FROM IMMUNOLOGY TO MRI DATA ANLYSIS:
PROBLEMS IN MATHEMATICAL BIOLOGY
by
Ryan Samuel Waters
A dissertation submitted in partial fullfilment
of the requirements for the degree
of
Doctorate of Philosophy
in
Mathematics
MONTANA STATE UNIVERSITY
Bozeman, Montana
April, 2015
c
COPYRIGHT
by
Ryan Samuel Waters
2015
All Rights Reserved
ii
Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The Role Interleukin-2 in Immune Response Regulation . . . . . . 5
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . . . .
Summary of Results . . . . . . . . . . . . . . . . . . .
Unpublished Experimental Results . . . . . . . . . . .
Dynamic Model of IL-2 Receptor (Model I) . . . . . . .
2.5.1 Parameter Selection . . . . . . . . . . . . . . . .
2.5.2 Results: Kinetics of Phosphorylation of STAT5
Cell Line Differentiation Model (Model II) . . . . . . .
2.6.1 Model . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Results: Conclusions of Model II . . . . . . . .
2.6.3 Results: Combining Model I and II . . . . . . .
IL-2 Receptor with Secondary Signal (Model III) . . .
2.7.1 Modeling Assumptions . . . . . . . . . . . . . .
2.7.2 New Parameters . . . . . . . . . . . . . . . . . .
2.7.3 Results . . . . . . . . . . . . . . . . . . . . . . .
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.5
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3 Kinetic Benefits of the Colocalization of Coupled Enzymes . . 54
3.1
3.2
3.3
3.4
3.5
3.6
Introduction . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . .
3.2.1 Bacterial Microcomparments .
3.2.2 Single Enzyme Encapsulation
3.2.3 Multi-Enzyme Encapsulation
Model Framework . . . . . . . . . . .
1D Model . . . . . . . . . . . . . . .
3.4.1 Model . . . . . . . . . . . . .
3.4.2 Rescaling . . . . . . . . . . .
Steady-state solution . . . . . . . . .
3.5.1 Discussion . . . . . . . . . . .
Rate of Convergence . . . . . . . . .
3.6.1 Linearization of the Boundary
3.6.2 Fourier Series Solution . . . .
3.6.3 Convergence Rate . . . . . . .
3.6.4 Discussion . . . . . . . . . . .
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Conditions
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54
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iii
Table of Contents - Continued
4 What is the Ideal Distribution of Viral Fragments on CRISPR? 73
4.1
4.2
4.3
4.4
4.5
Introduction . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . .
Analysis of Ideal CRISPR Distribution . . . . . . .
4.3.1 Discrete Case: Lagrange Multipliers . . . . .
4.3.2 Continuous Case: Euler-Lagrange Equation
Examples . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Linear Survival Function . . . . . . . . . . .
4.4.2 T (k) = a k n + c . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . .
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73
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5 MRI Data Analysis and Neurological Disease . . . . . . . . . . . . 87
5.1
5.2
5.3
5.4
5.5
5.6
Introduction . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . .
Development of Novel MRI Scale . . . . . . . . .
5.3.1 Physical/Cognitive Disability Scales . . . .
5.3.2 MRI Grading: Categories/Severity . . . .
5.3.3 Construction of CAMRIS-CTD . . . . . .
Mathematical Optimization of CAMRIS-CTD . .
Validation of CAMRIS-CDO and CAMRIS-PDO
5.5.1 MRI Score vs Disability Score . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . .
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. 87
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. 103
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References Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
iv
List of Tables
Table
Page
2.1
Model I: Binding Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2
Model I: Production/Decay Rates . . . . . . . . . . . . . . . . . . . . . 20
2.3
Model I: Conservation Equation Maximums . . . . . . . . . . . . . . . 20
2.4
Model I: Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5
Model III: Production/Decay Rates . . . . . . . . . . . . . . . . . . . . 49
2.6
Model III: Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1
MRI Grading Categories Continued . . . . . . . . . . . . . . . . . . . . 94
5.2
MRI Model Coefficients
5.3
Correlation Table - Discovery . . . . . . . . . . . . . . . . . . . . . . . 102
5.4
Correlation Table - Validation . . . . . . . . . . . . . . . . . . . . . . . 103
5.5
Comparison of Scores for Different Diagnostic Groups . . . . . . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . . 100
v
List of Figures
Figure
Page
2.1
IL-2 Receptor Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2
Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3
Model I Fit Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4
Model I Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5
Model II: Pitchfork Bifurcation . . . . . . . . . . . . . . . . . . . . . . 32
2.6
Model II: Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . . . . 32
2.7
Model II: Bistability in V and C . . . . . . . . . . . . . . . . . . . . . . 35
2.8
Model II: IL-2 and CD122 Supporting Bistability . . . . . . . . . . . . 40
2.9
Bound Receptors for Various Concentrations of CD25 . . . . . . . . . . 42
2.10 Model II: Daclizumab Bifurcation Example . . . . . . . . . . . . . . . . 44
2.11 Model III Cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.12 Intermediate vs High Affinity: Low IL-2 . . . . . . . . . . . . . . . . . 50
2.13 Intermediate vs High Affinity: Moderate IL-2 . . . . . . . . . . . . . . 50
2.14 Intermediate vs High Affinity: High IL-2 . . . . . . . . . . . . . . . . . 51
2.15 Affects of Daclizumab: Low IL-2 . . . . . . . . . . . . . . . . . . . . . . 53
2.16 Affects of Daclizumab: Moderate IL-2 . . . . . . . . . . . . . . . . . . . 53
3.1
Co-encapsulated Enzyme System . . . . . . . . . . . . . . . . . . . . . 57
3.2
Separate vs. Co-encapsulated Coupled Enzymes . . . . . . . . . . . . . 59
3.3
Michaelis-Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4
Diffusion between parallel plates . . . . . . . . . . . . . . . . . . . . . . 63
4.1
CRISPR: Acquisition and Interference . . . . . . . . . . . . . . . . . . 74
4.2
Viral Distribution and Survival Function . . . . . . . . . . . . . . . . . 82
4.3
Discrete Example: Ideal CRISPR Distribution . . . . . . . . . . . . . . 83
vi
List of Figures - Continued
Figure
Page
4.4
Continuous Example: Guassian Distribution . . . . . . . . . . . . . . . 85
5.1
MRI Grading Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2
Cross Validation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3
Correlation with NRS model . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4
Correlation with EDSS model . . . . . . . . . . . . . . . . . . . . . . . 101
vii
Abstract
This thesis represents a collection of four distinct biological projects rising from
immunology and metabolomics that required unique and creative mathematical approaches. One project focuses on understanding the role IL-2 plays in immune response regulation and exploring how these effects can be altered. We developed
several dynamic models of the receptor signaling network which we analyze analytically and numerically. In a second project focused also on MS, we sought to create
a system for grading magnetic resonance images (MRI) with good correlation with
disability. The goal is for these MRI scores to provide a better standard for large-scale
clinical drug trials, which limits the bias associated with differences in available MRI
technology and general grader/participant variability. The third project involves the
study of the CRISPR adaptive immune system in bacteria. Bacterial cells recognize and acquire snippets of exogenous genetic material, which they incorporate into
their DNA. In this project we explore the optimal design for the CRISPR system
given a viral distribution to maximize its probability of survival. The final project
involves the study of the benefits for colocalization of coupled enzymes in metabolic
pathways. The hypothesized kinetic advantage, known as ‘channeling’, of putting
coupled enzymes closer together has been used as justification for the colocalization
of coupled enzymes in biological systems. We developed and analyzed a simple partial
differential equation of the diffusion of the intermediate substrate between coupled
enzymes to explore the phenomena of channeling. The four projects of my thesis
represent very distinct biological problems that required a variety of techniques from
diverse areas of mathematics ranging from dynamical modeling to statistics, Fourier
series and calculus of variations. In each case, quantitative techniques were used to
address biological questions from a mathematical perspective ultimately providing
insight back to the biological problems which motivated them.
1
Introduction
Biological systems inspire diverse quantitative questions and diverse mathematical
techniques are needed to approach these problems. This thesis represents a collection
of four distinct biological projects rising from immunology and metabolomics that
each required unique and creative mathematical approaches.
Two of the projects focus on multiple sclerosis (MS), a degenerative, neurological,
autoimmune disorder, and developed from a collaboration with the Bielekova lab of
the National Institute of Neurological Disorders and Stroke at the National Institute
of Health. MS occurs when the effector T cells (white blood cells) mistakenly target
and attack the myelin sheath surrounding and protecting the neurons. The myelin
sheath acts as an insulator and helps neurons carry electrical signals, so its destruction
prevents neurons from effectively transmitting electrical impulses. The damage left
by these attacks (lesions) can cause a variety of neurological problems both cognitive
and physical ranging from minor numbness to paralysis.
The environmental or genetic factors that cause these T cells to target their own
body’s tissue are not fully understood. However, our collaborators have had success using Daclizumab, a humanized monoclonal antibody, for the treatment of MS.
Daclizumab binds to a portion the Interleukin-2 (IL-2) receptor forcing the cytokine
receptor to signal via a lower affinity pathway. The success of Daclizumab in managing
MS indicates that signaling through the IL-2 receptor likely plays an important role
in both the activation and maintenance of these effector T cell populations. Chapter
2 focuses on understanding the role IL-2 plays in immune response regulation and exploring how Daclizumab alters these effects. We developed several dynamic models of
the receptor signaling network which we analyze both analytically, using bifurcation
analysis, as well as numerically.
2
As previously mentioned, the symptoms of MS can be diverse in type and severity.
Not only does this make the initial diagnosis difficult, but this can also make assessments of disease progression difficult when testing drug efficacy. An accurate and
unbiased system for analyzing disease progression is required for effective evaluation
of clinical trials. Unfortunately, physical and cognitive disability scales inherently
have problems with grader and participant bias. Alternatively, up to this point there
has been poor correlation of physical or cognitive disability to neurological damage
measured via magnetic resonance imaging (MRI). The recently developed quantitative MRI scales have improved the potential correlation significantly, but they are
generally too sensitive leading to variability associated with the age (and type) of the
MRI scanner as well as being affected by small movements of the patient. In order
to make a compromise between these tradeoffs, we assisted our collaborators in the
development of a semi-quantitative MRI grading scale (Chapter 5). Our collaborators
chose a variety of categories to analyze on an MRI image and corresponding scores
for each category. This data was then used to choose an optimal set of categories,
which were in turn used to develop and optimize a linear model leading to a MRI
‘score’. The goal is for these MRI scores to provide a better standard for large-scale
clinical drug trials, which limits the bias associated with differences in available MRI
technology (because the categories were chosen to be easily discernible) and general
grader/participant variability (because of the quantitive nature of the MRI scale).
In addition to the two collaborations with the Bielekova lab, I have a third project
related to adaptive immunity involving the study of the CRISPR adaptive immune
system in bacteria and archaea (Chapter 4). Bacterial cells recognize and acquire
snippets of exogenous genetic material, which they incorporate into their DNA. This
portion of the DNA, known as the CRISPR (Cluster of Regularly Interspaced Short
Palindromic Repeats) locus, is characterized by short segments of viral DNA sep-
3
arated by a short repeated sequence. The CRISPR locus is transcribed and then
cleaved into short CRISPR RNA sequences (crRNA). The crRNAs are integrated
into a larger ribonucleo-protein complexes which scan the cell searching for foreign
DNA. Once located, the CRISPR complex targets the viral genetic material for destruction. In this project we explore optimal design for the CRISPR system given a
viral distribution to optimize its probability of survival. After developing a survival
probability function, ideas from multivariable calculus and calculus of variations were
used to determine the optimal CRISPR distributions.
Chapter 3 involves the study of the benefits for colocalization of coupled enzymes
in metabolic pathways. This project developed as a collaboration with Benji Schwarz
of the Douglas Biomimetic Chemistry Lab at the University of Indiana. If we consider
a simple metabolic pathway of two enzymes where the product of the first enzyme
is the substrate of the second, then simply based on concentration gradients, one
would assume that putting these enzymes closer together would result in increased
efficiency in the production of the final product (product of the second enzyme). This
hypothesized kinetic advantage, known as ‘channelling’, has been used as justification
for the colocalization of coupled enzymes in bacterial microcompartments (e.g. the
carboxysome). However, an experiment by our collaborators demonstrated that, for
some enzymatic pathways, colocalization does not result in kinetic enhancement of the
product formation. We developed and analyzed a simple partial differential equation
of the diffusion of the intermediate substrate between coupled enzymes to explore
the phenomena of channeling. Our model supports the experimental results that
the kinetic advantage of colocalization is not ubiquitous, but rather depends on the
specific parameters of the system.
Biological systems offer a collection of interesting quantitative problems. The four
projects of my thesis represent very distinct biological problems that required a variety
4
of techniques from diverse areas of mathematics ranging from dynamical modeling
to statistics, Fourier series and calculus of variations. In each case, quantitative
techniques were used to address biological questions from a mathematical perspective
ultimately providing insight back to the biological problems which motivated them.
5
The Role Interleukin-2 in Immune Response Regulation
2.1 Introduction
The immune system has many adaptive and dynamic components that are delicately regulated to ensure appropriate, precise, and rapid response to a foreign
pathogen. A delayed or inadequate immune response can lead to prolonged disease,
while an excessive or under-regulated response can lead to autoimmunity. Interleukin
2 (IL-2), and other signaling molecules (cytokines) in general, play an important role
in maintaining this balance. Through the use of several deterministic models and
bifurcation analysis, the goal of this project is to understand both the short term and
long term role of IL-2 signaling in maintaining this balance [101].
2.2 Background
Despite the fact that interleukin-2 (IL-2) and its receptors (Figure 2.1) represent
one of the most extensively studied cytokine signaling systems, unexpected findings
emerging from therapeutic manipulations of this system in-vivo cannot be explained
by simple conceptual models [19, 97]. Instead, mathematical modeling is likely required for predictive understanding of the varied effects IL-2 exerts on the immune
system.
On a functional level, growth-promoting effects of IL-2 on different lymphocytes,
which have been widely explored in in-vitro studies, lie in striking contrast to lymphoproliferation and autoimmunity that characterizes in-vivo genetic deletion of IL-2 or
its signaling chains [136, 150, 165]. As soon as this unanticipated in-vivo phenotype
was explained by the lack of FoxP3+ T regulatory cells (T-regs), the scientists were
again surprised by observations that the monoclonal antibody against CD25 (Daclizumab) that blocks signaling though high-affinity IL-2 receptor (IL-2R), potently
6
inhibits human autoimmune disorder multiple sclerosis (MS) [11, 61, 97, 98]. This is
surprising considering its in-vivo inhibitory effect on T-regs [100]. These data revealed
that IL-2 has both stimulatory and regulatory effects on the immune system, that go
beyond simple paradigm based on effector and regulatory T cell competition. On the
immuno-stimulatory side, recent animal studies identified role of IL-2 in development
of memory T cells and terminal differentiation of effector T cell [97]. In human data,
both genetic and Daclizumab-treatment related experiments demonstrated that inhibition of high-affinity IL-2 signaling is associated with increased susceptibility to
infections (even early in life), presumably due to inefficient T cell priming by dendritic
cells (DCs) [11, 29, 61, 132]. These cells trans-present IL-2 to primed T cells across
the immune synapse at the time when T cells do not yet express high affinity IL-2R
[168]. It is likely that this early IL-2 signal provides essential survival signal to primed
T cells. Human studies also revealed an important role of IL-2 signaling, through intermediate affinity IL-2R, in the differentiation of innate lymphoid cells, including
expansion and activation of immunoregulatory CD56bright Natural Killer (NK) cells
and inhibition of lymphoid tissue inducer (LTi) cells, which likely contribute to the
efficacy of Daclizumab in MS [13, 120]. Thus, at least in the human system, IL2 regulates immune responses not only via activation of FoxP3-Tregs, but also via
CD56bright NK cells, and by priming effector T cells activated via high-affinity IL-2R
to later apoptosis [90].
In order to understand these complex, and sometimes seemingly contradictory
effects of IL-2 signaling, we need to consider at least two levels of control of IL-2
signal transduction. Both of these depend on a complex structure of the IL-2 receptor,
which has three different subunits. The ‘intermediate affinity’ receptor is the dimer of
the IL-2Rβ (CD122) and the common γ-chain (γc ), while the ‘high affinity’ receptor
is the heterotrimer of those two subunits in addition to the IL-2Rα chain (CD25)
7
[161]. The γc subunit is known as the common γ-chain because of its incorporation
in the receptors for numerous cytokines (IL-2, IL-5, IL-7, IL-15, IL-21), while CD122
is shared by the two structural similar cytokines IL-2 and IL-15 [159, 169]. Each of
these two cytokines have their respective alpha chains which serve to increase the
affinity of IL-2 for the ‘intermediate affinity’ receptor.
The first level of control resides in inter-cellular competition for limited amount
of IL-2. The surface expression of different IL-2R signaling chains determines to a
large extent which cell receives IL-2 signal [28, 49]. The expression of IL-2R signaling chains, especially CD25 (IL-2Rα), varies greatly on different immune cells
and is generally linked to immune activation [49]. CD25 expression is induced on
adaptive immune cells by T cell- (TCR) and B cell (BCR) receptor signaling events
and on innate immune cells, such as DCs and macrophages, by pathogen associated
molecular patterns (PAMPs) and/or pro-inflammatory cytokines [19, 97] . Additional
complexity comes from the fact that activated immune cells can shed CD25 in form
of soluble CD25 (sCD25), which increases in pro-inflammatory (i.e. protease-rich)
environments. This may exert a decisive role on which immune cells ultimately win
the tug-of-war for the limited IL-2, because soluble sCD25 competes with cell-surface
expressed CD25 for IL-2 binding [19]. To the extent that the sCD25-IL2 complex can
bind to intermediate affinity IL2-R with the same affinity as IL2 itself, but not to high
affinity IL2-R, sCD25 can be viewed as a physiological equivalent of Daclizumab, since
it channels IL-2 signaling through intermediate affinity at the expense of high-affinity
IL-2R. Note that this cellular competition for IL-2 links innate and adaptive immune responses to the type and quantity of the pathogen that is inducing an immune
response, thus increasing a chance for effective pathogen clearance, while avoiding
excessive immune activation. Due to multiple positive (e.g. activating role of IL-2
on CD25 expression, IL-2-driven T cell survival signal) and negative (e.g. regulatory
8
role of T-regs and CD56bright NK cells on effector T cells, IL-2 driven programming
for T cell apoptosis) feedback loops, this inter-cellular aspect of IL-2 effects is well
suited for mathematical modeling [28, 49, 90]. New models will have to include more
cellular elements than previously studied T-effectors and T-regs and, ideally, should
also explore the pathogen-driven aspects of the immune response [49] .
The second level of IL-2 signal control, which, to our knowledge has not been
explored by any modeling approaches previously, resides in the intracellular compartment. The individual binding reactions determine the functional outcome of IL-2
signaling on a cellular level. Because signaling chains of the IL-2R (i.e. CD122, or
CD122 and IL-2Rγ, otherwise called common γ-chain, or CD132) do not contain
any intrinsic enzymatic activity, the outcome of IL-2 signaling is dependent on the
presence of intracellular signal transducers and adaptor molecules, see Figure 2.1.
The temporal and spatial availability of these varied molecules is affected by cellular
activating signals in a manner analogous to activation-induced changes in the availability and compartmentalization of IL-2R signaling chains. For example, resting
T cells generally lack CD25 (unless they are FoxP3+ T-reg cells), have limited surface expression of CD122 (i.e. only small proportion of T cells in peripheral blood
stains with anti-CD122 Ab in vivo and the proportion is higher for CD8+ T cells
in comparison to CD4+ T cells) and also lack the tyrosine kinase Jak3 (Janus kinase 3), which is essential for mediating IL-2-driven proliferation [13, 62, 97]. Thus,
only small proportion of resting (human) T cells can receive exogenous IL-2 signal
and this signaling (measured as phosphorylation of STAT5) is significantly inhibited
by Daclizumab [100]. In contrast, we observed that Daclizumab has no significant
effect on IL-2-induced STAT5 phosphorylation of polyclonally activated human T
cells, once they upregulated CD25 and CD122 signaling chains (unpublished data).
Nevertheless, T cells that received IL-2 signal under Daclizumab therapy have dif-
9
ferent functional outcome: while they enter proliferation cycle normally, they have
enhanced survival, linked to upregulation of anti-apoptotic molecule bcl-2 [132] (and
unpublished data).
We therefore asked the question what could be the possible mechanism(s) by which
presence of CD25 functionally affects the outcome of IL-2 signaling on activated T
cells, when high expression of CD122 assures that these T cells can phosphorylate
STAT5 in response to exogenous IL-2 with surprisingly similar kinetics when they receive IL-2 signal via intermediate affinity (i.e. when CD25 is blocked by Daclizumab)
versus high-affinity IL-2R.
Figure 2.1: Schematic depiction of IL-2R chains, their intracellular domains and their
putative function, assembled based on the literature review. The IL-2R complex is
not pre-assembled before IL-2 binding: CD25, IL-2 or CD122 alone (without bound
IL-2) have no measurable affinity for CD132; instead, IL-2 is required for the assembly of signaling complex. IL-2 can bind CD25 (low affinity IL-2R; Kd ≈ 10nM) and
CD122 (Kd ≈ 100nM), which, associated with CD132 upon IL-2 binding, form an
intermediate affinity IL-2R (Kd ≈ 1nM). When CD25 associated with CD122, it increases IL-2 binding to CD122 approximately 100 fold and this tertiary complex then
recruits CD132 to form high affinity IL-2R (Kd 10-50pM) [160]. In the quaternary
IL-2R structure, IL-2 makes separate contacts with IL-2Rα (CD25), -β (CD122) and
common γ-chain (CD132). CD25 makes no contact with either CD122 or CD132.
While exploring conceptual models of this problem, we realized that the current
10
knowledge of intracellular IL-2 signaling events is not sufficient to provide unequivocal
mechanistic explanation. Even though many studies investigated functional components of IL-2R signaling chains (see excellent reviews [19, 97] and Figure 2.1, which
depicts a schematic summary of current knowledge), these studies were performed in
varied (human and animal) cell lines or primary cells (resting or activated), which
differed in the level of expression of IL-2R signaling chains, their localization to lipid
rafts and availability and composition of signal transducers and adaptor molecules
[46, 62]. Without detailed understanding of these differences, generalization of observations from one cellular system to another is not currently possible.
The literature about IL-2-induced survival signaling to T cells is especially ambiguous [62, 97, 105]. In particular, it seems that the survival signal mediated by
high-affinity IL-2R (within the context of providing also proliferation signal to activated T cells) and the survival signal mediated by intermediate-affinity IL-2R (provided to resting T cells, but also to activated T cells in the presence of Daclizumab
or sCD25) are likely mediated by different mechanisms.
1. On one hand, there is a short-term survival signal, dependent on repeated IL2 signaling through high-affinity IL-2R, which also primes effector T cells to
apoptosis upon subsequent IL-2 withdrawal [90].
2. On the other hand, there is a long-term survival signal, observed in T cells that
downmodulated CD25 and consequently received predominantly intermediate
affinity IL-2R signal [19].
Furthermore, these two distinct signals through IL-2R have been shown to lead to
qualitatively different cell fates. Experiments have shown that cells receiving extended
‘high affinity’ signaling terminally differentiate into effector T cells, while cells that
have lower levels of CD25 preferentially differentiate into memory cells [19, 27, 79].
11
Interluekin-2 signaling in cells with high CD25 levels (typical of T effector cells) are
associated with activation and rapid proliferation followed by increased susceptibility
to apotosis [79]. On the other hand, IL-2 signaling in cells with relatively low CD25
levels (typical of memory T cells) leads to homeostasis and survival [19, 27].
Consequently, the current study seeks to present a series of models that provide
an explanation of the observed surprising phenomenon that the functional outcome
of IL-2 signaling is dependent on the presence of the receptor chain (CD25) that does
not elicit any distinct intracellular signaling function(s). This is true even under situation when the other two IL-2R signaling chains and all signal transducers/adaptor
molecules are unchanged between the conditions. We hope that proposed solutions
will spur further biological studies that would confirm, or refine the provided mathematical models.
2.3 Summary of Results
• We provide experimental evidence from our collaborators that the kinetics of
pStat5 production from IL-2 signal transduction through high affinity and intermediate affinity IL2-R are very similar. The kinetics of intermediate affinity
IL2-R is delayed with respect to that of high affinity IL2-R, but the concentrations of pStat5 at 30 minutes differ by only 11%, and we hypothesize that at
equilibrium the differences in concentration will be even smaller.
• We then propose a simple model of IL2 receptor, which properly parameterized recaptures quantitatively the experimental data (pStat5 levels) mentioned
above. This model (Model I) captures kinetics of the receptor on a time scale
of minutes.
• We then concentrate on the central question of this project, which is how can
12
two different functional outcomes and cell fates result from signaling through
the same receptor and with the same signaling molecule, and when the signals
only differ in their kinetics for 30 minutes after exposure. We propose a second
model that includes positive feedback of up regulation of CD25 production by a
dimer of pSTAT5. This model (Model II) builds on Model I, taking from it the
functional dependence of equilibrium pSTAT5 signal on amount of CD25 and
thus high affinity receptor in the membrane. Model II exhibits bistability where
at one stable equilibrium the cell produces low level of CD25 and thus receives
mostly intermediate affinity signal, while at the other stable equilibrium, the
cell produces high level of CD25 and thus receives mostly high affinity signal.
These two equilibria may correspond to two different cell fates where low CD25
cells receive long-term survival signal and becoming memory T-cells, while the
high CD25 cells receive short term survival and proliferation signal and terminally differentiating into effector T-cells. Unfortunately, our simple model only
exhibits bistability for a relatively small parameter range and only for fairly low
concentrations of IL-2 (less than 1.2 nM).
• We propose a third model (Model III) which, based on the structure of the IL2 receptor, explores a possibility that the pSTAT5 signal is purely short-term
survival and proliferation signal, and that a secondary signal serves as a long
term survival signal. We propose that when IL-2 is bound to IL-2R, pSTAT5
signaling takes place; but when subsequently IL-2 unbinds, the phosphorylated
intracellular chains transmit a secondary signal. We do not speculate on the
identity of the secondary signal, but remark that perhaps the Phosphoinositide
3-kinase (PI3K) cascade is a prime candidate, due to known downstream interactions with bcl2 [20]. The addition of a secondary signal can significantly
13
increase the range of IL-2 in which we observe bistability, and thus the cell
commitment to one of two fates.
2.4 Unpublished Experimental Results
We present here unpublished data from our collaborator on this project, Dr. Bibiana Bielekova, from National Institute for Nuerological Diseases and Stroke at the
National Institute of Health. In these experiments, CD4+ and CD8+ cells were
treated with either MA251 or Daclizumab, both of which are antibodies that bind
to the CD25 chain of the high affinity receptor. Daclizumab binds to the active site
of the CD25 chain, so it compets with IL-2 for binding, thus leading to increased
intermediate affinity signaling. Alternately, MA251 binds to another site on CD25
and serves as a control. Over the course of 8 days absolute cell counts for the separate cultures were determined using flow cytometry (Figure 2.2A). Additionally,
Carboxyfluorescein succinimidyl ester (CSFE) was used to demonstrate the lymphocyte proliferation of cells treated with MA251 and Daclizumab (Figure 2.2B). CFSE
is a fluorescent dye used to label proteins in lymphocytes and is used to determine
proliferation rates. After the initial staining the label is highly concentrated. As
the cells divide the label proteins are divided among the daughter cells leading to a
lower local concentrations and therefore lower fluorescences levels. The fluorescent
distribution can then be used to estimate proliferation rates. Again cell counts were
done using flow cytometry. On a shorter time scale, the kinetics for the production
of pStat5 were compared for the two cell treatments by comparing mean fluorescent
intensity (MFI) in both cases (Figure 2.2C). The third figure demonstrates the MFI
distribution while the first two figures plot the MFI mean (with error bars).
Notice that Figure 2.2B demonstrates that cells treated with Daclizumab have
indistinguishable rates of proliferation from those not treated with Daclizumab. How-
14
Cells Counts
Figure 2
CFSE
pStat5 MFI
pStat5 MFI
Time (days)
Time (min)
pStat5 MFI
Time (min)
Figure 2.2: Daclizumab does not inhibit proliferation of activated T cells, but prolongs their survival, while exerting little effect on IL-2 induced STAT5 phosphorylation. (A) Activated T cells were cultured with MA251 or Daclizumab (Dac) for 8
days. Absolute counts of cells were determined by flow cytometry. (B) CFSE-stained
activated T cells were cultured for 7 days. Proliferation of activated T cells with
high-affinity(MA251) or intermediate-affinity(Dac) IL-2R was determined by flow cytometry. (C) Phosphorylation of STAT5 by measuring mean fluorescent intensity
(MFI) at 10 and 30 minutes after exogenous IL-2 is given.
ever, Figure 2.2A demonstrates that cells treated with Daclizumab have increased cell
counts. Therefore, we infer that the IL-2 signaling through the intermediate affinity
15
receptor (cells treated with Daclizumab) versus high affinity receptor leads to prolonged survival. At the same time the pStat5 production is very similar with only a
slight delay in kinetics associated with the intermediate affinity signaling and possibly
a slightly lower equilibrium concentration of pStat5 (Figure 2.2.C).
2.5 Dynamic Model of IL-2 Receptor (Model I)
The goal of this model is to show that we can replicate the pStat5 dynamics seen in
Figure 2.2C with a simple model of IL-2 signaling. As mentioned in the introduction,
the IL-2 receptor has three different subunits. The ‘intermediate affinity’ receptor is
the dimer of the IL-2Rβ (CD122) and the common γ-chain (γc , CD132), while the
‘high affinity’ receptor is the heterotrimer of those two subunits in addition to the
IL-2Rα chain (CD25) [161]. We would like to emphasize for the benefit of the reader
that all the subchains indeed have two names that are both used in the literature: one
with the CD + number nomenclature and one using α, β and γ names for subunits.
The previous paragraph provides a translation, if one is needed for the rest of the
thesis.
The CD25 subunit has only a short cytosolic tail that is not capable of intercellular
signaling. Therefore, signal transduction must be mediated by CD122 and γc solely,
and so theoretically both the high affinity and intermediate affinity receptors share the
same signaling capabilities [27]. The tyrosine kinases, Jak1 and Jak3 are associated
with the cytosolic subunits of CD122 and γc respectively [27, 58, 87, 92]. Once the
IL-2 binds to the receptor, these tyrosine kinases autophosphorylate and, in turn,
phosphorylate tyrosine residues on the cytosolic region of CD122. In particular, the
phosphorylation of Tyr-338 leads to the activation of the Ras-Raf-MAP kinase and
PI3K pathways via the adapter protein SHC, while the tyrosine residues, Tyr-392
and Tyr-510, allow for the phosphorylation of STAT5 leading to the formation of the
16
dimer pSTAT5 [27, 58, 87].
In the absence of IL-2, the common γ-chain has no affinity for either CD25 or
CD122. While there is some interaction between CD25 to CD122, it is of very low
affinity and requires excess CD122 to drive the reaction [130]. The common γ-chain
has no affinity for IL-2, while CD25 and CD122 both have moderate affinity for
IL-2 individually, although CD25 has a dramatically faster binding rate [130, 160].
Additionally, the binding of CD25 to IL-2 significantly increases the affinity of IL-2
to CD122 [130]. Structurally, CD25 extends past CD122 above the surface of the
cell. Therefore, in the presence of sufficient CD25, we hypothesize that IL-2 will
predominantly first bind to CD25, and then to CD122 as a result of both proximity
and rate. For the purposes of the model, it will be assumed that this is the only
order in which the high affinity receptor will be formed. Additionally, since the the
common-γ chain does not have any independent affinity for IL-2, CD25, or CD122,
it will not be explicitly modeled.
Daclziumab is a humanized monoclonal antibody that binds to CD25 at a site
competing with IL-2 [125]. Introduction of Daclizumab does not affect T cell viability or the surface expression of CD122, CD25, or γc [60]. However, treatment
with Daclizumab has been shown to inhibit IL-2 mediated Stat5 activation, a fact
that is not particularly surprising considering Daclizumab competes with IL-2 for
CD25 binding [60]. While originally designed to prevent organ rejection in transplant
patients, Daclizumab is in phase III clinical trials for the treatment of relapsingremitting multiple sclerosis (RRMS) [11, 13, 15, 18, 113]. Using Model II, we will
explore the affects of the introduction of Daclizumab on the number and stability of
the equilibria of the system.
We therefore consider the following set of binding reactions which describe the IL2 signal transduction, as well as competition between intermediate and high affinity
17
receptors.
[IL2] + [α] *
) [IL2α]
(1)
[IL2] + [β] *
) [IL2β]
(2)
[IL2α] + [β] → [IL2αβ]
[IL2αβ] → [IL2] + [α] + [β]
(3)
(4)
The first two reactions are reversible reactions of binding and unbinding of IL-2
([IL2]) to CD25 ([α]) and CD122 ([β]), respectively. The third reaction forms a
high affinity receptor-ligand complex ([IL2αβ]). Upon unbinding of IL-2 the whole
complex disassociates (reaction 4). The assumption that the complex will completely
disassemble (4) when IL-2 unbinds is based on the understanding that CD25 and
CD122 have low affinity for each other in the absence of IL-2 [130].
To model these reactions we use mass action kinetics to arrive at a set of differential
equations
˙
[IL2β]
= kβ+ [IL2][β] − kβ− [IL2β]
˙ = k − [IL2α] + k − [IL2αβ] − k + [IL2][α]
[α]
α
α
αβ
+
−
˙
[IL2αβ]
= kαβ
[IL2α][β] − kαβ
[IL2αβ]
˙
[pStat5]
= p [IL2β] + [IL2αβ] 1 − [pStat5] − d[pStat5]
(2.1)
Here [pStat5] is the concentration of the signaling molecule produced by the IL-2 receptor when IL-2 is bound. Note that we assume that the intracellular signal pSTAT5
is generated at rate proportional to concentration of receptor-ligand complex, both
high and intermediate affinity ([IL2αβ] and [IL2β], respectively) and concentration
of un-phosphorylated pSTAT molecule. The latter assumption corresponds to the
assumption that there is a finite amount of Stat5 to be phosphorylated. Additionally,
[pStat5] is assumed to decay at a rate proportional to its concentration. We rescaled
18
the concentration of [pStat5] by the total concentration of Stat5, so that its value is
a percentage of its maximum value.
Note that we did not write a differential equation for concentrations of [β], [IL2α]
and [IL2]. These reactions would be redundant in the presence of conservation equations, where we assume that the total number of CD25 (α∗ ), CD122 (β ∗ ) and IL-2
(IL2∗ ) is conserved:
[IL2α] = α∗ − [α] − [IL2αβ]
[IL2] = IL2∗ − [IL2α] − [IL2β] − [IL2αβ]
(2.2)
[β] = β ∗ − [IL2β] − [IL2αβ]
Our final model, where we do not model the effects of Daclizumab is (2.1) where
the concentrations of [IL2α], [IL2] and [β] are replaced by right hand sides of (2.2).
When we model effects of Daclizumab on IL-2 transduction by IL-2 receptor we
assume that the binding reaction of Daclizumab with CD25 is much faster than the
other reactions and so the concentration of Daclizumab-CD25 complex is at equilibrium. The concentration of Daclizumab bound to CD25 will be estimated as a
function of CD25 by considering the simple reaction,
[Dac] + [α] *
) [Dα]
This reaction is assumed to be at equilibrium, and then the concentration of [Dα] can
be solved for explicitly as
[Dα] =
D∗ [α]
d
KDα
+ [α]
(2.3)
19
d
where D∗ is the total concentration of Daclizumab, and KDα
is the binding affinity
of Daclizumab to CD25. In the presence of Daclizumab the first conservation law in
(2.2) will be replaced by
[IL2α] = α∗ − [α] − [IL2αβ] − [Dα].
2.5.1 Parameter Selection
Below we list all the parameters for the system. For parameters not directly
cited from literature, the parameters denoted by * were determined by fitting data
of Ring [131], while those denoted with a “double dagger” (‡) were estimated using
techniques described below. The binding affinity for Daclizumab to IL2 was assumed
to be that same as that of the high affinity receptor to IL-2. The total concentration
of Daclizumab, denoted with a “single dagger” (†), was chosen to be as small as
possible while still exhibiting experimentally observed phenomena that the presence
of Daclizumab effectively changes IL-2 transduction from high affinity to intermediate
affinity even in the cells that have high affinity receptor chain, CD25. We demonstrate
that this value of dacliumab is sufficient in Section 2.7.3. The value denoted by ** is
not needed because of the equilibrium assumptions previously, see (2.3).
Table 2.1: Binding Rates
Reaction
Binding Affinity (nM)
[IL2] + [α] *
10[130, 160]
) [IL2α]
[IL2] + [β] *
0.9[121, 160]
) [IL2β]
[IL2α] + [β] → [IL2αβ]
[IL2αβ] → [IL2] + [α] + [β]
[Dac] + [α] *
0.01
) [Dα]
On-Rate (nM−1 min−1 )
0.6[160]
0.048[160]
0.57∗
0.006‡
-∗∗
20
Table 2.2: Production/Decay Rates
Signal Production Rate (nM−1 min−1 ) Decay (min−1 )
[pStat5]
0.45∗
0.028∗
Table 2.3: Conservation Equation Maximums
Conserved Variable
[IL2]
[α]
[β]
Maximum (nM)
1[131] , 5, 100, 500[131]
0, 1.77∗
0.51∗
Table 2.4: Initial Conditions
Variable
Initial Conditions (nM)
[α]
0, 1.77∗
[pStat5]
0.18∗
[β], [IL2β], [IL2αβ]
0
[Dac]
0, 2†
To estimate the unbinding reaction of IL-2 with the high affinity receptor (denoted
by ‡ ) we consider the following reaction.
[IL2] + [α] + [β] *
) [IL2α] + [β] *
) [IL2αβ]
This multistep reaction represents the sequential binding of IL-2 which ultimately
forms the high affinity receptor-ligand complex, [IL2αβ]. This combined reaction
has a reported Kd of 10 pM [130, 160]. Recall that we assumed that when IL-2
unbinds the whole complex IL-2−α − β disassembles based on the fact that CD25 has
a low binding of affinity for CD122 in the absence of IL-2 [130]. That is, we consider
the subset of reactions
k1+
k+
2
[IL2] + [α] + [β] −
)*
− [IL2α] + [β] −→ [IL2αβ]
k1−
k−
2
[IL2] + [α] + [β] ←−
[IL2αβ]
1
1
0.9
0.9
0.8
0.8
0.7
0.7
pStat5 (% MAX)
pStat5 (% MAX)
21
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
Fit
0
0
5
10
15
20
25
Fit
0
0
30
5
10
Time (min)
20
25
20
25
(b)
1
0.9
0.8
0.8
0.7
0.7
pStat5 (% MAX)
1
0.9
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
Fit
0
0
30
Time (min)
(a)
pStat5 (% MAX)
15
5
10
15
Time (min)
(c)
20
25
Fit
30
0
0
5
10
15
30
Time (min)
(d)
Figure 2.3: Model fits to the data from Ring, et al [131]. The experiments were run
in triplicate. The plots show the model curve (in blue) plotted with the experimental
data (red). Figures (a) and (b) show the production of pStat5 over time for cells with
no CD25 in the case of low (1 nM) and high (500 nM) IL-2, respectively. Figures
(c) and (d) also show the production of pStat5 over time except for cells with CD25,
again in the case of low (1 nM) and high (500 nM) IL-2. Note that 5 parameters and
two initial conditions were estimated by fitting this data (R2 = 0.9588).
For a general reversible reaction, Kd = kr /kf , where kf is the forward reaction rate
and kr is the rate of the reverse reaction. For this reaction, we approximate the value
of Kd , as Kd ≈ k2− /kf , where kf is the effective forward reaction rate of formation of
the high affinity receptor-ligand complex. If we consider each individual rate to be
mean rate for an underlying Poisson process, then these rates are reciprocals of the
22
mean time to transition from one state to the next.
k1+
k+
2
[A] −
)*
− [B] −→ [C]
k1−
Therefore we estimate kf as the reciprocal of the mean time to arrival to complex
[C] assuming we start at complex [A]. This can be computed explicitly by treating
this sequence of reactions as an absorbing Markov Chain with three states: A, B, and
absorbing state C. In particular,
kf ≈
k1+ k2+
k1+ + k2+ + k1−
and notice that this function has the following upper bound:
k1+ k2+
k1+ k2+
≤ min{k1+ , k2+ } ≤ k1+
≤
k1+ + k2+ + k1−
k1+ + k2+
In fact, when k1+ << k2+ , the second inequality becomes nearly an equality (i.e.
if k1+ and k2+ differ by an order of magnitude then the error is < 10%, and if they
differ by two orders of magnitude, then the error is < 1%) while the last inequality
becomes an equality. Therefore, when we consider this information in the context of
the formula Kd ≈ k2− /kf , we find that
k2− ≈ kf Kd ≤ k1+ Kd
So we chose to approximate k2− as k1+ Kd . Since this is an upper bound, it is important
to note that other choices for values of k2− must be less than or equal to the given
approximation. Values for k2− up to two orders of magnitude less than the value given
in the table were also tested in the model and differences in the level of [pStat5] were
23
on the order of 1%.
The parameters marked by
∗
were estimated by fitting experimental data for
pStat5 production at two concentrations of exogenous IL-2 and for cells either with
or without of CD25 [131]. In these experiments by Ring, et al [131], different CD4+
cells are divided into two groups based on the presence or absence of CD25. The cells
were then treated with either 1 or 500 nM of IL-2, and then the levels pStat5 are
measured at the time points 0, 1, 2.5, 5, 15, 30, 60, and 120 minutes. The level of
measured pStat5 begins decreasing after 30 min. This phenomena is likely the result
of downstream regulation that is not included in our model, and therefore we will
only consider the first six time points representing first 30 minutes. The parameters
were then chosen to minimize the sum of the square of the difference between pStat5
production for the model (pStat5m ) and the experimental data (pStat5exp ) for the
four conditions (cells with and without CD25 treated with low or high IL2 each in
triplicate). Let pStat5m (high, +) be the pStat5 concentration given by the model for
cells with CD25 treated with high IL-2, pStat5exp (low, −) be the pStat5 concentration
given experimentally for cells without CD25 treated with low IL-2, etcetera. Then,
we minimized
pStat5m (low, −) − pStat5exp (low, −)2
2
+pStat5m (high, −) − pStat5exp (high, −)
2
+pStat5m (low, +) − pStat5exp (low, +)
2
+pStat5m (high, +) − pStat5exp (high, +)
This was solved using the constrained optimization function, using a constrained
optimization method (Matlab function: fmincon), with constraint that all of the
parameters be positive. The results of the best fit along with the data are plotted in
24
Figure 2.3 (R2 = 0.9588).
1
1
0.9
0.9
0.8
0.8
0.7
0.7
pStat5 (% MAX)
pStat5 (% MAX)
2.5.2 Results: Kinetics of Phosphorylation of STAT5
0.6
0.5
0.4
0.3
0.2
0.5
0.4
0.3
0.2
0.1
0
0
0.6
0.1
High Affinity
Intermediate Affinity
10
20
30
Time (min)
(a)
40
50
0
0
High Affinity
Intermediate Affinity
10
20
30
40
50
Time (min)
(b)
Figure 2.4: This figure shows qualitative matches to data presented in Figure 2.2.C. In
particular, Figure (a) plots the production of pStat5 for high and intermediate affinity
IL-2 receptors for cells treated with 1 nM IL-2. Notice that there is delayed production of pStat5 associated with the intermediate affinity receptor with the equilibrium
concentration being only slightly lower (< 10%). Figure (b) shows the prediction of
the model if we instead treat with a moderate levels of IL-2 (5 nM). In this case the
differences in equilibrium concentration decrease significantly (< 2%) and the kinetic
differences become very small as well.
When comparing pStat5 signaling for intermediate affinity versus high affinity
IL-2R, the model demonstrates very similar equilibrium concentrations of phosphorylated STAT5 (pSTAT5) with slightly delayed kinetics associated with intermediate
affinity signaling, which matches the experimental results (Figure 2.4a and 2.2C).
These differences in kinetics quickly diminish in the presence of higher concentrations
(5 nM) of IL-2 (Figure 2.4b). Therefore, the model demonstrates that if pSTAT5
production is the primary factor in the differentiation of intermediate affinity versus
high affinity signaling the mechanism would have to be very sensitive, it would have
to work on scale of minutes to half an hour, and that it would most effective for lower
25
concentrations of IL-2.
While a full discussion of the results related to effect of Daclizumab will be reserved
for Model III, some of the results will be used for Model II. The pStat5 signaling for
Model I and III are identical, so any results shown here would simply be repeated.
2.6 Cell Line Differentiation Model (Model II)
To further examine the sensitivity of Model I and explore the possible long term
effects of IL-2 signaling, Model II explores the feasibility of a cell using different levels
of pStat5 signal as a proxy for differentiating IL-2 signaling through the intermediate
versus the high affinity receptor, ultimately leading to differentiation of cell fate.
Our first model indicates that for pStat5 concentration to be effectively used for the
differentiation of intermediate versus high affinity signaling, it must be very sensitive
and will be more effective for lower concentrations of IL-2 (compare Figure 2.4a at
1nM and (b) at 5 nM IL-2). The goal of the second model is to explore the longer time
effects of IL-2 signaling. The model will consider how a cell might use pStat5 signaling
for cell differentiation based on equilibrium concentrations of CD25. Specifically, two
of the major cell types, effector cells and memory cells, can be distinguished by the
levels of membrane bound CD25. Effector T cells are characterized by relatively high
levels of CD25, while memory cells have little to no membrane bound CD25 [19,
27, 79]. Biologically these two cell lines demonstrate two different stable equilibria
concentrations of CD25, which is instantiation of a well known property of dynamical
systems, namely, bistability.
Model Il incorporates the most important downstream positive feedback on IL-2
signal, CD25 up-regulation. Specifically, we will consider the fact that the pStat5
up-regulates production of CD25, which promotes the switch from intermediate to
high affinity signaling as CD25 gets incorporated into the cellular membrane. On the
26
time scale of the regulation of gene expression, the binding reactions associated with
the receptors will be considered to be at equilibrium and only production and decay
of CD25 and pSTAT5 will be modeled explicitly.
2.6.1 Model
We consider the longer time-scale of production and decay of both CD25 (α) and
pStat5, governed by the system of differential equations
˙
[pStat5]
= rs [Rb ][Stat5] − ds [pStat5]
˙ = pα [pStat5]2 − dα [α]
[α]
(2.4)
where [pStat5] is produced at a rate proportional to the number bound IL-2R ([Rb ])
and unphosphorylated Stat5 ([Stat5]). CD25 ([α]) is produced from its promoter
at a rate proportional to the concentration of its up-regulator, which is the dimer
of [pStat5] [103, 126]. Both [pStat5] and [α] decay linearly with rates ds and dα ,
respectively.
We estimate the concentration of bound receptor, [Rb ], by assuming that the
receptor binding equilibrates on a faster time scale. In general, we start with the
observation that the equilibrium concentration of bound IL-2 receptor (IL-2R) in the
form of either the intermediate or high affinity receptor is a function of IL-2, CD25
and CD122 concentrations.
[Rb ] = F ([IL2], [α], [β]).
We model the concentration of CD25 ([α]) as a dynamic variable, and consider
the concentrations of both [IL2] and CD122 ([β]) to be fixed. Then we assume that
27
the number of bound IL-2R can be modeled as a Hill function of the amount of
membrane bound CD25 ([α]) (see Figure 2.9). When there is no membrane-bound
CD25, all IL-2 receptors are intermediate affinity receptors and the number of bound
intermediate affinity receptors will be a function of the intermediate affinity receptor’s disassociation constant, the concentration of IL-2, and the number of membrane
bound CD122. Likewise, in the presence of excess CD25, all bound receptors will
represent high affinity receptors and the number of bound high affinity receptors will
be a function of [IL2], [β] and the high affinity receptors’s disassociation constant. So
we will assume that the concentration of IL-2 bound IL-2 receptors is given by,
[Rb ] = F ([α]) =
V0 [α]n
+ C0 ,
K0n + [α]n
(2.5)
where C0 represents the number of bound IL-2R for a specific fixed concentration of
IL-2 and a fixed number of membrane bound CD122 for a cell without CD25. The
number V0 represents the differential in the number of bound receptors between a
cell with excess CD25 and no CD25. Therefore C0 + V0 is the total number of bound
IL-2R in the presence of excess CD25 at the same fixed concentration of IL-2 and
CD122. The number K0 is the number of CD25 required for the increase in the
number of bound IL-2R to reach half of V0 , and n order of the Hill function. The
assumption that the number of bound IL-2R represents a Hill function with respect
to the concentration of CD25 was validated by Model I and the evidence will be
presented below in Section 2.6.3.
In this section we will analyze the dynamics of our system of two differential
equations as a function of Hill function parameters C0 , V0 , K0 , n, as well as other
parameters in (2.4). However, while [Rb ] is explicitly a function of [α], recall that
it is implicitly a function of the number of membrane bound CD122 and the total
28
concentration of IL-2. In particular, the parameters C0 , V0 and K0 all depend on
CD122 and IL-2 concentrations: C0 represents the concentration of membrane bound
receptors for a fixed total concentration of IL-2 and CD122 in the absence of CD25,
while C0 + V0 represents the concentration of membrane bound IL-2R in the presence
of excess CD25. Therefore,
[Rb ] := F [α]; IL2∗ , β ∗ .
In Section 2.6.3, we will use Model I to compute the dependence of parameters C0 , V0
and K0 on IL2∗ , β ∗ , which will allow us to explore the dependence of the dynamics
of (2.4) as a function of parameters IL2∗ and β ∗ .
Analysis of Model II (2.4) We will assume that the total concentration of Stat5
(pS ∗ ) which can be in the phosphorylated or the un-phosphorylated form, is conserved. Then we can rescale the concentration of [pStat5] by total concentration of
dα
. Then the
Stat5 (pS ∗ ) and the value of [α] by the collection of parameters
pα pS ∗
equilibria of the system (2.4) satisfy
V [α]n
+C
K n + [α]n
1 − [pStat5] − [pStat5] = 0
(2.6)
[Stat5]2 − [α] = 0
(2.7)
where we used new parameters defined by
V :=
rs
V0 ,
ds
C :=
rs
C0 ,
ds
K :=
dα
K0 .
pα pS ∗
29
The number and stability of equilibria can change as we vary the parameters V , C,
K, and n. In particular, we are interested in regions of parameter space that exhibit
bistability, that is, where there are two stable equilibria.
To simplify the notation we will denote by x the concentration of [pStat5]. We
will substitute (2.7) into the (2.6), and to give the single equilibrium equation
− V + C + 1 x2n+1 + V + C x2n = K n (C + 1)x − C
(2.8)
The left hand side is a polynomial, which we will denote by f (x), with one root
at zero, f (0) = 0, of order 2n and another root at
X0 :=
V +C
V +C +1
Notice that X0 < 1. Additionally, using some simple calculus, one can determine that
the maximum of f (x) occurs at xmax =
2n
X,
2n+1 0
and there is unique inflection point
xinf < xmax at
xinf =
2n − 1
X0 .
2n + 1
(2.9)
The right hand side which we denote by g(x), is a line with a single root at
X1 :=
C
C +1
(2.10)
Notice that X1 ≤ X0 . The intersection(s) of f and g represent the equilibria of the
system, so our goal is determine the number and stability of the equilibria, and additionally, how they depend on the parameters C, V , K, and n.
30
Lemma 1. If C and V satisfy
C
≥
C +1
2n − 1
2n + 1
2 V +C
V +C +1
(2.11)
then (2.4) has one equilibrium for all values of K.
Alternately, if
C
<
C +1
2n − 1
2n + 1
2 V +C
V +C +1
(2.12)
then there exist a range of values for K such that the model will have 3 equilibria.
Proof. We make several observations about the functions f (x) and g(x).
• f (x) does not depend on K;
• g(x) is a line whose intersection with x axis does not depend on K. So K only
affects the intersection of g(x) and the y axis, and therefore, the slope of g(x);
• the slope g 0 (x) = K n (C + 1) > 0 is always greater than zero.
The next key observation is that the central organizing center for the bifurcation
analysis is inflection point xinf . We make the following observations:
1. Since g 0 > 0, if the intersection X1 of g(x) with the x-axis is greater than xinf ,
that is,
X1 > xinf
(2.13)
then the functions g(x) and f (x) have a unique intersection (see Figure 2.6a).
2. On the other hand, we will show below that if X1 < xinf then there is always a
31
value of K, Kint = Kint (C, V ), such that
g(xinf ) = f (xinf ).
(2.14)
3. Finally, we show that when C and V satisfy the equation
C
=
C +1
2n − 1
2n + 1
2 V +C
V +C +1
(2.15)
then
g 0 (xinf ) = f 0 (xinf ).
(2.16)
4. We note that the parameter values where both (2.14) and (2.16) are satisfied
represent a locus of a family of pitchfork bifurcations. These parameter values
satisfy (2.15) and Kint = Kint (C, V ), where precise form of the latter function
is below (2.19). All panels in Figure 2.5 satisfy (2.14), while the middle panel
represents a locus of a pitchfork bifurcation.
5. Finally, we show that when (2.12) is satisfied, then g 0 (xinf ) < f 0 (xinf ) and
therefore at Kint there are three intersections of f (x) and g(x). Since existence
of 3 intersections is an open conditions, for any values of C, V satisfying (2.12)
there will be an open interval (Kmin , Kmax ) with Kint ∈ (Kmin , Kmax ), such
that there are three equilibria for all K ∈ (Kmin , Kmax ). It is also easy to see
from Figure 2.5 that (Kmin and Kmax ) are loci of saddle-node bifurcations of
equilibria.
We now compute the relevant quantities that we introduced above. By (2.9), the
32
0.05
0.05
0.05
0.04
0.04
0.04
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0
0
0.1
0.2
0.3
pStat5
pStat5
pStat5
(a)
(b)
(c)
0.4
0.5
Figure 2.5: The figures represent different plots of f (x) (blue) and g(x) (green) for
fixed V , three different values of C, and K chosen such that f (x) and g(x) intersect
at the point of inflection of f (x). (A) shows the plots for a value of C satisfying
(2.11) specifically as a strict inequality. (B) shows the plots for a value when (2.11)
is satisfied as an equality. (C) represents a value of C that satisfies (2.12).
K=5
K=1000
K=2
K=1
K=1
K=0.1
(a)
(b)
Figure 2.6: This figure plots f (x) (in green) and g(x) (in blue) for different values of
K. (a) Notice that when (2.11) is satisfied there is only one equilibrium for all values
of K. (b) Alternately when (2.12) is satisfied, then we see bistability for some values
of K.
definition of X0 , and (2.10), the statement (2.13) is equivalent to
C
≥
C +1
2n − 1
2n + 1
V +C
.
V +C +1
(2.17)
Therefore if C and V satisfy this inequality, for all K there will only be one point of
33
intersection of f and g (see Figure 2.6a.)
Now we discuss the case when (2.13) is not satisfied, i.e.
C
<
C +1
2n − 1
2n + 1
V +C
.
V +C +1
(2.18)
Then a straightforward calculation shows that when K = Kint (C, V ), where
2n−1 2n
2n+1
−
2n−1
(Kint )n :=
2n+1
2n−1 2n+1
2n+1
(C+1)(V +C) V +C+1
(V +C)2n+1
(V +C+1)2n
(2.19)
−C
then g(xinf ) = f (xinf ).
For K = Kint , we can compare derivatives g 0 (xinf ) to f 0 (xinf ). In particular, when
(2.15) is satisfied, then
g 0 (xinf ) = (Kint )n (C + 1)
2n−1 2n
2n+1
−
2n−1
= (C + 1)
2n+1
2n−1 2n
2n+1
−
2n−1
=
2n+1
2n−1 2n
2n+1
=
2n−1
2n+1
=
2n−1 2n+1
2n+1
(C+1)(V +C) V +C+1
2n−1 2n+1
2n+1
(V +C) V +C+1
−
V +C
V +C+1
−
2n−1 2n+1
2n+1
−
(V +C)2n+1
(V +C+1)2n
−C
(V +C)2n+1
(V +C+1)2n
C
C+1
(V +C)2n+1
(V +C+1)2n
2n−1 2
2n+1
V +C
V +C+1
2n − 1 2n−1 (V + C)2n
= f 0 xinf
2n−1
2n + 1
(V + C + 1)
Finally, we show that if C and V satisfy (2.12) then g 0 (xinf ) < f 0 (xinf ). This then
34
implies that at Kint (C, V ) there are three intersections of f and g, see Figures 2.5c
and 2.6b.
Indeed we compute,
g 0 (xint ) =
<
(V +C)2n
2n−1 2n
2n−1 2n+1
−
2n−1
2n+1
2n+1
V +C (V C+C+1)
(Kint )n (C + 1) =
2n−1
− C+1
2n+1
V +C+1
2n−1
2n − 1
(V + C)2n−1
= f 0 xinf
2n−2
2n + 1
(V + C + 1)
where again the inequality follows directly from (2.12).
To finish the proof we observe, that an analogous argument using the opposite
inequality (2.11) implies g 0 (xinf ) > f 0 (xinf ) and there is unique intersection of f and
g, see Figure 2.5a and 2.6a.
2.6.2 Results: Conclusions of Model II
We are interested to analytically describe the region of parameter space (in V , C,
K, and n) that can support multiple equilibria. To this end, we can solve for V in
(2.12) to get
V ≥
where n̂ =
C<
n̂
.
1−n̂
2n−1 2
.
2n+1
C(C + 1)
n̂
−C
1−n̂
Observe that (2.12) implies
(2.20)
C
C+1
< n̂, which in turn implies
Therefore, the right hand side of the (2.20) is always defined and positive.
We plotted the curve given by the equality of (2.20) for several choices of n with
the region where the inequality is satisfied marked by colored lines (see Figure 2.7).
Notice that as n increases the region where bistability is possible also increases. In
particular, notice that V has a vertical asymptote at C =
n̂
1−n̂
which increases as n
35
increases. Therefore, greater values of n provide larger regions of parameter space
supporting bistability.
10
9
8
7
V*
V
6
5
4
3
2
n=1
n=2
n=3
1
0
0
0.2
0.4
0.6
0.8
1
C*
C
Figure 2.7: The plot shows the regions of parameter values of V and C were bistability
is possible for different values of n. As n is increase, the size of the region expands.
Note, however, that this figure does not take describe the extend of the bistability
region in the direction of the parameter K.
In order to analyze the number and stability of equilibria in terms of the concentrations of IL-2 and CD122 directly (rather than V and C), we must first describe
the parameter space in terms of the original V0 and C0 rather than the rescaled quantities V and C, because V0 and C0 depend directly on the concentrations of IL-2 and
CD122 (see Section 2.6.1). However, we do not know the values of the parameters rs
and ds that are used to scale V0 and C0 to get V and C. Fortunately, V and C are
scaled by the same quantity
rs
,
ds
and so by looking at their ratio we are able to make
a statement about V0 and C0 directly as well. The ratio of V to C (and therefore V0
to C0 ) for V satisfying (2.20) is given by
V0
V
C +1
=
≥ n̂
C0
C
−C
1−n̂
(2.21)
By taking the derivative of the right hand side of (2.21) with respect to C, we see
36
that it is monotone increasing in C, and therefore has a minimum at C = 0, given by
V0
V
=
≥
C0
C
2n + 1
2n − 1
2
−1
(2.22)
In 2.6.3, we will combine the results from Model I with our results from Model II,
namely (2.21) and (2.22), to relate these conclusions to concentration of IL-2 (IL2∗ )
and CD122 (β ∗ ) explicitly.
Just from analysis of the (2.21) and (2.22) alone, some biologically relevant conclusions can be made. Specifically, notice that for small Hill coefficients (values of n),
the introduction of CD25 will need to significantly impact the concentration of bound
receptors in order for the model to exhibit bistability. For example, notice that for
n<
√
√2+1
2( 2−1)
≈ 2.91, the equation 2.22) gives an estimate
V0
C0
> 1. This means that for
the model to be capable of exhibiting bistability, we must require that
V0 +C0
C0
> 2. Bi-
ologically, this indicates that the model would only project the possibility of multiple
equilibria (at least for values of n < 2.91) when the concentration of IL-2 and CD122
are such that increasing the concentration of CD25 can double the concentration of
receptors bound to IL-2 (see (2.5)). This step-like response will likely only occur for
concentrations of IL-2 and CD122 near or below the binding affinity of the ‘intermediate’ affinity receptor (1 nM). In fact, we will use data from Model I to explore the
concentrations of IL-2 and CD122 satisfying (2.21) and (2.22) and further refine these
estimates for specific values of n (see Section 2.6.3, Figure 2.8).
Different Initial Conditions Allow Cells to Differentiate into Different Cell Lines.
If we consider the amount of membrane bound CD25 as a proxy for the differentiation
of effector T cells and memory T cells, then the bistability of this model indicates
that cells can use only pStat5 signaling along with the up-regulation of CD25 to
37
differentiate into these two cell types characterized by higher and lower equilibrium
CD25 concentrations. In particular, the initial concentration of membrane bound
CD25 will determine whether the cell will differentiate into an effector cell (higher
CD25 equilibrium) or a memory cell (lower CD25 equilibrium).
An alternative biological interpretation is that the initial level of T-cell ‘priming’,
or activation, may determine whether the cell becomes an effector T cell or a memory
T cell. Recall that after the infection the proteins from the infectious agent are
presented by antigen presenting cell (APC) to the naive T cells through temporary
creation of an enclosed communication channel between the two cells, known as a
‘synapse’. During activation, in addition to the antigen, CD25 is also presented by
APC to the T cell, thereby temporarily changing the intermediate IL-2 receptor to a
high affinity IL-2 receptor. The level of this CD25 presentation may be sufficient for
determining differentiation of cell type.
Therefore our model suggests that T cells could be differentiated by either their
level of membrane bound CD25 or level of ‘priming’ at TCR activation.
Affects of Daclizumab in Model II Daclizumab has shown efficacy in controlling
the rate of relapse in relapsing-remitting multiple sclerosis (RRMS) [10]. We are
interested to see how its presence affects the number and stability of the equilibria
of the model. Recall that Daclizumab is an antibody which binds to CD25 thereby
blocking high affinity IL-2Rxf signaling. The inclusion of Daclizumab should not
directly affect the kinetics of pStat5 or CD25 production (rs , ds , pα , dα ) and should not
affect the theoretical minimums or maximums of bound receptors (C0 or V0 ). However,
the introduction of Daclizumab will increase K0 , the concentration of CD25 required
for the increase in the number of bound receptors to reach half of the maximum
V0 of the increase in bound receptors, since some of the introduced CD25 will be
38
sequestered by Daclizumab. Additionally, the introduction of Daclizumab could have
an affect on n, but the exact effect on n is not obvious.
Our bifurcation analysis from the previous section shows that increasing K in
the parameter region which exhibits bistability, eventually results in a saddle-node
bifurcation leading to a loss of bistability. For values of K higher than the level at
the saddle-node bifurcation the system exhibits a single equilibrium with low CD25
expression (see Figure 2.6b). Since the low CD25 equilibrium corresponds to memory
cell phenotype and a lack of effector cell phenotype, this agrees with experimental observations and may explain the effectiveness of the Daclizumab therapy. Daclizumab
shifts the population away from effector T cells to memory T cell phenotype, lowering
the immune activity of the T cell population.
Without significant additional assumptions, the effects of varying n on the equilibria and bistability are not clear in the generality provided by Model II. We will,
however, use the information from Model I will to further investigate the affects of
Daclizumab on the equilibria in the next section.
2.6.3 Results: Combining Model I and II
All of the results of section (2.6.2) hold for arbitrary Hill functions (2.5). To make
additional observations about the effects of the concentrations of IL-2 and CD122
on Model II, we will use Model I to determine parameter choices for Model II. In
particular, we will compute explicitly values of parameters V0 and C0 from Model I for
specific concentrations of IL-2 and CD122. Recall that the parameters V0 and C0 are
explicitly related to the total concentrations of IL-2, CD122, and CD25. Specifically,
the value of C0 represents the number of bound IL-2 receptors at equilibrium in the
absence of CD25 (for fixed concentrations of IL-2 and CD122). Similarly, V0 represents
the increase in the number of bound receptors at equilibrium in the presence of excess
39
CD25.
To compute V0 , C0 from Model I for specific concentrations of IL-2 and CD122, we
simulate the Model I until equilibrium is reached. Then, the equilibrium concentration
of bound IL-2 receptors ([IL2β]+[IL2αβ]) can be determined as a function of different
concentrations of CD25 (see Figure 2.9). In this model, C0 represents the number
of ‘intermediate affinity’ receptors bound to IL-2 at equilibrium when there is no
available CD25 (the y intercept of the function in Figure 2.9a) while V0 +C0 represents
the total number of receptors (both intermediate and high affinity) bound to IL2 at equilibrium in the presence of excess CD25 (the horizontal asymptote of the
function in Figure 2.9a). Recall that K0 is given by the concentration of CD25 at
which the number of total bound receptors is half of the maximum (V0 + C0 ). For
various concentrations of IL2 and CD122, the values of V0 and C0 were estimated
by determining the equilibrium concentration of bound receptors in Model I for no
CD25 and excess CD25 (10 nM), respectively. In this way we use Model I to take
specific concentrations of IL-2 and CD122 to produce values of V0 and C0 that can
be analyzed for bistability in Model II.
In order to use our analysis leading to (2.22) we then take these values of V0 and
C0 and compute their ratio at different values of IL-2 and CD122. In addition, we
also include the parameter n. For different fixed values of n, we determine if the ratio
V0
C0
satisfies (2.22) to find areas of potential bistability in terms of concentrations of
IL-2 and CD122, see Figure 2.8. The regions where the bistability is possible form
a set of nested triangular regions, symmetric across diagonal, that are parameterized
by n. At low values of n (dark blue) the region is very small and it is centered at very
low concentrations of IL-2 and CD122. At n = 1.5 (union of yellow, light blue and
dark blue regions), the region extends to concentrations of either IL-2 and CD122 of
1 nM, but, interestingly, both concentrations must be at least 0.5 nM.
40
2
2
CD122 (nM)
1.8
1.5
1.6
n
1
1.4
1.2
0.5
1
0.5
1
1.5
2
IL2 (nM)
Figure 2.8: This image shows the regions of parameter space satisfying (2.21) for
different values of n. In particular, values to the left and below the curves represent
values of CD122 and IL2 where bistability is possible for different fixed values of
n. Specifically, the curves plotted are for n = 1, 1.25, 1.5, 1.75, and 2, as can be
seen on the color legend. In particular, the region in blue and yellow represents the
concentrations of IL2 and CD122 where bistability is possible for the model without
Daclizumab.
The contour lines in Figure 2.8 separating each region represent a specific value
of n. For example, the contour line between the yellow and orange regions represents
n = 1.5. For this value of n, to satisfy (2.21) and (2.22) the ratio of V0 + C0 to C0
must equal 4. Recall that this indicates that the model would only exhibit bistability
when the concentration of IL-2 and CD122 are such that increasing the concentration
of CD25 increases the concentration of bound receptors by more than 400%. The
regions colored in orange and red represent pairs of IL-2 and CD122 concentrations
where bistability is not possible for n = 1.5 while the regions which are yellow and
blue represent regions where the concentrations of IL-2 and CD122 satisfy (2.21) and
(2.22). Similarly, the line between the blue and yellow regions represents n = 1.25.
Therefore, the yellow, orange, and red regions represent concentrations of IL-2 and
CD122 where multiple equilibria are not possible, and the blue regions represent
concentrations of IL-2 and CD122 where bistability is possible.
41
While this analysis indicates that multiple equilibria are possible for certain concentrations of IL-2 and CD122 (given specific values of n), the precise extend of the
bistability region still depends on K0 as well as the ratios
rs
ds
and
dα
.
pα pS ∗
Affects of Daclizumab (Model I and II Combined) To better examine the effects
of Daclizumab on bistable behavior predicted by Model II, we will use Model I to
determine how the values of K0 and n change in response to increased concentrations
of Daclizumab. For this numerical experiment we will fix concentrations of IL-2 and
CD122 at 1 nM each, which is the same concentration used in the paper by Ring,
et al [131]. We then numerically simulate model I until an equilibrium is reached at
concentrations of CD25 ranging from 0 to 10 nM. The number of bound receptors,
which is a key input to Model II exhibits a Hill function response (figure 2.9a) with
n = 1.6,
K0 = 0.91
and
V0
= 2.20
C0
These numbers fall into the bistability region in Figure 2.8.
We then introduce varying concentrations of Daclizumab = {0, 1, 2, 3, 4 nM} and
at each concentration we repeat the computation of the Hill function (Daclizumab
concentration of 0 nM recapitulates exactly the Hill function in Figure 2.9a). We plot
these functions in Figure 2.9b and using the Matlab function, fmincon, we fit them
by Hill functions to estimate both K0 and n. We get
n = {1.49, 2.55, 3.61, 4.68, 5.8}
and
K0 = {0.91, 1.94, 2.91, 3.85, 4.77}.
Notice that both K0 and n increase linearly with the concentration of Daclizumab
as anticipated. This confirms the analysis in section 2.6.2 where we concluded that
the introduction of Daclizumab will lead to shift to a low CD25 equilibrium and hence
42
shift away from effector T call phenotype to memory T cell phenotype. Additionally,
notice that the values for C0 (the y-intercept) and V0 + C0 (the horizontal asymptote)
are not affected by the introduction of Daclizumab as we have predicted earlier. While
this analysis was done for IL-2 and CD122 concentrations of 1 nM, similar behavior
is observed at other values of IL-2 and CD122 (simulations not shown).
1
0.9
Receptors Bound (nM)
Receptors Bound (nM)
1
0.8
0.7
0.6
0.5
0.4
0.3
0
0.9
0.8
0.7
0.6
0.5
0.4
2
4
6
CD25 (nM)
8
10
0.3
0
(a)
2
4
6
8
10
CD25 (nM)
(b)
Figure 2.9: (A) Plotted is data generated by Model 1. The model was run until it
reached equilibria with 1 nM of IL-2, 1 nM of CD122, and concentrations of CD25
ranging from 0-10 nM. Then the equilibrium concentrations of bound receptors were
plotted as a function of the range of concentrations of CD25. Similar graphs can
be produced for different values of IL2 and CD122. (B) Effect of Daclizumab on
number of bound receptors. Similar graphs that in (A) for varying concentrations of
Daclizumab ({0,1,2,3,4} nM) The initial concentration and maximum concentration
of bound receptors do not change (C0 and V0 ), but the apparent half max (K0 ) and
order (n) of the Hill function both increase.
We note that the restriction on the ratio of V0 to C0 that permits bistability is more
relaxed at higher values of n (2.21). Therefore as the concentration of Daclizumab
is increased, and n increases ( see Figure 2.9b), the range of values V0 and C0 that
support bistability increases (see Figure 2.7).
Therefore, the introduction of Daclizumab, in addition to increasing value of K0
as discussed in section 2.6.2, also increases the region of concentration of IL-2 and
43
CD122 for which the existence of multiple equilibria is possible (see Figure 2.8). This
does not contradict our previous hypothesis that if we fix the ratio of V0 to C0 , as
well as the ratios
rs
ds
and
dα
,
pα pS ∗
then increasing Daclizumab (and therefore K0 and n)
will eventually lead to a saddlenode bifurcation and a single, lower equilibrium (see
Figure 2.10).
Bifurcation Example To better demonstrate the behavior of the model, we show
an example for specific parameter values which will demonstrate multiple equilibria
as well as the saddlenode bifurcation associated with the addition of Daclizumab.
For CD122 and IL-2 concentrations of 0.1 nM, values of n and K0 were estimated
for different concentrations of Daclizumab (ranging from 0 to 0.02 nM). Then a value
of
dα
pα pS ∗
was estimated to ensure that multiple equilibria existed for the case when
there was no Daclizumab. The solutions for (2.8) were solved numerically for the
corresponding values of n and K. Then the corresponding solutions given by (2.6),
(2.7), and (2.8) were plotted in Figure 2.10. Notice there is a saddle node bifurcation
for a concentration of Daclizumab slightly less than 0.015 nM.
In the absence of Daclizumab, the system has two stable equilibria. One lower
equilibrium near 0.1 nM of CD25 and another upper equilibrium with more than
double the concentration of CD25 (approximately 0.22 nM). The initial conditions
will determine which equilibria will be exhibited. However, after treatment with
Daclizumab, the system will transition to the lower equilibrium independent of the
starting equilibrium. Then even after the Daclizumab is removed, the system will
remain at the lower equilibrium until there is another perturbation.
This represents a hypothetical means by which Daclizumab might be used to manipulate the behavior of the system. One can imagine that the T cell population in
a person with multiple sclerosis may be biased towards the upper equilibrium and
44
0.22
4
CD25 (nM)
pStat5 (% of Max)
5
3
0.18
0.14
2
1
0.1
0
0.005
0.01
Daclizumab (nM)
(a)
0.015
0.02
0
0.005
0.01
0.015
0.02
Daclizumab (nM)
(b)
Figure 2.10: This figure shows a saddle node bifurcation when varying the amount of
Daclizumab in the system for specific values of IL-2 and CD122. Values for V0 , C0 ,
α
K0 , and n were chosen to match data produced by Model I. Additionally, drss and pαdpS
∗
were chosen to ensure the model demonstrated by stability when the concentration
of Daclizumab was zero.
thus have an excess of the effector T cells. The treatment with Daclizumab will shift
the population of T cells toward the lower equilibrium that represents the naive or
memory T cells. Even after the withdrawal of the treatment the population will stay
at the lower equilibrium, until presented with an additional immunological challenge
that will cause a perturbation of the system. Obviously, this analysis represents a
particular set of parameters for the model, and other outcomes exists. Depending
on the parameters chosen, there is the possibility of having a single equilibria for all
concentrations of Daclizumab as well as the possibility of two saddle node bifurcations. In the latter case, the termination of the Daclizumab treatment would lead
to population shift back toward the effector cells and thus the model would predict
re-emergence of the MS symptoms upon Daclizumab treatment termination.
45
2.7 IL-2 Receptor with Secondary Signal (Model III)
Examination of Models I and II showed that a cell can differentiate IL-2 signaling
through the intermediate affinity versus high affinity receptor based on the production
of pSTAT5 alone for relatively low concentrations of IL-2 (see Sections 2.5.2 and
2.6.3). In this section we propose a model, which we call a Model III, that examines
a hypothesis that there are two separate signal molecules transduced through the
IL-2 receptor, and that these signals correspond to presence of intermediate vs. high
affinity receptors. In particular, we hypothesize that pStat5 acts as a short-lived
survival and proliferation signal, while an alternative signal, which we will call S2 , acts
as a longer-term survival signal. We use detailed knowledge of the structure of IL-2
receptor in order to propose a mechanism for the production of S2 and this mechanism
will be shown to affectively differentiating signaling through the intermediate and high
affinity of IL-2R based on the levels of pStat5 and S2 (Figures 2.12, 2.13, and 2.14).
Since production of both signals will again depend on extracelluar concentration of IL2, the goal of the Model III is to exhibit the ability to differentiate signaling through
the intermediate and high affinity signaling at wider ranges of IL-2 signaling (∼10%
difference at 100 nM IL-2) than in Model I and II (see Section 2.7.3).
2.7.1 Modeling Assumptions
We assume that CD122 has two states, an ‘activated’ state ([βa ]) and a ‘deactivated’ ([βd ]) state. These states correspond to the state of the tyrosine kinases JAK1
and JAK3 on CD122 and the γ-chain, respectively. Different state here refers to
whether the tyrosine kinases are are phosphorylated or not. Recall, that when IL-2
is bound to CD122 (either in the form of the intermediate or high affinity receptor),
the receptor is ‘activated’ as a result of the auto-phosphorylation of JAK1/JAK3 on
46
CD122/γ, respectively. We make several assumptions about this second signaling
molecule, S2 . First, we assume that even if IL-2 becomes unbound, CD122 will remain ‘activated’ until the tyrosine kinases remain phosphorylated. Eventually, if IL-2
remains unbound, then the ‘activated’ CD122 will decay into the ‘deactivated’ state
when JAK1/JAK3 become dephosphorylated.
IL-2
+
BD
+
IL-2
d
BA
+
IL-2
+
s1
BA
pSTAT5
s2
Signal 2
Figure 2.11: When IL-2 is bound to CD122 (either in the form of high or intermediate
affinity receptor) as shown for the intermediate affinity receptor, then the complex
will preferentially produce pStat5. Alternately, when CD122 is ‘activated’, but IL-2
is not bound, then the complex will preferentially produce signal 2.
Our main hypothesis for the model is that the activated IL-2 receptor will take
on different configurations based on whether IL-2 is bound or not bound. When
IL-2 is bound to IL-2R (either in the form of high or intermediate affinity receptor),
then the complex will preferentially produce pStat5. Alternately, when IL2-R is
‘activated’, but IL-2 is not bound, then the complex will preferentially produce S2 .
We hypothesize that differential signaling is a result of a conformational change caused
by the presence and/or absence of the signaling molecule, IL-2. These assumptions
are supported by our current understanding of the IL-2 receptor as described in the
Background (Section 2.2) and in the description of Model I (Section 2.5).
47
As an example of a theoretical cause for this change in confirmation, consider that
CD122 and the common γ-chain have very low affinity for each other in the absence
of IL-2. If IL-2 unbinds the receptor, these two components will likely disassociate
[130]. If the time constant of this disassociation is shorter than the spontaneous
de-phosphorylation of Jak1/Jak3 kinases than there will be a state of the receptor
where CD122 and γ chain will be dissociated, yet the Jak1/Jak3 kinases will be
phosphorylated. It is possible that the disassociation of the CD122 and γ chain might
open new binding sites for adapter molecules, possibly leading to different preferential
signaling paths. The cartoon, Figure 2.11, demonstrates the proposed reactions while
only considering the intermediate affinity receptor. The high affinity receptor works
similarly, but was not included in the model cartoon, so that the unique features of
the model could be emphasized without complicating the figure.
To summarize, in addition to the assumptions made for Model I, we make the
following additional assumptions:
• The signaling portions of the receptor (CD122 and γc) have two different states,
specifically activated (phosphorylated) or deactivated (unphosphorylated).
• The binding of IL-2 to CD122 and the γ-chain leads to the activation of the
receptor.
• The receptor will remain activated for some time even if IL-2 disengages, eventually decaying back into a deactivated state. This means that dephosphorylation
is not instantaneous after IL-2 disengages.
• Only activated receptors are capable of signaling.
• The activated receptor has two different conformations based on whether IL-2
is bound or not leading to production of different preferential signals. If IL-2
48
is bound, the receptor preferentially phosphorylates STAT5 producing pSTAT5
as a signal. Alternately, if the receptor is activated, but IL-2 is not bound, then
S2 will be preferentially produced.
The differential equations associated with the model are described below with
some of the terms constrained by conservation equations (shown further down).
−
+
[β˙a ] = kβ− [IL2β] + kαβ
[IL2αβ] − kβ+ [IL2][βa ] − kαβ
[IL2α][βa ] − d[βa ]
˙
[IL2β]
= kβ+ [IL2] [βa ] + [βd ] − kβ− [IL2β]
˙ = k − [IL2α] + k − [IL2αβ] − k + [IL2][α]
[α]
α
α
αβ
+
−
˙
[IL2αβ]
= kαβ
[IL2α] [βa ] + [βd ] − kαβ
[IL2αβ]
˙
[pStat5]
= s1 [IL2β] + [IL2αβ] 1 − [pStat5] − d1 [pStat5]
[S˙2 ] = s2 [βa ] 1 − [S2 ] − d2 [S2 ]
For the purposes of the model, the concentrations of pStat5 and S2 were rescaled
by the (respective) total concentration of the signaling molecules and their precursors,
so that the value of each is reported as a percentage of their corresponding maximum
values. In addition, we assume that the overall number of CD25, CD122, IL-2 is
constant, which gives the following conservation relationships.
[IL2α] = α∗ − [α] − [IL2αβ] − [Dα]
[βd ] = β ∗ − [βa ] − [IL2β] − [IL2αβ]
[IL2] = IL2∗ − [IL2α] − [IL2β] − [IL2αβ]
49
As in Model 1, we did not write differential equations for some of the variables,
namely [IL2α], [βd ], and [IL2]. These quantities will be determined from the three
conservation laws above.
2.7.2 New Parameters
Because the presence of the secondary signal is being hypothesized, none of the
parameters associated with this signal are known explicitly. The decay and initial
conditions for for S2 were chosen to be the same as those for [pStat5]. The production
was chosen to be slightly higher to accentuate the differences in intermediate versus
high affinity signaling.
Table 2.5: Production/Decay Rates
Production Rate (nM−1 min−1 ) Decay Rate (min−1 )
[S2 ]
2.25†
0.028†
[βa ] → [βd ]
0.05†
Table 2.6: Initial Conditions
Initial Condition (nM)
[S2 ]
0.18†
2.7.3 Results
Differentiation of Intermediate vs High Affinity Signaling. While the equilibrium concentration difference in the secondary signal (S2 ) for intermediate versus
high affinity receptors (∼12%) shows only has a slight improvement over the difference in pStat5 (∼9%) for 1 nM concentrations of IL-2 (see Figure 2.12), the ability
of this mechanism to distinguish between IL-2 signaling through intermediate affinity
50
versus high affinity receptor does not diminish with increasing IL-2 as rapidly as the
1
1
0.9
0.9
0.8
0.8
0.7
0.7
S2 (% MAX)
pStat5 (% MAX)
difference in pStat5 signal.
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0
0
0.6
0.1
High Affiinty
Intermediate Affinty
20
40
60
80
0
0
100
High Affiinty
Intermediate Affinty
20
Time (min)
40
60
80
100
Time (min)
(a)
(b)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
S2 (% MAX)
pStat5 (% MAX)
Figure 2.12: For low levels of IL-2 concentration, difference in concentration of S2 ,
(B), for the intermediate affinity versus high affinity receptor (∼12%) does not provide
a significant advantages over the difference in pStat5 signaling, (A), between these
receptors (∼9%).
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0
0
0.6
High Affiinty
Intermediate Affinty
20
40
60
Time (min)
(a)
80
100
0.1
0
0
High Affiinty
Intermediate Affinty
20
40
60
80
100
Time (min)
(b)
Figure 2.13: For moderate levels of IL-2 concentration, the difference in concentration
of S2 , (B), for the intermediate affinity versus high affinity receptor (∼33%) provides
a significant advantages over the difference in pStat5 signaling, (A), between these
receptors (∼1%).
In fact, the difference in production of S2 between intermediate and high affin-
1
1
0.9
0.9
0.8
0.8
0.7
0.7
S2 (% MAX)
pStat5 (% MAX)
51
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0
0
0.6
High Affiinty
Intermediate Affinty
20
40
60
80
100
Time (min)
(a)
0.1
0
0
High Affiinty
Intermediate Affinty
20
40
60
80
100
Time (min)
(b)
Figure 2.14: For high levels of IL-2 concentration, the difference in concentration
of S2 , (B), for the intermediate affinity versus high affinity receptor remains ∼10%
while the difference in pStat5 signaling, (A), between these receptors is virtually
indistinguishable.
ity receptors actually improves ∼33% for 5 nM concentrations of IL-2 while at this
concentration if IL-2 the difference in pStat5 signaling has declined to ∼1% (see Figure 2.13). Furthermore, the S2 signal is still able to differentiate the two types of
receptors at very high concentrations of IL-2, where we still observe a difference of
∼10%
at 100 nM IL-2. At this high concentration of IL-2, the pStat5 production
through intermediate and high affinity receptors are virtually indistinguishable, see
Figure 2.14. Therefore, the signal S2 is able to differentiate intermediate and high
affinity signaling for much greater ranges of IL-2 concentration that the bistability
mechanism that was explored in Model II.
The signal S2 may represent a longer-term survival signal. As mentioned in the introduction, cells with higher levels of membrane bound CD25 terminally differentiate
into effector cells, and IL-2 signaling in effector T cells is associated with activation
and rapid proliferation followed by increased susceptibility to apotosis [79]. On the
other hand, IL-2 signaling in cells with lower CD25 concentrations, typical of memory
52
T cells, leads to homeostasis and survival [19, 27]. The levels of pStat5 signaling alone
are unlikely to be responsible for this difference long-term outcomes as the difference
in pStat5 concentration seems insufficient to elicit these long-term affects (see Figures
2.12a, 2.13a, and 2.14a). However, the balance of pStat5 and S2 could be sufficient to
produce these seemingly contradictory long-term outcomes. In particular, the pStat5
signal could act like a proliferation and short-term survival signal, while S2 provides
a long-term survival signal. Recall that difference in equilibrium level of S2 signaling
associated with the high and intermediate affinity receptors is signifcant for moderate
concentrations of IL-2 (see Figure 2.13b). Therefore, a cell with many high affinity
receptors would receive significantly less long-term survival signal potentially leading
to a higher rates of apoptosis. Additionally, some of the downstream signals of pStat5
and the other IL-2 associated pathways (PI3K, MAPK, Ras-Raf) have been shown
to demonstrate mutual inhibition (bcl-6 and Blimp1 for example), so the small initial
signaling differences could be exaggerated even further [20, 38, 43, 77].
Daclizumab Leads to Intermediate Affinity Signaling. The use of Daclizumab
to bind cellular CD25 and thus preventing IL-2 signaling through the high affinity
receptor has been used experimentally to represent intermediate affinity signaling
because cells treated with sufficiently high level of Daclizumab will signal exclusively
through intermediate affinity receptor [10, 125]. Therefore, this monoclonal antibody
allows for the comparison of intermediate affinity versus high affinity signaling in
identical cell types. We can use Model III to validate this experimental protocol. We
show that in the model both signals associated with cells only containing intermediate
affinity receptors are shown to be nearly identical to those of cells with high affinity
receptors treated with Daclizumab. To see this observe that the red plots (cells
containing CD25, but treated with Daclizumab) nearly overlay the green plot (cells
53
1
1
0.9
0.9
0.8
0.8
0.7
0.7
S2 (% MAX)
pStat5 (% MAX)
without CD25) in Figures 2.15 and 2.16).
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
High Affiinty
Intermediate Affinty
High Affinity + Daclizumab
0.1
0
0
0.6
20
40
60
80
High Affiinty
Intermediate Affinty
High Affinity + Daclizumab
0.1
0
0
100
20
Time (min)
40
60
80
100
Time (min)
(a)
(b)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
S2 (% MAX)
pStat5 (% MAX)
Figure 2.15: This figure plots the pStat5 signal (A) and S2 signal (B) for cells with
CD25, which will demonstrate primarily high affinity signaling (blue), cells without
CD25 (which can only have intermediate affinity signaling) in green, and cells with
CD25, but that have had their high affinity signaling blocked by Daclizumab (red).
All three cases are for extracellular concentrations of IL-2 of 1 nM.
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
High Affiinty
Intermediate Affinty
High Affinity + Daclizumab
0.1
0
0
0.6
20
40
60
Time (min)
(a)
80
100
High Affiinty
Intermediate Affinty
High Affinity + Daclizumab
0.1
0
0
20
40
60
80
100
Time (min)
(b)
Figure 2.16: This figure plots the pStat5 signal (A) and S2 signal (B) for IL-2 concentration of 5 nM for cells with CD25 (blue), cells without CD25 (green), and cells
with CD25 treated with Daclizumab (red).
54
Kinetic Benefits of the Colocalization of Coupled
Enzymes
3.1 Introduction
Compartmentalization is ubiquitous in biology from the organismal level, to organ
systems, organs, cells, organelles all the way to the molecular level. This compartmentalization is important to provide protection and separation of different systems
as well as for the creation of concentration gradients that form some of the fundamental basis for biological systems. Organization and compartmentalization of enzymes
play an important role in the control of enzyme kinetics as well as allowing for the
protection and localization of enzymes and intermediates. Enzyme encapsulation has
been mimicked in vitro to study the advantages that sequestration might provide in
vivo as well as for the potential benefits in various biochemical applications [45, 54,
154, 155, 158].
3.2 Background
3.2.1 Bacterial Microcomparments
‘Microcompartments’ are comon in bacteria and represent an example of enzyme
compartmentalization. The most famous example is the carboxysome, which contains
the enzymes responsible for carbon fixation (Ribulose-1,5-bisphosphate carboxylase
[RuBisCo] and carbonic anhydrase) in all cyanobacteria and many chemotrophs as
well. Carbon fixation represents the rate limiting step of the Calvin cycle, and the
carboxysome is thought to have developed to concentrate CO2 , the intermediate of
this enzyme cascade, to overcome the inefficiency of RuBisCo [31, 152]. Several other
examples of bacterial microcompartments have been characterized, most notable the
55
propanediol utilization (Pdu) and ethanolamine utilization (Eut) microcompartments
[31, 82, 170]. The enzymes within both of these microcompartments produce highly
diffusive and reactive intermediates [66, 139, 149]. Therefore, the sequestration of
these intermediates with an enzyme that transforms them into a more stable product
decreases the risk of unwanted reactions [66, 139]. All of these microcompartments
represent cases where localization of an intermediate is desired either to sequester
volatile substrates or to increase the regional concentration of a substrate to overcome
an inefficient enzyme.
In order to appreciate the magnitude of these effects, many experiments have
been designed to reproduce the hypothesized advantages of colocalization. Because
enzymes evolved to operate in such crowded conditions, typical batch experiments
do not provide accurate characterization of enzymes in biologically relevant conditions [47]. Enzyme encapsulation provides a way to mimic cellular macromolecular
densities (> 300 mg/mL) experimentally [47, 172]. While numerous structures have
been used for bioinspired encapsulation including polymerosomes, vesicles, and gel
matrices, protein cages have numerous advantages over these other systems. In particular, protein cages can be designed to fully self-assemble into highly homogenous
and symetric structures [45, 75, 76, 141]. These protein structures, generally referred
to as virus-like particles (VLPs), can be produced within bacterial or plant host [23,
24, 116, 138]. Additionally, proteins and enzymes can be tagged such that they are
incorporated into the interior of the protein cage architecture during self-assembly
to allow for the inclusion of cargo into the VLP in a uniform and controlled manner
[23, 50, 70, 84, 114, 142, 167]. These viral capsids have been used for many, varied
biotechnological applications ranging from sequestering toxic MRI contrasting agents
to development of chemotherapeutic agents and flu super-vaccines [45, 54–56, 128,
135, 154–158].
56
3.2.2 Single Enzyme Encapsulation
The existence of naturally occurring sequestration of metabolic pathways has been
considered evidence that these compartments lead to advantageous characteristics.
Therefore, most research focus on enzymatic sequestration has been on studying
beneficial catalytic properties resulting from encapsulation. These systems also have
been used to mimic the kinetics of enzymes in a setting with high enzyme density
emulating the affects of macromolecular crowding on enzyme behavior.
While increased enzymatic density (via encapsulation) was initially expected to
increase enzymatic turnovers, this has been mostly shown not to be the case. The
only exception is for the encapsulation of phenylalanine amonia lyase (PalB) which
was shown to have a nearly 5 times faster turnover for the encapsulated enzyme versus
the non-encapsulated [104]. However, this advantage was decreased as the number of
enzymes was increased, which contradicts the hypothesis that the increased turnover
is related to the evolution of this enzyme in ‘crowded’ environments. All other studies
have shown either no change in turnover [115, 118] or reduced activity [50, 70, 116]
upon encapsulation. So, for most enzymes, encapsulation has not been shown to
increase the turnover of these enzymes.
However, perhaps enzymatic turnover is not the most representative kinetic parameter for comparison of ‘free’ versus encapsulated enzymes as enzyme turnover is
dependent on the percentage of active enzymes in a sample. Therefore, in the case
that some portion of the enzymes may be inactivated by encapsulation possibly resulting from inability to fold properly the true enzymatic turnover can be difficult
to determine properly. On the other hand, the binding affinity (Km ) is independent
of the number of active enzymes, and thus may be a better indicator of the effects
of crowding and is independent of enzyme inactivation. Results of comparing Km
57
for encapsulated versus ‘free’ enzymes has shown diverse results ranging from significant increases in Km to decreases in Km [50, 116]. Hence, the effects of enzymatic
encapsulation vary between enzymes and packing density, and therefore cannot be
generalized.
3.2.3 Multi-Enzyme Encapsulation
More recent advances have lead to the ability to encapsulate multiple enzymes,
corresponding to the same pathway, within the same protein cage [119]. The encapsulation of multiple enzymes allows for direct comparison to a naturally occurring
bacterial microcompartments (like the carboxysome, Pdu, and EUT), which, as previously mentioned, all contain multiple enzymes. Despite the evidence that encapsulation of enzymes does not necessarily lead to increase in efficiency, colocalization
of coupled enzymes has been anticipated to enhance pathway kinetics as a result of
higher local concentration of the intermediate substrate [89].
A
B
Figure 3.1: (A) This figure demonstrates a theoretical viral cage containing a coupled
enzyme system with the different enzymes shown in red and blue respectively. (B)
Electron micrograph of assembled CelB-GLUK-P22 particles with an approximate
diameter of 60 nm. This image was reproduced from [119].
58
To explore the effects of colocalization of enzymes our collaborators from T. Douglas Biomimetic Chemistry group at Indiana University developed a system of coencapsulated coupled enzymes inside of VLPs [119]. As previously mentioned, the
ability to encapsulate various cargo, including single enzymes, had been previously
demonstrated [23, 24, 45, 54, 116–118]. Therefore, to explore the potential benefits of
colocalization, our collaborators looked at the kinetics of the production of the final
product of a coupled enzyme system for both separately encapsulated enzymes and
coencapsulated enzymes (see Figure 3.2).
For one experiment, the enzymes were individually encapsulated in their own
VLPs, which were then added to the same medium. The initial substrate for the
coupled system was added and a reporter protein for the production of the final
product was tracked as a function of time [119]. For a separate experiment, the
enzymes were encapsulated in the same VLP (see Figures 3.1 and 3.2). This system
was also treated with the initial substrate and the reporter protein was tracked as
a function of time [119]. The system was designed such that neither enzyme was
kinetically inhibited by encapsulation [119].
The anticipated experimental result was that colocalization of enzymes should,
in general, increase the efficiency of the pathway as measured by faster production
of the final product. This advantage bestowed by colocalization is called channeling.
The experimental system demonstrated some surprising kinetics. Depending on how
the system was assembled, the second enzyme had differing turnover resulting from
a slight inhibition from one of the confirmations, and this change affected whether
channeling was observed. In particular, the uninhibited system did not demonstrate
a channeling advantage for the colocalized enzymes [119]. However, for the system
in which the second enzyme was inhibited, the colocalization of the enzymes did
exhibited a channeling advantage [119]. The dependence of channeling on the rates of
59
E1
S
E2
U
P
(a)
E1
E2
Diffusion
S
U
U
P
R
(b)
Figure 3.2: To explore the potential kinetic advantage of colocalization of coupled
enzymes, our collaborators compared the system where coupled enzymes are coencapsulated in a VLP (A), with a system where the coupled enzymes are individually
encapsulated in separate VLPs (B). Our collaborators noticed that for certain systems colocalization would exhibit the anticipated advantage, while for other systems
this advantage was not present.
the individual enzymes required a more thorough analysis of the concept of channeling
beyond the simplistic hypothesis that colocalization should always lead to improved
kinetics. To further analyze these ideas, our collaborators solicited us to explore these
ideas using mathematical models. (The initial work on this project was published in
ACS, see the reference: [119].)
60
3.3 Model Framework
To begin our analysis we will consider a simpler system
E
E
1
2
S −→
U −→
P
where the production of P from S via the intermediate U is catalyzed by E1 and
E2 . We assume that E1 and E2 lie on two surfaces separated by some distance R
and there is simple diffusion of the intermediate U between them. A priori, it seems
logical that placing these two enzymes in close proximity would increase the efficiency
in the production of the final product and thus exhibit ‘channeling’. In particular,
we will consider channeling to be a difference in efficiency of the pathway between
placing the second enzyme in the sequence (E2 ) directly next to the first (E1 ) versus
some distance R from the first enzyme. But how should we define efficiency of the
pathway and thus channeling mathematically? Should we consider the steady-state
case or transients? Should we compare the production rate of the final product, total
production of the final product, or the lag time (or something else)? We will examine
some of these possibilities in different situations.
For the context of this problem, we will assume that both of our enzymes exhibit Michaelis-Menten kinetics. That is, the rate of product (P̂ ) formation is a Hill
function of the concentration of the substrate (Ŝ),
dP̂
Vmax Ŝ
=
dt
Km + Ŝ
(3.1)
Here Vmax ( mol
) is the maximum rate of production of the product of an enzyme and
s
is given by Vmax = kcat E0 , where kcat ( 1s ) is the enzyme turnover and E0 (mol) is
the total number of enzymes. Ŝ is the concentration of the substrate in the given
61
enzymatic reaction while Km is the concentration of the substrate at which the rate
of the equation is half of the maximum. Notice that for small values of Ŝ (Ŝ < Km ),
changes in Ŝ correspond nearly proportionally to changes in dP̂ /dt (see Figure 3.3).
Therefore, we will refer to concentrations of Ŝ less than Km as the linear regime of an
enzyme. Alternately, for Ŝ large (Ŝ >> Km ), even large changes in Ŝ will generate
very small changes in dP̂ /dt (see Figure 3.3). This region is considered the enzyme
limiting region. Concentrations of Ŝ within this region are sufficient to saturate all
of the available enzymes, so that increasing the concentration of Ŝ will not increase
the rate of turnover.
Michaelis-Menten Kinetics
No Channeling
dP̂
dt
Channeling
Ŝ
Figure 3.3: When comparing a system of colocalized enzymes versus enzymes separated by a more significant distance, a spatial gradient in the concentration of the
intermediate substrate at the two different locations of the second enzyme will only
affect the rate of formation of the final product, if the concentrations fall within the
‘linear’ region of the Michaelis-Menten equation.
Independent of our definition, channeling can only occur when the intermediate
substrate concentration (U ) differs at different spatial locations. In addition, if we assume that enzyme E2 exhibits Michaelis-Menten kinetics, the different concentrations
62
of U at different locations must fall within the ‘linear regime’ of enzyme E2 . As shown
in the Figure 3.3, even if there are substantial differences in substrate concentration as
we increase the distance between E1 and E2 , if both levels fall in the enzyme limiting
region of the Michaelis-Menten equation for enzyme E2 , then we would not expect
the system to exhibit channeling (See Figure 3.3).
3.4 1D Model
We will model the system described above
E
E
1
2
S −→
U −→
P
where we assume that both E1 and E2 follow Michaelis Menten kinetics (3.1) with
a maximum rate (Vmax ) of V1 and V2 and the substrate concentration at which the
reaction rate is half of Vmax (Km ) of K1 and K2 , respectively. We will assume that
the boundaries of the system are given by a square rectangular prism, such that, the
enzymes, E1 , lie on a membrane with area (A), and that the enzymes, E2 , lie on
the parallel membrane. On the other rectangular boundaries, we assume reflective
boundary conditions (that is, Uy = 0 and Uz = 0). If the initial conditions are
independent of y and z, then the diffusion within a (y,z) cross-section is uniform. To
this end, we will only consider the one dimensional diffusion problem,
Ut (x, t) = D Uxx (x, t)
where D is the diffusion coefficient of the intermediate substrate, U .
(3.2)
63
E2
z
E1
y
x
Figure 3.4: The model we are assuming is diffusion between finite parallel plates. We
have a rectangular prism with the enzymes, E1 and E2 , located on opposite ends. We
assume the other boundaries are reflective reducing the problem to one dimensional
diffusion.
At the boundary, we will prescribe the flux to be
Ux (0, t) = −
M
DA
(3.3)
A similar boundary condition was used in [26] and [25]. Additionally, at the other
boundary will will prescribe the flux to be
Ux (R, t) = −
1 V2 U (R, t)
DA K2 + U (R, t)
(3.4)
3.4.1 Model
The choice of the boundary conditions (3.3) and (3.4) supplement the linear diffusion equation (3.2). With initial conditions given by the function φ̂(x), we arrive at
the model. For the initial model we will use Û , x̄, and t̄, so that the simpler notation
U , x, and t can be used after the equations are rescaled.
64
Ût̄ (x̄, t̄) = D Ûx̄x̄ (x̄, t̄)
IC:
BCs:
Û (x̄, 0) = φ̂(x̄)
M
DA
U (R, t)
V2
Ûx̄ (R, t̄) = −
DA K2 + U (R, t)
Ûx̄ (0, t̄) = −
3.4.2 Rescaling
We can simplify our partial differential equation (PDE) along with its corresponding initial conditions and boundary conditions by making the given substitutions:
D
t̄
R2
DA
φ̂(x)
φ(x) =
RM
x̄
R
DA
Û
U=
RM
t=
x=
Then our PDE simplifies to,
Ut (x, t) = Uxx (x, t)
IC:
U (x, 0) = φ(x)
BCs:
Ux (0, t) = −1
Ux (1, t) = −
where the only parameter is V =
V2
M
and K =
V U (R, t)
K + U (R, t)
K2 DA
.
RM
65
3.5 Steady-state solution
To find the equilibrium of this linear PDE, we set Ut = 0. Then, we have Uxx = 0,
which implies that Ueq = A x + B. The slope has to be the same across the entire
domain, including the endpoints. Therefore, we can determine the constants A and
B from the boundary conditions,
Ueq (x) =
K
+ 1 − x.
V −1
Notice that this equilibrium is only feasible when V > 1. We will show in section 3.6.3
that when V > 1, then this equilibrium is stable.
If we consider the original (non-rescaled) concentration, Û (but leaving x scaled),
we have that at equilibrium, the concentration is given by:
Ûeq (x) =
RM
M K2
+
(1 − x)
V2 − M
DA
(3.5)
We are ready for the analysis of the channeling effect based on the equilibrium
solution. To quantify the effects of various parameters on channeling, we will compare
the concentration of the intermediate substrate at both endpoints. From the equation
(3.5), the difference in the concentration at the endpoints, x = 0 and x = 1, is
RM
.
DA
We can ask now for what values of M , V2 , K2 , R, and D, will this difference
remain in the ‘linear regime’. This is equivalent to asking when is the concentration
at x = 1 smaller than K2 . This leads to
Ûeq (1) =
M K2
≤ K2
V2 − M
66
Therefore, we should expect to see channeling (at equilibrium) when
2M
≤1
V2
with a difference in concentration at the endpoints of
(3.6)
RM
.
DA
3.5.1 Discussion
From analysis of the steady-state solution, we only expect a system to exhibit
‘channeling’ at steady-state if (3.6) is satisfied. In particular, this implies that ‘channeling’ will only occur whenever M is less than half of the value of V2 . Recall that,
M =
V1 S
,
K1 +S
and therefore is the rate of production of the first enzyme. Then, (3.6)
can be interpretted intuitively. For a steady-state to occur the rate of production
of the intermediate (flux into the system) must be less than the rate of consumption (flux out). Additionally, for channeling to occur we also have to assure that the
rate of the reaction at the exit wall remains in the linear regime and thus must be
bounded above by K2 . If (3.6) is not satisfied, and M >
V2
2
then the concentration
will exceed K2 , such that the consumption at the boundary will fall in the enzyme
limiting region.
3.6 Rate of Convergence
We have shown that there are certain conditions where a system can exhibit channeling at steady-state, biological systems are rarely at equilibrium. Cells are dynamic
systems which are constantly adapting and responding to their ever-changing environment. Therefore, of greater interest than steady-state behavior is the transient
behavior of a metabolic pathway. In particular, we are interested to explore how colocalization of enzymes might enable a pathway to respond more rapidly to changes
67
in their environment. To analyze this response rate, we will look at the rate of convergence of an arbitrary solution to the steady-state solution by solving the equation
explicitly using Fourier series.
3.6.1 Linearization of the Boundary Conditions
Consider the perturbation of our solution U from the equilibrium, Ueq by W (x, t),
U (x, t) = Ueq (x) + W (x, t)
In particular, let
U (x, t) =
K
+ 1 − x + W (x, t)
V −1
Plugging (3.7) into the PDE gives
Wt (x, t) = Wxx (x, t)
BCs:
Wx (0, t) = 0
V Ueq (x) + W (x, t)
Wx (1, t) = −
K + Ueq (x) + W (x, t)
Taking the derivative of everything with respect to and letting → 0 yields
Wt (x, t) = Wxx (x, t)
BCs:
Wx (0, t) = 0
Wx (1, t) = −C ∗ W (1, t)
and C ∗ =
(V −1)2
.
KV
(3.7)
68
3.6.2 Fourier Series Solution
We will solve the linearized partial differential equation analytically using Fourier
series. If all of the eigenvalues have negative real parts, the solution converges to
the steady-state solution, and the rate of the convergence is of the same order as
the smallest eigenvalue of the linear operator. In addition, we will use the rate of
convergence to compute a characteristic time-scale for the convergence of the solution.
Our solution is separable, that is, W (x, t) = X(x)T (t). Then our PDE becomes,
X(x)T 0 (t) = X 00 (x)T (t)
which can be separated as,
X 00 (x)
T 0 (t)
= −λ =
T (t)
X(x)
We get two differential equations
T 0 (t) = −λ T (t)
(3.8)
X 00 (x) = −λ X(x)
(3.9)
Solving (3.8), we get T (t) = Ae−λt . We discuss three cases: λ negative (1), zero
(2), and positive (3).
√
1. If λ < 0, then the solution to (3.9) is given by X(x) = C1 e−
λx
√
+ C2 e
λx
. To
satisfy the boundary conditions, Wx (0, t) = X 0 (0)T (t) = 0. Assuming T (t) is
not identically zero (i.e. solutions are non-trivial), this implies X 0 (0) = 0. That
69
is,
√
√
X 0 (0) = −C1 λ + C2 λ = 0
which implies that C1 = C2 . Then, by considering the second boundary condition, we see that Wx (1, t) = −C ∗ W (1, t), and thus T (t)X 0 (1) = −C ∗ T (t)X(1).
Again assuming T(t) is not identically zero, this implies X 0 (1) = −C ∗ X(1), or
√
√
√ √
√
√ √
√
−C1 λe− λ + C2 λe λ = C1 λe− λ + C2 λe λ
which simplifies to −C1 = C1 . Therefore, C1 = C2 = 0.
2. If λ = 0, then the solution to (3.9) is X(x) = C1 x + C2 . Again if we consider
the boundary conditions, we see for X 0 (0) = 0, then C1 = 0. And to satisfy
X 0 (1) = −C ∗ X(1), then C2 = 0 also.
3. So to generate nontrivial solutions, we are left with λ > 0. Then, we can solve
(3.8) and (3.9) independently as
T (t) = B e−λ t
√
X(x) = C1 sin
√ λ x + C2 cos λ x
Now we seek C1 , C2 , and λ that satisfy the boundary conditions. To satisfy the
boundary condition at zero, we require that
√
√
√
X 0 (0) = C1 λ cos(0) − C2 λ sin(0) = C1 λ = 0
which implies that C1 = 0 or λ = 0. Recall from above, that we demonstrated
that to generate nontrivial solutions, λ 6= 0, so let C1 = 0. Now applying the
70
boundary condition at x = 1, we get
√
√ √ −C2 λ sin λ = −C2 C ∗ cos λ
So in order to have nontrivial solutions we will assume C2 6= 0, such that
√
λ
must satisfy
tan
√ C∗
λ =√
λ
(3.10)
Notice that (3.10) has infinitely many solutions, λn . Therefore, the Fourier
series solution of our PDE is given by
V (x, t) =
∞
X
An e−λn t cos
p
λn x
n=1
where An are the standard Fourier series coefficients.
3.6.3 Convergence Rate
Notice that since λn > 0, all of the eigenvalues (given by −λn ) are negative, and
therefore Fourier series converges to V = 0 for all initial conditions. This proves that
the equilibrium solution computed previously is globally asymptotically stable. Now
we wish to know how fast the Fourier series solution converges. In general the rate
of convergence should be exponential with order approximately equal to the smallest
eigenvalue, λ1 . In particular,
V (x, t) =
∞
X
n=1
An e
−λn t
X
∞
p
p
−λ1 t
−(λn −λ1 ) t
λn x = e
λn x ≤ e−λ1 t C
cos
An e
cos
n=1
Now we aim to more precisely determine the size of λ1 . Note that 0 < λ1 ≤
π 2
2
71
as can be seen by analyzing the intersection of the left hand side and right hand of
(3.10) for C ∗ ∈ [0, ∞).
Because we cannot solve (3.10) explicitly, we will use the first two terms of the
Taylor approximation of the tangent function to approximate λ1 in terms of C ∗ . In
particular,
λ1 ≈
3C ∗
3 + C∗
(3.11)
This estimate has accuracy of approximately than 1.3% for C ∗ < 1, and 13% for
C ∗ < 10, with a maximum error of less than 22% as C ∗ → ∞. For better accuracy,
we can estimate λ1 with the first three terms of tangent to get,
45
λ1 ≈
2
s
3 + C∗
3C ∗
2
3 + C∗
4
−
+
45
3C ∗
with an error of approximately 2% for C ∗ < 10, and a maximum error less than
4%. Unfortunately this formula is not particularly enlightening, so we will use (3.11)
for the rest of the analysis. The units on λ are inverse time, so we can define a
characteristic time-scale of convergence as th = 1/λ1 . Therefore, we can use our
estimate for λ1 , to find
th ≈
1
1
K
1
+
=
+
1
C∗ 3
V (1 − V )2 3
and therefore,
R2
K2 DA
K2 RA
1
R2
t̄h ≈
+
=
+
D V2 R(1 − VM2 )2 3
V2 (1 − VM2 )2 3D
72
3.6.4 Discussion
Notice that as R increases so does the characteristic time scale of relaxation to
steady state. In fact, the characteristic time-scale increases quadratically with R.
Additionally, the relaxation time scale depends on the ratio
V2
K2
which is the derivative
of Michealis Menten reaction rate at zero and characterized the steepness of the linear
regime of the second reaction. As
to its lower bound of
R2
.
3D
V2
K2
→ ∞ the characteristic time scale converges
As expected, this lower bound is proportional to the
characteristic time-scale of pure diffusion. Therefore, if the rate of consumption is
high, then the reaction will be diffusion limited.
On the other hand, a
V2
K2
→ 0, then the characteristic time-scale approaches
infinity. Specifically, this implies that if we have a metabolic pathway with a slow
secondary enzyme (as in the carboxysome for example), then colocalization might be
necessary to increase the responsiveness of the pathway.
73
What is the Ideal Distribution of Viral Fragments on
CRISPR?
4.1 Introduction
An adaptive immune system is important to ensure appropriate, precise, and rapid
response to foreign pathogens. While adaptive immune response has traditionally
only be attributed to vertebrates, it has fairly recently been discovered that some
bacteria (and archaea) also demonstrate a form of adaptive immunity. The CRISPR
(Clustered Regularly Interspaced Short Palindromic Repetitions) system is made up
of short phage homologs which with the help of Cas (CRISPR-associated) proteins
work to recognize and silence exogenous genetic material. Additionally, when a new
virus is recognized, a spacer associated with the virus can then be incorporated into
the CRISPR sequence. The mechanisms for acquisition and recognition are still not
fully understood. We are interested to look at ‘ideal’ CRISPR distributions for a
given distribution of viral genomes and to explore how different acquisition strategies
might lead to different CRISPR distributions. In particular, we will demonstrate that
the optimal CRISPR distribution might not be one that would be chosen via ‘naive’
acquisition strategies.
4.2 Background
CRISPRs represent a portion of the DNA loci in some bacteria and archaea characterized by snippets of exogenous DNA ‘spacers’ separated by repeated segments.
This region was initially discovered in E. coli in 1987, and was later named for the
repeated segments of DNA separating the spacers [72, 73, 106]. Eventually several
research groups concurrently recognized that several CRISPR spacers were exactly
or very nearly viral homologues and were likely derived from extra-chromosomal ge-
74
netic material [17, 108, 123]. These discoveries lead to the natural hypothesis that
this DNA region might be responsible for an acquired immune system in prokaryotes
possibly functioning similar to RNA interference in eukaryotes.
Figure 4.1: Cas proteins help the cell to recognize foreign genetic material leading
to inclusion of a novel spacer into the CRISPR locus. The spacers are transcribed
and processed by different Cas proteins into crRNAs. Finally the spacer joins a
Cas protein complex to scan the genetic material looking for a match. When the
Cas/crRNA recognizes a sequence, the accompanying genetic material is rendered
inactive. Credit for the image goes to Horvath, et al [67]
The CRISPR locus is usually associated with several other genes (CRISPR-associated
or cas genes) which help the CRISPR to function effectively. The Cas proteins are
important for the acquisition of new CRISPRs as well as playing a key role in CRISPR
targeted interference and regulation of the CRISPR system [2, 94, 95, 151, 163, 164,
171]. For the CRISPR system to effectively carry out its immune response functions,
it must be able to efficiently manage the two primary mechanisms of adaptive immune
75
response, namely, acquisition and interference, see Figure 4.1.
The acquisition stage of the CRISPR system is responsible for making the system
adaptive and dynamic. Without efficient acquisition, the CRISPR system can not
effectively evolve to match the changing viral milieu. Although it is known that the
Cas1 and Cas2 proteins are involved in spacer acquistion, the exact role these proteins
play is still not completely understood [21, 151, 171]. However, the structure of many
of the Cas1 proteins has been resolved, and they are strikingly similar, leading to the
logical conclusion that these proteins probably all function analogously [2, 164].
Spacers are added in an ordered fashion primarily (although not exclusively) adjacent to the leading end [44, 48, 107]. The mechanism for recognition and targeting
of specific newly acquired spacers is not well understood although it is known that
spacers are not chosen at random. Rather the excised spacers are generally located
next to short DNA motifs labeled PAMs (protospacer adjacent motifs) [17, 42, 107,
143, 144]. Additionally, there is evidence that suggests that if a cell has spacers
that target a specific phage (even if slightly mismatched), then the cell is more likely
to acquire more spacers associated with this phage [40, 151]. This idea, known as
‘priming’, has been used to justify the phenomenon that CRISPRs have often been
shown to contain multiple spacers associated with same virus or phage [40, 129, 151].
In addition to allowing the bacteria to favor building immunity to rapidly evolving
viruses, we demonstrate that this strategy also leads to the optimal CRISPR spacer
distribution.
The second mechanism of the CRISPR-cas system is expression and interference.
The CRISPR locus is typically transcribed as a single long strand that is subsequently
cleaved at the ‘repeat’ regions into the corresponding spacers [1, 21, 64, 65, 93, 143].
The cleaved CRISPRs then typically fold to form a ‘hairpin-loop’ created by the
palindromic repeat region that is characteristic of mature CRISPR crRNA as well as
76
other small RNA [65]. After the processed crRNA sequence joins a Cas protein, the
corresponding complex is prepared to scan for exogenous genetic material (with the
appropriate spacer sequence).
Single base pair mismatches in the spacer and PAM region have been shown to
be sufficient to prevent recognition [42, 59]. However, the crRNA mediated silencing
has been shown to be effective even in the case of multiple base pair mismatches
particularly when the mismatches are located far from the PAM sequence [59, 143].
Once the crRNA recognizes a sequence of foreign genetic material, the associated
Cas protein recruits endonucleases (e.g. Cas9) to cleave the exogenous DNA [74,
163]. The eventual illumination of this simple and effective mechanism for genetic
regulation naturally lead to the use of engineered CRISPR/Cas9 systems for genetic
modification and genome silencing [68, 140].
Because of their fairly recent discovery and even more recent characterization, relatively little mathematical modeling has been done associated with the CRISPR/Cas
system. The mathematical modeling that has been done to this point has primarily
focused on stochastic analysis of the co-evolutionary dynamics of the host/pathogen
interaction [9, 32, 33, 71, 85, 91, 162]. The aim of this project is to look at ‘ideal’
CRISPR distributions and appropriate acquisition strategies that might enable bacteria to optimize their probability of survival.
4.3 Analysis of Ideal CRISPR Distribution
The question of particular interest for our analysis is how should a bacteria (individual or population) distribute its CRISPR locus to maximize the probability of
survival? To address this question, we will start by making a few assumptions. First,
the population of viruses in the environment will be assumed to be fixed, and therefore the probability of encountering a given virus (pi ) is also fixed. In general viruses
77
may enter and leave an environment, and viruses can mutate rapidly, but for our
analysis, changes in the viral population will not be considered.
Additionally, we will assume that the probability of surviving an attack by a given
virus (Ti ) depends on the identity of the virus (i) and is a function of the number of
spacers (ki ) associated with that virus. Additionally, Ti is assumed to increase monotonically as a function of the number of spacers. Therefore, each virus may have a
different virulence, and more spacers associated with a given virus implies greater immunity. More spacers associated with a given virus corresponds to greater immunity
because viruses mutate frequently [166]. Therefore, if a CRISPR has multiple spacers
associated with a given virus, then it allows the bacteria a greater chance to match
the viral sequence even if a mutation event has occurred. Finally, we will assume that
the survival function, T (v, k), will be concave down with respect to the number of
CRISPRs, that is,
δ2T
(v, k) < 0
δk 2
(4.1)
Biologically, this means that increasing the number of spacers for a specific virus
increases immunity more rapidly if the virus only has a limited number of spacers
already associated with that virus, compared to the lower increase in immunity when
there are already numerous spacers associated with the virus.
4.3.1 Discrete Case: Lagrange Multipliers
Consider an environment with a fixed, unchanging, finite number of viruses (N )
with a (discrete) probability distribution, pi , indicating the probability of encountering the i-th virus. In addition, consider that a given bacteria has a fixed number of
CRISPR spacers (C) each of which are associated with at most one virus.
78
We denote this immunity function for the i-th virus by Ti (ki ) where ki is the
number of spacers in the CRISPR associated with the i-th virus. Then the probability
of surviving the next attack (S) is given by the sum over the probability of interacting
with any given virus multiplied by the likelihood of surviving such an attack. That
is,
S :=
N
X
pi Ti (ki )
(4.2)
i=1
Recall pi represents a probability distribution and, additionally, ki has the property
N
P
that
ki = C. We rescale Ti and ki such that
i=1
N
X
ki = 1.
(4.3)
i=1
We are interested to determine, for a given viral distribution {pi } and a set of immunity functions, Ti , can we find an optimal distribution for ki to maximize the
probability of survival as given by (4.2). The optimization of (4.2) subject to the
constraint, (4.3), represents a basic constrained multivariable optimization problem.
We can solve this problem using Lagrange multipliers to determine the optimal distribution, k [86, 148]. In particular, the solution is given implicitly by the system of
equations
pi Ti0 (ki ) + λ = 0
and
N
P
i=1
(4.4)
0(−1)
ki = 1. If Ti0 (x) is invertible with inverse Ti
(·), then we can solve for the
79
ki s explicitly as
0(−1)
Ti
ki =
−λ
pi
(4.5)
λ is computed by substituting (4.5) back into the constraint (4.3).
4.3.2 Continuous Case: Euler-Lagrange Equation
While there are a finite number of viruses, and therefore, the discrete case is
the most accurate representation of actual viral distributions, to make more general
statements, it will be useful to consider continuous probability distributions. For N
sufficiently large, a continuous distribution should be a reasonable approximation of
the discrete case. To this end, consider the continuous approximation of the same
problem. Given a fixed continuous probability distribution, p(v), and a fixed immunity function, T (v, k), can we find a continuous probability distribution for our
CRISPR spacers, k(v), that maximizes,
Zb
S(v) :=
p(v) T v, k(v) dv
(4.6)
a
The constraint that k(v) be a probability distribution can be explicitly written as
Zb
k(v) dv = 1
(4.7)
a
This problem represents a standard problem from calculus of variations with an
isoperimetric constraint. Therefore, if we assume T (v, k) is twice continuously differentiable with respect to k, then the general maximum of (4.6), if it exist, is achieved at
a function k(v) that satisfies the Euler-Lagrange equations [57]. That is, the solution
80
is given implicitly by,
p(v)
δT
v, k(v) − λ = 0
δk
(4.8)
where λ is chosen to satisfy the constraint that k(v) is a probability distribution,
(4.7).
Theorem 1. If
δ2 T
δk2
< 0, then k(v) satisfying (4.8) is a local maximum of (4.6).
Proof. Let f be a function satisfying (4.8). For small and v independent of , let
g (v) = f (v) + v
d2
{S
d2
v, g (v) }=0 < 0, then f represents a local maxi d2
mum. Therefore we now compute d
2 {S v, g (v) } =0 .
be a perturbation of f . If
Zb
2 d2
d
T
S v, g (v) = p(v) 2 dv < 0
d2
d =0
=0
(4.9)
a
The total derivative of T (v, k) is
dT
δT dx δT dk
δT dk
δT
=
+
=
=
v
d
δv d
δk d
δk d
δk
and the second derivative is
2
d2 T
2δ T
=
v
d2
δk 2
(4.10)
81
Given p(v)v 2 ≥ 0 for all v, if (4.1) is satisfied, then
Zb
2
d2
2δ T
S v, g (v) = p(v)v
dv < 0
d2
δk 2
=0
(4.11)
a
4.4 Examples
We now provide some specific examples for the optimal CRISPR distribution given
specific forms of the immunity function (T ) and particular viral distributions (p). Let
a and c be arbitrary, non-negative constants.
4.4.1 Linear Survival Function
Theorem 2. If T (k) = c k, then the maximum of (4.6), subject to (4.7), is given by
k(x) = δ(x − xpm ) where xpm is the the value of x where p(x) attains its maximum.
Proof. Let T (k) = c k. Then (4.6) is given by
Zb
p(v) T k(v) dv = c
a
Zb
p(v)k(v) dv
(4.12)
k(v) dv ≤ pmax
(4.13)
a
Notice that, for all k(x)
Zb
Zb
p(v)k(v) dv ≤ pmax
a
a
82
Select k(x) = δ(x − xpm ). Then
Zb
p(v)δ(x − xpm ) dv = p(xpm ) = pmax
(4.14)
a
thus proving the result.
Biologically, this implies if the survival function is linear, then optimal probability
of survival occurs when the entire CRISPR locus contains only spacers associated with
the virus which the cell is most likely to encounter.
4.4.2 T (k) = a k n + c
To ensure that T (k) is not convex, let 0 < n < 1. Additionally, since T (k)
represents a probability of survival, a + c ≤ 1.
A Discrete Example Consider the discrete example where we let a = 1, c = 0,
√
and n = 12 , then our survival function is T (k) = k (see Figure 4.2b).
1
Propability of Surving Attack
Probability of Attack
30%
20%
10%
0%
0
1
2
3
4
5
Viruses
(a)
6
7
8
9
0.75
0.5
0.25
0
0.0
0.2
0.4
0.6
0.8
1.0
Percentage of CRISPR
(b)
Figure 4.2: (A) The discrete viral distribution represents the probability of encountering one of ten viruses labeled {0,1,2, ..., 9 }. (B) The survival function is monotone,
1
concave and given by T (k) = k 2 .
83
CRISPR Viral Distribution (%)
40%
Viral Distribution
CRISPR Distribution
30%
20%
10%
0%
0
1
2
3
4
5
6
7
8
9
Virus
Figure 4.3: Using Lagrange multipliers, the optimal CRISPR distribution can be
computed from the viral distribution and the survival function. Notice that the
optimal CRISPR distribution is more concentrates on the high probability viruses
in the population. That is, the the CRISPR spacers are highly concentrated on the
viruses that are most common. We will show that true in general for this form of the
survival function T (k) = a k n + c for the Gaussian and exponential (continuous)
probability distributions.
If we have a discrete distribution of viral particles given by Figure 4.2a, then using
Lagrange multipliers, we can compute the optimal CRISPR distribution (see Figure
4.3).
General Continuous Case Recall that for the continuous case the maximum of
(4.6) is given by k(v) satisfying (4.8). Therefore, the optimal CRISPR distribution
for the continuous case given the survival function is of the form T (k) = a k n + c, is
given by
k(x) =
a n p(x)
λ
1
1−n
So if the viral probability distribution, p(x), is a Gaussian or exponential distribution, then k(x) is given by the same type of distribution with a different variance.
84
For example, if p(x) is a normal distribution with standard deviation σ, i.e.
x2
1
p(x) = √ e− 2σ2
σ 2π
then k(x) is also a normal distribution
k(x) =
2
1
√x
−
√ e 2(σ 1−n)2
(σ 1 − n) 2π
√
√
√
but with standard deviation σ 1 − n. Recall that 0 < n < 1, so 1 − n is well√
defined and 1 − n < 1. Therefore, in general, if p(x) is a normal distribution,
then k(x) is also a normal distribution but with smaller variance. Figure 4.4 shows
examples of different optimal CRISPR distributions for different values of n where
p(x) is given by the standard normal distribution with a standard deviation of 1
(shown in black).
Similarly, if p(x) is given by an exponential distribution with standard deviation
1
,
α
i.e.
p(x) = α e−α x
then
k(x) =
α
α
e− 1−n x
1−n
such that k(x) is still an exponential distribution, but with standard deviation
1−n
.
α
Therefore, again the CRISPR distribution has the same form as the viral distribution,
but with a smaller standard deviation.
85
1
1.0
Survival Probability
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.0
0.2
0.4
0.6
0.8
1.0
Percentange of CRISPR
n=0
n = 0.2
n = 0.4
n = 0.6
0.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
Ideal CRISPR Distribution
n = 0.8
n=1
Viral Distribution
(a)
n = 0.2
n = 0.4
n = 0.6
n = 0.8
n=1
(b)
1
Figure 4.4: (A) The different survival functions of the form T (k) = k n are plotted
for various values of n including the extreme cases of n = 1 (red) and n → ∞ which
is shown as the (unit) Heaviside step function centered at zero (black). (B) The
optimal CRISPR distributions corresponding to each of these functions are plotted in
the same corresponding color. The black distribution represents the viral distribution,
p(x), and also the optimal CRISPR distribution for the Heaviside step function. At
the other extreme, the optimal CRISPR distribution for the linear T is the Dirac
delta function (represented by the red circle) centered at the most common virus.
4.5 Discussion
We have shown that, at least for survival functions of the form T (k) = ak n + c
and viral populations given by either Gaussian or exponential distributions, that
the optimal CRISPR distribution is a probability distribution of the same type, but
with smaller standard deviation. This implies that to maximize the probability of
survival (depending some on the survival function of the individual cell) the bacteria
should focus most of its attention on those viruses most likely to occur. This is not
the CRISPR distribution that a bacteria is likely to acquire using ‘naive’ acquisition
strategies. If, for example, a cell is equally likely to acquire a spacer from every
virus that it encounters, then the CRISPR distribution should exactly mirror the
86
viral distribution. Even worse, if a bacteria is more likely to acquire a virus that it
does not recognize, then this strategy would lead to a CRISPR distribution with even
greater variance than that of the viral population.
For a bacteria to be able to acquire an optimal CRISPR spacer distribution, the
cell should choose to acquire spacers associated with viruses that are most common. In
order to acquire spacers against the most common viruses, the cell needs a mechanism
for sampling the viral population distribution. To some extent this is an impossible
task. However, a cell does already have some information about the viral distribution.
The cell’s own CRISPR locus acts as a form of memory of the viral attacks that the
cell has already encountered. If one assumes that the cell’s individual history provides
some insight into overall viral population, then an ideal acquisition strategy might
be for a cell to be more likely to acquire spacers against viruses already represent
in its CRISPR locus. Therefore, the phenomenon of ‘priming’ which causes a cell
to acquire spacers related to those already located in its CRISPR locus may in fact
be the ideal acquisition strategy for choosing spacers to create an optimal CRISPR
distribution and maximize the probability of survival.
87
MRI Data Analysis and Neurological Disease
5.1 Introduction
“Clinical-radiological paradox” is the term for poor correlations observed between
quantitative magnetic resonance imaging (MRI) measures and disability scales in
multiple sclerosis (MS) [6, 7]. The goal of this project is to use current knowledge of multiple sclerosis and mathematical tools to develop a MRI grading system
that correlates well with established disability scoring systems (expand disability status scale, Scripps neurological rating scale, symbol digit modalities test). We aim
to establish biologically-meaningful, categorical MRI elements characterized by high
signal-to-noise ratio with corresponding grades and combine these in a mathematically optimized linear model to provide a reliable, universal tool for measuring central
nervous system (CNS) tissue destruction. This is a collaborative project with Dr.
Bibi Bielekova of the National Institute of Neurological Disorders and Stroke at the
National Institute of Health and her lab leading to a paper currently submitted to
Annals of Clinical and Translational Neurology.
5.2 Background
Multiple sclerosis (MS) is degenerative, neurological, autoimmune disease characterized by the destruction of the myelin sheath of neurons by the body’s own immune
system forming lesions (damaged and inflamed tissue) on the brain and spinal cord
[8, 35, 36, 110]. The type and severity of the physiological symptoms depends on
the size and location of these lesions [35, 36]. The symptoms are highly varied ranging from the fairly innocuous (tingly skin) to extreme (full paralysis) [35]. Despite
the seemingly direct correlation between these symptoms and specific lesions and the
significant improvements in imaging, there is not, as of yet, a significant correlation
88
between MRI analysis and severity of disability [6, 41, 63]. The poor connection
between MRI analysis and severity of disability has lead to difficulty in providing a
universal measure of disease severity which might aid in the development of large-scale
clinical trials.
Different clinical scales have played a unifying role in multi-center clinical trials;
however, these scales are rarely documented in broad clinical practice [22, 134]. On
the other hand, magnetic resonance imaging (MRI) of the CNS is routinely performed
by virtually all neurologists in developed countries. Yet, in most neurological diseases,
including multiple sclerosis (MS), an unsatisfactory correlation between the extent of
brain damage measured by quantitative MRI (qMRI) metrics and clinical disability
is observed and termed the clinical-radiological paradox [7]. qMRI data are also
critically influenced by hardware, sequence parameters and post-processing methods.
Consequently, data generated by different groups are often not readily comparable
[30, 78].
Therefore, the purpose of this study was to develop and validate a simple scale
of CNS tissue destruction. First, our collaborators determined a set of biologicallymeaningful and disability-guided categories of MRI elements that we wished to consider along with a categorical grading system for each variable. Then we used mathematical tools to optimize the linear combination of these elements. We report the
construction of an MRI scale we named CAMRIS-CTD (Combinatorial Age-adjusted
MRI Scale of CNS Tissue Destruction) and its optimization/validation process.
5.3 Development of Novel MRI Scale
Our collaborators at the NIH devised a set of categories of MRI imaging that
reflect CNS tissue destruction along with a scoring scale for each category based on
published knowledge (Figures 5.1 and Table 5.1) [12, 16, 51–53, 153, 173].
89
90
Figure 5.1: Categories for analysis were chosen based on literature. Above demonstrates examples of how individual MRIs were scored for each category.
91
Then they retrospectively rated MRIs from 305 untreated subjects from the discovery cohort while being blinded to their prospectively acquired clinical scores. All
MRI elements were found to positively correlate as measured by any one of the widely
used clinical measures of disability (expand disability status scale [EDSS], Scripps
neurological rating scale [SNRS], symbol digit modalities test [SDMT]), with the exception of contrast-enhancing lesions (denoted [o] below) and qualitative description
of predominance of deep white matter lesions in supratentorial brain (denoted [g]).
5.3.1 Physical/Cognitive Disability Scales
We now describe in detail clinical disability test. Our goal was to develop a
combined MRI score was designed to corollate with these scores. The expanded
disability status scale (EDSS), which represents the standard scale for disability in
MS is a score ranging from 0 to 10 with the only allowed values being fractions
with base 2. The EDSS score is determined based on the level of disability in eight
functional systems, specifically: pyramidal, cerebellar, brainstem, sensory, bowel and
bladder, visual, cerebral, and other. Then a score is given based on the degree of
disability in one or a number of these categories ranging from mild to severe. A score
of five or less indicates that the patient is fully ambulatory without aid (maybe for
only a limited distance) while a score between 5 and 6.5 indicates that the patient is
ambulatory with aid. A patient with a score above a 6.5 is non-ambulatory while a
score of 8.5 indicates that the patient is fully bed-ridden.
The more universal Scripps neurological rating scale (SNRS) offers a greater range
of scores (100 compared to 20 for EDSS) with more linear behavior [145]. The Scripps
rating scale includes 22 categories ranging from mentation and mood to motor function in each individual hand and foot. In each category the patient is given a categorical rating (i.e. normal, mild disability, moderate disability, severe disability) with a
92
corresponding numerical value. These values are summed across all of the categories
to give the patient a score between -10 and 100 with lower scores corresponding to
higher levels of disability.
MS functional composite (MSFC), which consists of measures of ambulation (25
foot walk), fine motor function (9 hole peg test), and cognition (paced serial auditory addition test; PASAT) [34, 39]. Symbol digit modalities test (SDMT), which
has been proposed as more reliable cognitive scale than PASAT [146]. The SDMT
scale measures the time required for a patient to pair abstract symbols with specific
numbers that have been previously assigned. The test combines patient attention,
visual processing, memory, and reaction time with some fine motor processing.
5.3.2 MRI Grading: Categories/Severity
Our collaborators chose 15 categories of MRI elements that reflect CNS tissue
damage as demonstrated by the literature. Specifically, the scoring of the cerebrum
focused on T2 lesion load (T2LL; [a]), T1 black hole fraction (BHFr; [b]), cerebral
atrophy [c] and qualitative identification of periventricular (PV) lesions [d], juxtacortical/cortical (JC) lesions [e], involvement of corpus callosum (CC) [f], predominance
of deep white matter (WM) lesions [g], and presence of deep gray (GM) matter lesions
[h]. Evaluation of infratentorium comprised grading of lesion load (LL) and level of
atrophy for brainstem [i,l], cerebellum [j,m], and medulla and upper cervical spinal
cord (SC) [k,n]. Contrast-enhancing lesions (CELs; [o]) were quantified in reference
to pre-contrast T1- and T2-weighted scans (more details in figure 1 legend).
T2LL [a] was defined as number of lesions (larger than 3mm in diameter) showing
high signal on T2 weighted images (T2WI) or fluid-attenuated inversion recovery
(FLAIR) images in the supratentorial brain. The number of T2LL was categorized
as: 0 = None, 1 = 1 to 4 lesions, 2 = 5 to 10 lesions, 3 = more than 10 lesions, and
93
MRI Category
[a] T2 lesion load (T2LL)
[b] T1 black hole fraction
[c] Atrophy
[d] Periventricular (PV) lesions
[e] Juxtacoritcal/cortical (JC) lesions
[f] Corpus Callosum (CC) involvement
[g] Mostly deep white matter (WM) lesions
[h] Deep grey matter (GM) lesions
[i] Brainstem lesion load (LL)
[j] Cerebellum lesion load
Verbal Description
Category Score
None
0
Mild (<5 small lesions)
1
Moderate (5-10 lesions)
2
Severe (>10 lesions)
3
Severe and confluent
4
None
1
<50%
2
>50%
3
None
0
Mild
1
Moderate
2
Severe
3
Absent
0
Present
1
Absent
0
Present
1
Absent
0
Present
1
Absent
0
Present
1
Absent
0
Present
1
None
0
Mild (1 lesion)
1
Moderate (2-4) lesions
2
Severe (>4 lesions/confluent)
3
None
0
Mild (1 lesion)
1
Moderate (2-4) lesions
2
Severe (>4 lesions/confluent)
3
94
MRI Category
Verbal Description
[k] Medulla & Upper Cervical lesion load
[l] Brainstem atrophy
[m] Cerebellum atrophy
[c] Medulla & Upper Cervical atrophy
[i] Contrast-enhancing lesion load
Category Score
None
0
Mild (1 lesion)
1
Moderate (2-4) lesions
2
Severe (>4 lesions/confluent)
3
None
0
Mild
1
Moderate/Severe
2
None
0
Mild
1
Moderate/Severe
2
None
0
Mild
1
Moderate
2
Severe
3
None
0
Mild (1 small lesion)
0.5
Moderate (1-5) lesions
0.75
Severe (>5 lesions/confluent)
1
Table 5.1
4 = more than 10 and confluent lesions.
BHFr [b] was defined as the percentage of T2 lesions that have appearance of
black holes on T1WI. 1 = None (no black holes present), 2 = less than 50% of lesions
are black holes, and 3 = 50% or more lesions are black holes.
To assess cerebral atrophy [c], we considered both ventricular size and width
of cortical sulci, best appreciated on axial cuts through insular cortex or vertex.
Presented examples focus on difference in ventricular size. 0 = None (No visible
atrophy of ventricles or sulci), 1 = Mild (mild increase in ventricular size or visible
atrophy, but often only focal increase in sulci width), 2 = Moderate (considerable
95
enlargement of lateral ventricles, definite enlargement of the third ventricle or more
wide-spread broadening of the sulci) and 3 = Severe (concomitant severe atrophy of
ventricles and sulci that leads to diffuse reduction of visible brain tissue). Presence
of deep gray matter lesions (GML) [h] was evaluated in thalamus, lenticulate nuclei
and caudate nuclei (both head of the cause and tail) by using T2WI/FLAIR as well
as T1WI/MP- RAGE, although only FLAIR images are depicted. 0 = Absent, 1 =
Present.
Contrast-enhancing lesions (CEL; [o]) were identified on post-contrast T1WI in
reference to pre- contrast T1- and T2WI/FLAIR and quantified as 0 = None, 0.5 =
1 CEL, 0.75 = 2 to 5 CELs, 1 = more than 5 CELs.
Evaluation of infratentorial lesion load for brainstem [i], cerebellum [j], and medulla
and upper cervical spinal cord [k] were defined on T2WI/proton density images and
T2WI/STIR images (for CS lesions). Lesions were verified on T1WI/MP-RAGE. 0
= None, 1 = 1 lesion, 2 = 2 to 4 lesions, 3 = more than 4 lesions or large/confluent
lesions.
Level of atrophy of brainstem [l] and cerebellum [m], were evaluated on axial and
sagittal of T1WI/MP-RAGE images. For brainstem rating: 0 = None, 1 = Mild
(mild flattening of the pons with visible enlargement of 4th ventricle or widening of
the interpeduncular cistern with mild flattening of cerebral peduncles), 2 = Moderate/Severe (severe atrophy of the mesencephalon with small cerebral peduncles and
large interpeduncular cistern and definite flattening of pons with enlarged cisterna
pontis). For cerebellar rating: 0 = None, 1 = Mild (mild widening of cerebellar sulci
and narrowing of middle cerebellar peduncles), 2 = Moderate/Severe (clear widening
of cerebellar sulci, prominent atrophy of 4th ventricle and middle cerebellar peduncles) Atrophy of the medulla and upper cervical spinal cord [n] was evaluated on
sagittal and axial T2WI and T1WI/MP-RAGE and rated as 0 = None, 1 = Mild
96
(segmental atrophy at a single level) 2 = Moderate (mild diffuse or 1-3 focal atrophy
segments visible by naked eye, but with > 50% of thickness preserved compared to
non-affected area), 3 = Severe (diffusely atrophied spinal cord or > 3 focal atrophy
segments or < 50% of thickness preserved).
5.3.3 Construction of CAMRIS-CTD
We now describe the process for the construction of a composite score from the
MRI scoring categories, that correlates with the disability scores. First, our collaborators considered the simplest linear model, the composite MRI score (S1) as a sum
of the individual scores in each MRI category, with the two elements that showed
negative correlation with disability being subtracted:
S1 = a + b + c + d + e + f − g + h + i + j + k + l + m + n − o
This composite MRI score correlated with clinical measures (the Spearman correlation coefficients 0.64, -0.65, -0.56, and -0.54 for EDSS, SNRS, SDMT, and MSFC,
respectively).
5.4 Mathematical Optimization of CAMRIS-CTD
At this point of the process, our collaborators sought our help in the process of
designing the best possible composite score. We started by multiplying the scoring
of those elements with negative correlation by negative one so that all of the elements would have positive correlation with all of the disability scales tested (EDSS,
SNRS, SDMT). Then, the goal was to determine the best linear model between MRI
measurements and age as explanatory variables with the different disability scores
(Scripps NRS, EDSS, SDMT) as response variables using the least-squares method-
97
ology with the constraint that all coefficients must be of the same sign. We seek to
determine the coefficients αi that optimize the general linear model
R=
X
αi ui
where ui are individual measurements
ui ∈ {Age, a, b, c. . . . , −g, . . . , m, n, −o}
Let M to be the matrix of MRI category measurements with the rows representing
different patients and the columns being the different categories, and let d be the
vector of a given disability score for each patient (NRS, EDSS, or SDMT). Then we
seek to find the vector α = {αi } that minimizes
|M α − d|2
(for each disability score) subject to the constraint that αi ≥ 0 since each of the
individual elements were positively correlated with disability (after the sign of [o] and
[g] had been changed). This constrained least squares problem was solved using the
Matlab function, lsqnonneg.
Next, we sought to find a subset of the explanatory variables that explained the
explanatory variables as well, or better, than the full set. Since we cannot consider
every subset of 16 variables (216 combinations), we ordered the response variables
(MRI categories) by the following procedure. We preformed LOOCV (Leave-OneOut-Cross-Validation) of least-squares approximation. This method works by leaving
one data point out, fitting the least square linear model to the remaining data and
calculating the square of the difference between the predicted value for the removed
98
point and the actual value. We repeat this for the entire data set by removing, with
replacement, each point of the data set (of size n). We considered the collection of
n values that each coefficient of an explanatory variable attains during the LOOCV
process.
We then computed the mean, variance and coefficient of variation, CV (the standard deviation divided by the mean), of each such collection. We viewed the coefficients that have low CV as those which are better specified by the data, and those
with a large CV as not being specified well by the data. The measurements with
coefficients with high CV may be redundant for achieving a good fit. We therefore
ordered the explanatory variables by the CV of their coefficient from LOOCV procedure in descending order. This order was different for each of the three response
variables/scores. We used this ordering to construct nested sets of variables, where
the k-th set consisted of first k variables in the order. We then performed LOOCV
again on each of these sets of explanatory variables to compute the cross-validation
value (CVV) as the mean square error (MSE) from the square error for each subset
(of the n total). For example, if the ordering by CV from lowest to highest yields
an order of explanatory variables {Age, k, n, i, c, ...}, then we perform LOOCV on
classes of variables {Age}, then {Age, k}, then {Age, k, n}, etc. Finally, we plotted
the cross-validation value (CVV) as a function of the number of explanatory variables
to determine the model that achieves the minimal CVV, see Figure 5.2.
We noted that there was a broad minimum of CVV between 5 and 8 variables for
each score. We considered these to be the most informative variables, as including
more variables beyond this 8 actually worsens the cross-validation value. As expected,
there is a strong overlap between the sets of variables for NRS and EDSS measures,
and less overlap with SDMT measure. In the interest of determining a single set of
explanatory variables for the prediction of physical disability (NRS and EDSS), we
99
EDSS
NRS
SDMT
5
260
145
140
240
4.5
135
200
180
130
CVV (MSE)
CVV (MSE)
CVV (MSE)
220
4
3.5
125
120
115
160
110
3
140
105
2.5
120
0
2
4
6
8
10
12
Number of Response Variables
(a)
14
16
100
0
2
4
6
8
10
12
Number of Response Variables
(b)
14
16
0
2
4
6
8
10
12
14
16
Number of Response Variables
(c)
Figure 5.2: Plots of the Cross-Validation as a function of number of ordered set of
explanatory variables. The order of the variables is used as the x-axis. Note that T
represents Age. The first plot (A) shows the MSE for each consecutive model using the
ordering associated with the NRS score as described above. Notice that the minimum
occurs at five variables, vNRS:={Age, k, n, i, c}. The plots B and C represent the
same except for the EDSS and SDMT scores respectively. Notice that for the EDSS
score the minimum occurs at seven variables, vEDSS:={Age, k, n, o, i, m, c}, while
the minimum for the SDMT score occurs at six variables, vSDMT:={Age, c, a, i, k,
j}. Notably the vNRS is contained in vEDSS. Additionally, notice that {n} is part
of the vNRS and vEDSS, but not in vSDMT; at the same time {a, j} are in vSDMT,
but not in either vNRS or vEDSS.
considered the union of the variables for the best models associated with the NRS (c,
i, k, n, Age) and EDSS (c, i, k, m, n, o, Age) measures, where the best is determined
by each model’s minimum CVV. So the combined model includes the explanatory
variables, {c, i, k, m, n, o, Age}. Since we did not test every possible combination of
explanatory variables (> 65, 000 possibilities), our claim that this is the best model
is limited within this ordered set of variables.
After combining the two sets of explanatory variables, we determined the best
linear model between this combined set of explanatory variables and the NRS score
and EDSS score, individually, using least-squares methods. The same was done for
the explanatory variables associated with the SDMT model and the SDMT score.
This provided us with three linear models, two associated with the same explanatory
variables and a third with its own set of explanatory variables. The coefficients
100
associated with these models can be seen in Table 5.2. Additionally, estimates for
the accuracy of the fits and predictions, can be found in Table 5.3. In Figures 5.3
and 5.4, we show scatter plots for both estimates and predictions associated with the
linear models for the combined explanatory variables (i.e. for NRS and EDSS).
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Age
0
0
4.45
0
0
0
0
1.86
1.12
0
5.30
0
0.00
5.50
1.42
0.26
EDSS
0
0
0.20
0
0
0
0
0
0.43
0
0.85
0
0.36
0.64
0.25
0.04
SDMT
2.14
0
2.45
0
0
0
0
0.02
0.33
3.82
0
0
0
0
0
0.08
NRS
Table 5.2: This table represents the coefficients associate with the explanatory variables of a linear model fit to the labeled score. The zero scores represent variables that
were eliminated by the methods described above because they were determined to be
redundant or not well specified by the data. In particular, the combined (physical)
model is made of the explanatory variables {c, i, k, m, n, o, Age} and the cognitive
model is made up of the variables {a, c, h, i, j, Age}.
We also evaluated quadratic models. Although quadratic models improved correlations with disability (fitting dataset Pearson r = 0.89), subsequent validation
demonstrated that models that included all quadratic terms over-fitted data (testing
dataset Pearson r = 0.64) and were therefore abandoned.
Intriguingly, age ranked as the top variable in every optimized model, which our
collaborators considered strong support of the ‘age-penalty’ concept. Out of five remaining critical variables in EDSS and SNRS models (which overlapped significantly)
infratentorial MRI parameters were the strongest determinants of physical disability,
representing 4 out of 5 top MRI elements, followed by brain atrophy (Table 5.2). The
main difference between EDSS and SNRS resided in selecting cerebellum atrophy
[m] in the EDSS model, whereas deep GM LL [h] was selected in the SNRS model.
Because of the significant correlation between EDSS and SNRS (Spearman r = 0.93,
p < 0.0001) and the historical preference for EDSS, we selected a single mathematical model for ‘physical’ disability, consisting of the 6 highest ranking parameters for
EDSS (table 1) and named it CAMRIS-PDO (Physical Disability Optimized). In
101
10
100
9
90
8
80
7
EDSS
NRS
70
60
6
5
4
50
3
40
2
30
20
40
1
50
60
70
80
90
0
40
100
50
60
70
80
90
100
NRS projected
NRS projected
(a)
(b)
Figure 5.3: The combined model was fit to the NRS score. The first scatter plot
was produced to plot the NRS predicted score (based on the model) versus the actual
score. The closer the circle lies to the red line the better the prediction. The standard
deviation for this model was 11.35, so using the model and a patients MRI score the
NRS score can be estimated +/- 11.35 points with 68% confidence, and within +/22.7 points with 95% confidence. The other scatter plot shows the projected NRS
score against the EDSS score with a line fit through the plot.
10
100
9
90
8
80
6
EDSS
EDSS
7
5
4
70
60
3
2
50
1
0
0
2
4
EDSS projected
(a)
6
8
40
0
2
4
6
8
NRS projected
(b)
Figure 5.4: These scatter plots were formed similarly to the previous plot by plotting
the EDSS predicted score by the combined model versus the actual score (which the
model was fit too). Again, the closer the circle lies to the red line the better the
prediction.
contrast, different MRI parameters were selected by the model optimized for SDMT,
namely brain atrophy [c], supratentorial T2LL [a], brainstem LL [i], cerebellum LL
102
[j], and supratentorial deep GM lesions [h] (table 1; figure e-1B,). Thus, we named
this model CAMRIS-CDO (Cognitive Disability Optimized).
Table 5: The degree of correlation between composite MRI scores and clinical disability scales – DISCOVERY SET
EDSS
CAMRIS-CTD
MSFC
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
correlation
correlation
correlation
correlation
correlation
correlation
correlation
0.67
0.69
-0.69
-0.71
-0.58
-0.59
-0.58
-0.58
95% CI
0.60 to 0.73
0.63 to 0.75
-0.74 to -0.62
-0.76 to -0.65
-0.66 to -0.50
-0.67 to -0.50
-0.65 to -0.50
-0.65 to -0.50
R square
0.45
P value
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
0.75
0.75
0.76
0.76
0.54
0.54
0.56
0.60
0.70 to 0.80
0.69 to 0.79
0.70 to 0.80
0.71 to 0.81
0.45 to 0.62
0.45 to 0.62
0.47 to 0.63
0.52 to 0.67
r
CAMRIS-PDO
SDMT
correlation
r
95% CI
0.47
0.34
0.56
P value
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
-0.60
-0.61
0.63
0.65
0.65
0.65
0.54
0.55
-067 to -0.52
-0.68 to -0.53
0.55 to 0.69
0.57 to 0.71
0.57 to 0.71
0..57 to 0.71
0.46 to 0.62
0.47 to 0.63
95% CI
0.57
0.33
R square
r
CAMRIS-CDO
SNRS
0.29
0.39
0.31
R square
0.36
P value
< 0.0001
< 0.0001
< 0.0001
< 0.0001
< 0.0001
0.42
< 0.0001
< 0.0001
0.30
< 0.0001
305
305
305
305
266
266
292
292
Number
of subjects
Table 5.3: The CAMBRIS-PDO (Physical Disability Optimized) correlates highly
with the two physical disability scales: EDSS (ρ = 0.75) and NRS (ρ = 0.76). The
CAMBRIS-CDO (Cognitive Disability Optimized) correlates highly with the cognitive disability scale, SDMT (ρ = 0.65). These correlations are against the patient
groups used to train the model.
Mathematical modeling established that utilized clinical scales reflect only partially full disability, with some function(s) likely not captured. For the purpose of
biomarker discovery, our collaborators aimed to develop a universal MRI scale of CNS
tissue destruction that integrates all functional/disability elements. Therefore, they
took those elements from their model that were biologically meaningful (i.e. T1 BHFr,
which was shown to be an important determinant of CNS tissue destruction [3, 4,
12] and CELs, which may obscure underlying atrophy due to associated edema), and
combined it with information gained from the three mathematical models by omit-
103
ting the remaining MRI elements [d-g], which were found redundant by mathematical
models. The resulting model was named CAMRIS-CTD.
5.5 Validation of CAMRIS-CDO and CAMRIS-PDO
We validated the physical- or cognitive-disability mathematically optimized models, as well as the CAMRIS-CTD, with an independent cohort of 114 untreated subjects (not used for the optimization of the models). All three MRI scores show highly
statistically significant (p < 0.0001) Spearman correlation coefficients for EDSS,
SNRS, SDMT, and MSFC (Table 5.4) comparable to those calculated for the discovery cohort.
Table 5X: The degree of correlation between composite MRI scores and clinical disability scales – VALIDATION COHORT
EDSS
CAMRIS-CTD
SDMT
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
Pearson
Spearman
correlation
correlation
correlation
correlation
correlation
correlation
correlation
0.69
0.70
-0.74
-0.73
-0.45
-0.44
-0.44
-0.50
95% CI
0.57 to 0.77
059. to 0.79
-0.81 to -0.64
-0.81 to -0.63
-0.60 to -0.28
-0.60 to -0.25
-0.59 to -0.27
-0.64 to -0.34
R square
0.44
P value
< 0.0001
< 0.0001
< 0.0001
< 0.0001
0.71
0.71
0.75
0.73
0.60 to 0.79
0.60 to 0.79
0.66 to 0.82
0.63 to 0.81
< 0.0001
< 0.0001
0.54
0.21
0.20
< 0.0001
< 0.0001
0.52
0.49
95% CI
0.35 to 0.65
0..31 to 0.63
R square
0.27
P value
< 0.0001
< 0.0001
92
92
r
95% CI
R square
0.50
P value
< 0.0001
< 0.0001
< 0.0001
101
101
0.57
< 0.0001
r
CAMRIS-CDO
MSFC
correlation
r
CAMRIS-PDO
SNRS
Number of
114
114
114
114
subjects
Table 5.4: The models were validated against an independent patient cohort. The
CAMBRIS-PDO (Physical Disability Optimized) still demonstrated high correlation
with the two physical disability scales: EDSS (ρ = 0.71) and NRS (ρ = 0.75). The
CAMBRIS-CDO (Cognitive Disability Optimized) correlates highly with the cognitive disability scale, SDMT (ρ = 0.52).
104
For the research groups only interested in MS, we assessed performance of CAMRISCTD in an MS sub-cohort (N=277): we observed comparable, significant (adjusted
p < 0.0001) correlations with clinical measures (Spearman correlations of CAMRIS
PDO with EDSS/SNRS 0.73/0.75 and CAMRIS-CDO with SDMT 0.68) as in the
combined cohort.
To address potential influence of scanner/field/sequences, we re-analyzed correlations between the three CAMRIS models and disability scales for the subgroup of
subjects that were acquired on the 1.5 T scanner (3-5mm cuts, N=91) versus 3T
scanners (1mm3 resolution, N=105). In both sub-cohorts, we validated statistical
significance of correlations between CAMRIS-CTD and clinical scales, even though
the strength of correlations was lower for 1.5T cohort (Spearman coefficients between
CAMRIS-PDO and EDSS/SNRS 0.58/0.58 for 1.5T cohort and 0.82/0.85 for 3T cohort, for CAMRIS-CDO and SDMT r = 0.44 for 1.5T and r = 0.67 for 3T cohorts).
5.5.1 MRI Score vs Disability Score
As stated in the introduction, in order to facilitate multicentric translational studies, our goal was to devise a tool that can reliably estimate physical and cognitive
disability of patient cohorts based on retrospective analysis of acquired, clinical-grade
MRI images. Therefore, we compared discriminatory power of CAMRIS-predicted
clinical scores versus measured clinical scores in detecting significant differences between diagnostic subgroups. We observed that CAMRIS- predicted EDSS, SNRS and
SDMT differentiated tested diagnostic groups with analogous power as prospectivelyacquired clinical scales (Figure 5.5; analogous results were observed for discovery and
validations cohorts, so only combined analysis is presented).
Multiple sclerosis has several different phenotypes, however, we will focus on two
major phases of the disease. First, the relapsing-remitting (RRMS) phase is charac-
105
terized by an acute phase of the disease, lasting a few weeks to a few months, followed
by a healthy period of no clinical symptoms of the disease. The relapsing-remitting
phase can last for the life of the patient or can develop (over the course of years or
decades) into the secondary progressive phase (SPMS) of the disease, where clinical symptoms continuously worsen and patient progresses quickly toward disability
and death. Additionally, primary progressive (PPMS) differs from SPMS and RRMS
in that the patient never experiences periods without symptoms (even during early
phases of disease).
In search for the parameters that may identify or forecast development of progressive MS, we observed an intriguing dichotomy between CAMRIS-predicted and
clinically-measured physical disability: the difference between CAMRIS-projected
and measured EDSS and SNRS scores were consistently higher in RRMS in comparison to progressive MS subjects (Figure 5.5). In other words, RRMS patients
had generally lower physical disability in comparison to the amount of visible CNS
tissue destruction on MRI, whereas progressive MS subjects had consistently higher
disability in comparison to visible CNS tissue damage.
5.6 Discussion
The presented study originated from the need to objectively quantify CNS tissue
destruction in patients with neuroimmunological diseases for the purpose of biomarker
development and validation, especially in the context of international consortia such
as those being formed under the Progressive MS Alliance. While MRI of the CNS
tissue seemed to be the logical vehicle for this purpose, the use of different scanners
and sequences represented a formidable challenge. Based on the experience of our
collaborators with MRI analyses in natural history cohorts [12, 99] and in clinical
trials, [14, 18] we recognized that semi-quantitative MRI measures, such as number
106
Figure 3
Discovery
***
***
0
-20
-40
SP
M
S
S
PP
M
S
R
R
M
D
C
IS
/R
IS
O
IN
D
-60
S
S
S
20
SP
M
PP
M
R
M
R
C
IS
/R
IS
IN
D
-6
**
N
IN
-4
40
D
projected SNRS – actual SNRS
-2
SP
M
S
S
R
M
PP
M
S
R
C
IS
/R
IS
D
IN
D
O
N
IN
H
D
0
0
O
20
2
H
D
40
4
N
CAMRIS‐CTD
**
***
***
IN
D
projected EDSS – actual EDSS
***
*
6
H
***
60
Validation
***
***
**
20
0
-20
-40
S
S
SP
M
PP
M
S
M
R
R
IS
D
C
IS
/R
IN
O
IN
D
-60
D
S
S
SP
M
PP
M
S
M
R
IS
R
IS
/
C
R
D
-6
*
H
-4
40
N
-2
D
S
S
SP
M
PP
M
S
M
R
R
IS
D
/R
C
IS
D
IN
O
IN
N
H
D
0
0
IN
20
2
O
40
**
4
D
***
6
N
IN
projected EDSS – actual EDSS
CAMRIS‐CTD
*
H
***
projected SNRS – actual SNRS
**
60
Table 5.5: The discrepancy in CAMBRIS-CTD projected EDSS and SNRS scores and
actual EDSS and SNRS scores were plotted for different diagnostic groups. Interestingly we observed a strong dichotomy for a few diagnostic categories. In particular,
the difference between the projected and measured EDSS and SNRS scores were
consistently higher in RRMS in comparison to progressive MS subjects. This difference potentially represents an interesting area of research as similar levels of visible
CNS tissue destruction on MRI corresponds to different levels of disability for these
different patient groups.
of CELs or T2 lesions, have an excellent signal-to-noise ratio but limited correlation
with clinical outcomes. On the other hand, qMRI measures suffer from considerable
non-biological variability [37] and are affected by hardware (scanner/field strength,
coil), sequence parameters [111], and post-processing methods. At least part of qMRI
variability is due to movement, which is inevitable even in the most cooperative
subjects due to the long duration of the MRI in comparison to the small changes
107
that qMRI metrics can discern. Even when research groups remove MRIs due to
‘poor technical quality’, quantitative changes of censored MRI data are reliable across
several years but not in shorter intervals [3, 69, 80, 83, 133, 147].
Therefore, we hypothesized that thoughtful combination of several semi-quantitative
MRI measures may provide correlation with clinical outcomes that is comparable to
correlations obtained by qMRI markers [51, 96, 127, 137]. Using a grading system of
large changes (i.e. discernable by naked eye) for each selected MRI element limits its
sensitivity, but increases specificity of the grading (i.e. increases the signal-to-noise
ratio). Sensitivity of the CAMRIS-CTD is then re-established due to meaningful
combination of multiple non-redundant MRI elements.
CAMRIS-CTD is not a diagnostic scale; on the contrary, as a scale of CNS tissue
destruction, it should be applicable to more diseases than just MS. While MS MRI
studies [12, 16, 51–53, 153, 173] guided selection of semi-quantitative elements used
for mathematical modeling, we saw no reason to restrict patients to MS or to apply
limitations of thresholds used by MS diagnostic criteria [102, 122]. Instead, to promote broad utilization, simplicity became the decisive consideration. For example,
quantification of supratentorial T2LL beyond 10 lesions takes disproportionately more
of the rater’s effort in relation to benefit provided. In patients with supratentorial
T2LL above 10, the magnitude of brain atrophy is a more important determinant of
disability than presence of additional lesions, as confirmed by modeling. Thus, the
emphasis towards lower lesion counts provides discriminatory power for patients who
have not yet developed brain atrophy. Similarly, for difficult to assess MRI elements,
such as BHFr, we used a large rating step with proven utility [12]. We verified that
each included MRI element individually correlated with disability, and mathematical
modeling trimmed redundant elements in CAMRIS-CTD.
Modeling also established that different MRI elements predict physical versus cog-
108
nitive disability. In fact, we were impressed by how well the mathematical models
selected those anatomical structures that underlie physical versus cognitive disability
[81, 88, 124]. Modeling also confirmed that, likely through its effects on recovery from
tissue injury, age is a non-redundant determinant of disability. Functional recovery
(neuronal plasticity) likely underlies the disproportionate preservation of neurological
functions in comparison to accumulated CNS tissue damage observed in younger (i.e.
RRMS) patients [5]. Conversely, the lack of repair seems to be the major determinant of whether (or when) RRMS patients develop progressive disease (Figure 5.5).
This hypothesis can be tested by prospective follow-up of RRMS patients who have
high/low differences in CAMRIS-predicted scores versus clinically-measured disability
scores, for their future development of SPMS.
There are several limitations here. First, we selected semi-quantitative MRI elements and their rating steps a-priori, and different MRI markers or rating steps may
provide better performance. We expect that this study will spur further developmental work from interested research groups, which may test a broader combination of
MRI elements and potentially improve the scale, analogous to the improvements in
MS diagnostic criteria [122]. Nevertheless, we also believe the upper limit of correlation between MRI and clinical scales is around ρ = 0.8, because any neurological scale
is only an imperfect approximation of the true disability and most of the brain tissue
is ‘clinically silent’. Additionally, subclinical disability, which becomes apparent at
the extremes of functional requirements such as prolonged, intense activity, is not
captured by neurological examination. In view of these considerations, the current
version of CAMRIS-CTD predicts clinical disability exceptionally well.
Second, although semi-quantitative MRI markers may be less susceptible to scanner/sequence effects than qMRI measures, CAMRIS-CTD may still be affected by
technical (scanning) parameters. Fortunately, our comparison of results between 1.5T
109
and 3T cohorts indicates that while higher field strength and enhanced sequences increase performance of the CAMRIS-CTD, results obtained from older 1.5T cohorts
remain valid.
Finally, the semi-quantitative nature of CAMRIS-CTD may lead to inter-rater
differences. Analogous to the excellent standardization of EDSS, we addressed this
issue by providing a rating guide, with representative MRI images for each element
in Figures 5.1 and Table 5.1. The strength of independent validation (in terms of
statistical significance, but also the precision of regression slopes), performed on a
cohort three times smaller than the modeling cohort, provides unequivocal evidence
for the robustness of CAMRIS-CTD, if the rater(s) follow included guidelines.
In conclusion, CAMRIS-CTD represents a simple, universal, and robust MRI rating scale of CNS tissue destruction. The correlations observed in pilot and validation
cohorts, in MS-only subcohort and in subcohorts of patients scanned at 1.5 versus 3T
MRIs are fully comparable to those previously reported for qMRI measures, [52, 112,
127, 137] including composite qMRI scales [4, 96, 109]. Thus, we expect that as a tool
that can readily bridge previously-acquired patient cohorts to diverse translational research projects, CAMRIS-CTD will facilitate international multicenter studies, such
as those that are being formed under the Progressive MS Alliance.
110
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