pathway fitted within our definitions of additive, synergistic, and antagonistic
combination behaviors. These definitions are explicitly demonstrated in Figure 3-2. Of the
465 target behaviors explored, majority showed additive behaviors, a small but non-negligible
targets combinations showed synergistic behaviors. Likewise, there were decent number of
target combinations with antagonistic behaviors.
Qualitatively antagonistic, independent,
suppression (Figure 3-1 E) are subtly three different classes of target behaviors where the
total concentration of inhibitor needed to produce an output effect in combination is more
that would be needed if this amount of inhibitor was given to one of the two targets alone.
However, our current metric cannot distinguish this subtle difference and classifies all three
as antagonistic targets.
A summary of the combination outcome for all the target combinations in EGFR pathway
is shown in Figure 3-4. The two axis represent the corresponding targets in the model and the
color metric is the combination metric that was used to summarize the combination behavior.
This metric can range from -oc
(in theory) to +0.5 (see section 3.2.4). The negative values
say that the total effective inhibitor needed in the system to produce an output effect is higher
in combination compared to when the inhibitor is given as one of the two drugs alone at the
same effective concentration (linear gradient of the color from blue to white). This represents
67
an antagonistic combination behavior. Color white corresponds to combination metric value
of 0 and represents additive combination target behaviors. The gradient from white to red
represents combination metric values from 0 to +0.5 corresponding to a synergistic target
behaviors of various strengths. Here, the effective inhibitor concentration needed to produce
an output effect is less in the combination compared to the case where the inhibitors are
given as one of the inhibitors for single targets.
3.3.2
Additive Targets
Most of the (summarized in Figure 3-4) drug combinations in this pathway show additive
behaviors. A representative additive combination result is shown in Figure 3-5A. It is a
figure like this that the combination metric summarizes in a single number. But looking at
this pre-processed data helps to get a sense of what is going on when the two targets are
inhibited simultaneously. This figure results from simultaneous inhibition of Raf and EGFR
protein in the signal transduction cascade. EGFR is the receptor that initiates the signal
transduction in this pathway by binding with its ligand, EGF. Raf is the first protein of the
Mitogen-Activated Protein Kinase (MAPK) cascade. Each of these targets were inhibited
at 100 different inhibitor levels creating a matrix of 1e4 combination inhibitions. The two
axis of the plot represent actual amount of inhibitors in the systems and the corresponding
percentage of target effect or percentage of target inhibited are shown with arrows (see
'Target Effect' section 3.2.3) . The color map represents the effect of these inhibition on the
output (ERK-pp area for 5000 seconds). Blue means that the inhibitor combination had
no effect in changing the output and red means that the inhibitor combination was able
to completely block the model output, hence producing no ERK-pp signal when stimulated
by an EGF input. The black lines are the lines of constant effective inhibitor levels in the
systems and different lines represent different constant values.
There are a number of key features that can be extracted from this figure, including the
single target behaviors of the individual. First, the two targets have different potencies on
68
Overall Combination Behavior
0.5
03
Afj -
0.4
''.25
0.3
0
.9-0.
25
-0.
5
-0.
-0-
-0.4
010
0-0.5
5
10
IM -20
15
&Wee9
UL25
30
Ta rgets
Figure 3-4: Summary of all target combinations behaviors in EGFR pathway studied: Value
of 0 describes an additive target combination behavior, negative values describes antagonistic behavior
and positive value describes synergistic behavior. The magnitude of these values describe the degree of
antagonism or synergism. There is no lower limit on the negative numbers but the maximum positive
number that a combination can have is 0.5 [see section 3.2.4]. Because both the drugs are introduced in the
model simultaneously, this is a triangular matrix. The broad classes of targets are labeled in the axis and
the detailed targets and the corresponding indexes can be found in the supplementary material.
their own. About 97% inhibition of Raf protein produces a 50% effect on the output ERKPP but to get the sarme effect from EGFR inhibition, 99% of EGFR has to be inhibited
(this was focus of our work described in Chapter 2 of this thesis). A key observation that is
of interest for our current analysis is that as one tracks the lines of constant total effective
69
inhibitors in the system (the over-layered black lines) we see that the color (output effect)
on the image plot remains the same. This means that, given the constant effective inhibitor
level in the system, output effect remains the same and does not depend on how inhibitors
are distributed among the two targets. This is our definition of additive targets and hence
these two target combinations show an additive behavior. We emphasize that the overlayered black lines are the lines of constant "effective" inhibitor concentration to account
for the fact that two targets can (and usually do) have different potencies as was the case
in this particular example that we explored. A purely constant total inhibitor would not
account for this as a more potent inhibitor would mask the effect of a less potent one, hence
producing the case that the combination is always worse than one of the single target effects
at a constant concentration level.
Figure 3-5A shows a case where we can see for ourselves that the target combination
is additive. This method of tracking the output effect under constant "effective" inhibitor
concentration is a manual one presented here to make the arguments explicitly. However,
this manual method is not a viable one to evaluate all the combinations, and a automated
approach was developed for this classification (section 3.2.4). Analysis of all the plausible
combinations behaviors in this pathway revealed that majority of the target combination in
this pathway show a behavior that is similar to that of EGFR and Raf combination. The
summary of the results for all the targets in the cascade and how they behave is shown in
Figure 3-4.
3.3.3
Synergistic Targets
Although most of the target combinations in this pathway are additive (section 3.3.2), there
were some target combinations that deviated from the additive trends along a few of the
contours of constant total effective concentration levesl. An example of target combination
that shows this deviation is shown in figure 3-5B. This figure shows the combination results
arising from simultaneous inhibition of Ras-GTP protein and Raf protein in the EGFR
70
pathway. The axis, as described in section 3.3.2, is a measure of fraction of each of target
inhibited and the color surface is a measure of the effect of this simultaneous inhibition on
the area of ERK-pp protein for 5000 seconds.
In Figure 3-5B, as we follow the contours of constant total effective inhibitor concentration
we see that the overall effect obtained at the combination of the two is better than that would
be achieved if either of the inhibitor was given alone. This target combination hence falls into
our definition of synergistic target. It is important to point out that the synergistic behavior
is seen only for a few contours of constant 'effective total inhibitor concentration' and not all
of them. These synergistic behaviors are observed at effect levels where individual targets
on their own have some partial effects on the output ERK-PP (as opposed to none or full
effects).
Area ERK-PP
Area ERK-PP
99.99-
0.8
99.99
99.9
0.6
99
99.9=
02
2
~
00
n
0
0.2
2
0
M-
0
6
900.4
0.4
90
0.8
2
2
00
8
99 99.99
90 99.990
8
99 99.99
99.9
log1 O[Ras-GTP #/cell]
log1 0[EGFR #/cell]
Figure 3-5: Representative additive and synergistic targets in EGFR pathway: (A) Additive
target resulting from simultaneous inhibition of EGFR (x-axis) and Raf protein (y-axis). Each of the
inhibitor were sampled in a log-scale from 1 pico-molar to 1 milli-molar concentrations. The axis number
represents the percentage of total target inhibited. The output metric (color map) is the measure of fractional
change in the area of the ERK-pp protein in the presence of inhibitors relative to case when there were no
inhibitors in the model. The black lines are the lines of constant total effective inhibitor level. As we track
the output effect metric along each of these black lines we see that this metric (same color) is constant along
the line. (B) Synergistic target resulting from simultaneous inhibition of Ras-GTP protein (x-axis) and Raf
protein (y-axis). Each inhibitor was sampled in a log-scale from 1 pico-molar to 1 milli-molar concentrations.
Here, as we track one of the black lines in the range where the output effect is changing (the rainbow region)
we see that the output effect at some combination of the two targets produces output effect that is better
than giving either one of the inhibitor at higher concentrations.
71
3.3.4
Antagonistic Targets
Similar to the synergistic case, a careful evaluation of each combination target behavior
revealed that there are non-negligible number of antagonistic targets within this EGFR
pathway. It is important to point out that these targets that we classified as antagonistic
targets were antagonistic only in some intermediate range of concentrations of the two inhibitors of interest (as was the case with synergistic targets in section 3.3.3). In other ranges
(at very low and very high inhibitor concentrations) these combinations were still additive.
For, most practical purposes (in terms of dose selection) it is the intermediate values that
are more interesting, hence our analysis has a practical implications.
Figure 3-6 shows three different ways in which a target can be antagonistic. Figure 36A is a case that would fit the definition of a antagonistic target combination in straight
forward sense. The figure shows that as we follow the lines of constant effective inhibitor
concentration (black lines over layered on the image surface), for a number of these contours
we see that inhibition of either of the target alone produced a greater effect compared to
the case when the two inhibitors are given in some intermediate combination. Figure 3-6B
shows a second case of antagonistic target combination.
In a strict sense this is a case of
an independent target behavior where the presence of the second inhibitor does affect the
amount of first inhibitor needed to achieve a given output effect. This particular case could
equally be called an additive target as one of the targets requires infinite amount of inhibitor
to produce any output effect. By definition [24]additive targets satisfy:
D1
DX1
Where, D,
+
D~2
Dx2
=
(3.3)
= amount of inhibitor 1 contributing to the combination therapy
Dxi = amount of inhibitor 1 alone required to produce an output effect that is being attempted to achieve through the combination
D2
= amount of inhibitor 2 contributing to the combination therapy
72
Dx 2 =
amount of inhibitor 2 alone required to produce an output effect that is being at-
tempted to achieve through the combination
Here,
D,
lim D1 = Di; VD2 .
(3.4)
2 - +00
But we classify this as an independent target (and subsequently antagonistic) - first, because
if fits the definition of independent target behavior, and second, because it is an additive
one only in the limD,2 - oo.
Independent targets refer to a combination behavior scenario
where presence of the second inhibitor does not affect the amount of first drug needed to
produce a given output effect. This consequently means that in combination the second drug
is essentially just being wasted in the system without producing any effect on the output.
So, the presence of second inhibitor just increases the effective amount of total inhibitor in
the system, which fits our definition of antagonistic target behavior.
Figure 3-6C shows the third common way in which inhibition of two targets in combination can be antagonistic. Again, in a strict sense, this type of combination behavior is
called suppression. The first target on its own has, at inhibition levels of 98% is able to
completely block the ERK-pp signal producing an output effect of 1. However, the presence
of the second inhibitor in the system - the one that inhibits She protein, takes away (or
suppresses) the output effect exerted by the presence of first inhibitor in the system. Again,
this is a special case of antagonistic target behavior where combination of administering the
inhibitor by distributing it over two targets has worse effect than when the whole inhibitor
dose was given to inhibit only one of the targets.
Careful observation of the relationship between targets that showed a common relationship for antagonistic targets as was observed for synergistic targets. The target combinations
that showed antagonistic behaviors were the one of the binding partners and the downstream
complex it forms (in contrast to the synergistic case where the effect arises from inhibiting
both the binding partners simultaneously). This is neither to say that all the antagonistic
targets fall under this class nor does it imply that all the target combination that obey this
73
rule are antagonistic. The point we make is that a large subset of antagonistic targets obeyed
this relationship of being one of the reactants and the subsequent product in forms. This
observation is depicted in the figure 3-6B.
Area ERK-PP
Area ERK-PP
8
999
V0
99
)0 -C
90--
99.99
0.6
99.9
0
0
90--
0.2
2
-4
t
90
99.9
0
0.4
2
0
8
0.6
99-
0.4
22>
0
-
8
-0
0
0.2
0
2
8
99 9.99
99
90
0
99.99
99.9
log 1 O[Ras-GTP #/cell]
log 1 0[(EGF-EGFRi*)2-GAP-Grb2-Sos #/cell]
Area ERK-PP - pure
8
0.8
=99.99
U
99.9
99
.
00.
90
o
0.4
2
00
0.2
8
2
0
99|99.99
90
99.9
log10[(EGF-EGFRi*)2-GAP-SHC #/cell]
Figure 3-6: Representative antagonistic targets in EGFR pathway: All the three sub-plots in this
figure fall within our definition of antagonistic target. (A) This is an example of antagonistic target where
both the targets on their own are able to attenuate the output completely, but in combination they produce
on intermediate effect. (B) This is an example of antagonistic target where the one target does not have any
effect on the output and it does not affect the effect produced by the second drug to the output in anyway.
One could think of this as an independent target behavior. (C) This is an example of antagonistic target
where one target on its own can completely attenuate the output completely when given in high enough
concentration, the second target (Shc) on its own does not do much, but in combination in masks the effect
of the first drug. One could think of this as an example of suppression target.
74
3.3.5
Parameter Variability Analysis
Two method used to analyze the effect of parameter variability in the models are detailed in
method (section 3.2.5). The results from the first method showed that combination behavior
that was synergistic in normal published remained synergistic for most of the ensemble models generated. Combination target that showed additive behavior in the normal published
model, on average, showed a shift towards synergistic behavior. These results are summarized in figure 3-7. Likewise, antagonistic target combination, on average, showed a behavior
that was more additive when looking at the model ensemble.
(B)
(A)
Additive in normal model
20
80
15
60
V~o
40-
5
20
(C)
Synergistic in normal model
Antagonistic in normal model
15
,a
0
0
-1
0
-0.5
Combination Metric
0
0.5 -1
10
5
0
-0.5
Combination Metric
0.5 -1
0
-0.5
Combination Metric
0.5
Figure 3-7: Distribution of combination metrics for an additive, synergistic, and antagonistic targets when
the parameters in the model was allowed to vary.
Overstimulated variant models showed combination behavior that was similar to what
was observed in ensemble models. Depending on the degree of over-stimulation (see section
1.3.2) there were on average, more synergistic targets overall (figure 3-8). The fact that two
methods agree on overall behavior is perhaps not that surprising given that generation of
ensemble models (described in section 2.2.6) biases in favor of models with over-stimulated
pathway signaling akin to one of the cancer variant models. A careful evaluation of there
synergistic behaviors revealed that these targets that went from being additive in normal
model synergistic in overstimulated models, were in fact synergistic in much higher concentrations of inhibitor. Their behaviors at lower concentrations were still additive and likewise
for antagonistic targets.
75
Variant I Model: Overall Combination Behavior
0.5
30
0.4
Q
0
0.3
25
A.-0
1-
0.2
10
1
10-
20-0.3
F5
-0.4
'-0.
"-0.3
00.
"bN
5
10
20
15
25
30
Targets
Figure 3-8: Combination behavior summary for one of the three over-stimulated model variants of EGFR
pathway.
3.4
Discussion
With an emergence of combination therapy as a standard point of care for disease like cancer
[88, 35, 58], finding combination pairs that show desired outcome is an important question
in the drug development process.
Understanding the landscape of combination outcome
behavior such that combination target selection decisions can be biased towards a choice
that increases the probability of any two target combination to have a desired effect is an
important step in finding effective combination targets. In this work we exhaustively explored
76
that landscape using a computational method that uses an ordinary differential equation
(ODE) based model of a biochemical network described using mass-action kinetics.
We
showed that most target combinations are additive. Synergistic and antagonistic behaviors
are rare to find on whole. In doing so, we also demonstrate a way in which computational
methods can be used on a fairly well calibrated biochemical model to aid identification of
suitable combination behaviors.
In the work described here, the choice of the biochemical network, the output we decided
to track, and a demonstrated role of this pathway in cancer mean that we were mostly
interested in synergistic target behaviors. However,. synergism may not always be a desired
effects one aims for. The same approach we have developed here can evaluate drug toxicity
instead of drug efficacy. In such case, antagonism, where the toxic effects of two drugs are
less than additive, may be a more favorable behavior to aim for. Given that computational
methods are much more accessible and less time intensive compared to experiments, they
may provide a good first pass to narrow down the number of experiments that need to be
done to confirm observations and trends observed in these studies.
We understand that most of our analysis is based on the definition of additivity that
we chose based on loewe additivity criteria developed as combination index by Chou et al
[24]. We recognize that there are alternative definitions of what is means to be an additive
target. One of the most commonly used alternative is called bliss independence [43, 60, 36].
This uses probability like thinking to evaluate an additive combination behavior of the two
drug targets by assuming that their sites of actions work independent of each other and
are combined only at effect levels.
The later definition of additivity is preferred for its
practical application but has often been criticized for its lack of statistical rigor. The former,
on the other hand, has a sound mathematical basis, but needs full characterization of the
single target behavior before the definition can be used to make a meaningful deduction
[23].
Further to the mathematical basis, and statistical rigor, we chose to use the former
definition of additive because it can be easily to expanded to a combination of any number of
77
drugs. A comparative study of how the general landscape of the combination behavior change
with later definition of additivity would be an interesting study for future work. There is,
however, an implication that the later definition leads to a lot of false positive result. That
may account for why we, along with others who use the loewe based definition find synergistic
and antagonistic target behaviors to be a rare traits [76 although 'synergistic' combinations
are reported about quite frequently in literature. One of the reasons for popularity of bliss
independence in research and literature arises from ease with which they can be applied to
quantify experimental measurements as it does not need full characterization of single drug
dose response curve (which is required in the case of loewe definition) to make an inference
on combination behavior. However, a rigorous comparison will be needed to fully understand
the strengths and limitations of these two definitions of additivity.
We made an effort to understand common features between targets that resulted in
antagonistic or synergistic target behavior as opposed to the ones that showed additive
behaviors. Our observation of the synergistic and antagonistic targets showed that most of
the synergistic targets are binding partners of each other. This is not to say that inhibition
of binding partners lead to synergistic behavior, neither do we want to imply that only
binding partners can exhibit a synergistic behavior. The later does not even hold within
the scope of our model. We just want to report a class of synergistic targets that we found
on this pathway are binding partners of each other. Likewise, there was a distinct class
of molecular relationship that exhibited an antagonistic behavior. Inhibition of one of the
reactant and the resulting product it forms lead to antagonistic target behavior. However,
as in the case of synergistic behavior, this class of antagonistic behavior is a subset of all the
antagonistic targets we saw. Further study within this and other biochemical networks are
needed to either strengthen or refute these observations. Furthermore, these generalizations
of synergistic and antagonistic behaviors need further constraints to understand exactly what
relationships between molecular targets give synergistic and antagonistic target behaviors.
In our efforts to understand synergistic targets, a second class of molecular targets stood
78
out. This second class consisted of three molecular targets: (1) EGFR, (2) (EGF-EGFR*)2GAP-Grb2, and (3) (EGF-EGFR*)2-GAP-SHC*-Grb2.
These three targets showed syner-
gistic behavior with a large number of other targets. A subset of these behavior does fit into
the binding partner hypothesis but not all do. An analysis of the amount of material inhibited by inhibitors of these targets seem to suggest that, at intermediate concentrations, one
of the inhibitor is able to inhibit more material then it was able to when the total inhibitor
in the system was purely made up of this inhibitor. This implies that.the presence of second
inhibitor changes the model dynamics in such a way that the first inhibitor gets to do more
work in inhibiting the target than would be the case if only the first inhibitor was present
at equivalent concentration.
All the molecular targets inhibited in this work were treated as competitive inhibitors.
Treating these targets as uncompetitive, or non-competitive targets may change the landscape of combination behavior that we see. This could be an interest in future studies. Unlike
in the single target inhibition case (Chapter 2) where an inhibitor of a particular target was
able to inhibit all the functions of that target, here inhibitor were treated as more selective
entities that only targeted one particular function of a protein. This decision was made to be
realistic in the way that combination targets used in clinical setting. The detailed molecular
entities inhibited by each inhibitor is given in Appendix B.
Furthermore, our work here and the subsequent deductions made were based on the
analysis of a single biochemical pathway. Study of multiple biochemical networks of the same
pathway, and other chemical pathways will be needed to get a comprehensive understanding
of general trends in combination target behaviors. Efforts are underway within our research
group in this area.
79
Chapter 4
Therapeutic Design Strategies for
Safety and Efficacy
Abstract
Current molecular therapeutics generally serve as inhibitors, antagonists or, less frequently,
agonists that exert relatively crude control over the biology with which they interact. Looking forward, systems biology offers the potential of gaining a much deeper understanding
of cellular control mechanisms. One early application of this knowledge is the development
of molecular therapeutics personalized to be effective in particular categories of patients.
Another might be to alter the set point of fundamental biological processes, such as affecting
the balance between the tendency for cell proliferation versus apoptosis, which could have
important implications for cancer prevention and neurodegenerative disease. Further applications, however, could involve the incorporation of personalized and multifactorial effects
into therapeutic design. Future therapies could sense characteristics of the patient and interact differently as a result, essentially producing a personalized effect tuned across a variety
of categories of patients. Likewise, therapeutics able to sense and respond appropriately
may be useful to distinguish diseased from normal tissue and to generate a type-appropriate
response. In this project we explore the potential of such therapeutics using biochemical
pathways of signaling processes together with optimization techniques with multifactorial
objective functions. The results show great potential for effecting programmed responses
that operate robustly across wide ranges of conditions.
80
4.1
Introduction
Major efforts in computational and mathematical systems biology have been focused on calibrating biological models using experimental data. An important goal of this endeavor is
to propose detailed mechanistic models that are not only able to faithfully simulate experimental data, but are also able to predict system behaviors under contexts that are either
hard or expensive (in terms of time, or resources, or both) to do experimentally. Moving
forward, the major motivation for calibrating the models is to explore and exploit their predictive potentials to design therapeutic intervention strategies that have desired optimized
properties. Here, we use a well calibrated Epidermal Growth Factor (EGF) induced EGF
Receptor (EGFR) pathway [94, 50, 6] (details in section 1.3.1) to computationally explore
some of the therapeutic design principles. The aim of this work is to explore simple design
strategies that can differentially regulate the normal and diseased (mutated) pathways.
Most of the intervention strategies in drug design industry are limited to small molecule
inhibitors (or antibodies) which exert crude control over biology with which they interact.
They exert their effects by inactivating target activity (or activities) at some fixed proportions.These proportions are generally based on number of experimental assays in animal
models and other pre-clinical studies. However, these are still static calculations where the
proportion of the targets to inhibited is pre-determined and not calculated depending on
how the system is responding to the intervention once inside a human body or cell.
An
implicit assumption here is that the environment in which the intervention needs to exert
its effect is similar to environments in the assays in which drug dosage (and hence the proportional target inhibition values) were determined.
Even when dose conversion is taken
into account to go from animal models to human subjects, an assumption used is that these
conversions are well calibrated and known. So, the motivation for this work comes from the
quest for more controllable or tunable circuits that can act as regulators of the biology they
interact with rather than switches that turn events on or off. In this context, we propose
two desirable goals that would be suitable to differentially regulate normal and diseased
81
models or cells and propose systems level design strategies as a comparison with the typical
inhibitors that are ubiquitously present in the field. A combination of design, optimization,
and simulation frameworks are used to propose three different intervention strategies. Depending on the design goal chosen, we show that some strategies have clear advantages over
others. These advantages mainly come from their efficacy in number of different operating
conditions, interaction kinetics with the targets, and mutation status of the pathway under
study.
4.2
4.2.1
Materials and Methods
Model Details and Setup
A biochemical network of EGFR Pathway proposed by Schoeberl et al. [94], modified by
Hornberg et al. [50] and Apgar et al. [6] was modeled with Ordinary Differential Equations
(ODEs) and integrated using odei5s function in Matlab 2009a (Mathworks Inc.) was used
to evaluate the normal system behavior.
An input to the system is extracellular Epider-
mal Growth Factor (EGF). An activated form of (i.e. doubly phosphorylated) extracellular
regulated kinase (ERK-pp) signal is taken as an output of the system.
ERK is the most
downstream protein of the pathway considered here and is an important transcription factor
that can trans-locate into the nucleus and can regulate the expression of growth or proliferation associated genes.
A detailed description of the signaling proteins involved in this
pathway and nature of signal propagation is described in section 1.3.1.
Furthermore, this
work also uses variant 3 model described in section 1.3.2 such that therapeutic designs can
be tested for their capability to alter the over-stimulated signals in variant model and leave
the same signal mostly intact in the normal model.
82
4.2.2
Objectives
Given a normal and a cancer phenotype as described in section 1.3.2, two distinct goals are
considered in designing the interventions strategies. These are (1) intervention introduced
in the model should minimally affect the dynamics of the normal model while changing the
dynamics of the cancer model to follow the normal cell dynamics. In other words, the goal
here is to design intervention strategy that does not affect the normal cells but regulates
the dynamic of the cancer cells such that they start signaling like the normal cells even in
the presence of a mutation in the pathway and (2) intervention introduced in the model
should minimally affect the normal model while completely blocking the growth signal from
the cancer cells. In other words, this second objective aims for an intervention strategy that
minimally affects the signaling dynamics in normal cell while not allowing any signal to pass
to the end of the signal transduction cascade in the over-stimulated cancer variant of the
model. These objectives are described in optimization mathematical framework in equation
4.1 and are shown schematically in figure 4-1.
bJ = min(
al
t=o
(Xdesiredt - Xnorma,(X, p))2
OnormaIt
+
ti
t=2
2
(Xdesiredt - Xcancert (X, p))
(4.1)
cancer2
For objective I, the desired signal for the normal model (first summation in the objective
function) is that of the normal model before any intervention was introduced. The same
signal dynamics is desired for the cancer model under intervention. In other words, we want
an intervention strategy that does affect the normal model dynamics much while bringing
to aberrant nature of the cancer model dynamics to normal levels. For objective II, we
want the desired signal for the normal model to be the same as in objective I, but for the
cancer model, instead of driving its dynamics to that of a normal one we want it to go to
zero. In summary, with first objective we want both cancer and normal model to signal as
normal. However, for objective II, we want normal to signal as in the case where there was
83
(A)
6
10 X 10
6
(B)
x10 Design Objective 11
-Z10
.b=
Design Objective I
75
_9 8
w8
6
E
0
-0
4
2
2C
-
,ill0
CJ000- 11000 21000
2000 3000 4000 5000
Time (s)
for cancer
2000 3000 4000 5000
Time (s)
Objectives of therapeutic design strategies considered in this work. (A) Describes objective I
and cancer models need to optimize to achieve normal models dynamics (B) Describes
normal
where both
tries to completely block the signal in cancer model with minimal effect on
optimization
where
II
objective
the dynamics of normal model
Figure 4-1:
no intervention whereas the over-stimulated cancer model to not signal at all. In some sense,
Objective II can be thought of as a more demanding one among the two. In a more control
engineering sense, the problem comes down to designing a common controller for normal and
cancer model such that the dynamics of each of the model are governed as stated in the two
objectives. In ideal world, we do not want to affect the normally behaving cells in anyway,
but when an intervention is applied to human tumors, there is no way to ensure that they
are targeted exclusively to the disease cells. So, the design framework here actively considers
the efforts to minimize the side effects in the process of therapy design.
4.2.3
Design of Intervention Strategies and Optimization Framework
Three intervention design were explored mainly inspired by the common regulation motifs
that are observed in most, if not all, of the biological systems. These are : (1) Kinetically
Tuned Inhibitors, (2) Feedback Loops and (3) Feed-forward Loops. A schematic of how each
of these three designs are implemented within biochemical model is shown in figure 4-2.
84
B
A
Upstream Signals
Ip
Tuned Inhibitors
(small molecule)
naaf
Ptasel
:M E K PP
-6
Downstream Signal
0
Ptase2
Ptase3
Output
ERK
OUTPUT
Tuned Inhibitor
Intervention strategies
D
C
AC
A
RafD
(.Laf
-PRaf
Ptasel
K-PP
Ptase2
Ptase2
Ptase3
OUTPUT
OUTPUT
Feed-forward
Feedback
Figure 4-2: Schematic representation of three design strategies explored to evaluate their ability to meet objectives described in equations 4.1. (A) Summary of three designs considered (B) Molecular implementation
of kinetically-tuned inhibitors (C) Molecular implementation of feedback loops (D) Molecular Implementation of feed-forward loop
Kinetically Tuned Inhibitors
Kinetically Tuned inhibitors here refer to slight variants of the typical inhibitors that are
widely used in the field. The difference arise from the fact that the kinetics of interaction
of these inhibitors with their targets are chosen such that they do not necessarily bind to
the target as tightly as possible to block the downstream signaling. Instead, the kinetics
are optimized to minimize the objective function given in equation 4.1. The fact that the
objective function contains two opposing aspects, meaning that the normal cell dynamics
should not be changed much while affecting the cancer cell dynamics, mean that the inhibitor
85
kinetics with the target with which they interact reflects this aspect of the design. Molecular
level implementation schematic of this design is shown in figure 4-2B. The inhibitor, Ir, can
bind to its target to form an inactive inhibitor-target complex. This inhibitor target complex
is allowed to dissociate to back to inhibitor and target. So, this design can optimize the
association and dissociation rate constants of the inhibitor with the target. The optimization
parameters range were chosen to be physically realizable though not necessarily the typically
accessible values.
Feedback Loops
As the name suggests these refer to biochemical wiring from a downstream species in the
pathway to an upstream one. The goal is to regulate the signal flux through upstream species
according to the amount of downstream species that is already present in the system. These
circuits, by the very nature of the design, contain time delay factor between the response
and the regulation. As the present response in the downstream species is used to regulate
the future influx from the upstream species, such time delays, if long enough, can introduce
oscillations in the systems. The mathematical basis of how the oscillations actually come
about is beyond the scope of this work here. Here, the delays introduced by our feedback
interventions are not long enough to cause oscillations in the system as such behavior were
not observed in extensive simulation works. Molecular level implementation schematic of
this design is shown in figure 4-2C. This design acts at two states. First the intervention
A is enzymatically activated by the molecule it is sensing (ERK-pp in this case) forming a
product A-p. This A-p then enzymatically acts on the molecule at which the feedback is
acting (Raf-p as shown in the diagram). Intervention A here can be thought of as an inactive
phosphatase that is activated by an enzyme.
86
Feed-forward Loops
As opposed to feedback circuits where response of a down stream component is used to
regulate the future influx of signal through upstream species, feed-forward is equivalent to
establishing a short circuit in the pathway. Here the upstream signal is used as a sensor for
the pathway activity, and dependent on this sensor information downstream component of
the pathway is regulated to obtain a desired system dynamics such that the objective established in equation 4.1 is achieved to the best of the ability. Molecular level implementation
schematic of this design is shown in figure 4-2D. This design acts at two states. First the
intervention A is enzymatically activated by the molecule it is sensing (Raf-p as shown in
the figure) forming a product A-p. This A-p then enzymatically acts on the molecule at
which the feed-forward is acting (ERK-pp as shown in the diagram). Intervention A here
can be thought of as an inactive phosphatase that is activated by an enzyme.
4.2.4
Targets Design
All the intervention were designed in the normal model and model containing mutation at Raf
protein level. This was to capture the most conservation design strategies as this is the most
downstream of the three mutations that were recognized as the common mutations of this
pathway. The targets for kinetically-tuned inhibitors were chosen to be Raf (Phosphorylated
form), MEK (doubly phosphorylated form) and ERK (doubly phosphorylated form). For
the feedback models 6 possible combinations arising from Raf-p, MEK-pp and ERK-pp were
considered. First three are ERK-pp feeding back the information to itself, MEK-pp to itself
and Raf-p to itself. The other three are (i) from ERK-pp to Raf-p (ii) from ERK-pp to
MEK-pp and (iii) MEK-pp to Raf-p.
The same would also be true for the feed-forward
expect that there is no difference between the feedback and the feed-forward systems when
the both the targets are the same (i.e.
ERK-pp to itself or Raf-p to itself) so only the
remaining three (i) from Raf-p to MEK-pp (ii) Raf-p to ERK-pp and (iii) MEK-pp to ERKpp were considered. These designs were then optimized using the equation 7 using fmincon
87
(constrained optimization routine) function in Matlab using Active set as the algorithm for
the optimization - a constraint optimization is needed to reflect the physical constraints
on the interaction kinetics like the diffusion limits or positive interaction constants. These
optimized intervention strategies are then tested for number of different operating conditions
from the ones that they were designed and optimized for. These include, different input levels
of the ligand and the models with two other mutations (EGFR/endocytosis and Ras) instead
of Raf mutation.
4.3
Analysis of the Optimized Designs
For the first objective all the three designs could find at least one set of design parameters
that satisfactorily met the design objective. A set of optimized trajectories are shown in
figure 4-3. Figure 4-3A shows ERK-pp dynamics for normal model cancer variants models
with an without the presence of an inhibitor for MEK protein for which the binding kinetics
were optimized. This shows that in the presence of the optimally design inhibitor, normal
model is affected minimally by the normal model while, the over-stimulated is affected such
that it tracks the trajectories of the normal model without intervention well at later times.
At initial times the signaling in cancer model with inhibitor is still over-stimulated.
This
compromise is needed to rescue the signaling in the normal model in the presence of the
intervention. A further analysis of this design showed that the objectives were met only for
a very finely tuned values of inhibitor kinetics and deviation from this meant that either the
inhibitor did nothing to both the normal and cancer model or it did bring the over-stimulated
signal in the cancer model down, but at the cost of blocking the signal in the normal model.
Similar results were observed when ERK was inhibited.
A representative feedback model that was able to track the first objective of making both
normal and over-stimulated cancer variant model to signal like the normal model is shown
in figure 4-3B. This particular figure demonstrates the case of feedback from ERK-pp to
88
MEKpp. Like in the case of kinetically tuned inhibitors, presence of optimized feedback
loop both in normal cell has minimal effect, but the same design when present in the overstimulated cancer variant model is able to track the trajectory of normal cell signaling very
well. Further exploration of the optimized designs showed that feedback design was more
tolerant to changes in the model parameters. Small changes in the model parameters did
not affect the ability of this design to meet the objective it was designed for.
A representative feed-forward model that was able to track the first objective well is
shown in figure 4-3C. The figure shown here is a optimized feed-forward loop from MEK-pp
to ERK-pp. This design meets the objective function in a manner that is very similar to the
feedback loop. Like in the feedback case, exploration of the optimized feed-forward parameters revealed that the design was not sensitive to small changes in the design parameters
providing a design window for molecule design.
106 Optimized nhiiin of MEK
(A)
'W
(B)
X10ptimized
-
1
-cr
-
-
-
X
(C)
feedback from ERKpp to MEKpp
-
-
0
ptimized feedforward from MEKp to ERKpp
-
-
1.~
-10
g-8
T8
6
-6 6n
4
4
4
0
2
2
-6
00-
0
1000
2000
3000
Time (s)
4000
5000
0
1000
2000 3000
Time (s)
4000
5000
0
1000
2000 3000
Time (s)
4000 5000
Figure 4-3: Optimized designs for three design strategies explored to make both normal and cancer cells
signal as a normal cell. (A) System behavior with optimized MEK inhibitor, (B) System behavior with
optimized feedback design from ERK-pp to MEK-pp, and (C) System behavior with optimized feed-forward
design from MEK-pp to ERK-pp.
On the other hand, when these three designs were challenged with the second objective of
keeping the signaling in the normal model intact while blocking the signal in a over-stimulated
cancer model, only the feed-forward design was able to decently meet the objective. Model
simulations with optimized designs are shown in figure 4-4A. It was not surprising that an
inhibitor, with limited degree of freedom in parameters that could be optimized, could not
meet this more challenging objective of introducing the same intervention in both normal
89
and cancer cell, and telling it to do nothing in the the case of normal but block the signal
completely in the case of cancer cell. What was more surprising was that even feedback design
with exact same number of design parameters as the feed-forward design was not able to
meet the objective. An analysis into why this was the case revealed that by the time feedback
was exerted in the model proportional to the amount of signal at the output, some of the
signals had already passed compromising its ability to block the signal completely. However,
in the case of feed-forward, sensing occurs on the upper part of the chemical pathway and
effect is exerted at the bottom, hence, feed-forward effect exerted had an influence on the
output helping the design achieve the objective. While we have not tested this hypothesis in
this work, is it plausible that the feedback design would be able to meet the design goal if the
kinetic rate constants at the bottom of the cascade was slower compared to the time-scale
at which feedback was exerted. In this case we tested the robustness of this feed-forward
design by taking the design optimized in our model, and applying it to a biochemical model
of the same biochemical process with different mathematical details. In doing so, the design
was able to meet the objective without having to re-optimize for parameters. The results of
this analysis is shown in figure 4-4B.
(A)
12
Training Model
X 106
~
.
Test Model
(B)x 105
#--------------6
10
1
1
1
cancer
222
4
8
'Mcg
- -
- - - - - --
-
uJ
J
2
normal+
Clu
1000
- - --
-
- ---
4
0
1
1
-
2000
3000
4000
0
5000
20
40
60
80
100
120
140
160
180
200
Time (min)
Time (s)
Figure 4-4: Of the three explored, only feed-forward strategy was able to meet the second objective. The
design showed robustness to design parameters and biochemical details of the model. (A) System behavior
with optimized feed-forward design from MEK-pp to ERKpp. (B) Behavior of a different system of same
biochemical process with feed-forward design optimized for systems in (A).
90
4.4
Summary
In this chapter we analyzed some simple therapeutic design strategies that can be explored to
meet multi-factorial objectives that interventions should ideally meet. By equally weighting
efficacy and toxicity of a drug intervention in our objective function, we showed that if we
know the exact state of the system in advance, we may get away with typical inhibitor like
designs by tuning their kinetic parameters just right. However, these designs fail to meet the
design goals when the precise knowledge of the system is missing. Furthermore, we showed
that more involved designs of feedback and feed-forward are more adaptive to the systems
because of their ability to exert an effect according to the state of the system. This ability also
makes these intervention designs more robust to the changes in precise details of the model
state. In addition, timing at which these interventions can exert an effect compared to the
time scale at which the model signals can mean that some of these designs are more adaptable
than others. We saw this difference in adaptability in terms of differences in performance of
feedback and feed-forward circuits when presented with a demanding objective of blocking
the signal in over-stimulated model while minimally affecting the normal model.
91
Chapter 5
Summary and Future Directions
Research Projects described in this thesis demonstrate ways in which decently well-calibrated
biochemical models can be used in early stages of drug discovery process, namely target
identification and therapeutic design strategies. Our work on single target analysis (Chapter
2) show that there are, in fact, better and worse places to intervene. The nature of signal
propagation from amplification and saturation viewpoint was explored to explain some of the
trends on why targets are sub-linear or super-linear. One of the most striking result from the
single target identification work is that, despite the drug being 'perfectly' present at the site
of action, the nature of signal transduction in the pathway mean that most targets have to
be inhibited very strongly to see any effect on output. Need for high inhibitor concentrations
to achieve efficacy has sometimes been attributed to inefficiencies in drug delivery systems
or drug metabolism.
While these are certainly very crucial components of drug design
processes, we propose from this analysis that the very nature of signal transduction could
also be limiting steps.
In the pathway analyzed in this thesis, we identified saturation
and amplification as key factors contributing to target behaviors. A more extensive study
in more elaborate versions of this biochemical pathway, and other biochemical pathways
of different cellular processes are needed to reinforce this hypothesis and discover further
network properties that can be used as a proxy for better and worse target behaviors. Some
92
of these efforts are currently underway within research group where this thesis work was
developed.
Another important aspect of drug discovery is concerned with making design and target
decisions in the face of biological uncertainty. It is very important to model these uncertainties such that any deductions made appreciates the biological ambiguities present in the
system. In this work we incorporated these model uncertainties in two different ways. One
was by creating three over-stimulated variants of the published model and evaluating the
target behaviors in the resulting models. The second approach was to create an ensemble
of models by varying all the model parameters simultaneously. This ensemble of models
was then used to quantify agreements between the models on target behaviors of proteins or
protein complexes in the pathway. If most models agree on particular behavior for target,
we can make a confident decision on how those targets really behave. In this regards, model
and parameter uncertainty are going to be an integral part of biological networks. These
uncertainties not only encompass our lack full understanding of the system, but also the
fact that at fundamental level biochemical reactions and processes are stochastic in nature.
Hence, no matter how perfectly we collect and analyze the data, there is going to be some
stochasticity that we need to account for in addition to uncertainties that arise from our
limited understanding and observation of biology of interest. Modeling these uncertainties
and making decisions by embracing these uncertainties is going to be an important way
forward for this field.
Our method for holistic exploration of combination targets behavior in a biochemical
network described in chapter 3 of this thesis gives some insight into the overall landscape
of combination target behaviors that exists in a biochemical results. We show that most
target combinations are additive. Synergistic and antagonistic behaviors are rare to find,
and even when they do exist, they do so in only a limited range of inhibitor concentrations.
Again, like in the case of single target behaviors, understanding this landscape in potential
range of parameter variability is important in developing a robust picture of where suitable
93
combination are and what their exact mdlecular identities are. We approached this using
two methods akin to that described for the single target behavior. Significant progress in the
field can be made by exploring combination landscape of a number of biochemical networks
to deduce a set of general rules (these are more likely to be heuristics rather than hard and
fast rules) that can bias a combination choice toward synergistic or antagonistic behaviors.
Computational framework that we have developed in this work can facilitate this effort.
Our work on therapeutic design strategies in chapter 4 demonstrates the limitations of
inhibitor like therapeutics when challenged with even the simplest of design objectives. We
suggest and explore simple alternatives to inhibitors that could exist and the potential that
they may provide. But this is a very simple exploration of the problem. This merely surfaces
the issue here. Small molecule inhibitors have played a significant role in combating disease
processes and advancing medical science as a field. They were great when our understanding
of biology and disease process were limited and we wanted to patch one or two flaws that
we identified as causal events for disease processes.
However, our current understanding
of biological and disease process has leaped far ahead. We have gone from a reductionist
approach to a top-down systems level approach and have unraveled some of the intricate
network level deregulations that contribute to complex disease like cancer. However, the
tool set that we use to try to combat this process had remained fundamentally the same.
We have gone from single therapy to combination therapy paradigm but have not really
explored alternative design strategies and their potentials. This is one area of focus for the
future work with a tremendous amount of challenges and potentials.
In this work we used a protein within the biochemical network analyzed as a proxy for
disease phenotype or an output that we wanted to regulate or control. In reality, disease
phenotypes usually manifest at tissue or organ level. Models that link biological processes
at varying levels of organization details are going to be crucial to further our understanding of health and disease processes. These models are will be able to capture the effects
of intervention therapies more accurately then the approach we have taken in this thesis,
94
which focuses on a signature within a sub-cellular network. There are large number of efforts
currently underway within biomedical research community to develop quantitative mathematical models of tissues or organs in normal physiology and disease. What is missing is a
link between these higher level models and the molecular level models like the ones analyzed
here. Most therapies act at a molecular level and their phenotypes are measured or observed
at tissue and organ levels. A way to model this whole process quantitatively is going to be
crucial in understanding modes drug action and mechanisms for failure, paving a way for a
more systematic and less failure prone approach to drug discovery.
95
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Appendix A
Single Target Intervention
A.1
A.1.1
Target Effect Metric
Single Substrate Enzymatic Reactions
For the case of a single substrate in the absence of the inhibitor, the enzymatic reaction has
the form:
E+S
k
ES -t
(A.1)
E+P
k-1
(A.2)
Making the Quasi-Steady State Approximation (QSSA) for [ES].
(A.3)
Using, [E], = [E] + [ES]
dP
dt
-
kcat [ES] = kcat[E]o[S] ; KM
Km +S
105
-
kcat + k_ 1
k1
(A.4)
In the presence of Competitive Inhibitor:
k1
E-+-S "=ES
kcat
(A.5)
E+P
k-i
E+I
kil
(A.6)
-EI
ki-1
(A.7)
Using QSSA for [EI] and [ES] and[E], = [E] + [EI] + [ES]
dP
kcat [E]o[S]
dt=
kcat
[
ES]
=
K,,I
dt
Km(1 +j)I)+ + S
(A.8)
= kcat
k[f [E]]o& ; Wheref = 1
(A.9)
KM + ISf
+
[[]
K1
In the presence of Non-Competitive Inhibitor:
E + Skct
(A.10)
E+P
ES
k-1
E + IN
kil
ki-1
EI + S
k1
ESI
(A.11)
ESI
(A.12)
k-1
kil
ES + I -
ki-1
Using QSSA for [EI], [ES], [ESI] ,and[E], = [E] + [EI] + [ES] + [ESI]
dP
= kcat[ES] =
dt
(1+
kcat[E]o[S]
f(Km + [S])
From these equations we see that the factor
f
keat [E]o[5]
-L')(Km + [S])
(A.13)
(A.14)
(A.15)
LI) is a measure of the decrease
= (1 + 4
in the initial rate of product formation in the case of the non-competitive inhibitor and a
decrease in effective substrate concentration in the case of the competitive inhibition. So,
this factor or its transformed forms as appropriate (e.g., log(f) or 1-1/f or 1/f) is taken as
the measure of the effect of inhibitor on the target.
106
Two or More Substrate Enzymatic Reactions
A.1.2
While
f
= 1 + 2l
K
1
seems to be a suitable measure for the target effect when the enzyme
on the pathway has a single substrate, an important question is whether it is scalable to
the cases where there are multiple substrates for an enzyme on the pathway or when the
full reaction needs more than one enzymatic step before it reaches completion.
This is
particularly applicable in the case of our cascade as downstream kinases like Raf or MEK
in their activated (and hence phosphorylated) forms take two steps for reaction completion.
This can be modeled as though the enzyme (kinase) has two substrates.
In the case when there is no inhibitor in the system:
E + Si,
k1
k-1
E +S
2
-k2
k-2
keati
ES1 -
E+P
(A.16)
1
' ES 2 -cat2 ) E+P
(A.17)
2
Using QSSA for [ES 1], [ES 2], and[E] = [E] + [ES 1 ] + [ES 2]
dP
dt
dP 2
dt
107
(A.18)
kcati[E]o[Si]
Kj1+K-1[S2] +[S1]
_
kcat2 [E]o[S 2]
Km2 +
K-2
[S1]+[S 2]
A.20)
In the presence of a Competitive Inhbitior:
E + SSca
E +S
2
k1
ES1
k-i
kcatl1
E+P
ES2 kcat 2
7k-2
(A.21)
1
(A.22)
+P2
E+I N
(A.23)
EI
ki-1
(A.24)
Using QSSA for [ES,], [ES 2], [EI], and [E]O = [E] + [ES 1 ] + [ES 2]
dP1
kcat1 [E]o[S1]
-
dt
[Ti
-
Km1(1+
A, c)r
yL1.L~J)
- ~.,-.,
f )+
[22 + [S1]
kcat1 [E] o Is
Km1 + K.
Km 2 f
S2
dP 2
dt
+
+
Eff
fA-
kcat2 [E]0 [S 2 ]
_
Km 2 (1+-)+r
[S]+[S2 ]
(I+
[Si
Km 2 kcat'2[E]
o +~2+
2
m2
f
S
[I]
K1
(A.26)
+[2
f
(A.27)
In the case of Non-Competitive Inhibition:
S kcatl1
E + S, 7k1S~~
ES,
ca1 E + P
k-1
E+S
k2
2
k-2
2
ES 2
kcat2k
a2E+P
2
+EI
E+I
ki-1
E+ S
EI + S2
-- ES1I
k-1
k2
7-
ES21
k-2
ES1 +I N LES1 I
ki-I
ES 2 +I
Nki:+
ki-1
ES 2 1
Using QSSA assumption for all the intermediate complexes.
108
] [ES1I]
[-ES1 = [E][S
1 ] [ES2] = [E][S2
[ES 2]'
[ES1]]'
=
[ES 1][I] ; [ES 2 I]
K1
-
[ES 2][I]; [EI]
K1
=
[E][I]
K1
(A.28)
(A.29)
[E]O = [E] + [ES 1] + [ES 2 ] + [ESI] + [ES 2I] + [EI]
=E
+ [S1][I] + [2][I]
KmiKi
dP= kca [ES1] =kcat [E][S 1 ]
Km1
dt
[E]O[S 1 ]
kcat1
)(i
)+(1+
(1+
Kmi
Km 1
Km
2
(A.30)
+
[E]0 [ 1 ]
kcati
[I]
K1
Km 2 KI
S)
_
f
(1 + -I)
(Kmi + K-1 [S 2 ] + [S1])
2
k
dP2 k. 2 E 1
=_kcat2[ES = c]at2 [E] [S 2]
-
dtc
[E]o[S 1 ]
kcatI
(Km1 + K
K. [S2]
KI
+
[Si])
(A.31)
Kmn2
kcat 2
Km 2
[E]o[S 2 ]
) + (1+
(1+
+
[E] 0 [5 2]
kcat2
(i+
I)(s2
I)
(Km2 +
2)
_
2[S1]+[S 2])
kcat2
f
[E]O[S 2]
±
(Km2 +rn,[S1]+[S2])
(A.32)
This analysis shows that the single substrate case is generalizable to case when there are
two substrates for an enzyme in system. As in the case of a single substrate for an enzyme,
the effect of competitive inhibitor on the initial rate of product formation is the same as
effective reduction in the substrate concentration by a factor of
f
= 1 +
i . Likewise,
the presence of non-competitive inhibitor in the system reduces the initial rate of product
formation by a factor
f.
109
A.2
Biochemical Network
All the biochemical reactions describing the models in used in this work is provided in a
tabular form below. The 2 columns represent the reactants, the third column describe the
product they form. The fourth and the fifth columns describe the identities of forward and
reverse reaction constants.
Reactant1
GAP
Grb2
Sos
Ras-GDP
Shc
Grb2-Sos
EGFRi
EGFi
(EGFEGFRi*)2
.(EGFEGFRi*)2-GAP
(EGFEGFRi*)2GAP-Grb2
Reactant2
0
0
0
0
0
0
0
0
0
Product
0
0
0
0
0
0
0
0
0
kForward
kd214
kd222
kd224
kd226
kd231
kd230
k60
k61
k60
kReverse
k200
k200
k200
k200
k200
k200
kd60
kd61
kd60
0
0
k60
kd60
0
0
k60
kd60
(EGF-
0
0
k60
kd60
0
0
k60
kd60
0
0
k60
kd60
0
0
k60
kd60
0
0
k60
kd60
EGFRi*)2-
GAP-Grb2-Sos
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP
(EGFEGFRi*)2-
GAP-Grb2-SosRas-GTP
(EGFEGFRi*)2-
GAP-SHC
(EGFEGFRi*)2-
GAP-SHC*
110
(EGFEGFRi*)2GAP-SHC*Grb2
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGTP
ERK-PP
ERK-P
phosphatase3
ERK
ERKi-PP
ERKi-P
phosphatase3
ERK
ERK
MEK-PP
MEK-PP
ERK-PP
ERK
ERKi-P
MEKi-PP
ERKi-PP
(EGFEGFR*)2GAP-Grb2
(EGFEGFRi*)2-
0
0
k60
kd60
0
0
k60
kd60
0
0
k60
kd60
0
0
k60
kd60
phosphatase3
phosphatase3
ERK-P
phosphatase3
phosphatase3
phosphatase3
ERKi-P
phosphatase3
MEK-PP
ERK-P
ERK-P
MEK-PP
MEKi-PP
MEKi-PP
ERKi-P
MEKi-PP
Prot
ERK-PP-phosphatase3
ERK-PP-phosphatase3
ERK-P-phosphatase3
ERK-P-phosphatase3
ERKi-PP-phosphatase3
ERKi-PP-phosphatase3
ERKi-P-phosphatase3
ERKi-P-phosphatase3
ERK-MEK-PP
ERK-MEK-PP
ERK-P-MEK-PP
ERK-P-MEK-PP
ERKi-MEKi-PP
ERKi-MEKi-PP
ERKi-P-MEKi-PP
ERKi-P-MEKi-PP
(EGF-EGFR*)2-GAPGrb2-Prot
k56
k57
k58
k57
k56
k57
k58
k57
k52
k53
k52
k55
k52
k53
k52
k55
k4
kd56
kd57
kd58
kd57
kd56
kd57
kd58
kd57
kd44
kd53
kd44
kd55
kd44
kd53
kd44
kd55
kd4
Proti
(EGF-EGFR*)2-GAPGrb2-Prot
k5
kd5
0
0
EGFRi
(EGF-EGFRi*)2
k6
k6
kd6
kd6
GAP-Grb2
EGFR
(EGF-EGFR*)2
111
(EGFEGFR*)2GAP-Grb2
Proti
(EGFEGFR*)2-GAP
(EGFEGFR*)2GAP-SHC
(EGFEGFR*)2GAP-SHC*
(EGFEGFR*)2-
0
(EGF-EGFRi*)2-GAPGrb2
k
kd6
0
0
Prot
(EGF-EGFRi*)2-GAP
k15
k
kdl5
kd6
0
(EGF-EGFRi*)2-GAPSHC
k6
kd6
0
(EGF-EGFRi*)2-GAPSHC*
k6
kd6
0
(EGF-EGFRi*)2-GAPGrb2-Sos
k
kd6
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-Prot
k4
kd4
(EGFEGFRi*)2GAP-Grb2-Sos
0
(EGF-EGFR*)2-GAPGrb2-Sos-Prot
k5
kd5
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP
k6
kd6
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-Prot
k4
kd4
(EGFEGFRi*)2GAP-Grb2-Sos-
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GTP
k6
kd6
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GTP-Prot
k4
kd4
GAP-Grb2-Sos
(EGFEGFR*)2GAP-Grb2-Sos
Proti
(EGFEGFR*)2GAP-Grb2-SosRas-GDP
(EGFEGFR*)2GAP-Grb2-SosRas-GDP
Proti
Ras-GDP
(EGFEGFR*)2GAP-Grb2-SosRas-GTP
(EGFEGFR*)2GAP-Grb2-SosRas-GTP
112
Proti
(EGFEGFR*)2-GAPSHC*-Grb2
(EGFEGFR*)2-GAPSHC*-Grb2
Proti
(EGFEGFRi*)2GAP-Grb2-SosRas-GTP
0
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GTP-Prot
k5
kd5
(EGF-EGFRi*)2-GAPSHC*-Grb2
k6
kd6
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*-
(EGF-EGFR*)2-GAPSHC*-Grb2-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos
k
kd6
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*-
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP
k6
kd6
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP
0
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-Prot
k5
kd5
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGTP
k6
kd6
Grb2
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
Proti
Grb2-Sos
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP
Proti
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GTP
113
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GTP
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGTP
MEK-PP
MEK-P
phosphatase2
MEK
MEKi-PP
MEKi-P
phosphatase2
MEK
ERK-PP
ERKi-PP
ERK-PP
ERKi-PP
ERK-PP
ERKi-PP
ERK-PP
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGTP-Prot
k4
kd4
Proti
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGTP-Prot
k5
kd5
phosphatase2
phosphatase2
MEK-P
phosphatase2
phosphatase2
phosphatase2
MEKi-P
phosphatase2
(EGFEGFR*)2GAP-Grb2-Sos
(EGFEGFRi*)2GAP-Grb2-Sos
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
Sos
Sos
0
MEK-PP-phosphatase2
MEK-PP-phosphatase2
MEK-P-phosphatase2
MEK-P-phosphatase2
MEKi-PP-phosphatase2
MEKi-PP-phosphatase2
MEKi-P-phosphatase2
MEKi-P-phosphatase2
(EGF-EGFR*)2-GAPGrb2-Sos-ERK-PP
k48
k49
k50
k49
k48
k49
k50
k49
k126
kd48
kd49
kd50
kd49
kd48
kd49
kd50
kd49
kd126
(EGF-EGFRi*)2-GAPGrb2-Sos-ERKi-PP
k126
kd126
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-ERKPP
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-ERKipp
k126
kd126
k126
kdl26
Sos-ERK-PP
Sos-ERKi-PP
(EGF-EGFR*)2-GAP-
k126
k126
k127
kd126
kd126
kd127
k127
kd127
k127
k127
kd127
kd127
k127
kd127
Grb2-Sos-ERK-PP
ERK-PP
0
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-ERKPP
ERK-PP
ERKi-PP
Sosi
0
ERKi-PP
0
Sos-ERK-PP
(EGF-EGFRi*)2-GAPGrb2-Sos-ERKi-PP
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-ERKiPP
114
ERKi-PP
Phosphatasel
Sosi
Raf*
Sos-ERKi-PP
Raf*-phosphatasel
k127
k42
kd127
kd42
Raf
Phosphatasel
Raf
MEK
MEK-P
MEK-P
MEK-PP
MEK
MEKi-P
Rafi*
Rafi*
Ras-GTP
Ras-GTP*
Ras-GTPi
Ras-GTPi*
Ras-GDP
Phosphatasel
Rafi*
Phosphatasel
Raf*
Raf*
Raf*
Raf*
Rafi*
Rafi*
MEKi-P
MEKi-PP
Raf
Raf*
Raf
Rafi*
(EGFEGFR*)2GAP-Grb2-Sos
(EGFEGFR*)2GAP-Grb2-Sos
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
Ras-GTP
Raf*-phosphatasel
Rafi*-phosphatasel
Rafi*-phosphatasel
MEK-Raf*
MEK-Raf*
MEK-P-Raf*
MEK-P-Raf*
MEK-Rafi*
MEK-Rafi*
MEK-P-Rafi*
MEK-P-Rafi*
Raf-Ras-GTP
Raf-Ras-GTP
Raf-Ras-GTPi
Raf-Ras-GTPi
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP
k43
k42
k43
k44
k45
k44
k47
k44
k45
k44
k47
k28
k29
k28
k29
k18
kd43
kd42
kd43
kd52
kd45
kd52
kd47
kd52
kd45
kd52
kd47
kd28
kd29
kd28
kd29
kdl8
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP
k19
kdl9
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP
k18
kdl8
k19
kdl9
k18
kdl8
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP
k19
kdl9
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP
k18
kdl8
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP
k19
kdl9
Ras-GTP
Ras-GDP
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
Ras-GDP
Ras-GTPi
Ras-GDP
(EGFEGFRi*)2GAP-Grb2-Sos
(EGFEGFRi*)2GAP-Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
Ras-GTPi
115
(EGFEGFRi*)2
(EGFEGFR*)2GAP-Grb2-Sos
(EGFEGFR*)2GAP-Grb2-Sos
(EGFEGFR*)2-GAP-
GAP
(EGF-EGFRi*)2-GAP
k8
kd8
Ras-GTP*
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GTP
k20
kd20
Ras-GDP
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GTP
k21
kd2l
Ras-GTP*
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-Ras-
k20
kd20
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGTP
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GTP
k21
kd2l
k20
kd20
Ras-GDP
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GTP
k21
kd2l
(EGFEGFRi*)2GAP-SHC*-
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGTP
k20
kd20
Ras-GDP
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGTP
k21
kd2l
EGFR
EGF-EGFR
0
EGFi
EGF-EGFRi
0
0
GAP
(EGFEGFR*)2-GAP
(EGFEGFR*)2GAP-Grb2
(EGFEGFR*)2-GAP
EGF-EGFR
(EGF-EGFR)2
(EGF-EGFR*)2
EGF-EGFRi
(EGF-EGFRi)2
(EGF-EGFRi*)2
EGFR
(EGF-EGFR*)2-GAP
(EGF-EGFR*)2-GAPGrb2
(EGF-EGFR*)2-GAPGrb2-Sos
ki
k2
k3
kiOb
k2
k3
k13
k8
k16
kdl
kd2
kd3
kdlO
kd2
kd3
kdl3
kd8
kd63
k17
kdl7
(EGF-EGFR*)2-GAPSHC
k22
kd22
GTP
SHC*-Grb2-Sos
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
Ras-GTPi*
Ras-GDP
(EGFEGFRi*)2GAP-Grb2-Sos
(EGFEGFRi*)2GAP-Grb2-Sos
Ras-GTPi*
Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
EGF
EGF-EGFR
(EGF-EGFR)2
EGFRi
EGF-EGFRi
(EGF-EGFRi)2
0
(EGF-EGFR*)2
Grb2
Sos
She
116
(EGFEGFR*)2GAP-SHC
Grb2
Sos
(EGFEGFR*)2-GAP
Shc*
(EGFEGFR*)2-GAP
Sos
Shc*
(EGFEGFR*)2-GAP
Grb2
(EGF-
0
(EGF-EGFR*)2-GAPSHC*
k23
kd23
(EGFEGFR*)2GAP-SHC*
(EGFEGFR*)2-GAPSHC*-Grb2
Shc*-Grb2-Sos
(EGF-EGFR*)2-GAPSHC*-Grb2
k16
kd24
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos
k25
kd25
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos
Shc*-Grb2-Sos
(EGF-EGFR*)2-GAPGrb2-Sos
Grb2-Sos
She
(EGF-EGFR*)2-GAPSHC*
Shc*-Grb2
(EGF-EGFR*)2-GAP-
k32
kd32
k33
k34
kd33
kd34
k35
k36
k37
kd35
kd36
kd37
k16
k37
kd24
kd37
Shc*-Grb2-Sos
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos
k40
k41
kd40
kd4l
(EGF-EGFRi*)2-GAPGrb2
(EGF-EGFRi*)2-GAPGrb2-Sos
k16
kd63
k17
kdl7
(EGF-
(EGF-EGFRi*)2-GAP-
k22
kd22
EGFRi*)2-GAP
SHC
0
(EGF-EGFRi*)2-GAPSHC*
k23
kd23
(EGFEGFRi*)2GAP-SHC*
(EGFEGFRi*)2GAP-SHC*Grb2
(EGF-EGFRi*)2-GAPSHC*-Grb2
k16
kd24
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos
k25
kd25
Grb2-Sos
Grb2-Sos
Grb2
0
Shc*
Shc*
Shc*-Grb2
SHC*-Grb2
EGFR*)2-GAP
Sos
Grb2-Sos
(EGFEGFRi*)2-GAP
Sos
Shc*-Grb2
(EGFEGFR*)2GAP-SHC*
Grb2
(EGFEGFRi*)2GAP-Grb2
She
(EGFEGFRi*)2GAP-SHC
Grb2
Sos
117
(EGFEGFRi*)2-GAP
(EGFEGFRi*)2-GAP
(EGFEGFRi*)2-GAP
(EGFEGFRi*)2-GAP
Grb2-Sos
GAP
GAP
GAP
GAP
GAP
GAP
GAP
GAP
Sosi
Shc*-Grb2-Sos
Grb2-Sos
Shc*
Shc*-Grb2
(EGFEGFRi*)2GAP-SHC*
Ras-GTP
Ras-GDP
Ras-GTPi
Ras-GDP
Ras-GTP*
Ras-GDP
Ras-GTPi*
Ras-GDP
Ras-GDP
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos
(EGF-EGFRi*)2-GAPGrb2-Sos
(EGF-EGFRi*)2-GAPSHC*
(EGF-EGFRi*)2-GAPSHC*-Grb2
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos
k32
kd32
k34
kd34
k37
kd37
k37
kd37
k41
kd41
GAP-Ras-GTP
GAP-Ras-GTP
GAP-Ras-GTPi
GAP-Ras-GTPi
GAP-Ras-GTP*
GAP-Ras-GTP*
GAP-Ras-GTPi*
GAP-Ras-GTPi*
Sos
k300
0
k300
0
k300
0
k300
0
k300
kd20
kd21
kd20
kd21
kd20
kd21
kd20
kd21
0
Model Parameters
1
Exact values of the parameters in the model. The first order rate constant have units of s-
The second order rate constants have a unit of (#mlcIes)- s1.
Name
Value
kO
kdO
0.000000e+000
0.000000e+000
kI
kd1
k10b
kdl0
k2
kd2
3.000000e+007
3.840000e-003
5.430000e-002
1. 100000e-002
1.660000e-005
1.000000e-001
k3
1.000000e+000
kd3
1.000000e-002
118
k4
kd4
kd5
kM
k
kd6
k8
kd8
k13
kdl3
k15
kdl5
k16
kd16
k17
kd17
k18
kd18
k19
kd19
k20
kd20
k21
kd2l
k22
kd22
k23
kd23
kd24
k25
kd25
k28
kd28
k29
kd29
kd32
k32
kd33
k33
kd34
k34
kd35
1.730000e-007
1.660000e-003
1.480000e-002
0.000000e+000
5.0000OOe-004
5.0000OOe-003
1.660000e-006
2.000000e-001
2.170000e+000
0.000000e+000
1.000000e+004
0.000000e+000
1.660000e-005
0.000000e+000
1.660000e-005
6.0000OOe-002
2.500000e-005
1.300000e+000
1.660000e-007
5.0000OOe-001
3.500000e-006
4.0000OOe-001
3.660000e-007
2.300000e-002
3.500000e-005
1.000000e-001
6.000000e+000
6.0000OOe-002
5.500000e-001
1.660000e-005
2.140000e-002
1.660000e-006
5.300000e-003
1.170000e-006
1.000000e+000
1.000000e-001
4.0000OOe-007
2.000000e-001
3.500000e-005
3.0000OOe-002
7.500000e-006
1.500000e-003
119
k35
k36
kd36
kd37
k37
k40
kd40
k41
kd4l
k42
kd42
kd43
k43
kd44
kd45
k45
kd47
k47
k48
kd48
kd49
k49
k50
kd50
kd52
kd53
k53
kd55
k55
kd56
k56
kd57
k57
k58
kd58
k52
k44
k60
kd60
k61
kd6l
kd63
7.500000e-006
5.000000e-003
0.000000e+000
3.000000e-001
1.500000e-006
5.0000OOe-005
6.400000e-002
5.0000OOe-005
4.290000e-002
1.180000e-004
2.0000OOe-001
1.000000e+000
0.000000e+000
1.830000e-002
3.500000e+000
0.000000e+000
2.900000e+000
0.000000e+000
2.380000e-005
8.0000OOe-001
5.800000e-002
0.000000e+000
4.500000e-007
5.000000e-001
3.300000e-002
1.600000e+001
0.000000e+000
5.700000e+000
0.000000e+000
6.0000OOe-001
2.350000e-005
2.460000e-001
0.000000e+000
8.330000e-006
5.0000OOe-001
8.910000e-005
1.960000e-005
5.500000e-003
0.000000e+000
6.700000e-004
0.000000e+000
2.750000e-001
120
0.000000e+000
1.660000e-007
2.000000e+000
1.0000OOe-004
0.000000e+000
2.170000e+000
1.810000e-004
1.970000e-004
8.250000e-005
3.0100OOe-005
3.0000OOe-005
5.430000e-005
1.000000e-007
k63
k126
kd126
kd127
k127
k200
kd214
kd222
kd224
kd226
kd231
kd230
k300
Variant Models
Variant I
Following changes were made to the original model to obtain model variant I:
Parameter
k13
kd6
Value
2.170000e+001
5.0000OOe-004
Variant II
Following changes were made to the original model to obtain model variant II:
1) These reactions on removed from the model.
Reactant1
Reactant2
Product
kForward
kReverse
Ras-GTP*
Ras-GTPi*
Raf*
Rafi*
Raf-Ras-GTP
Raf-Ras-GTPi
k29
k29
kd29
kd29
2) These reactions were added to the model.
Reactantl1
Reactant2 I Product
121
I kForward I kReverse
Ras-GTP
Ras-GTPi
Raf*
Rafi*
Raf-Ras-GTP
Raf-Ras-GTPi
k29
k29
kd29
kd29
Variant III
Following change was made to the original model to obtain variant III:
Parameter
k42
A.3
Value
1.180000e-007
Molecular Identities of Targets Inhibited
Reactant1
EGFR
EGFRi
EGFR-Il
Reactant2
I1
I1
0
Product
EGFR-Il
EGFRi-Il
EGFRi-11
kForward
kil
kil
k6
kReverse
kdil
kdil
kd6
EGFRi-Il
0
0
a1k60
kd60
EGF-EGFR
EGF-EGFRi
(EGF-EGFR)12
12
12
0
(EGF-EGFR)-12
(EGF-EGFRi)-12
(EGF-EGFRi)-12
ki2
ki2
k6
kdi2
kdi2
kd6
(EGF-EGFRi)-
0
0
a1k60
kd60
12
(EGF-EGFR)2
(EGF-EGFRi)2
(EGF-EGFR)2-
13
13
0
(EGF-EGFR)2-I3
(EGF-EGFRi)2-13
(EGF-EGFRi)2-13
ki3
ki3
k6
kdi3
kdi3
kd6
0
0
a1k60
kd60
14
14
(EGF-EGFR*)2-I4
ki4
kdi4
(EGF-EGFRi*)2-I4
ki4
kdi4
0
(EGF-EGFRi*)2-I4
k6
kd6
0
0
alk60
kd60
13
(EGF-EGFRi)2-
13
(EGF-EGFR*)2
(EGFEGFRi*)2
(EGFEGFR*)2-14
(EGFEGFRi*)2-14
122
(EGF-
15
(EGF-EGFR*)2-GAP-I5
ki5
kdi5
I5
(EGF-EGFRi*)2-GAP-
ki5
kdi5
EGFR*)2-GAP
(EGF-
15
EGFRi*)2-GAP
(EGFEGFR*)2-
0
(EGF-EGFRi*)2-GAP15
k6
kd6
0
0
alk60
kd60
15
0
16
GAP-15
0
(EGF-EGFR*)2-GAPSHC-16
ki5
akd214
ki6
kdi5
0
kdi6
16
(EGF-EGFRi*)2-GAPSHC-I6
ki6
kdi6
0
(EGF-EGFRi*)2-GAPSHC-16
k
kd6
0
0
a1k60
kd60
16
0
17
Shc-16
0
(EGF-EGFR*)2-GAPSHC*-I7
ki6
akd231
ki7
kdi6
0
kdi7
17
(EGF-EGFRi*)2-GAPSHC*-I7
ki7
kdi7
0
(EGF-EGFRi*)2-GAPSHC*-I7
k6
kd6
0
0
alk60
kd60
17
18
Shc*-I7
(EGF-EGFR*)2-GAPGrb2-18
ki7
ki8
kdi7
kdi8
GAP-15
(EGFEGFRi*)2GAP-15
GAP
GAP-15
(EGFEGFR*)2GAP-SHC
(EGFEGFRi*)2GAP-SHC
(EGFEGFR*)2GAP-SHC-16
(EGFEGFRi*)2GAP-SHC-16
She
Shc-16
(EGFEGFR*)2GAP-SHC*
(EGFEGFRi*)2GAP-SHC*
(EGFEGFR*)2GAP-SHC*-I7
(EGFEGFRi*)2GAP-SHC*-I7
Shc*
(EGFEGFR*)2GAP-Grb2
123
(EGFEGFRi*)2-
18
(EGF-EGFRi*)2-GAPGrb2-18
ki8
kdi8
Prot
(EGF-EGFR*)2-GAPGrb2-18-Prot
k4
kd4
(EGFEGFRi*)2-
(EGF-EGFR*)2-GAPGrb2-18-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPGrb2-18
k6
kd6
0
0
alk60
kd60
18
(EGF-EGFR*)2-GAPSHC*-Grb2-I8
ki8
kdi8
18
(EGF-EGFRi*)2-GAPSHC*-Grb2-I8
ki8
kdi8
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-I8-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*-
(EGF-EGFR*)2-GAPSHC*-Grb2-I8-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPSHC*-Grb2-I8
k
kd6
0
0
alk60
kd60
18
I8
0
19
Shc*-Grb2-I8
Grb2-I8
0
(EGF-EGFR*)2-GAPGrb2-Sos-19
ki8
ki8
akd222
ki9
kdi8
kdi8
0
kdi9
GAP-Grb2
(EGFEGFR*)2GAP-Grb2-18
Proti
GAP-Grb2-18
(EGFEGFR*)2GAP-Grb2-I8
(EGFEGFRi*)2GAP-Grb2-18
(EGFEGFR*)2-GAPSHC*-Grb2
(EGFEGFRi*)2GAP-SHC*Grb2
(EGFEGFR*)2-GAPSHC*-Grb2-I8
Proti
Grb2-18
(EGFEGFR*)2-GAPSHC*-Grb2-I8
(EGFEGFRi*)2GAP-SHC*Grb2-18
Shc*-Grb2
Grb2
Grb2-I8
(EGFEGFR*)2GAP-Grb2-Sos
124
(EGFEGFRi*)2-
19
(EGF-EGFRi*)2-GAPGrb2-Sos-19
ki9
kdi9
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-19-Prot
k4
kd4
(EGFEGFRi*)2GAP-Grb2-Sos-
(EGF-EGFR*)2-GAPGrb2-Sos-19-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPGrb2-Sos-I9
k
kd6
0
0
alk60
kd60
19
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-I9
ki9
kdi9
19
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-I9
ki9
kdi9
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-I9-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*-
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-I9-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-I9
k
kd6
0
0
alk60
kd60
19
0
19
Sos-19
0
Grb2-Sos-19
ki9
akd224
ki9
kdi9
0
kdi9
GAP-Grb2-Sos
(EGFEGFR*)2-GAPGrb2-Sos-I9
Proti
19
(EGFEGFR*)2-GAPGrb2-Sos-I9
(EGFEGFRi*)2GAP-Grb2-Sos19
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
(EGFEGFR*)2GAP-SHC*Grb2-Sos-19
Proti
Grb2-Sos-19
(EGFEGFR*)2GAP-SHC*Grb2-Sos-19
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-I9
Sos
Sos-19
Grb2-Sos
125
Grb2-Sos-19
Shc*-Grb2-Sos
(EGFEGFR*)2GAP-Grb2-Sos-
0
19
110
0
Shc*-Grb2-Sos-I9
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-I10
akd230
ki9
kilO
0
kdi9
kdil0
I10
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP-I10
kilO
kdilO
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-I10Prot
k4
kd4
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP-I10
0
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-I10Prot
k5
kd5
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP-110
k
kd6
0
0
alk60
kd60
110
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-I10
kilO
kdil0
I10
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP-110
kilO
kdil0
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-I10-Prot
k4
kd4
Ras-GDP
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP
(EGFEGFR*)2GAP-Grb2-SosRas-GDP-110
Proti
(EGFEGFR*)2GAP-Grb2-SosRas-GDP-I10
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP-I10
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP-I10
126
Proti
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP-I10
(EGFEGFRi*)2-
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP-I10
0
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-I1O-Prot
k5
kd5
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP-I1O
k
kd6
0
0
alk60
kd60
Ill
Ill
Ill
0
112
112
112
113
113
113
113
113
114
114
114
114
114
Ras-GTP-11
Ras-GTPi-Ill
Ras-GDP-I11
0
Raf-112
Raf*-I12
Rafi*-Il2
MEK-113
MEK-P-I13
MEKi-P-13
MEK-PP-113
MEKi-PP-1l3
ERK-114
ERK-P-I14
ERKi-P-114
ERK-PP-114
ERKi-PP-114
kill
kill
kill
akd226
kil2
kil2
kil2
kil3
kil3
kil3
kil3
kil3
kil4
kil4
kil4
kil4
kil4
kdill
kdill
kdill
0
kdil2
kdil2
kdil2
kdil3
kdil3
kdil3
kdil3
kdil3
kdil4
kdil4
kdil4
kdil4
kdil4
GAP-SHC*Grb2-Sos-RasGDP-I10
Ras-GTP
Ras-GTPi
Ras-GDP
Ras-GDP-I11
Raf
Raf*
Rafi*
MEK
MEK-P
MEKi-P
MEK-PP
MEKi-PP
ERK
ERK-P
ERKi-P
ERK-PP
ERKi-PP
Parameter
ki*
kdi*
alk60
akd214
akd231
akd2224
akd230
Value
1.660000e-006
1.000000e-003
oz *k60
a* kd214
* kd231
a * kd224
* kd230
127
akd226
a
a * kd226
1.0
a here represents the stability of the target-inhibitor complex compared to the stability
of the target protein alone. Binding with an inhibitor is likely to increase the stability of the
protein (hence a decrease its degradation rate). The results shown in this case are for the
case where there is no increase in the stability of the protein because of its association with
inhibitor, hence, a value of 1. The target behavior was not affected by the choice of value
for a as long as it was above 0.75.
128
Appendix B
Combination Target Intervention
B.1
Molecular Targets Inhibited
Reactant1
EGFR
EGFRi
EGFR-Il
EGFRi-Il
EGF-EGFR
EGF-EGFRi
(EGF-EGFR)12
(EGF-EGFRi)12
(EGF-EGFR)2
(EGF-EGFRi)2
(EGF-EGFR)2I3
(EGF-EGFRi)213
(EGF-EGFR*)2
(EGFEGFRi*)2
(EGFEGFR*)2-14
Reactant2
I1
I1
0
0
12
12
0
Product
EGFR-Il
EGFRi-Il
EGFRi-I1
0
(EGF-EGFR)-12
(EGF-EGFRi)-12
(EGF-EGFRi)-12
kForward
kil
kil
k6
a1k60
ki2
ki2
k
kReverse
kdil
kdil
kd6
kd60
kdi2
kdi2
kd6
0
0
a2k60
kd60
13
13
0
(EGF-EGFR)2-I3
(EGF-EGFRi)2-13
(EGF-EGFRi)2-I3
ki3
ki3
k6
kdi3
kdi3
kd6
0
0
a3k60
kd60
14
14
(EGF-EGFR*)2-I4
(EGF-EGFRi*)2-I4
ki4
ki4
kdi4
kdi4
0
(EGF-EGFRi*)2-I4
k6
kd6
129
(EGFEGFRi*)2-14
(EGFEGFR*)2-GAP
(EGFEGFRi*)2-GAP
(EGFEGFR*)2GAP-15
(EGFEGFRi*)2GAP-15
(EGFEGFR*)2GAP-Grb2
(EGFEGFRi*)2GAP-Grb2
(EGFEGFR*)2GAP-Grb2-16
Proti
0
0
a4k60
kd60
I5
(EGF-EGFR*)2-GAP-I5
ki5
kdi5
15
(EGF-EGFRi*)2-GAP15
(EGF-EGFRi*)2-GAP15
ki5
kdi5
k6
kd6
0
0
a5k60
kd60
16
(EGF-EGFR*)2-GAPGrb2-16
ki6
kdi6
16
(EGF-EGFRi*)2-GAPGrb2-16
ki6
kdi6
Prot
(EGF-EGFR*)2-GAPGrb2-I6-Prot
k4
kd4
(EGFEGFRi*)2-
(EGF-EGFR*)2-GAPGrb2-16-Prot
k5
kd5
0
(EGF-EGFRi*)2-GAPGrb2-16
k6
kd6
0
0
a6k60
kd60
17
(EGF-EGFR*)2-GAPGrb2-Sos-17
ki7
kdi7
17
(EGF-EGFRi*)2-GAPGrb2-Sos-17
ki7
kdi7
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-17-Prot
k4
kd4
(EGFEGFRi*)2GAP-Grb2-Sos17
(EGF-EGFR*)2-GAPGrb2-Sos-17-Prot
k5
kd5
0
GAP-Grb2-16
(EGFEGFR*)2GAP-Grb2-16
(EGFEGFRi*)2GAP-Grb2-I6
(EGFEGFR*)2GAP-Grb2-Sos
(EGFEGFRi*)2GAP-Grb2-Sos
(EGFEGFR*)2-GAPGrb2-Sos-I7
Proti
130
(EGFEGFR*)2-GAPGrb2-Sos-17
(EGFEGFRi*)2GAP-Grb2-Sos17
(EGFEGFR*)2GAP-Grb2-SosRas-GDP
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP
(EGFEGFR*)2GAP-Grb2-SosRas-GDP-18
Proti
(EGFEGFR*)2GAP-Grb2-SosRas-GDP-18
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP-18
(EGFEGFR*)2GAP-SHC
(EGFEGFRi*)2GAP-SHC
(EGFEGFR*)2GAP-SHC-19
(EGFEGFRi*)2-
0
(EGF-EGFRi*)2-GAPGrb2-Sos-I7
k6
kd6
0
0
a7k60
kd60
18
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-18
ki8
kdi8
18
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP-18
ki8
kdi8
Prot
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-I8Prot
k4
kd4
(EGFEGFRi*)2GAP-Grb2-SosRas-GDP-18
0
(EGF-EGFR*)2-GAPGrb2-Sos-Ras-GDP-I8Prot
k5
kd5
(EGF-EGFRi*)2-GAPGrb2-Sos-Ras-GDP-18
k6
kd6
0
0
a8k60
kd60
19
(EGF-EGFR*)2-GAPSHC-19
ki9
kdi9
19
(EGF-EGFRi*)2-GAPSHC-19
ki9
kdi9
0
(EGF-EGFRi*)2-GAPSHC-19
k6
kd6
0
0
a9k60
kd60
GAP-SHC-19
131
(EGFEGFR*)2GAP-SHC*
(EGFEGFRi*)2GAP-SHC*
(EGFEGFR*)2GAP-SHC*-I10
(EGFEGFRi*)2-
110
(EGF-EGFR*)2-GAPSHC*-I10
kilO
kdilO
110
(EGF-EGFRi*)2-GAPSHC*-I10
kilO
kdilO
0
(EGF-EGFRi*)2-GAPSHC*-I10
k
kd6
0
0
a10k60
kd60
Ill
(EGF-EGFR*)2-GAPSHC*-Grb2-11
kill
kdill
Ill
(EGF-EGFRi*)2-GAPSHC*-Grb2-11
kill
kdill
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-I11-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*Grb2-Il1
0
(EGF-EGFR*)2-GAPSHC*-Grb2-I11-Prot
k5
kd5
(EGF-EGFRi*)2-GAPSHC*-Grb2-11
k6
kd6
0
0
allk60
kd60
112
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-I12
kil2
kdil2
112
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-II2
kil2
kdil2
GAP-SHC*-I10
(EGFEGFR*)2-GAPSHC*-Grb2
(EGFEGFRi*)2GAP-SHC*Grb2
(EGFEGFR*)2-GAPSHC*-Grb2-111
Proti
(EGFEGFR*)2-GAPSHC*-Grb2-I11
(EGFEGFRi*)2GAP-SHC*Grb2-I11
(EGFEGFR*)2-GAPSHC*-Grb2-Sos
(EGFEGFRi*)2GAP-SHC*Grb2-Sos
132
(EGFEGFR*)2GAP-SHC*Grb2-Sos-I12
Proti
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-I12-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*-
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-I12-Prot
k5
kd5
(EGFEGFR*)2GAP-SHC*-
0
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-I12
k
kd6
0
0
a12k60
kd60
113
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-113
kil3
kdil3
113
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP-113
kil3
kdil3
Prot
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-113-Prot
k4
kd4
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP-113
0
(EGF-EGFR*)2-GAPSHC*-Grb2-Sos-RasGDP-I13-Prot
k5
kd5
(EGF-EGFRi*)2-GAPSHC*-Grb2-Sos-RasGDP-113
k6
kd6
0
0
a13k60
kd60
Grb2-Sos-I12
Grb2-Sos-I12
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-112
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP-113
Proti
(EGFEGFR*)2-GAPSHC*-Grb2-SosRas-GDP-113
(EGFEGFRi*)2GAP-SHC*Grb2-Sos-RasGDP-113
133
Ras-GTP
Ras-GTPi
Ras-GDP
Ras-GDP-115
Raf*
Rafi*
Raf
MEK-PP
MEKi-PP
MEK-P
MEKi-P
MEK
ERK-PP
ERKi-PP
ERK-P
ERKi-P
ERK
GAP
GAP-124
Grb2
Grb2-I25
Sos
Sos-126
She
Shc-127
Grb2-Sos
Grb2-Sos-128
Shc*
Shc*-Grb2-Sos
Shc*-Grb2
114
114
115
0
116
116
117
118
118
119
119
120
121
121
122
122
123
124
0
125
0
126
0
127
0
128
0
129
130
131
Ras-GTP-114
Ras-GTPi-114
Ras-GDP-115
0
Raf*-I16
Rafi*-I16
Raf-117
MEK-PP-118
MEKi-PP-118
MEK-P-119
MEKi-P-119
MEK-I20
ERK-PP-121
ERKi-PP-121
ERK-P-122
ERKi-P-122
ERK-I23
GAP-124
0
Grb2-125
0
Sos-126
0
Shc-127
0
Grb2-Sos-I28
0
Shc*-I29
Shc*-Grb2-Sos-I30
Shc*-Grb2-I31
134
ki14
ki14
kil5
akd226
kil6
kil6
kil7
kil8
kil8
kil9
kil9
ki20
ki2l
ki2l
ki22
ki22
ki23
ki24
akd214
ki25
akd222
ki26
akd224
ki27
akd231
ki28
akd230
ki29
ki30
ki3l
kdil4
kdil4
kdil5
0
kdil6
kdil6
kdil7
kdil8
kdil8
kdil9
kdil9
kdi20
kdi2l
kdi2l
kdi22
kdi22
kdi23
kdi24
0
kdi25
0
kdi26
0
kdi27
0
kdi28
0
kdi29
kdi30
kdi3l
Appendix C
Effects Exerted by Interventions
C.1
Mathematical Basis
This section provides a mathematical basis for the effects exerted by kinetically tuned inhibitors, feedback, and feedforward circuits on the rate of change of the target.
C.1.1
Kinetically-Tuned Inhibitors
Idealized reaction for this process can be written as:
Tp + I
-1
Tp:I (inactive complex)
Here, Tp is the target at which the inhibitor acts and I is the inhibitor introduced in the
system. Inhibitors are treated as inputs and maintained at constant level in the implementations.
In the absence of the inhibitor:
Let,
dTp
d__ = X(x, t)
dt
(C.1)
(C.2)
135
Where, X is a function of x and t. x represents all the other species in the model that
contribute to the rate of change of Tp.
Then, in the presence of the inhibitor:
dTp= X(x,t) - k1 x I x Tp
dt
(C.3)
Because I is constant, the effect of introducing inhibitor to the rate change of the target
that we are trying to modulate is a linear function of the target itself.
C.1.2
Feedback and Feed-Forward Loops
Idealized reactions for this process can be written as:
A + Sp k2
Ap + Tp -
Ap + Sp
Ap+T
OR
Ap + Tp
>Ap +<D
Here, A is the inactive protein therapy introduced that is treated as a input and maintained at a constant level, Sp is where the 'sensing' part of the feedback or feedforward takes
place, Ap is the activated form of A (activation carried out enzymatically), Tp is the active
target that Ap either deactivated or degraded enzymatically.
In the absence of the intervention:
dT p = X(x,
t)
dt
136
(C.4)
In the presence of the intervention:
dTp
=X(xt) - k3 x Ap x Tp
dt
(C.5)
dAp
dt
(C.6)
=
k2 x A x Sp
(C.7)
A is constant
Ap = k2 x A x
Sp
dt
=
f(x, t), Sp
I
(C.8)
Sp dt
is a function of x which includes Tp
X(x, t)-k3 x k2 x Ax
Spdt x Tp
Quadratic effect on rate of change of Tp through f Sp dt x Tp
137
(C.9)
(C.10)