The Application of Signaling Networks to Cancer Metastasis and
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Cellular Motility through the EGFR Pathway
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Submitted to the MIT Department of Mechanical Engineering in Partial Fulfillment of the
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June 2015
Massachusetts Iinstitute of Technology 2015. All rights reserved.
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MIT School of Engineering
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Linda G. Griffith
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Thesis Supervisor
Douglas A. Lauffenburger
Thesis Supervisor
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Chairman, Committee on Graduate Students
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Ranjeetha Bharath
S.B., Mechanical Engineering
Massachusetts Institute of Technology, 2013
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The Application of Signaling Networks to Cancer Metastasis and
Cellular Motility through the EGFR Pathway
by
Ranjeetha Bharath
S.B., Mechanical Engineering
Massachusetts Institute of Technology, 2013
Submitted to the MIT Department of Mechanical Engineering in Partial Fulfillment of the
Requirements for the Degree of
S.M. in Mechanical Engineering
at the
Massachusetts Institute of Technology
June 2015
Submitted on May 8, 2015
Abstract
This thesis explores the problem of cancer metastasis by analyzing the various downstream
components of the epidermal growth factor receptor (EGFR) pathway. This work develops a
mathematical model that consists of partial differential equations and signaling networks.
Analysis techniques for these nonlinear reaction-diffusion equations included a study of the
biological background and motivation, along with computational simulation of the various sets
of models developed. The modeling effort combined biochemical reaction-diffusion equations
for various species with mathematical descriptions of the mechanical machinery of the cell to
characterize the foundations of cell movement in response to stimuli. By quantifying and
qualifying the signaling networks and molecular pathways involved in cellular signaling and
linking intracellular signaling to the mechanical machinery of the cell, it is possible to quickly
check in silico the effects of changing various feedback parameters and signaling molecule
concentrations. By creating a model of this process, it is possible to perform rapid tests of
different pharmaceuticals on the biochemical and biomechanical pathways, in order to assess
how they would affect cell motility and cancer metastasis on a large scale.
2
Table of Contents
Abstract ...........................................................................................................................................
2
Table of Figures ..............................................................................................................................
4
Acknow ledgem ents.........................................................................................................................
7
Chapter 1: Introduction ...................................................................................................................
8
Chapter 2: Background .................................................................................................................
11
2.1 Small GTPases ........................................................................................................................
15
2.2 Epiderm al Growth Factor Receptor (EGFR) ......................................................................
17
2.3 Actin N etw ork.........................................................................................................................
19
2.4 Contractile M achinery and Integrins ...................................................................................
22
Chapter 3: M athem atical M odel and M ethods.........................................................................
24
3.1 M athem atical and Biochem ical Background .......................................................................
24
3.2 M odules and Descriptions......................................................................................................
28
3.3 Relational Analysis .................................................................................................................
41
3.4 Software and Com putational M odeling .............................................................................
42
3.5 M odeling Progression .............................................................................................................
46
3.6 Challenges to M odeling ..........................................................................................................
48
3.7 Models.....................................................................................................................................
49
Chapter 4: Results .........................................................................................................................
52
4.1 Cell Polarization......................................................................................................................
52
4.2 Param eterization to M atch Experim ental Data..................................................................
54
4.3 PIP 2 Concentration Test ..........................................................................................................
56
4.4 Initial Conditions and Boundary Conditions ......................................................................
57
4.5 EG F and M enai .....................................................................................................................
58
4.6 EGF-Induced Polarization ..................................................................................................
62
4.7 3D Results...............................................................................................................................
64
...............................................................................
72
Chapter 5: Applications and Future W ork .................................................................................
80
Chapter 6: Conclusion...................................................................................................................
82
References.....................................................................................................................................
83
4.8 Double-Peak in the Barbed
n
e
3
Table of Figures
Figure 1 (a) The process of cancer cells entering and exiting the bloodstream (Lee, 2007). (b)
Cellular mechanical components (Taylor, 2011).........................................................................
9
Figure 2. The three stages of cell crawling across a surface. (Cooper, 2000) .......................... 12
Figure 3. 3D Imaging of invadopodia (Albiges-Rizo, 2009)...................................................
12
Figure 4. Methods of cellular invasion in cancer metastasis and the key components involved
(N umberg, 20 11)...........................................................................................................................
13
Figure 5. Epithelial to mesenchymal transition (Kalluri, 2009). .............................................
14
Figure 6. Intravasation and extravasation (Reymond, 2013)...................................................
15
Figure 7. Small GTPase activation-inactivation process, mediated by GEFs (EtienneM anneville, 2002).........................................................................................................................
15
Figure 8. Spatial organization of RacI, Cdc42, and RhoA using biosensors (Machacek, 2009).16
Figure 9. Spatial organization of Rac, Rho, and CDC42 in a cell to improve modeling efforts
(Maree, 2006)................................................................................................................................
17
Figure 10. Key cellular receptors and selected pathways in metastasis (Ciardiello, 2008)......... 18
Figure 11. (a) Electron microscopy showing actin cytoskeleton (NIH) (b) highlighting
individual families of branching filaments in actin (Svitkina) ..................................................
19
Figure 12. Arp2/3 creating actin branching (Nurnberg, 2011) ................................................
20
Figure 13. (a) Normalized elastic modulus for actin (Chaudhuri, 2007). (b) Structures composed
of actin filam ents (Lodish, 2000)..............................................................................................
20
Figure 14. Fit to actin polymerized using model described above (Sept, 2001)...................... 22
Figure 15. Myosin proteins walk along the actin fibers. (Berg, 2002) ...................................
22
Figure 16. Integrins embedded in the plasma membrane, linking intracellular proteins with the
extracellular m atrix. (Cooper, The Cell. 2001).........................................................................
23
Figure 17. Michaelis-Menten kinetics; reaction rate plotted against substrate concentration
(B erg, 2002)..................................................................................................................................
24
Figure 18. Diffusion process depicted graphically; particles diffuse through random walks...... 26
Figure 19. Partial differential equations evaluated in ID vs 3D...............................................
27
Figure 20. Sample constant values from literature (Dawes, 2007)..........................................
28
Figure 21. Steps involved in MATLAB implementation of algorithms..................................
42
Figure 22. Sample finite element mesh for COMSOL implementation. ................................
43
Figure 23. Example input page for COMSOL implementation...............................................
44
Figure 24. Proposed MATLAB Graphical User Interface (GUI) ............................................
44
Figure 25. Simple network relating key species (Holmes, 2012). ...........................................
46
Figure 26. Different levels of coverage and intricacy in models. (a) (Maree, 2006) (b) (Holmes,
2012) (c) N etw ork used in this w ork. ......................................................................................
47
Figure 27. Control feedback to represent the black box that is being explored....................... 49
Figure 28. Species can be active in the cytosol region, membrane region, or both.................. 50
Figure 29. N etw ork used in this w ork......................................................................................
51
Figure 30. Cellular polarization diagram . ................................................................................
52
Figure 31. (a) Localization of GTPases (Mayor, 2010). (b-d) Simulation results showing cellular
polarization for Rho, Cdc42, and Rac........................................................................................
53
4
Figure 32. (a) Original Rae result (b) Desired model result shape [not actual result] (c) Rac
54
ELISA data (Talento) ) (d) Parameter match to fit data (Talento) ..........................................
Figure 33. (a) Experimentally matched simulation (b) Original simulation Results .............. 55
Figure 34. (a-d) Set of plots displaying early model results for stochastic initial conditions..... 56
Figure 35. Relationship between PIP2 hydrolysis and EGFR (Haugh, 1999).......................... 57
58
Figure 36. Sample view screen for COMSOL platform. ........................................................
Figure 37. Mena affects cellular mechanical machinery (Gertler, 2011). ................................ 59
Figure 38. Change in PIP2 concentration for the absence (a) or presence (b) of EGF
60
concentration from sim ulation results........................................................................................
Figure 39. Change in barbed end metric with PIP3 to Rac feedback parameter. ..................... 61
Figure 40. Parameter variation to fit Rac data trend from ELISA experiment......................... 61
Figure 41. (a) EGF induced cell polarization. (b) Pathway diagram highlighting P13K linkage.62
Figure 42. The effect of PIP 3 to Rac feedback on polarization using an EGF gradient stimulus. 63
Figure 43. Variety of early 3D modeling efforts. A, B, and C are COMSOL, D is Mathematica.
64
.......................................................................................................................................................
Figure 44. COMSOL implementation from first stages of 3D modeling process. ................... 65
Figure 45. (a) Rae and (b) Rho in 3D and ID for a rectangular prism geometry.................... 65
Figure 46. (a) Rae concentrations in 3D (b) Rho concentrations in 3D; present at both ends as
66
experim entally predicted...............................................................................................................
Figure 47. (a) Myosin phosphatase. Black arrows show ring-like formation. (b) Different
67
view ing angle................................................................................................................................
Figure 48. a-b. Barbed end time series (0 and 30 seconds). Insignificant change in concentration.
.......................................................................................................................................................
68
Figure 49. Predictive approach to cellular geometry. (a) Initial cell shape. (b) Output to
68
actomyosin network (Mak, 2014). (c) New cell shape. .............................................................
Figure 50. (a) PLC Gamma simulation result, spiking begins in center and localized to one side.
(b) PLC gamma stain in HeLa cell to show membrane ruffles (Santa Cruz Biotechnology). ..... 69
Figure 51. (a) P IP 3 . (b) PIP 2 .......................................................
. . . .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . .. . .
Figure 52. PIP 3 polarization without use of EGF stimulation to induce polarization; fails to
capture essential biological concepts ........................................................................................
Figure 53. Cdc42 stochastic initial conditions........................................................................
Figure 54. Stochastic initial conditions translated while moving downstream to ROCK. ..........
Figure 55. Rac with original diffusion coefficient..................................................................
Figure 56. Rae with increased diffusion coefficient (two orders of magnitude) ......................
Figure 57. The two key components of protrusion are shown. (a) The early peak based on
cofilin. (b) The late peak based on Arp2/3. (c) The sum of both as the barbed end protrusion
m etric . ...........................................................................................................................................
69
70
70
71
71
71
73
Figure 58. (a) WAVE, WASP, and Arp2/3 spatiotemporal plot with low EGF stimulation
(PI3Kinase pathway, the late pathway). X axis: Time 0 to 100 seconds; Y axis: Position across
20pm cell. (b) EGF increase with Pl3kinase increases protrusion and cellular polarization....... 74
Figure 59. Variation in barbed end metric in response to changing the feedback between Cdc42
and N -W A S P . ...............................................................................................................................
5
75
Figure 60. Variation in barbed end metric in response to changing the feedback between PIP 2
and N -W A SP . ...............................................................................................................................
75
Figure 61. (a) Barbed end assay for MTln3 cells (Mouneimne, 2004). (b) Model results for
varying PIP 2 to barbed end feedback. ........................................................................................
77
Figure 62. (a) Barbed end assay for MTln3 cells, control and PLC inhibitor (Mouneimne, 2004).
(b) Model results for PLC component and PLC inhibition. ....................................................
77
Figure 63. A ctin peaking..........................................................................................................
78
Figure 64. Capping rate and timescale analysis of first peak (normalized)............................. 78
Figure 65. The effect on the barbed end/actin parameter of increasing the model Mena parameter
through a mechanistic inhibition mechanism on capping and actin elongation. ....................... 79
Figure 66. Actin and myosin Brownian dynamics simulation (Mak, 2014)............................ 80
6
Acknowledgements
I would like to thank Professor Doug Lauffenburger and Professor Linda Griffith for their
incredible advice and support throughout this process. Without their invaluable guidance, none of
this work would be possible. I want to thank them for this amazing opportunity and for taking the
time to help me out with this.
I would also like to thank the National Institutes of Health for funding this work on cancer
metastasis.
In addition, I would also like to thank Professor Roger Kamm, Professor Muhammad Zaman, Dr.
Shannon Hughes, Dr. Michael Mak, Dr. Vivi Andasari, Dr. Fabian Spill, Ms. Suzanna Marie
Talento, and Dr. Steve Wasserman for their support.
Finally, I would like to thank my family and friends for their support throughout this process.
7
Chapter 1: Introduction
More than half a million Americans are expected to die each year from cancer; that is
approximately 1,600 people per day. After heart disease, cancer is the leading cause of death in
the United States. In fact, nearly one in four deaths is caused by cancer (ACS, 2014). Cancer
metastasis is the process by which cancer spreads from one location in the body to multiple
locations around the body. For example, a breast cancer tumor can potentially metastasize to the
brain, lungs, bone, and other locations. Once cancer has spread, it becomes much more difficult to
treat since it is no longer localized. Tumor metastasis accounts for approximately 90 percent of
suffering and death in cancer patients (ACS, 2014).
When cancer metastasizes, cells leave their location of origin at the primary tumor site and spread
to other parts of the body, traveling through the blood stream to get there. Cancer cells exert and
experience many forces from their microenvironment as they go through this process. The cell
must integrate various chemical and mechanical stimuli through its signaling network and then
respond accordingly. Understanding, quantifying, and eventually controlling this process could
potentially help with the development of effective therapeutics. On the way to a secondary tumor
site, cells often pass through various microenvironments, including the "stroma, the blood vessel
endothelium, the vascular system and the tissue at a secondary site" (Wirtz, 2011). Each
microenvironment exhibits a new set of challenges for the cell to overcome. For example, a
mammary tumor cell may encounter dense networks of collagen I and fibronectin, where
crosslinking enhances integrin signaling and bundling (Wirtz, 2011). These different
microenvironments provide a complicated array of interactions with the tumor cell as it
metastasizes.
Thus, the feedback pathways between signaling and movement are intricate and difficult to
characterize. In the process of modeling cancer metastasis, it becomes vital to identify key
pathways and make simplifications in order to mathematically capture the most important steps
that allow cancer cells to proliferate. It is helpful to understand and quantify metastasis in order to
discover and target potential therapeutic and diagnostic targets. When modeling the signaling and
motility pathways involved in metastasis, there are many challenges. For example, cells in threedimensional environments have features that are different from motility on flat surfaces, such as
pore size, fiber orientation, and structural components. In addition, the processes and linkages are
extremely complicated and involve many different species that each behave differently when
placed under various biological conditions. While it is useful and interesting to begin to understand
the entire process of cancer metastasis, it is even more vital to select the key system players to
model mathematically.
During cancer metastasis, cells break away from the tumor of origin, squeeze through the epithelial
wall of blood vessels, and move around the body. Figure 1a below shows the key components of
8
this process. Cancer cells leave the primary tumor site and invade the basement membrane, travel
through the bloodstream, and adhere to a distant site. In the event that a tumor cell successfully
implants at a new location and starts a secondary tumor, then the cancer has metastasized.
Metastasis poses significant clinical challenges for treating cancer, because when the cancer has
implanted at multiple sites throughout the body, it is much more difficult to locate and destroy.
Many cells may leave a primary tumor, but each individual cell faces a large number of barriers
before successfully implanting at a new tumor site. Cells must invade the basement membrane,
chew through the extra-cellular matrix (ECM), enter the blood stream, adhere at a new location,
and successfully implant and begin dividing again. Cells often use devices to get past the obstacles
they face. For example, matrix metalloproteases (MMPs) to chew through surrounding matrix.
Most importantly for this work, there are factors that can make a cell more motile. Thus, the goal
here is to identify and quantify the activity of these factors and their impact on cancer cell motility.
For example, the Menainv isoform of a protein Mena involved in cell motility is more effective at
driving cancer metastasis (Philippar, 2008). Once again, it is important to remember that
metastasis is a complicated process, and choosing the right places to start modeling it can begin to
yield insights about the nature of vital parts of this process.
a
btw
ECM
FRopodium
e
*0
e~
9OWWCoica
-a\ctinat
Rufi
Figure 1 (a) The process of cancer cells entering and exiting the bloodstream (Lee, 2007). (b)
Cellular mechanical components (Taylor, 2011).
9
Creating a model of the key parts of this process can make it possible to perform rapid tests of
different pharmaceutical linkages and pathways. This would allow the user to be able to assess
how changes would affect cell motility on a large scale. The set of models developed throughout
the course of this thesis are developed in a modular fashion with various interlinks between key
components. It is simple to remove or add to existing sets of equations and system components
by choosing which modules are being assessed and creating connections to new species. As new
pathways are discovered or existing pathways modified, the modular nature makes it simple for
the user to easily change the model. This process of trying to characterize the signaling networks
and molecular pathways involved in cellular signaling and the mechanical machinery of the cell
allows the user to try to quickly check the effects of changing various parameters and signaling
molecule concentrations, key factors in testing new pharmaceuticals. Namely, this model of
interconnected signaling pathways and cellular processes can provide a quick in silico analysis of
how changes in various protein concentrations or feedback strengths between system players can
affect overall function. The long term goal of this modeling effort is to provide a tool to test out
various drugs or effectors on a comprehensive three-dimensional model of a cell.
10
Chapter 2: Background
There are many biological processes involved in cancer cell metastasis. Delving into the details
underlying these biomechanical processes can help to frame the work presented here and put it
into the appropriate biomechanical context. Figure 1 B shows a cell and some of its key mechanical
components. A cell's shape is defined and modulated by its actin cytoskeleton. Actin in
combination with myosin serve as the most important components of cellular mechanical
machinery; these proteins work together to make a cell expand, contract, and move. As the cell
reacts to external stimuli in its environment, all the components of the mechanical machinery act
in concert to extend protrusions or contract in order retract a particular portion of the cell. Myosin
proteins, also labeled in Figure lB, form a superfamily of motor proteins that move along actin
filaments and hydrolyze adenosine triphosphate (ATP), helping the cell contract.
Actin is the most abundant protein in many eukaryotic cells; even in non-muscle cells, actin
composes 1-5% of cellular protein (Lodish, 2000). A human cell is composed of a cytosolic portion
within its plasma membrane, containing a nucleus, other organelles, and a mechanical structure
with components such as actin and myosin, which help give the cell structure and allow it to move.
This is obviously a very complicated and multi-faceted system. Focusing on specific intracellular
signaling molecules and how they impact the cell's mechanical machinery (primarily, actin and
myosin) can provide a valuable path to effectively modeling such a complex system.
When the cell moves along a flat surface, as shown in Figure 2 below, it extends protrusions, such
as filopodia or lamellipodia. By extending protrusions, grabbing onto the extracellular matrix, the
area that surrounds the cell, and contracting at the trailing edge, cells can move around their
environments. This process of cellular motility is at the center stage in cancer metastasis, because
cancer cells must leave their primary tumor site and move to a new location. There are three main
stages that coordinate cellular movement on a substrate.
This three-step process is shown graphically in Figure 2 below. First, the leading edge of the cell
extends forward in the direction of movement. Then, the part of the cell that extend attaches itself
to the substrate with focal adhesion kinases. Finally, the back of the cell retracts into the cell body
using myosin-mediated contraction. This three-step extension, attachment, and retraction process
is responsible for a cell's movement over a substrate. The cell moves by attaching itself to a
substrate and "pulling" itself along in the direction of persistence. In a three-dimensional
environment, however, the mechanics and pathways can get more complicated.
11
t
xtenslon of 1...d1ng edge
t
Attadimml to St1bstratum
t
Re1racdo11 ol llalllng odge
Figure 2. The three stages of cell crawling across a surface. (Cooper, 2000)
Figure 3. 3D Imaging of invadopodia (Albiges-Rizo, 2009).
The metastasizing tumor cell is subjected to a three-dimensional environment, with a different set
of mechanical restrictions. In Figure 2, the cell is shown moving across a flat substrate using a
simple three-step process. A cell trying to leave a tumor must often extend protrusions in all three
directions. It experiences a different set of mechanical factors in its environment than it would
while moving along a simple flat substrate.
12
Figure 3 shows an image of a cell protrusion in 3D. This particular protrusion is known as an
invadopodia. Notice the intricate shapes of cellular protrusion shown in Figure 3; these extensions
are not simply definable or predictable mathematically. Characterizing cellular movement in threedimensions requires the development of a new toolbox. This work serves as a step toward attaining
that goal, by synthesizing a set of models to represent key cellular pathways in I D and extending
them in a simplified manner to 3D.
Figure 4 shows a series of cellular protrusions in 3D that a cancer cell can use as it becomes more
motile and begins to invade the surrounding area. As cancer cells invade the basement membrane,
they are subjected to a variety of mechanical constraints that limit their motion. In response, cells
extend protrusions that come in many unique sizes and shapes.
Basement membrane
ExtrwAe utar matrix
tamellipodia Or
Protruding bleb
Invadopodia
pudopodi
ROCK
Myosin
C11WAVE
N-WP
Cortactin
Natwe Reviem Icancer
Figure 4. Methods of cellular invasion in cancer metastasis and the key components involved
(Numberg, 2011).
13
The precise set of reasons behind cancer metastasis are unknown, but there are many reasonable
hypotheses. It is known that cancerous tumors often exist in a hypoxic microenvironment, or an
environment where there is a lack of oxygen. "Prolonged hypoxia increases genomic instability,
genomic heterogeneity, and may act as a selective pressure for tumor cell variants.. .alter nonspecific stress responses, anaerobic metabolism, angiogenesis, tissue remodeling, and cell-cell
contacts" (Subarsky, 2003). Essentially, tumors provide cells with poor living environments and
mounting evidence suggests that this could potentially motivate the tumor cells to leave in search
of a better environment. A poor or hypoxic environment, along with external signaling factors, can
cause cells to begin the process of metastasis.
Cancer cells can undergo a process called epithelial-mesenchymal transition (EMT) that can
initiate metastasis. In EMT, a polarized epithelial cell undergoes biochemical changes that allow
it to exhibit a mesenchymal phenotype where it loses its polarity and adhesion to neighboring cells.
These phenotypic changes lead to enhanced invasiveness, migratory capacity, and resistance to
apoptosis, among other things (Kalluri, 2009). Figure 5 shows the process of epithelialmesenchymal transition, and lists important markers used at each step.
E-sh"
Oyp
Intmendakt ph-
N
a ells vuaonS
phenom"
MOsnc
j
~
T
S
Figure 5. Epithelial to mesenchymal transition (Kalluri, 2009).
After cancer cells leave the primary tumor location, they can enter the bloodstream in a process
called intravasation, and re-enter tissue in extravasation. Figure 6 depicts these two processes. A
variety of different mechanical processes are highly relevant to intravasation and extravasation.
The cell must first squeeze through a very tight junction in the endothelial wall of the bloodstream.
Then, it must survive the shear stress of the flow through the blood stream. Finally, the cell must
leave the bloodstream to adhere to tissue and squeeze back out of the bloodstream (Reymond,
2013). While in the bloodstream, the cells are subjected to shear-stresses from blood flow. In
addition, their size can be a limiting factor. Organelles within the cell also play a role in
intravasation and extravasation. For example, the nucleus must often bend and deform in order to
squeeze through the tight space available in order to enter the bloodstream. The nuclear size can
14
be a bottleneck, or limiting factor, as the nucleus is stiffer than other parts of the cytoplasm. In
fact, the nucleus can be 3-10 times stiffer than the surrounding cytoplasm (Lammerding, 2007),
acting as a further barrier to metastasis.
throughSi
ionod~y!
______________________C___
Figure 6. Intravasation and extravasation (Reymond, 2013)
2.1 Small GTPases
The identification of Rho GTPase proteins began in 1985 (Ridley, 2012). Small GTPases are a
family of hydrolase enzymes that can hydrolyze guanine triphosphate. Note that the acronym GTP
in "GTPase" stands for guanine triphosphate. The Rho GTPases are a family of signaling proteins.
They are small, approximately 21-30 kD (Yang, 2002), and are subfamily of the Ras superfamily.
The G proteins are molecular switches and cycle between an active and inactive state. There are
twenty Rho GTPases described for mammalian cells (Heasman, 2008). Guanine nucleotide
exchange factors catalyze the GTPase activation process. When active, GTPases interact with
target effector proteins to mediate signaling network processes. GTPases are found in all
eukaryotic organisms. The GTPases impact filopodia, actin stress fibers, and lamellipodia. Certain
linkages and combinations of the GTPases are key in 3D protrusions, and others are on flat surface
protrusions.
Plasma membrane
GTP
GDP
Effectors
Figure 7. Small GTPase activation-inactivation process, mediated by GEFs (EtienneManneville, 2002).
15
Guanine nucleotide exchange factors (GEFs) and GTPase activating proteins (GAPs) allow
GTPases to switch between the active and inactive state. GEFs catalyze the GDP to GTP exchange
to activate Rho proteins. On the other hand, GAPs allow the hydrolysis of GTP to GDP (EtienneManneville, 2002). This process is depicted in Figure 7 above. This sort of active to inactive state
switching plays an important and interesting role in the downstream signaling network for EGFR
and the activation of the cellular mechanical machinery, and a key role in the work presented here.
Characterizing the relationship between the GTPases and the downstream network species is key
to understanding and quantifying cellular motility. For example, a downstream effector of RhoA
is Rock], which helps cells contract. All three of the GTPases studied here are shown to be
activated at the leading edge of the cell (Machacek, 2009); see Figure 8 below. This presents a
complicated spatiotemporal relationship for which the exact dynamics is not completely
understood. A simplified representation with cell polarization (Rac and Cdc42 on one side, Rho
on the other) provides a more approachable and easier route to modeling basic behavior, shown in
Figure 9. Despite these useful simplifications, biosensor studies show that in the cell protrusion
process, all three GTPases are activated at the front of migrating cells (Machacek, 2009). The
results in 3D given below in this work show that Rho GTPase exists at both the leading and trailing
edge of the cell, while Rac GTPase is polarized toward at the leading edge, which is consistent
with this notion. The complicated dynamic spatiotemporal relationships between these species,
along with the dozens of other important species in metastasis, is very difficult to capture in a
model.
a0
Fzs
Figure 8. Spatial organization of Rac 1, Cdc42, and RhoA using biosensors (Machacek, 2009).
16
C&-42
RaR
Figure 9. Spatial organization of Rac, Rho, and CDC42 in a cell to improve modeling efforts
(Maree, 2006).
2.2 Epidermal Growth Factor Receptor (EGFR)
EGFR is a transmembrane receptor to which ten different ligands can bind selectively. Once
bound, the receptor forms a dimer that activates autophosphorylation through tyrosine kinase
activity. This process can trigger a series of pathways resulting in cancer-cell proliferation and the
activation of invasive or metastatic behavior (Ciardiello, 2008). There are two kinds of EGFR
antagonists that have been successfully tested in phase 3 clinical trials and are currently in use:
anti-EGFR monoclonal antibodies and small-molecule EGFR tyrosine kinase inhibitors
(Ciardiello, 2008).
Anti-EGFR monoclonal antibodies bind to the extracellular EGFR domain and block ligandinduced EGFR tyrosine kinase activation. The second, EGFR tyrosine kinase inhibitors (e.g.
gefitinib or erlotinib), compete with ATP to bind to the intracellular catalytic domain and inhibit
the autophosphorylation and downstream signaling. This method is effective because the EGF
signaling pathway leads to important key biological effects such as invasion, metastasis, and cell
proliferation. There exist on the order of 50,000 to 100,000 EGFR receptors per cell. However, in
cancerous cells, sometimes EGF receptors can be overexpressed, up to 1.7 to 1.9 times
(Zimmerman, 2006).
17
Epidermal growth factor is important for cell proliferation and differentiation. When it binds to
the cell surface receptor, it induces dimerization that activates the tyrosine kinase activity that
initiates a signal transduction cascade. Tyrosine kinase is a cellular enzyme useful to turn
biological functions on or off, by transferring a phosphate group from adenosine triphosphate to a
cell protein.
If protein kinases mutate to stay functionally "on," that can cause cell growth without proper
regulation, a key aspect of cancer. Figure 10 below shows different sets of transmembrane
receptors that are embedded inside the cellular plasma membrane. The EGFR receptor is shown in
relationship to the pathway that connects to cell proliferation, cell survival, invasion and
metastasis, and tumor-induced neoangiogenesis. Most importantly for the work here, EGFR is
strongly connected to the metastasis pathway in cells.
Figure 10. Key cellular receptors and selected pathways in metastasis (Ciardiello, 2008).
18
2.3 Actin Network
The cellular cytoskeleton is composed of actin filaments which link together to form an actin
network. Actin filaments can be capped or uncapped, allowing further polymerization. Actin and
myosin form the basic mechanical machinery that causes changes in cell shape and motility. Using
this mechanical machinery, the cell can examine its surroundings by pushing on the extracellular
matrix and restructuring itself to move around. As discussed above, motility is essential to cancer
metastasis.
When cancer progression or metastasis is initiated, the cell must leave its original tumor site, make
it through the tissue to the bloodstream, and get carried around the body until it implants at a
secondary tumor site. By studying the signal process that leads to the activation of a cell's
mechanical machinery, and coupling that with a description of how the mechanical machinery
works, it is possible to develop an understanding of how a cancer cell can biophysically
metastasize.
Globular actin, or G-actin, can polymerize to form F-actin filaments. The addition of ions like
magnesium, sodium, and potassium induces this polarization (Lodish, 2000). The F-actin filaments
can then bundle together to form larger structures that help provide structure in the cytoskeleton,
shown in Figure 1 Ia below. Figure II b shows the actin cytoskeleton and individual families of
branching actin filaments. As the actin filaments branch and extend, they give rise to forces within
the cytoplasm that can push the cell membrane outward to create cell movement. This branching
process is mediated by a protein Arp2/3, which is a fundamental key player in the mathematical
models presented in this work. Arp2/3 stimulates the formation of actin assembly and branching,
which is important for cell protrusion and metastasis. The function of this protein is shown in
Figure 12.
a
b
Figure 11. (a) Electron microscopy showing actin cytoskeleton (NIH) (b) highlighting
individual families of branching filaments in actin (Svitkina)
19
WCA domain
from NPF
C
F-actin
binding
F-actin
binding
Figure 12. Arp2/3 creating actin branching (Numberg, 2011)
a
b
I
O-
8-
-M
I
[
EI~
o50
E Efx -
500
q(Pa)
z
Emin
0.1
1
Figure 13. (a) Normalized elastic modulus for actin (Chaudhuri, 2007). (b) Structures composed
of actin filaments (Lodish, 2000).
20
The actin filaments join together to form actin bundles by using a fascin cross-linker (see Figure
13b); the scale of space between cross linkers is approximately 36 nanometers (Lodish, 2000).
There are several ways to model actin filaments in a cell, in order to mathematically characterize
the actin cytoskeleton.
Networks composed of actin exhibit interesting mechanical properties that provide insight into the
mechanical structure of a cell. Actin networks exhibit stress stiffening and softening. There are
three regimes of elasticity, as shown in Figure 13a. The first regime is the linear regime, followed
by stress stiffening and then softening. In fact, the stress softening of an actin network is reversible.
Both the reversible elastic behavior and the large elastic modulus suggest that actin network
architecture is good for high compressive loads. These measurements apply to lamellipodia
(Chaudhuri, 2007).
The thermodynamics and kinetics of actin nucleation are fundamental to the entire cytoskeletal
process. Analyzing these factors can predict the behavior for actin monomers over a range of
concentrations. Characterizing the nucleation-elongation process for actin can yield interesting
insights (Sept, 2001). Modeling protein-protein interactions using Brownian dynamics simulations
is an effective way to approach this process. The equation that forms the basis of Brownian
Dynamics simulations is given:
DAt
R (t + At) = R (t) + kTF + S
kT
1
Here, R gives the protein position, D is the diffusion constant, At is a time step, k is Boltzmann's
constant, and T is the temperature. S is a stochastic term to describe Brownian motion from solvent
interactions (Sept, 2001). "The Brownian dynamics simulations assume that the formation of each
protein- protein complex is controlled by diffusion and electrostatic interactions. We know this to
be the case for barbed-end actin polymerization, but here this assumption also applies to the
nucleation phase" (Sept, 2001).
Structure
DT
(A/ps)
DR
(rad2/ps)
Monomer
0.0103
1.23 X 10~
Dimer (a)
0.00798
Dimer (b)
Trimer (c) or (1)
0.00805
5.91 X
4.69 X
3.54 X
2.41 X
Tetramer (i)
0.00707
0.00638
10-6
10-6
10-6
10-6
Table 1. Translational and rotational diffusion coefficient for ellipsoid structures used in
monomer polymerization model (Sept, 2001).
21
1
0.81
j0.6
0.
0.4 I
LL
0
0
500
1000
Time (s)
1500
2000
Figure 14. Fit to actin polymerized using model described above (Sept, 2001).
2.4 Contractile Machinery and Integrins
The contractile machinery of a cell is extremely important to its ability to move. As a cell moves,
the trailing end has to contract in order to pull it back towards the center of motion. The key players
in contractility are ROCK, Rho, and Myosin.
yo~n~ir-v
(
ap
-4 0
AW®V
*4-066
Figure 15. Myosin proteins walk along the actin fibers. (Berg, 2002)
22
ROCK serves many functional purposes with in the cell. Not only is it important for cell
contraction, but is also very important for the assembly of the actin network, focal adhesions, and
intermediate filament mechanics. Rho is extremely important for cellular function. For this work,
the focus is on its function in mediating cellular contractility. Myosin serves as a molecular motor
that moves along the actin filaments in a cell, hydrolyzing ATP. The proteins within the myosin
family are key for muscle contraction and cellular motility. Using an exchange of ADP to ATP,
they move along the actin filament in order to create contractile forces within the cell. This process
is shown in Figure 15; it is important both for cellular movement along a flat substrate, and also
for cellular movement in a 3D environment, such as in cancer metastasis.
Cells use focal adhesions in order to attach themselves to the substrate they are moving through or
placed on. As a cell moves, there is a "tug of war" between the focal adhesions at each end of the
cell, with the side that has the stronger pull moving the cell forward. Figure 16 displays how actin
filaments attach to vinculin and talin, which connect to integrins that are attached inside the
extracellular matrix.
Figure 16. Integrins embedded in the plasma membrane, linking intracellular proteins with the
extracellular matrix. (Cooper, The Cell. 2001)
23
Chapter 3: Mathematical Model and Methods
3.1 Mathematical and Biochemical Background
There are multiple types of reaction mechanisms that occur within the cell. Describing the kinetics
to model these reactions is important to developing a model of metastasis and the signaling
networks it employs.
Michaelis-Menten kinetics are often used to describe enzyme reaction kinetics. Assuming the
following reaction:
E +S
ES ->
E+P
(2)
Then the reaction kinetics can be described as:
d[P]
Vmax[S]
dt
Km+[S]
The rate of the reaction increases with the concentration of the substrate, eventually plateauing at
Vmax, as shown in the plot below in Figure 17:
Subetrte concentration IS] -+
Figure 17. Michaelis-Menten kinetics; reaction rate plotted against substrate concentration
(Berg, 2002).
24
Another important type of reaction kinetics is the Hill function, which describes the binding of a
ligand, accounting for the presence of other ligands. The Hill equation is given as:
=
[L]"
K + [L]n
(4)
This describes the fraction of bound ligand sites over total ligand sites as a function of the unbound
ligand concentration and the dissociation constant, K.
A key element here is n, which is the Hill coefficient. If n>1, the interaction is characterized as
positively cooperative binding, where a bound ligand increases the affinity for other molecules.
For n<l, this is negatively cooperative binding, where binding decreases affinity. In the n=1 or
non-cooperative case, the enzyme affinity to ligand is independent of whether the others are bound.
Reaction rates can be of zero-order. This means that the reaction rate is independent of the reactant
concentrations. There is a simple rate law to describe this sort of reaction:
d[A] = k
dt
(5)
A first-order reaction's rate depends on the concentration of one reactant, and is zero-order for the
other reactants involved.
d[A](6
= k[A]
dt
(6)
Progressively increasing to the nth-order reaction, the equation becomes:
d[A] = k[A]
(7)
Using combinations of these reaction kinetics, it is possible to produce sets of equations called
reaction-diffusion equations that then describe the mechanistic interactions in a physical system.
Diffusion is a key element of this process, shown graphically in Figure 18. Molecules can diffuse
through a cell and spread through the cytoplasm. Diffusion is a slower process than active transport
of molecules. For example, molecules must be actively transported to the end of long neurons
because a diffusive process would be too slow for them to reach the end in time to conduct cellular
processes. Nevertheless, for the case of metastasis, cellular diffusion of molecules like Rac, Cdc42,
and Rho are important to model. Diffusion is the process by which particles will move from a
region of high concentration to that of low concentration. Brownian motion and random
fluctuations cause the particles to engage in a random walk due to thermal energy. An expression
for the diffusion constant is given below:
RT
D =6MrN
25
(8)
.
*
Here, R is the ideal gas constant, T is the temperature, il Is the viscosity, r is the particle radius,
and N is Avogadro's number.
Figure 18. Diffusion process depicted graphically; particles diffuse through random walks.
Differential equations can be used to describe a system mathematically. The structure of a
differential equation provides linkages between different variables and their derivatives. In the
case of this work, the time derivatives of a set of state variables are related to the first and second
spatial derivatives along with other source terms.
Partial differential equations are a class of differential equations that have multiple independent
variables. For example, a differential equation in space and time would be a partial differential
equation due to the independent variables to describe space and time, which could be x,y,z,t in
rectangular coordinates. Analyzing cellular behavior thoroughly requires a 3D description because
metastasizing cells in a cancer have important behavior in all three spatial dimensions and time.
While a flattened ID and time perspective can lend enormous insight into what is happening,
progress toward a 3D description is extremely important for developing a complete description of
cancer metastasis and signaling networks. However, partial differential equations in three
dimensions and time, a total of four independent variables, are complex and difficult to interpret.
For example, examine a reaction-diffusion equation for a ID and time case (Eqn. 9), and contrast
it with a 3D and time case (Eqns. 10- 11).
26
a2 A
aA
= DV 2 A + S
A
-D
at
(10)
/a 2 A
a 2A
2
(x
ay 2
az 2
-+-++
2
A
+S
(1
The 1 D case provides the solution on a line of space as shown below, where the 3D case provides
the solution on a generalized 3D shape (Figure 19b). As shown in figure 19a, in the ID and time
case, there is a solution for each dependent variable that varies with space for a single time
snapshot. In the case of this work, this would be the concentration of a particular species. In the
3D case, the solution is given at every point, with a profile for each of the three spatial dimensions
at every time slice. Figure 19b below shows a particular time slice and a family of plots, while
Figure 19a shows the ID case.
a
Figure 19. Partial differential equations evaluated in I D vs 3D.
27
3.2 Modules and Descriptions
Timescales are important for cellular function, because the order of various molecular events
determines what kind of action will occur on a cellular and organism level. For example, cellular
protrusions are governed by a set of molecular events that occur in a particular order, with various
events occurring at different times. One molecule's concentration might peak at a certain time,
which causes a cascade of a different molecule, and then the pathways engages the mechanical
machinery of the cell in a particular location where the peak occurred.
Trying to map out these timescales hinges on the values of constants within the model. There are
many unknown constants, but this can be accounted for using a parameter variation or sensitivity
analysis. The results for some of these sensitivity analyses are shown below in Chapter 4. The
constant parameters within the model (of which there have been hundreds) are variable depending
on the particular cell and experiment being performed. Many of the parameters are available in
literature, and those which were not available were estimated using parameter variation analysis.
Sample parameters are given in Figure 20. Selected parameter variation plots are given in Chapter
4.
k2,
kP13K
PIP, to PIP2 baseline conversion rate (by P15K).
PIP 2 to PIP, conversion rate.
PIP2 to PIP3 baseline conversion rate (by P13K).
kpTEN
PIP3 to PIP2 baseline conversion rate (by PTEN).
D,
PI diffusion rate.
kp,,K
0.84 s'
1.4 s0.0072 s~
4.3 s-
0.5-5 im2 S I
Figure 20. Sample constant values from literature (Dawes, 2007).
Different versions of the models developed have different equations and mechanisms to represent
different processes. Some are more or less accurate, or developed for different biological purposes.
Note that the set of equations presented here does in no way represent a full description of all
models developed; that would be too large to include here. This is a collection of some of the most
important biochemical pathways and suggested mechanisms. The annotations for each presented
equation are given in tables below, with the term numbering corresponding to the terms listed in
the right side of the equation. Every equation is presented in differential form with the time
derivative given as a function of the spatial derivatives and species state. Note that the key goal in
this work is to develop an effective means for relational analysis; the units of various terms are
less important than the relational trends and feedback mechanisms, which are the focus here.
28
Module I: Small GTPases
Small GTPases are an important part of the cellular networks involved in intracellular signaling
and motility. They play a key role in cell polarization, with concentration gradients and localization
they create being vital for the reorganization of the actin cytoskeleton. The equations for the small
GTPase module are adapted from (Holmes, 2012). This first equation is provided for active Rac,
a GTPase. The equation explanations are given as an approximate example of possible
mechanisms at work in the complex network of interactions.
(12)
aRac
a 2 Rac
Rac,
42
1
at -x 2 + [PP3-RacP 3 + PCd42-RacCdC ) + aRac + S] RaCT ~ Rac-RacRac
Explanation
Term
1
Rac is assumed to diffuse through the cytosol
2
The reaction between PIP3 and Rac is assumed to be first order and proportional to the
concentration of PIP 3 but also the ratio of Inactive Rac to Active Rac
3
A first order reaction between Cdc42 and Rac, tempered by the ratio of Inactive Rac to
Active Rac
4
Constant Term
5
Gradient: Not used in EGF stimulation based model
6
Degradation of Rac
Table 2. Description of terms for active Rac.
The following equation describes the inactive Rac species, which is described by cycling between
the two states.
(13)
dRac1 a
t
Term
a 2 Rac a
=Da2 21 n [P3-RacP 1 P3 +
PCdc42-RacCdC4 2 )
x
+
l R]
Rac1 a
+ IRac-RacRac
RaCTot
Explanation
1
Inactive Rae is assumed to diffuse through the cytosol
2
The reaction between PIP3 and Rac is assumed to be first order and proportional to the
concentration of PIP3 but also the ratio of Inactive Rae to Active Rac. The effect is
opposite for Active Rac.
3
A first order reaction between Cdc42 and Rac, tempered by the ratio of Inactive Rac to
Active Rac. Effect opposite for Active Rac
4
Constant Term
Degradation of Rac yields an increase in inactive Rac
5
Table 3. Description of terms for inactive Rac.
29
This equation describes the active Rho GTPase species.
aRho
a2 Rho
IRho
Rholn
-t = D
+ +
racn Rho
at
Term
(14)
IoRho-RhoRho
Explanation
1
Rho is assumed to diffuse through the cytosol
2
Unidirectional signaling from Rac to Rho. Rae is assumed to antagonize and inhibit Rho.
Degradation of Rho
3
Table 4. Description of terms for active Rho.
ORho
at
=
a 2 Rho
I Rho
+xa
-
Rho1 n
Rhoot + PRho-RhoRho
(15)
________
a22)
Term
1
Explanation
Inactive Rho is assumed to diffuse through the cytosol
2
Unidirectional signaling from Rac to Rho. Rac is assumed to antagonize and inhibit Rho.
The effect is the opposite as for Rho (Active).
3
Degradation of Rho yields an increase in Inactive Rho
Table 5. Description of terms for inactive Rho.
Cdc42
at(+t = D
Term
[
a 2 cdc42
+
MCc42
Cdc42 1 n
a) rho aCdc42Tt
Explanation
1
Cdc42 diffuses through the cytosol
2
Inhibition of Cdc42 by Rho
Degradation/conversion to inactive Cdc42
3
Table 6. Description of terms for active Cdc42.
30
Iac
42
-cdc4
2
Cdc4 2
(16)
at
= D
a2
Cdc42
aX 2
-
aCdc42n
rCdC42
]
Cdc42 T1
(1+
+
scdc
42
-cdc
42
Cdc 4 2
(17)
Cdc42T t
Explanation
Term
1
Diffusion
2
3
Opposite effect than Cdc42
Conversion from Cdc42
Table 7. Description of terms for inactive Cdc42.
Module II: Phosphoinositides & PLC Pathway
There are two key overarching pathways that lead to protrusion within a cell; here they are denoted
as the cofilin pathway and the arp2/3 pathway. Both are responsible for the development at barbed
ends, and different upstream factors are responsible for each, while there is some overlap. The
phosphoinositide and PLC pathway module consists of a wide variety of system players that are
involved in the cofilin pathway and the Arp2/3 pathway of protrusion.
The following equations provided in Module II were adapted, integrated, and expanded from the
following sources: (Holmes, 2012), (Haugh, 1999), and alternative selections from (Tania, 2011).
The role of species like the phosphoinositides in cell motility is complicated and multi-faceted. A
variety of base models needs to be integrated together in order to fit in its different pathways
throughout the cell. The equation for species PIPi (Phosphatidylinositol 4-phosphate) is given here.
This species interacts with other phosphoinositides and this interaction is Rac-mediated.
(18)
a2PIP
2PIP, 1
t
D
y
+
pip 1
- Ppip'-pip,
+
1
Rac
+
a
kp15 K
Term Explanation
1
Diffusion through cytosol
2
Constant Term
3
Degradation term
4
Conversion to PIP2 dependent on Rac
5
Conversion from PIP2
Table 8. Description of terms for Phosphatidylinositol 4-phosphate.
31
PIP1 + @PP
2
-P
1
PIP 2
A potential overarching equation to describe PIP 2 (Phosphatidylinositol (4,5)-biphosphate) is given
here as:
(19)
aPIP2
at
PIP 2
+
1
R)k
1 +
Rho
RhoTotkPEPI
The term: -(k_
(Tania, 2011):
Rac
Rac -)
D a 2 pIp2
2
IP 2 =D a
8xRaCTot
3
PIP -
1
+kP-(k
+
RaCTot
kpI 3 KPIP2
+kL+kE)PP+
I2
+ kPLC + kCE)PP 2 (Haugh, 1999) can also be described in the form used in
-dhyd
(PLC
- PLCrest PIP 2
PLCrest
Both adaptions might serve to describe the same physical phenomena, the PIP2 -PLC connection.
Term
Explanation
1
Diffusion
2
3
Degradation of PIP2
Conversion from PIPi, influenced by Rac, proportional to PIPi concentration
4
Conversion to PIP 3 , influenced by Rac, proportional to PIP 2 concentration
5
Conversion from PIP 3, influenced by Rho and proportional to PIP 3 concentration
6
7
Conversion from PI to PIP 2, away from PIP 2 to PI to describe the PLC gamma PIP 2 cycle
Term to include constant in kinetics
Table 9. Description of terms for Phosphatidylinositol (4,5)-biphosphate.
PIP 3 (Phosphatidylinositol (3,4,5)-triphosphate)
(20)
dPIP3
at
Term
= D
a 2 PIP2
ax2
2
/
+ 1 +
Rac
RacTot
Rho
'~(
f(EGF, kPI 3 K)PIP 2 -
1 +
RhoTot)
kPTENPIP 3
Explanation
1
Diffusion
2
Conversion from PIP 2, proportional to Rac
3
This term, using the function of EGF and Pl3Kinase, brings in the EGF interaction to
amplify this pathway. In some versions of this model, the EGF interaction is simulated
mathematically with a Hill function.
4
Conversion to PIP2, proportional to Rho
Table 10. Description of terms for Phosphatidylinositol (3,4,5)-triphosphate.
32
This is the equation for Phosphatidylinositol (PI), a species involved in the phospholipase-C
pathway (Haugh, 1999).
OPI
at
=
P1(O) =
SrPP(k-1 + k
k(kgL+
kkPLC
Term
(21)
rp*I, + k_ 1 PIP2 - k*PI
+E)
k)
E)
Explanation
1
Constant Term
2
3
Conversion from PIP 2
Degradation of PI
4
Initial Condition
Table 11. Description of terms for Phosphatidylinositol. (Asterisks indicate dependence on
receptor activity; zero superscripts indicate absence of receptor activity values) (Haugh, 1999)
Inositol Phosphate (IP) (Haugh, 1999)
IP
t=
Term
k*PC
2
(22)
+ PITP/PLC
Explanation
1
Conversion from PIP 2 to IP
2
Constant Term
Table 12. Description of terms for Inositol Phosphate.
k+ = k(1 - Di + X+*)
Gi + Ki + fa* -
(Fai+ Ki + fa*)2
- 4aifa*
PLC-yI is an important system player which mediates cell motility. It plays a large role in
mediating the EGFR pathway linking it to the downstream factors that lead to cell movement and
potentially metastasis.
33
dPLC
dt = Stimulus + 'pIc - dpicPLC
(23)
Stimulus = ISO(H(t - ton) - H(t - toff))
Term
Explanation
1
Mimics EGF stimulation
2
Constant Term
3
Degradation of PLC
Table 13. Description of terms for PLC-yl.
This form is presented in Tania et al. 2011. It uses a stimulus function in PLC to represent EGF
stimulation, which is extremely important.
It can be possible to alternatively represent this pathway as a function of an EGF species.
PLC-yl (Alternative-EGF Dependent, not Stimulus Dependent), and PLC as a Hill function
dependent on EGF.
dPLC
dt=
dt
EGF-PLCEGF + 'pic
- dpicPLC
PLC = f(EGF stimulus gradient,n, K)
Term
Explanation
1
Linkage between EGF stimulation and PLC
2
Constant Term
3
Degradation of PLC
Table 14. Alternate description of terms for PLC-yl.
34
(24)
Module III: Contractile Machinery
Equations 26-27 adapted from citation: (Kaneko-Kawano, 2012)
Myosin-Phosphatase
dMP
dt
kcat(MP)(MPtot - MP)
Kml + (MPtot - MP)
kcat 2 (ROCK)(MP) + ki(MPt0 t
-
Mp) (25)
Km 2 + MP
Term Explanation
1
Relating the myosin phosphatase concentration and inactive MP (this form of MP
contains a phosphorylated MYPTI) using Michaelis-Menten kinetics
2
Relating ROCK (active form of Rho-kinase) concentrations and MP using MichaelisMenten kinetics
3
Describes the activation by other pathways
Table 15. Description of terms for Myosin-Phosphatase.
Phosphorylation of Myosin Light Chain
(26)
dpMLC
dt
kcat 3 (ROCK)(MLCtot - pMLC)
Km 3 + (MLCtot - pMLC)
kcat 4 (MP)(pMLC)
Km 4 + pMLC
Term Explanation
1
ROCK directly influences the conversion to pMLC (Michaelis-Menten kinetics)
2
MP increases the reaction from pMLC to MLC
3
Describes the activation by other pathways
Table 16. Description of terms for Phosphorylation of Myosin Light Chain.
Linking together the above equations with the GTPase Module:
aRock
at
a2 Rock
Sck = D X+@Rho-RockRho - kock-RockRock
Term Explanation
1
Diffusion
2
Feedback from Rho to ROCK
3
Degradation of ROCK
Table 17. Description of terms for ROCK.
35
(27)
Module IV: Cofilin
Equations 28-32 adapted and integrated from: (Tania, 2011)
Cofilin-PIP2
dPC
Term
PIP2
= k
C - d 2 PC
dt
pip2 PI2,rest P
2C
-
dhyd
PLC - PLCrest
y
PLCrest
PLCrest
p
(28)
Explanation
I
Dependence on phosphorylated cofilin
2
Degradation of PC
3
Describes the activation by other pathways
Table 18. Description of terms for Cofilin-PIP2.
Active Cofilin
(29)
= dpcPC + dhyd PLC
dt
Term
-
=dCCPLCre
PLCrest PC - k'nFCa + koffCf kmpCa + kpmCp
)
m
5
Explanation
1
PIP2 bound cofilin changes to active cofilin
2
Change from PIP 2-bound cofilin to active cofilin
3
Degradation tempered by F
4
F-actin bound cofilin to active cofilin
5
Transitioning to phosphorylated cofilin
6
Phosphorylated cofilin changing to active cofilin
Table 19. Description of terms for Active Cofilin.
F-Actin Bound Cofilin
= k' nFCa - koffCf - Fsev(Cf)
Fsev(Cf) = ksevCfrest
Term
Cf
C,
f, rest
Explanation
1
Active cofilin to f-actin bound cofilin
2
Degradation
3
Severing of actin bound cofilin
Table 20. Description of terms for F-Actin Bound Cofilin.
36
"-sev
(30)
G-Actin Bound Cofilin
dCM = Fsev(Cf) kmp Cm + kpmCp
Term
(31)
Explanation
1
Severing of actin bound cofilin
2
Transition to phosphorylated cofilin
3
Phosphorylated cofilin to G-actin bound cofilin
Table 21. Description of terms for G-Actin Bound Cofilin.
Phosphorylated Cofilin
dCP
=kmp(Ca+Cm
Term
-2kpm
p -k
pip2
PIP2
PIP2,rest C
(32)
Explanation
1
Transition from active cofilin and G-actin bound cofilin
2
Leaving phosphorylated cofilin state
3
Leaving proportional to PIP2 concentration ratio
Table 22. Description of terms for Phosphorylated Cofilin.
A simplified model for cofilin is developed in this work, capturing the key PLC linkage and
degradation.
aCof /
Rac
t=11
+ Rat-) kpI3 PIP2
at
Ractot
Term
1
-
Pcof-cofCOf
Explanation
Feedback from PIP 2 - Cofilin pooling breakdown
2
Degradation and leaving separated state; lumped parameter description
Table 23. Description of terms for simplified cofilin metric.
37
(33)
Module V: Protrusion
Barbed Ends (Tania N. , 2013)
dB
d = AFsev(Cf) - kcapB - @Cap-BCap +
Term
PArp-BArp
2/ 3
(34)
Explanation
I
Severing of F-actin bound cofilin
2
Degradation of barbed ends
3
Capping protein inhibits barbed ends
4
Positive feedback from Arp2/3 to the creation of barbed ends*
Table 24. Description of terms for Barbed Ends.
The cofilin parameter Cof [t > t1] incorporated at a particular time point into the actin metric
equation is very important. It is known that the first barbed end peak is not dependent on P3Kinase
(Mouneimne, 2004) (Rheenen, 2007). Thus, incorporating cofilin into this actin-protrusion metric
for all time would be inappropriate for a modeling effort. Rather, it must be incorporated after the
first peak, as is done here.
aA
Term
=D
a2 A
kcapA + kpip2-API
2
PLC + Cof [t > t1 ]
(35)
Explanation
I
Monomer diffusion
2
Negative feedback from capping to protrusive process and nucleation
3
PLC and PIP2 pathway, direct from EGFR
4
First barbed end peak is independent of P13K
Table 25. Description of terms for actin nucleation metric.
The cytoplasmic actin monomer diffusion coefficient is 6 pm 2/s (Schaus, 2007).Using the actinprotrusion metric in addition to the Arp2/3 barbed end metric can yield a powerful tool to analyze
overall cell protrusion in the form of new barbed ends in the actin species population. Then, the
different biological factors underlying the two peaks can be broken down and analyzed in depth.
In addition, this sort of model set-up can be used to perform sensitivity analysis of upstream
parameters (for example, N-WASP, Wave, and PLC parameters) that are unknown from literature.
This final metric, the protrusion or barbed end metric, is useful for analysis and as a simulation
tool to yield interesting insights.
*This term connects Arp2/3 to Barbed Ends. The relative timescales of this portion and the rest of
this feeding in from the cofilin pathway can represent the two varying time scales observed in
experimental barbed end production data. This interplay can be used to meaningfully tune the
model to biologically represent the motility timescales.
38
(36)
dCap
dt
[Cap1
= -kCap-Cap
cCap
-n1+
J
a3 )
Term
ICap
Cap
n cap1 CapTot
Cap
1+ Mena n-Cap CaPTot
Explanation
I
Degradation of capping proteins
2
Feedback from PIP2 to inhibit capping. When PIP2 levels drop, this protein might be
dissociated from the membrane (Citation)
3
Feedback from Mena to inhibit capping
Table 26. Description of terms for Capping Protein.
Module VI: The Arp2/3 Pathway
dArp2/3
-BArp-ArpArp2
dt
3
+ PWave-
Wave
+ PNW
NW
(37)
1ArpArp2/3
n-cap]Arp2 /3normalization
(1+ Actin)
as.
dArp2/3
BArpArpArp 2 / 3 + fwaveArp2/3Wave +PNW-Arp/23NW
factor
- PActin-Arp2/3Actin
dt
The second alternative form is a useful mathematical modeling tool because it is less complex than
the proposed form that uses actin inhibition, but can recreate the lowering of Arp2/3 through the
use of a linear negative feedback term.
Term
Explanation
1
Degradation of Arp2/3
2
Positive feedback from Wave to Arp2/3
3
Positive feedback from N-Wasp to Arp2/3
4
Inhibition of Arp2/3 by actin
Table 27. Description of terms for Arp2/3.
39
dNW
dt=-BNW-NWNW
dt
Term
+
@Cdc4 2 -NWCdC 4 2
+ PPIP2-NwPI2
(38)
Explanation
1
Degradation of N-WASP
2
Positive feedback from Cdc42
3
Positive feedback from PIP 2
Table 28. Description of terms for N-WASP.
dWave =
Term
Bwave-waveWave +
Explanation
I
Degradation of WAVE
2
Positive feedback from Rac
Table 29. Description of terms for WAVE.
40
pac-waveRac
(39)
3.3 Relational Analysis
With a model at this particular level of complexity, it is important to have a way to test it for
conceptual accuracy and analyze how it can be vetted and used for predictive power. A useful way
to tune and test this complicated model is to try to replicate relational trends. This process involves
recreating trends like peaks or troughs, applied in combination with the correct chemical
mechanisms to describe reactions can help to build and create this model.
For example, this model strives to replicate trends in concentrations of several key species. In
experimental results, the total active Rac in the cell, determined using a Rac ELISA experiment,
was shown to have an initial peak and then drop off over time. Another set of experimental results
showed that barbed ends displayed two peaks, each mediated by different experimental factors.
When recreating relational trends, it makes more sense to focus on relative effects rather than
absolute values of concentrations. For example, looking for spatial or temporal changes instead of
focusing on the actual values of the numbers. A key relational analysis metric to study for
protrusion is the protrusion barbed end metric:
BM = f(A) + f(Arp)
(40)
This is a function of the cofilin based pathway and the Arp2/3 pathway, both which lead to
protrusion. Obviously, a long term goal of any modeling effort is to eventually describe and predict
reality exactly. However, as a first step presented here, the goal is only to produce relational trends
and create a first-generation large scale model in 1 D and a very basic 3D representation of key
factors.
41
3.4 Software and Computational Modeling
The MATLAB pdepe solver is used extensively in order to model and characterize the system in
1D and time. There is no native ability in the MATLAB environment to solve the partial
differential equations in 3D and time; for that purpose, the COMSOL computing software was
used. The pdepe solver is capable of inputting parabolic-elliptic partial differential equations in
ID and solving them for initial-boundary value condition problems. The underlying algorithm is
based on a piecewise nonlinear Galerkin/Petrov-Galerkin method which is second-order accurate
in space (Skeel, 1990). Implementation of a ID and time partial differential equation solver in
MATLAB requires a series of important inputs. The framework of a MATLAB implementation is
given in Figure 21.
-- PDE and Source Term Definition Function
-- Initial Conditions
Boundary Conditions
Figure 21. Steps involved in MATLAB implementation of algorithms.
The COMSOL software package uses finite element analysis to solve the sets of partial differential
equation over defined boundaries. Namely, the large 3D problem is broken into smaller parts or
finite elements to approximate the solution over a large domain. COMSOL creates a mesh in 3D,
as shown in Figure 22. The user can change the mesh size depending on the level of accuracy
desired. However, decreasing the mesh size too much would increase the calculation time.
42
'M Go"
Figure 22. Sample finite element mesh for COMSOL implementation.
The implementation in COMSOL involves incorporating generalized partial differential equations
and the constraints imposed on them into different modules that comprise the model. For example,
the Rho species can be split into a Rho cytosol component and a Rho membrane component (Spill,
Fabian 2014). This method was implemented in the original COMSOL model (Spill, Fabian 2014).
Equation 41 connects the different parts of the reaction-diffusion equation in 3D, in the form used
in COMSOL.
a2
Rho
ea 0t2 +da
oRho
at +V-F=f
[Rho
r = -'
aRho PRho
y
(41)
(41
(42)
az
This is the means through which equations are incorporated in 3D. A particular species might be
incorporated in the cytosol and the membrane, or only in one of those. By taking the equations
given above, or a simplified form of them, and incorporating them into a 3D representation, it is
possible to simulate a 3D cell. It is assumed that reactions and diffusion are described by the same
relationships in x, y, and z, which is a reasonable assumption to make because there is no
directional bias to the way these molecular mechanisms work.
43
ge em
M 5Sna a
4
r
e
ADM~dh
bnsw tasbpsum+.d3+v
D.W
a
r
X
r
LAo-p.
ly
is
.ateoc'
d.
-6,
-
-rSm~i
Figure 23. Example input page for COMSOL implementation.
-
.eimmaTm~5m)
PP 2 COVOMWEDF SWMMM
12
-
io
WTI
7
Cv
to
SpWW k*Wsfion-
40
OD
TkM
OOIN
POBMM
Figure 24. Proposed MATLAB Graphical User Interface (GUI)
44
00
Early versions of the model implementations included a graphical user interface (GUI) component.
The GUI component was for the software user to input a set of desired parameters and easily run
the implementation without a strong understanding of its inner workings. By working with current
and potential users, this platform was planned in order to make it as simple and straightforward as
possible to use. Figure 24 shows the proposed GUI format.
A vital input for the user to have is the simulation time. In a previously implemented GUI, the user
had the ability to also implement the cell length in addition to the time. This feature turned out to
be less useful for the user than the time, because the cell lengths used during a particular study are
usually fixed. The cell length most often used in the analysis was 20 Pm. A potentially useful
feature to have in a GUI is a menu bar to select which species is to be studied, or even to have the
option to simultaneously view all species.
In addition, the user should be able to decide whether the species being studied should be spatially
integrated or given at a particular time point and location. In some versions of the software
developed here, the user is able to input this information in response to in-line questions, instead
of a user interface. A user interface that can graphically receive this information would be far
superior. In addition, the ability to view the plot results in real time within the GUI window would
be helpful.
Additional features that are not completely necessary but which would be helpful would be the
ability to plot families of curves by choosing a particular parameter to iterate through. In addition,
it would be helpful to the user to export the data to a spreadsheet or the figures to a picture file
format such as jpeg.
Creating a user interface for this software would be helpful to the end user because it would allow
him or her to use this tool as a quick diagnostic to test out new linkages, parameters, or therapeutic
inputs.
45
3.5 Modeling Progression
The model has been through many sets of iterations to add various components and increase the
complexity and coverage of the signaling network. The biophysical network of signaling
molecules and feedback pathways is incredibly complex. The process of determining which
particular networks to include in the modeling pathway is involved and requires both judgment
and experimental guess-and-check. Early models include linkages between the Rho GTPases and
the phosphoinositides, in the style of the network presented in Figure 25. The different species
within the system interact with each other to create cellular motion. The signal S represents a
gradient caused by EGF stimulation. Rac then inhibits Rho, which converts PIP3 to PIP 2 and
inhibits Cdc42. Rac also causes conversion from PIP2 to PIP3 and from PIPI to PIP2. These and
other feedback linkages within this system result in cellular polarization, with Rac and Cdc42
concentrated toward the leading edge of the cell, and Rho concentrated mostly toward the trailing
edge.
Modl 4
d)
S(xt
CDC 42
fil
M3K
MK
PIP
PP3
PIP2 0
FN
Figure 25. Simple network relating key species (Holmes, 2012).
The models link various signaling networks together in order to create a web of interacting proteins
that eventually lead to actin and myosin changes. Some examples are shown in Figure 26. There
are a variety of early relationships linking important actin characteristics as well. For example, it
is possible to link the number of barbed ends in the membrane and the lamellipodium protrusion
speed (Maree, 2006) where vo is polymerization speed, and b is the barbed end density/unit length,
and w is the membrane resistance in force/unit length.
46
W
(43)
v = voe
a
C
b
>
CONTRACTION
ModlW 4
d)
-lb
CDC 42
PIPI
-------------- W
rIM
PROTRUSION
Figure 26. Different levels of coverage and intricacy in models. (a) (Maree, 2006) (b) (Holmes,
2012) (c) Network used in this work.
There exist different levels of complexity in the modeling process. A sampling of the evolution of
networks studied in this modeling effort is shown in Figure 25. Figure 26a, b, and c each show key
system players that are interconnected by a variety of mechanisms. Choosing whether or not to
include intermediate connections is an important modeling decision. This modeling effort began
with some of the original models presented above. This work aims to piece together and then
expand existing models to begin an effort to comprehensively link signaling networks to cancer
metastasis, linking in components like N-WASP, EGF, among others.
47
3.6 Challenges to Modeling
There exist many challenges in the modeling process for a system of this level of complexity. The
species modeled here represent only a small fraction of the variables that exist in this system
overall. For example, many other signaling pathways and cascades are involved in cancer
metastasis. Studying the EGF model system described here narrows the field of analysis to make
it easier to model important behavior and signaling network linkages, but it leaves out many
factors.
There are many state variables involved in this system. The behavior as modeled is an educated
guess, as with all models, of what is actually going on in the physical system. The linkages
proposed might not be the most accurate description of each of the mechanistic connections
between species. Inaccuracy in capturing the form of a mechanistic linkage can result in model
errors and room for improvement.
The biological system being modeled also has many unexpected twists. There are many factors
that cannot be accounted for as they are unknowns. There is variability in these factors based on
the cell type, particular patient situation, and external factors that are completely out of control.
By simplifying the problem and focusing on a specific set of manageable system variables, it is
possible to start building up a model and adding complexity over time.
Not all the biological pathways involved are modeled, and even within the modeled pathways,
there are many unknown constants. Parameter sensitivity analysis was performed to try to deal
with this situation and determine how changing a particular parameter could affect the model
results and outcome relative to experimental data and biological relational trends. Many constant
parameters are unknown and variable from case to case.
This entire process is multi-step, from tumor cell detachment to the time it reattaches at a secondary
site. Only certain portions of that overall process can be modeled successfully with the resources
and timeframe available here.
48
3.7 Models
The general form of a reaction-diffusion equation used in this model, given in three dimensions,
is given below. This form describes the first time derivative of a species as a function of a diffusion
term and a source term. For the work contained here, the source term is often a function of other
species within the system depending on how they influence the particular species under question
and a degradation term proportional to the concentration of the particular species. The full threedimensional form is given:
2 A+S
-=DV
(44)
at
aA = /a 2 A
at
D-
a 2A
a 2A\
(45)
++
Starting with the equations for Rac, Rho, and Cdc42, which are small GTPases, it is possible to
work outwards and connect in significant system players for the cell's mechanical machinery to
be activated. This sort of system description can be used to identify missing links or pathways and
pinpoint important linkages that control significant operations. By sewing together a network of
growth factor signals, second messengers, and pathway connections linking the players together,
it is possible to create a mathematical model to describe the basics of cancer metastasis. The
network used here is given Figure 29. A simplified description to illustrate the goal of this system
modeling involves a black box setup for the cell's signaling network, with a growth factor signal
being the input to the black box, and the cell's movement the output, shown in Figure 27.
Growth Factor
Movement
Figure 27. Control feedback to represent the black box that is being explored.
After the cell is exposed to a growth factor distribution, its internal machinery begins to respond.
Different pathways are activated throughout the cell when molecules diffuse or are actively
transported to particular locations. For example, the phosphorylation of certain proteins can affect
their behavior and cause particular results, changing a cell's response to stimuli. Molecules within
a cell can react with other molecules or diffuse throughout the cell. Thus, a "reaction-diffusion"
style model is chosen here for most of the species. Species can be most active in the membrane,
or can have active roles in the membrane or cytosol, as depicted in Figure 28.
49
I
OV
now
Figure 28. Species can be active in the cytosol region, membrane region, or both.
The network shown in Figure 29 displays the set of signaling networks used to create the current
sets of models. Arrows can indicate up regulation, reaction, or down regulation, through inhibition.
While the diagram below currently does not delve into that level of detail or specification, the
model equations presented in the various modules above and model equations below strive to take
into account most of the particular interactions between species, attempting to account for the
chemical kinetics. In some cases, the actual chemical mechanism is described by a simplified
mathematical description which has the same effect (increase or decrease in the species of interest)
in order to aid the modeling effort.
Some of the coefficients for these model equations are currently available in literature for particular
experimental conditions, and others are determined by fitting relational trends in existing data, for
example the relative size and structure of two barbed end peaks which are experimentally
observed. The relationships presented in Figure 29 are the result of a careful study of a wide
number of sources, given in the reference section.
It is possible to simplify the overall set of equations described above for specific cases and to
perform in-depth analysis. With almost thirty coupled partial differential equations, it becomes
difficult to study overall results in great detail. While individual modules can be studied, it is hard
to look for overall system trends and recreate experimental results.
50
MLC
[
R"a
W
L
RacRh
Cdc42
pl
EGFR Heterodl merI
| Meo/P--o'V
won
PROTRUSION
Figure 29. Network used in this work.
51
Chapter 4: Results
Chapter 4 summarizes key and interesting model results that shed a light on its predictive and
descriptive capacity. Each section below highlights and explains the results while linking them in
to the big picture goals.
4.1 Cell Polarization
A vital result of the model is that of cell polarity. When cells move, the different signaling proteins
polarize the cell so that one end of the cell moves forward and the other end retracts. This behavior
is essential to cell movement. In particular, the concentrations of Rac and Cdc42 are high towards
the front of the cell, and the concentration of Rho is high towards the back of the cell. This is
because Rho is associated with the contractility mechanical machinery of the cell, while Rac and
Cdc42 are associated with the actin polymerization and barbed ends formation which push out
protrusions. Figure 30 shows graphically what is meant by cellular polarization, in a simplified
manner.
Figure 30. Cellular polarization diagram.
Each end of the cell is polarized with the correct species in order to recruit the contractile and
motility machinery within the cell. Downstream effectors of the GTPases are used to engage actin
and myosin within the cell in order to make the cell components expand and contract so the cell
can move around. In Figure 31 a below, the key components of cellular polarization on a substrate
are shown. At the leading edge of the cell, Rac and cdc42 are more prevalent, and their downstream
effectors lead to the formation of lamellipodium and filopodia through the activation of actin
polymerization and actin turnover. At the trailing edge of the cell, on the other hand, RhoA is more
concentrated. There, the ROCK and myosin light chain networks cause the trailing edge of the cell
to contract. At both ends of the cell, there are focal adhesions which tug on the cell and provide
extra mechanical force to make the cell move. Figures 31 b-d show a series of polarization results
from the models. There is a clear distinction with Rac and Cdc42 at the front end of the cell at
20im, and Rho at the retracting end. This polarization was set up using a Rac gradient. Results
provided in section 4.5 use an epidermal growth factor stimulus to induce cellular polarization.
52
d b
a
I
I.'A
1I'
(tucuAJO)
d
C
n~
0sjaxia
C4
[77---
7
r
-
I
I
I
*1
(.uw3I~q) .~usw~o
Figure 31. (a) Localization of GTPases (Mayor, 2010). (b-d) Simulation results showing cellular
polarization for Rho, Cdc42, and Rac.
53
4.2 Parameterization to Match Experimental Data
In the early stages of modeling, experimental data was matched using the initial conditions to
reproduce similar trends to what was seen in the data, namely a spike in spatially integrated active
Rac in the cell followed by a drop-off, in response to EGF stimulation. By changing PIP 2
concentration, the shape of the Rac curve was modulated (Figure 32d) in order to match the
expectation of a spike in active Rac from experimental data, instead of the initial result, a drop-off
(Figure 32a). Figure 32b shows the desired trend, not an actual modeling result.
a It-bI
0.9
0.95
0.6
0.9
0.6
0.4
076
0
0.7
0.1
Z
0.2
____________0.
50
0
c
100
10
200
TIm (Second%)
300
250
OAS
05
15
n10
d
*
_
.75-.
C
.7
C00
C.I:t1*s
10
0.1
II
4
so 4 1U
10140
19
150
Tbm
to.)
Figure 32. (a) Original Rac result (b) Desired model result shape [not actual result] (c) Rac
ELISA data (Talento) ) (d) Parameter match to fit data (Talento)
Unfortunately, simply the increase of PIP 2 concentration in biological terms does not correlate to
EGF stimulation, despite the fact that it gave the correct modeling result. Instead, the relative
concentrations of PIP2 and PIP3 represent specific realistic expectations for a biological system.
54
Realistic values of PIP 2 and PIP3 in a cell are 30 pM and 0.05 pM, respectively (Dawes, 2007).
This is a lesson in modeling that while changing a particular parameter might result in the correct
result, that parameter variation might actually not be physically or biologically motivated itself.
In this particular case, it was not the absolute value of PIP 2 that was the solution as believed, but
rather the relative values. The value for the PIP2 concentration within a cell is almost 103 larger
than PIP 3 . This process reveals some of the pitfalls of blindly fitting data. For example, starting
with a set of data points and running large loop iterations and mean squared error minimization
algorithms to fit data exactly would not necessarily have biological meaning. Changing specific
parameters that explain the biological mechanisms and chemical reactions to recreate relational
trends are more valuable in the long run for the modeling process.
Figure 33a and Figure 33b show two spatiotemporal heat maps of the actin species within the cell
with and without experimental matching of the Rac parameter. After the Rac ELISA data was
matched with the model, a spike in actin at about 180 seconds occurred, instead of a sharp dropoff (Figure 33b).
a
b
Figure 33. (a) Experimentally matched simulation (b) Original simulation Results
A similar relational result is shown below in Figure 34, using stochastic initial conditions. By using
stochastic initial conditions to represent randomness and uncertainty in the system, it is possible
to run simulations in order to study the relationship between active Rac and actin filament density
over space and time. Figures c and d show the relationship between a spike in active Rac at 5-80
seconds that yields a spike in actin a certain time later, both occurring at the leading edge of the
cell.
55
Active
ab
3
UU
AcnFM
Rau
i m
I
As shown below in Figure 35 (Haugh, 1999), the concentration Of PIP2 must decrease with time in
the presence of EGF stimulation. The biological mechanistic reason for this is that under EGF
stimulation, PIP 2 hydrolysis increases. Figure 35 below depicts that relationship, showing the
increase in PIP 2 hydrolysis as a function of the EGF fraction maximum.
On the other hand, without EGF stimulation, PIP2 concentration would simply continue to rise
without being curbed by the feedback. However, there is also flow from the PIP 2 species to PIPI,
to be considered and factored into the model equations. Nevertheless, controlling for this factor, it
is biologically expected to see either an increase or decrease in PIP 2 concentration based on the
presence or absence of EGFR stimulation.
These biological results are reproduced by the model, as expected, and displayed in Figure 38. The
parameter changes to generate this result are biologically motivated. The linkage of EGF into the
Phosphoinositide module is through its link to Pl3Kinase (P13k). The results of this analysis are
shown in section 4.5 below.
56
1.0
0.8
-.
6-0
0.61
E
0.2-
0.0
0.0
0.2
0.6
0.4
0.8
1.0
pY-EGFR, Fraction Maximum
Figure 35. Relationship between PIP 2 hydrolysis and EGFR (Haugh, 1999).
4.4 Initial Conditions and Boundary Conditions
The initial conditions vary according to the study being performed, but are generally based on
values from literature for expected resting cellular concentrations. For example, expected realistic
concentrations for the phosphoinositides PIP 2 and PIP 3 in a cell are 30 pM and 0.05 pM (Dawes,
2007).
In 3D, species can be separated into the cytosol or the membrane and consequently given initial
and boundary conditions to allow flux from the cytosol to the membrane and vice versa. Figure 36
shows a sample COMSOL screenshot to enter the equations. In ID, the boundary conditions are
no-flux to avoid flow out of the cell.
57
r
oirs
06000"m
Figure 36. Sample view screen for COMSOL platform.
Stochastic initial conditions are used in some studies and applications of the models. Within a
biological system, there are often fluctuations in concentrations of a particular species, and these
fluctuations can lead to larger changes through signal amplification and feedback. Using stochastic
conditions can be very helpful to model such applications. A challenge to modeling and a potential
avenue for further work is to analyze in great detail stochastic simulations in 3D. In section 4.7,
Figures 53-56 show the application of stochasticity in 3D and how it can be affected by parameters
within the system, for example, the amount of diffusion present and the size of the diffusion
coefficient.
4.5 EGF and Menail
kP13K-f actor
-
P13K
Rac
EGF nm
)kPIMKfactor
Cofilin = PIP2( +
(1+Racloal
(46)
(47)
EGF based stimulation is incorporated through the use of a Hill function dependent on P13kinase,
EGF concentration, and a large K. value, and Hill coefficient n. EGF is also incorporated in a
similar manner for the PLC pathway. It is very important to distinctly separate the PLC and the
PI3K based pathways that link in EGF and Mena. By modulating the constant parameter that links
into the P13K factor, or the PLC factor, it is possible to simulate the existence of Menai"v. Figure
36 shows a graphical depiction of how Menai"v works mechanistically, and interferes with the
process of capping actin filaments.
58
e
AW"na-
Cofflin
9
4
EVH1
Cofilin
Severs
capped filaments
p
G
T
Elongates and protwcts
lei
Figure 37. Mena affects cellular mechanical machinery (Gertler, 2011).
An interesting check on the model against biologically expected results is its predictive capacity
for PIP2 concentration over time. As mentioned in section 4.3, which discusses the PIP 2
concentration test, when incorporating EGF into the model, the concentration of PIP2 should
decrease. On the other hand, in the absence of EGF, the PIP2 concentration should rise over time.
Figure 38 shows exactly that effect. In this case, the result is obtained by incorporating EGF
through a Hill function that affects the PI3Kinase parameter in the model. As is biologically
expected, PIP2 concentration rises from its initial concentration (in this case, 10 pM, but the effect
was shown for other concentrations such as 50 pM among others, not shown) in the absence of
EGF stimulation, and decreases from its initial concentration in the presence of EGF stimulation.
Another interesting feedback pathway in this network is the PIP3 to Rac linkage, which represents
the linkage between EGF stimulation and the Rho GTPases. Figure 39 displays the effect of
varying this feedback on the double peak in the barbed end metric. This parameter has an effect
on both the peaks, but appears to more strongly affect the second transient peak.
59
PIP 2 Concentration-No EGF Stimulation
70
---------
50-
N
20
10
20
10
O
30
Time (Seconds)
405
60
PIP 2 Concentration-EGF Stimulation
b
12
C 10
0
W
0
042
%
O20
304000s
Time (Sec)
Figure 38. Change in PIP2 concentration for the absence (a) or presence (b) of EGF
concentration from simulation results.
60
-
1200
K=5
K=10
-K=15
1000-
1000
800
600
400
200
10-
40
30
20
50
Time (sec)
90
80
70
s0
100
Figure 39. Change in barbed end metric with PIP3 to Rac feedback parameter.
Figure 40 shows how the parameter space can be used to fit the relational trend in Active Rac data
obtained using a Rac ELISA study, which was experimentally performed by Suzanna Talento.
Note that these parameters being plotted are normalized. While the initial rise and asymptote can
be recreated by this model, the timescale of the Rac peak is not fully recreated by varying
biologically relevant parameters. The integrated Rac concentration achieved in this simulation is
the 1 D Rac concentration in the model integrated over the length of the cell to represent a full-cell
concentration, since the experimental setup included lysing the cells and measuring overall
concentration.
0.9
0.8
0
, 0.7,
o*
~0.6N
0.5
01 0
20
300
400
500
600
700
800
Time (Sec)
Figure 40. Parameter variation to fit Rac data trend from ELISA experiment.
61
900
4.6 EGF-Induced Polarization
In the earlier versions of the model, cell polarization was obtained using the application of a Rac
gradient. However, it is now possible to induce cell polarization using purely an EGF stimulus.
Figure 41a shows the effect of this stimulus on active Rac and active Rho. The level of polarization
and the spatiotemporal profile is dependent on the EGF gradient applied to the cell.
a
No
7
6
46
2b
Active Rho
Active Rac
bPi
EGFR Heterodimer
PLC-y
IP
IMena/Mena""'
P13K
PIP2
PIP3
Figure 41. (a) EGF induced cell polarization. (b) Pathway diagram highlighting P13K linkage.
Cellular polarization can be achieved in silico through the use of an applied Rac gradient, as shown
in the simulation results above, e.g. in Figure 31. Nevertheless, a more biologically accurate
description would induce cellular polarization through the use of a gradient in EGF that is linked
into the model. Figure 41 a shows the result of such a linkage. Here, the P13Kinase parameter is
used to represent the EGF stimulation. Then, through the whole parameter space, the EGF gradient
is translated in a more realistic manner to cause polarization in the GTPases. Namely, instead of
62
purely applying a suggested gradient in the GTPase species, it is induced using a realistic EGF
gradient from outside the cell. The Rac species is polarized to the "front" end of the cell or location
of EGF greatest EGF stimulation, and the Rho species is polarized to the "rear" end of the cell.
a
2.5
.5
100
.5
Active Rac
Back of cell
2.)-
.5
15
10
51
I
10
I
I
20
I
30
I
I
40
50
Active Rho
b
I
60
70
80
90
II
Iii
- .5
100
Active Rac
Active Rho
Figure 42. The effect of PIP3 to Rac feedback on polarization using an EGF gradient stimulus.
Figure 42 displays a nuanced look into EGF-induced polarization described in Figure 41. This
figure highlights the importance of the PIP 3 to Rac feedback in the GTPase species polarization in
response to EGF stimulation. This has important biological ramifications; the functions of the Rac
and Rho species within the cell for motility are very important. Without the EGF gradient causing
the GTPase species polarization, further effects in motility through their linkage with the cell's
mechanical machinery would be impossible. Figure 42a shows the case without PIP3-Rac
feedback, and Figure 42b shows the case with PIP3-Rac feedback; it is clear that there is spatially
graded polarization only in the second case (front and back of the cell are labeled in Figure 42a).
63
4.7 3D Results
Applying a simplified version of this model in three dimensions yields rudimentary but interesting
results. Results from early model applications in 3D are shown in Figure 43. Figures 43a-c display
these basic results for sample cell geometries, implemented in COMSOL, and Figure 43d shows
an implementation in Mathematica. Mathematica was not a viable option as a platform for this
work because it does not contain the ability to solve nonlinear PDEs over 3D regions. This
capability is fundamental to the implementation of this model in 3D.
F3dt.
Figure 43. Variety of early 3D modeling efforts. A, B, and C are COMSOL, D is Mathemnatica.
64
r u
I-
Figure 44. COMSOL implementation from first stages of 3D modeling process.
Figure 44 shows a result from an implementation in COMSOL from early on in the modeling
process, for the species Rac. This figure displays a combination of 3D slice and surface plots to
give a full 3D depiction of results from the differential equation solution. The slices are crosssections from the cytosol, while the surface plot is the area of interest in polarization. The surface
concentration in Rac ranges from 5.4 iM to 0.31 ptM, a polarization in concentration between the
leading edge and trailing edge of the cell.
aR
b
Figure 45. (a) Rac and (b) Rho in 3D and 1 D for a rectangular prism geometry
65
Figure 45a shows a plot for Rac in 3D for a rectangular prism geometry. The concentration in 3D
ranges from 5.5 pM to 0.31 pM. Figure 45b shows a plot for Rho. The concentration for Rho
varies from 1.28 pM to 1.2 pM. Rac is more polarized than Rho.
Figure 46 displays the polarization of Rac toward the end of the cell, but resulting from a gradient
in EGF stimulus. The graphs in Figure 36 are also for an ellipsoid geometry. This EGF-based
stimulation is similar to the analysis performed in section 4.6 for the 1 D case. The gradient in Rac
is less pronounced than the one in Figure 44 or Figure 45, since the results are translated
downstream from the EGF gradient. The Rac concentration now ranges from 1.85 PM to 0.3 PM.
Instead of an approximate factor of 18 times, the polarization is approximately 6 times greater at
one end of the cell than the other, a more reasonable result. Figure 45b shows the graph for Rho.
The concentration of Rho is very similar throughout the cell (at 1.2 pM), without a clear gradient.
Namely, the apparent and reproducible polarization patterns are actually consistent with
experimental observation that Rho is at the leading edge as well, as: "recent biosensor studies have
shown that all three GTPases are activated at the front of migrating cells" (Machacek, 2009). In
order to fully develop a 3D cellular polarization effort, more analysis is needed, however.
a
b
Figure 46. (a) Rac concentrations in 3D (b) Rho concentrations in 3D; present at both ends as
experimentally predicted.
Figure 47 displays the concentration of myosin phosphatase over the cellular surface for the 3D
model. Notice how myosin phosphatase forms what appear to be fibers (light blue and green
patches) that run down the cell. The concentration ranges from 0.55 piM to 0.08 pM, a difference
of 6.9 times. It is useful to note that the model, although simple, is beginning to form some
interesting characteristics that could be studied and enhanced in further analysis.
Notice the formation of rings (black arrows in Figure 47a) in the myosin phosphatase
concentration. This sort of behavior can be used in the future to model cell division of metastatic
cells at their site of implant, since cytokinesis in eukaryotes uses actomyosin contractile rings
66
(Calvert, 2011). These formations in Figure 47 could be the effect of noise, but for future work it
is a valuable area to investigate, since the concentrations are approximately an order of magnitude
greater in the red and orange areas. The concentration ranges from 0.55 pM to 0.08 piM.
b
Figure 47. (a) Myosin phosphatase. Black arrows show ring-like formation. (b) Different
viewing angle.
Figures 48a and 48b show a barbed end time series over the course of 30 seconds in the cell. The
model appears to predict a polarized cell. However, what is really happening is less insightful.
While the cells below appear polarized, indeed they are either not polarized or only very slightly
polarized, on a scale much smaller than what is of relevance. The color bar simply rearranges to
areas of greatest concentration, but the concentrations are very similar at both ends of the color
bar. A similar effect was observed in a WASP time series plot (not shown here).
67
b
a
Figure 48. a-b. Barbed end time series (0 and 30 seconds). Insignificant change in concentration.
It is possible to locate and import real geometries of cells, though this was not possible in this
work. Then, the model can be applied to these geometries to yield potentially interesting results.
However, a much more insightful use would be to apply the model to a simple geometry, as was
done above, and then extrapolate changes in the mechanical machinery of the cell to predict a new
shape, instead of simply applying the model to protrusive shapes. This predictive concept is shown
in Figure 49.
C
a
b
Figure 49. Predictive approach to cellular geometry. (a) Initial cell shape. (b) Output to
actomyosin network (Mak, 2014). (c) New cell shape.
68
Figure 50. (a) PLC Gamma simulation result, spiking begins in center and localized to one side.
(b) PLC gamma stain in HeLa cell to show membrane ruffles (Santa Cruz Biotechnology).
Figure 50a shows the distribution of PLC gamma, with a variation from maximum to minimum
concentration of 0.41 pM. The PLC gamma spiking begins at the center of the cell and is localized
to one side. Figure 50b shows a PLC gamma stain in HeLa cells to show membrane ruffles, with
a nuclear counterstain. This kind of distribution would be the eventual goal, but the basic aspects
are visible in the 50a plot. Note that PLC Gamma concentration varies from 1.4 pM to 1.0 PM.
Figure 51 shows the simulation results for an elliptical shape for PIP2 and PIP 3 . Figure 51 b shows
PIP 2 distributed throughout the cell, not localized to any one section. The same is true for PIP 3, as
shown in Figure 51b. In this model, it appears that the phosphoinositides are more evenly
distributed than Rho, which is more evenly distributed than Rac, which is very clearly polarized.
Rho is important at both ends of the cell, while Rac and Cdc42 are primarily most active at the
front end of the cell. Nevertheless, at this early stage, it is nearly impossible to draw conclusions
about polarization.
b
Figure 51. (a) PIP3 . (b) PIP 2
It is important to note that when EGF is not used to induce cell polarization, it appears that PIP3 is
indeed polarized, as shown in Figure 52. There is a delta of 0.041 pM. This seems like a small
69
number relative to the polarization in Rac, but recall that generally PIP 3 is present in the cell at
concentrations of around 0.05 ptM (Dawes, 2007), so a fluctuation of 80% that size is large. This
is an interesting discrepancy and might mean that it is in fact essential to induce polarization using
EGF. This is because EGF-based polarization captures the notion that PIP3 is actually involved not
just at the leading edge but also in cell retraction. In fact: "...cortical accumulation of PIP3 was
often correlated with local retraction of the periphery" (Asano, 2008). PIP 3 fluctuations can lead
to spontaneous polarizations as well.
-I
Figure 52. PIP 3 polarization without use of EGF stimulation to induce polarization; fails to
capture essential biological concepts.
In order to further test the functioning of the model, it is useful to study how these stochastic initial
conditions play out through the downstream effectors. This experiment was carried out for a
rectangular prism and an ellipsoid. Figure 61 shows the downstream effect on ROCK (Rho kinase)
of fluctuations in Cdc42. The concentration of ROCK varies from 0.7pM to 0.02pM, but with the
majority of fluctuations being far less than that range.
Figure 53. Cdc42 stochastic initial conditions.
70
Figure 54. Stochastic initial conditions translated while moving downstream to ROCK.
Figure 55. Rac with original diffusion coefficient.
Figure 56. Rac with increased diffusion coefficient (two orders of magnitude)
Increasing the diffusion coefficient by two orders of magnitude spreads out the stochastic
fluctuations rapidly and removes their effect quickly. The 3D analysis here provides a simple first
step towards a wide range of possibilities for 3D motility analyses. Fine-tuning the concepts of
71
polarization and concentration gradients in 3D and improving the accuracy of the modeling process
can add a great deal of insight. The analysis in 1 D presented here for parameter variation can
eventually be performed in 3D. In addition, the network linkages used can be refined and a
parameter sensitivity analysis performed based on data in 3D if it becomes available. The next
steps should aim to accurately reproduce polarization in 3D, and connect that to movement.
4.8 Double-Peak in the Barbed End Metric
The next set of figures and results show how the various components of the model can act in
concert to produce an important biological relational result, the double peak in barbed ends. In
order to help parameterize new parts of the model for which constants are unknown, it is useful to
perform a parameter variation analysis. For example, Figure 59 shows the variation in barbed end
metric for a change in the feedback between cdc42 and N-WASP, two key species in the barbed
end pathway. As a reminder, a sample set of proposed equations are reproduced here:
a2Wasp
aWasp
Sat
=
D
2
x
(48)
s-wasp-WaspWasp + $Cdc42 -waspCdc
4 2
+ IPIP2-wasp PIP2
As expected, in Figure 59, varying the parameter that represents feedback from Cdc42 to WASP
changes the second peak which is representative of the Arp/23 pathway, not the Cofilin pathway.
The Cofilin pathway is not dependent upon N-WASP, and as expected, is hardly affected by this
change. It is useful to measure how other key parameters from the N-WASP and WAVE module
affect the barbed end peak, a metric downstream within the pathway. Shown below in Figure 60
is the variation in barbed ends for a change in the feedback between PIP2 and N-WASP. Once
again, as expected, it is the second peak that is affected. In this case, for a feedback parameter of
K=0.5, the change is very extreme and that affects the first peak, but only slightly, as is clear in
Figure 60. Figure 57 illustrates how the two main pathways within the model can work in concert
to produce a double peak in the key protrusion metric. The early peak based on cofilin combines
with the later peak based on Arp2/3 to create the double peak. The first two plots are spatiotemporal
plots for cellular location (x-axis) and time (y-axis). In order to create a metric to study the overall
trend, the species are integrated over the length of the cell in order to study an overall concentration
metric.
Figure 58 displays spatiotemporal heat maps for three species, WAVE, WASP, and Arp2/3, in
order to show the details of the model's second peak. As the double peak in barbed end metric can
be broken down into the cofilin and the Arp2/3 peak, the Arp2/3 peak can further be broken down
into its key components, N-WASP and WAVE. Figure 44a shows the case for which there is low
EGF stimulation. This stimulation is through the P13K pathway, in order to represent the
downstream effects of the full model through the GTPases. Namely, EGF stimulation causes a
72
downstream effect that travels from the phosphoinositides through the GTPases and eventually
leads to changes in the WASP, WAVE, and Arp2/3 peak. Figure 58b shows the case of high EGF
stimulation, and an increase in protrusion and cellular polarization, as displayed in the intensity
and shape of the heat map.
20
a
110
00
2
20
40
60
80
100
20
I 40
60
80
100
I
I
0
0~10i
0
I
C 1000
500
n
0
I
I
I
I
20
40
60
80
100
Time (sec)
Figure 57. The two key components of protrusion are shown. (a) The early peak based on
cofilin. (b) The late peak based on Arp2/3. (c) The sum of both as the barbed end protrusion
metric.
73
Figure 58. (a) WAVE, WASP, and Arp2/3 spatiotemporal plot with low EGF stimulation
(Pl3Kinase pathway, the late pathway). X axis: Time 0 to 100 seconds; Y axis: Position across
20ptm cell. (b) EGF increase with Pl3kinase increases protrusion and cellular polarization.
Figure 59 shows the variation in barbed end metric for a change in the Cdc42-WASP feedback
parameter. This parameter affects primarily the second peak, or Arp2/3 peak, as would be expected
biologically. From this parameter sensitivity analysis, which shows the downstream effects on a
key system output for the variation in a specific constant parameter, it is clear that the value range
of Kcdc42-WASP is approximately 0.01-1. A similar parameter analysis is performed for the feedback
between PIP2 and WASP; here, the acceptable range is also similar, 0.01-1.
74
-KO0.01
-
Iw9
-K=0.05
1400
-K=O.l
1200
1000
---I
400
200
10
20
30
40
50
Time (sec)
60
70
80
90
100
Figure 59. Variation in barbed end metric in response to changing the feedback between Cdc42
and N-WASP.
-K0.01
K=0.05
-K=0.1
-KnO.5
1200
1000
400
2W0
10
20
30
40
50
Tme (Sec)
60
70
80
90
100
Figure 60. Variation in barbed end metric in response to changing the feedback between PIP2
and N-WASP.
75
Feedback Parameter
Rac to Wave
Cdc42 to Wasp
PIP 2 to Wasp
Wasp to Arp
Wave to Arp
Actin to Arp
PIP 2 to Barb
EGF (PLC)
EGF (Pi3k)
Approximate
Range
-0.1-1 [1/s]
-0.1-1 [1/si
-0.1-1 [1/s]
~0.1-1 [i/s]
-0.1-1 [1/si
-0.9 [1/s]
-1.9-2.5 [s iMi 1
~1
~0.01 to 1
Table 30. Tabulation of feedback parameter ranges. The EGF parameters correspond to the hill
function scales.
Table 30 tabulates the result of performing a series of sensitivity analyses for some of the
representative feedback network linkages in the system that have not yet been characterized
experimentally or mathematically.
The species PIP2 plays a key role in the overall model. In response to EGF stimulus, it is greatly
affected and leads to downstream changes in the pathway that cause increased motility. The
feedback between PIP 2 and barbed ends is extremely important when tuning the model. A
parameter analysis of this feedback in the model yields an interesting set of insights and allows the
model to more closely match experimental data. The data shown in Figure 61a are from a barbed
end assay in MTln3 cells (Mouneimne, 2004). Notice that while the peak heights can be tuned to
the data, the time scales are still different. Improving the model to recreate accurate timescales is
an area for further research. Because this is a relational analysis study in order to characterize a
very complicated system, the concentration absolute values are ignored, and the relational trends
between different parts of the curve are analyzed instead.
Figure 62a shows the results of a barbed end assay, but for the case of control cells and a PLC
inhibitor (Mouneimne, 2004). Knocking out PLC affects the early transient but not the late
transient. The model simulation gives a similar result for the knockdown of the PLC parameter. In
addition, the first transient peak scales with increasing PLC (plot not shown). The second transient
is slightly affected, but not nearly as much as the first transient.
Notice how the experimental results from the barbed end assays and the simulation results occur
on different timescales. The model can be tuned to change the timescale of the peaks to an extent,
but this tuning parameter is not biologically relevant, so the peaks were left at the current locations.
Nevertheless, from a relational-analysis perspective, it is easy to see how the model can be
compare to the experimental results in a general fashion.
76
a
b
710.
00
-KNO.1
- -K00.5
-Kul
-KI
ROW
-KO1.7
-K81.9
700
90O
-K02.5
40W.
3NW0
400
300
IWO
0
0
0
120
10
246
30
40ime
20
36il
100
(Sec)so
Tim (S)
Figure 61. (a) Barbed end assay for MTln3 cells (Mouneimne, 2004). (b) Model results for
varying PIP2 to barbed end feedback.
2-1w
aam
7M
b
-PLC Inhblian
ow
Control
150W.
400
,00.
300
GOW.
2W0
-
12000.
VIC
30W.
PLC inhibitor
S
40
1
180
240
3W0
360
20
40
(0
Timen ("eC)
so
100J
Thae s
Figure 62. (a) Barbed end assay for MTln3 cells, control and PLC inhibitor (Mouneimne, 2004).
(b) Model results for PLC component and PLC inhibition.
77
Figure 63 displays a spatiotemporal plot of the cofilin actin peak. This does not include the Arp2/3
pathway. Once again, notice how early the first transient peak, based on cofilin, is situated. This
parameter is variable and the peak can be shifted in the model. This is a matter for further research
and study because the peak can be shifted using many of the parameters in the model, but the
choice must be biologically motivated for it to have meaning. For example, the variation of the
capping rate can indeed shift this peak, as shown in Figure 64. However, as discussed throughout
this work, blindly fitting data to the expected value can lead to many modeling pitfalls. In this case,
the experimentally expected value for kcap is 1 I/s, but that is not necessarily the value that yields
the correct location for the peak. There are other factors in the model that must account for peak
shifting, because it is known what the value of kcap is. Nevertheless, relationally, it provides an
interesting insight and could potentially serve as a lumped parameter choice to represent other key
factors.
4
.5
1.
.5
2
10
20
30
40
0
Time (moo)
0
70
60
00
100
Figure 63. Actin peaking.
o~
- k =I
-k =01
0.-
0.
zC
2
-
0.1
Time (sec)
Figure 64. Capping rate and timescale analysis of first peak (normalized).
78
One way to incorporate the EGF stimulus is through the P13K and PLC gamma parameters;
however, it is also possible to mechanistically incorporate the Mena and Menanv isoform parameter
through the use of a new set of module equations. That process and results are shown here. The
mechanism of Mena is to promote filament elongation with direct monomer transfer and also to
protect the filaments from capping proteins (Gertler, 2011). These mechanistic behaviors can be
incorporated into the model equations by linking in a capping protein species and connecting it to
the actin and barbed end metrics.
The Mena" isoform can sensitize the motility response to EGF for tumor cells, and can increase
sensitivity by 25-50 fold (Gertler, 2011). This pathway and stimulatory effect precedes the Arp2/3
accumulation, but acts on the cofilin severing path. Figure 65 shows these effects from the model
simulation. By increasing the parameter representing Mena in the model by multiple test factors,
it is clear that an increase in the range of 25-50 fold causes a change in the peaking, and that is
most pronounced in the cofilin-based peak, rather than the Arp2/3 peak, as is expected. There is
almost a 50% increase in the first transient, and only a 20% increase in the second transient. Further
modeling efforts could aim to decrease the effect on the second transient by improving the
mechanism. In addition, experimentally there is actually an 80% increase in free barbed ends
(Gertler, 2011) than control cells or cells with MenacIssic within the first 20 seconds, whereas in
this simulation result there is only a 50% increase. This is another point of improvement for further
work.
1200--
naid
-1.3x
-1.6x
-2.Ox
1000-
Increase
Increase
Increase
Mena''" (30x fold increase)
800
Boo
600O
400
200
10 20
30
40
50
Time (sec)
60
70
80
90
100
Figure 65. The effect on the barbed end/actin parameter of increasing the model Mena parameter
through a mechanistic inhibition mechanism on capping and actin elongation.
79
11
NO -
Chapter 5: Applications and Future Work
The next phase of this work is its improvement in three dimensions and its application to a cellular
mechanical model based on an actin-myosin network. This model can output concentrations of
these species which feed in to an actin and myosin system, based on Brownian dynamics. This
model can then predict cellular movement. A significant challenge will be to feedback those results
iteratively with this model because this model requires a state description of many more variables
than are outputted from a mechanical model; for example, concentrations of the various species.
Eventually, this modeling approach can be fully integrated into the 3D realm and fed back into an
actin-myosin Brownian dynamics simulation such as the one shown in Figure 66.
Figure 66. Actin and myosin Brownian dynamics simulation (Mak, 2014).
The current 3D model is rudimentary and a simple first step toward a world of expansion in to the
3D realm. Getting the cellular polarization results in 3D to work accurately in concert for all the
Rho GTPases is a challenge. In addition, the dynamics and signaling networks are very different
in three dimensions than on a flat surface. Developing the networks presented here and adapting
them specifically for new pathways in 3D is a key future step. Another key direction for future
80
- --
-.;Md
work is improving the timescales of the model, such that they match experimental data more
accurately. In addition, moving the model to 3D while at the same time using experiments in 3D
would be very helpful to the 3D modeling process.
Future work can incorporate an even more intricate 1 D model that accounts for the correct time
scales and exact concentrations for the species involved. This effort creates a large scale modeling
effort spanning many areas involved in cellular motility and develops a set of predictive relational
trends that can reproduce biologically expected results. Taking this effort to the next level would
involve an in-depth iterative process with an experimental team that can perform targeted data tests
on key species. A bio-sensing experiment and modeling approach could prove to be very helpful
as real time spatial and temporal data would be invaluable in such a modeling effort.
81
Chapter 6: Conclusion
In summary, this work brings together many signaling networks in cancer metastasis and creates
a series of models that describe this process in quantitative detail. This work builds on existing
formulations and develops new sets of modular relationships that can be used to predict and
analyze cellular behavior under metastatic conditions, such as in the presence of the Menainv
isoform. This work mathematically represents different aspects of the EGFR signaling network
that are involved in cancer metastasis. It strives to tie together different chemical and mechanical
species within the cell that are important in connecting the overarching signaling network with the
mechanical machinery of the cell.
Each module and formulation represents many iterations of computational simulation and careful
attention to biochemical mechanism. The family of models can be used to predict certain aspects
of cellular behavior and have been tuned to recreate experimental results. This modular setup has
advantages because individual modules can be studied in detail. This work also begins a simple
and rudimentary move to three-dimensions, as a stepping stone to creating a 3D biochemicalbiomechanical model of a cell.
The models also have limitations because the physical system is extremely complicated. There are
a vast number of species and interactions at play. The models are just abstractions of an extremely
complex physical system that cannot be fully modeled with current knowledge and methods.
Nevertheless, the models are able to recreate interesting and important biological results. The next
steps will be to refine the mechanisms used in the equations and improve the 3D accuracy. Then,
the goal will be to tie the improved model in with a large-scale actin and myosin network to
reproduce more aspects of cellular motility in cancer metastasis.
The description of the signaling networks underlying cancer metastasis presented here are modular
and provide an expansive depiction of the underlying processes. The system provides a quick test
in silico to determine how factors like a new pharmaceutical could potentially affect the feedback
between different system components. This model can show how eliminating or amplifying a
certain parameter would impact the overall system. This work is a step toward a large-scale
biochemical and biomechanical mathematical model of the metastasizing tumor cell.
82
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