University of Trieste
Masters Thesis
A Three Dimensional
Computational Study of
Multicellular Dynamics in
Viscous Extracellular
Environment
Supervisor:
Author:
Prof. Edoardo Milotti
Chota Shore Salle
A thesis submitted in fulfilment of the requirements
for the degree of Masters of Science in Physics
in the
Research Group of Computational Biophysics
DEPARTMENT OF PHYSICS
March 2014
Declaration of Authorship
I, Chota Shore Salle, declare that this thesis titled, ’A Three Dimensional Computational Study of Multicellular Dynamics in Viscous Extracellular Environment’
and the work presented in it are my own. I confirm that:
This work was done wholly or mainly while in candidature for a Masters
degree at this University.
Where any part of this thesis has previously been submitted for a degree or
any other qualification at this University or any other institution, this has
been clearly stated.
Where I have consulted the published work of others, this is always clearly
attributed.
Where I have quoted from the work of others, the source is always given.
With the exception of such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have
made clear exactly what was done by others and what I have contributed
myself.
Signed:
Date:
i
UNIVERSITY OF TRIESTE
Abstract
FACULTY OF MATHEMATICAL, PHYSICAL AND NATURAL SCIENCES
DEPARTMENT OF PHYSICS
Masters of Science in Physics
A Three Dimensional Computational Study of Multicellular Dynamics
in Viscous Extracellular Environment
by Chota Shore Salle
Cell migration is a fundamental process of biological systems in animals which
results from various phenomena like embryonic morphogenesis, wound healing
and cancer development. Three dimensional motion of cells in inhomogeneous
extracellular environment is studied using off-lattice hybrid discrete-continuum
computational modeling, where cells are treated individually while the extra cellular environment where the cells live in is treated as continuum by making use
of Langevin Equation. By considering different cell mechanical properties corresponding to cells, the effect of Brownian motion is found to decrease on the cells
dynamics as viscosity of the environment increases from 0.001 Pa.s (water like
environment) to 1.0 Pa.s ( hyaluronate and collagen in extracellular environment)
and hence one can neglect the effect of Brownian motion of cells in case of high
viscous environment.
iii
RIASSUNTO
Computazionale della dinamica multicellulare in un ambiente extracellulare viscoso
La migrazione delle cellule è un processo biologico fondamentale negli animali,
ed è associato a vari fenomeni come la morfogenesi embrionale, la guarigione delle
ferite e lo sviluppo dei tumori. In questa tesi viene studiato il movimento delle cellule in tre dimensioni, in un ambiente extracellulare non omogeneo, utilizzando un
modello computazionale in cui gli elementi base sono cellule che si muovono in un
mezzo continuo utilizzando un’equazione del moto che è formalmente un’equazione
di Langevin. Considerando diverse proprietà meccaniche delle cellule, si trova che
l’effetto del moto Browniano diminuisce al crescere della viscosità ambientale, passando da 0.001 Pa·s (ambiente acquoso) a 1 Pa·s (la viscosità dell’acido ialuronico e
del collagene nell’ambiente extracellulare), e dunque l’effetto del moto Browniano
può venire trascurato nel caso di cellule in ambiente ad alta viscosità.
Acknowledgements
I am deeply indebted to my Advisor Professor Edoardo Milotti for his continuous
support and encouragement throughout my graduate research study. He gave me
increasing freedom to pursue my own ideas and his guidance is very unique. Every
time I got stuck in my projects and came to him for help, he provides a valuable
direction for me to go with.
Academic Professors, non-Academic staffs (specially secretaries) and officials both
at University of Trieste and ICTP were so helpful and lovely that I would like
to thank them all, of course whom I could not pass with out mentioning is Prof.
Maria Peressi.
Among friends, the help of Enrico Monachino is unforgettable that he is among
my bests.
This study has been possible thanks to a full grant from Abdus Salam International
Center for Theoretical Physics, in the program of Joint Laurea Magistralis in
Physics. Thank you.
iv
Contents
Declaration of Authorship
i
Abstract
ii
Acknowledgements
iv
Contents
v
1 Introduction
1.1 The Biology of Cell Migration . . . . . . . . . . . . . . . . . .
1.2 Computational Studies in Biological Systems . . . . . . . . . .
1.2.1 Modeling Techniques: Discrete, Continuum and Hybrid
1.2.2 Hybrid modeling . . . . . . . . . . . . . . . . . . . . .
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Stochastic Motion of Cells
2.1 Stochastic Differential Equation . . . . .
2.2 Forces Between Cells . . . . . . . . . . .
2.3 Numerical Solution of Langevin Equation
2.4 Description of The Simulation Program .
2.4.1 Visualizing Using Paraview . . .
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3 Results and discussions
3.1 Single cell dynamics . . . . . . . . . . . .
3.2 Cell Pair dynamics . . . . . . . . . . . .
3.3 Multicellular Dynamics . . . . . . . . . .
3.3.1 When Young’s Modulus is 400 Pa
3.3.2 When Young’s Modulus is 34 Pa
3.3.3 Change of Adhesion Parameters .
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4 Conclusion and future directions
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25
Contents
vi
A C++ Code
26
Bibliography
30
For those who lose their life by Cancer!
vii
Chapter 1
Introduction
Cancer is one of the most common diseases in adulthood with millions of people are
suffering from it around the world and hence has gained more and more attention
in research of different scientific disciplines in recent years [1]. It distributes in
the body from the origin of abnormality in the process called Metastasis, which
involves the escape of the cancer cells from the tumor mass, migration, invasion of
tissue and colonization [2]. Cell migration plays a fundamental role, not only in
cancer progression but also in processes as various as its development, embryonic
morphogenesis and wound healing [3]. The development of cancer involves various
temporal and spatial scales as well as multiple level of molecular and biological
organization- from genes to cells, to tissue and to the organism.
Unlike that of non living material, biological cells are dynamic system, whose continuously changing chemical and physical characteristics cannot be characterized
by fixed mechanical properties. They also have a complex micro structure that
evolves in response to their chemomechanical environment as well as disease state
[2].
The application of experimental and theoretical concepts from physics and engineering to study biological problems has provided deeper understanding of organ
function, mechanical response of tissue, physical and mechanical process associated
with blood flow, etc.
This study focuses on better understanding of dynamics of biological cells making
use of mathematical and Computational techniques described in section 1.2. After
giving a brief qualitative introduction to biology of migration of cells in the next
1
Chapter 1. Introduction
2
section, the description of a method that is used to study the migration of cells
in this thesis is outlined in section 1.2.2. At the end of this brief introduction
chapter, general objectives and the structure of this thesis is outlined.
1.1
The Biology of Cell Migration
Motions of Cells in vivo usually takes place in a three-dimensional environment
and in the presence of significant amounts of extracellular matrix (ECM), which is
the complex medium that surrounds cells and with which they interact. The ECM
interacts with the cell mainly through specific cell receptors. These receptors are
responsible for growth and differentiation of cell and also mediate cell attachment,
polarization and migration. Integrins are the major cell surface receptors involved
in cell-ECM interactions (adhesions and detachments) [4]. The interplay between
the mechanical properties of cells and the forces that they produce internally or
that are externally applied to them play an important role in maintaining the normal function of cells. Among other intrinsic sources, the cytoskeleton, which lies in
the cytoplasm and consists of net work of filamentous proteins, acts as a dynamic
structure that resists, transmits and generates cellular forces. Cytoskeleton also
plays many other roles including maintaining cell shapes, whose deformation is is
known to affect cell migration [5].
In addition to ECM, cells also experience external forces directly applied to the
cell or transmitted via cell-cell interfaces. Any alterations to the cell function due
to biochemical processes occurring within the human body, invasion of foreign
organisms or disease development can significantly alter the mechanical properties
of the cell. Consequently, measures of the mechanical properties of cells could
be used as indicators of their biological state and could offer valuable insights
into the pathogenic basis of diseases (like cancer, asthma, osteoporosis, deafness,
atherosclerosis, glacoma, etc [5]), including the possible identification of one disease
from another [2]. The specific mechanisms by which mechanical irregularities lead
to disease states, as well as how or if they can be remedied, are still unclear.
Cells are able to feel a certain number of external signals capable of influencing
their movement. These are, for example, the presence of a soluble chemical or of
a gradient thereof (chemotaxis), a particular distribution of adhesion molecules
Chapter 1. Introduction
3
on the substratum (haptotaxis), the presence of an electrical field (galvanotaxis)
or directed movement of cell motility via mechanical cues (mechanotaxis), for
example fluid shear stress. Cell migration guided by gradients in substrate rigidity
(i.e. stiffness), a subset of mechanotaxis, in a process called durotaxis [6].
Observation show that cells perform a random motion similar to Brownian motion,
which suggests that a mathematical equation analogous to that is used in theory of
Brownian motion can be used to describe cell migration, although the mechanism
by which cells move are radically different from the thermally originated movement
of particles suspended in a fluid [6]. The active (random) motion of cells thus can
take a similar functional relation as in Einstein relation, but with emphasis that
its cause is different than purely thermal motion, it is called Cellular- Einstein
relation. In this study the term kB T in Brownian motion is thus replaced by
experimentally motivated value of active random motion with a value of 10−16
Nm as in [7, 8].
1.2
Computational Studies in Biological Systems
Experimental studies of biological systems have advanced with unprecedented resolution and are allowing us to peek in to the world of genes, biomolecules and
cells and flooding us with the data of immense complexity that we are just barely
beginning to understand. On the other hand, treatment strategies are determined
by tumor response at the tissue scale in space over long scales in time. Thus,
there is a huge gap that separates our knowledge of molecular components of the
cell and what is known from our observation of its physiology- how these cellular
components interact and function together to enable the cell to sense and respond
to its environment, to grow and divide, to differentiate, to age, or to die. This gap
is hampering progress for example, in individualized cancer care to extrapolate patient specific micro scale quantitative information in to formulation of therapeutic
and intervention strategies at clinically relevant scales [9, 10]. To close this gap,
one can resort to mathematical modeling, where biological hypotheses are put in
to mathematical equations with the help of physics and chemistry, and then make
use computers to solve the resulting equations which are usually too complex to
be solved analytically.
Chapter 1. Introduction
4
In silico Modeling allows one to assess the contribution of specific aspects of a
biological phenomenon that are difficult to control experimentally. For example,
the role of cell adhesion forces between cells in healthy and tumor progression
individually and /or in combination with other factors can be studied. Because
the models are explicitly defined, in silico experiments and the resulting kinematic can provide insights into which aspects of the biology are most important.
These tools can suggest new research directions by identifying aspects of the biology that most affect the overall mechanics and behavior of the cells and tissue [11].
The best simulations and models should capture the behavior of existing experiments based on fundamental facts about the biological systems under consideration
including cells and tissues. Not only should it capture the existing experiments,
but also should be able to predict biological behavior and correlates with new
experiments as the purposes of modeling are to capture all that we know about
disease and to develop improved therapies tailored to the needs of individuals [12].
1.2.1
Modeling Techniques: Discrete, Continuum and Hybrid
Current modeling of multi cellular dynamics in general is of three kinds; discrete,
continuum, or hybrid, i.e., the integration of both discrete and continuum. A continuum model deals with phenomena on larger spatial and temporal scales, and
it is the easier model to analyze both analytically and computationally, and benefits from existing fundamental physical principles. Despite these advantages, the
averaging over space realized in continuum formalisms often cannot fully account
for the diversity of cellular and sub-cellular dynamical features. There fore this
method is not good choice if one is interested in cell properties that varies over
small spatiotemporal scales, for example epithelial–mesenchymal transition and
tumor heterogeneity [13].
Discrete models on the other hand captures cellular and sub-cellular properties
very well and are a natural choice in the small scales which includes random and
biased cell migration (kinesis and taxis) and cell- cell and cell-ECM interactions.
The major disadvantage of this method being its computational demand which
limits the number of cells being considered. Also these methods can not take in
Chapter 1. Introduction
5
to account, eg. tissue scale mechanics better than continuum modeling. Discrete
models can be either lattice based ( where cells are positioned on fixed lattice or
lattices and the interaction direction is limited) or lattice free (cell positions and
interaction directions are arbitrary.)
The advantage of discrete modeling is the disadvantage of continuum modeling,
and vice versa. And there fore, current researches focus on making use of hybrid
techniques that utilize the benefits of both continuum and discrete descriptions
and minimizes drawbacks of these methods. Thus, hybrid based methods are the
best alternative providing a more comprehensive understanding of cell migration
which in turn leads to more accurate and efficient prediction of tumor morphology
and potential for invasion [14].
1.2.2
Hybrid modeling
Different definitions of hybrid modeling exist in the field. Authors in [13] divided roughly in to composite hybrid modeling and adaptive hybrid modeling.
In composite hybrid modeling individual cells are treated discretely but interact
with other chemical and mechanical continuum fields. In adaptive hybrid models,
both discrete and continuum representations of cells are chosen dynamically and
adaptively where appropriate. This study uses composite hybrid modeling.
There have been many similar studies. For example; Anderson et al. [15] used
hybrid modeling for tumor growth and invasion, and obtained a result of tumor morphology variation that depends upon tumor microenvironment conditions
suggesting that differentiating therapy aimed at cancer-micro-environment interactions might be more useful than making the microenvironment harsher (e.g.,
by chemotherapy or antiangiogenic therapy). But their model was lattice based,
and suffers from the disadvantage lattice based modeling have in describing cell
migration. Kim et al. [16] used lattice free descrete-continuum modeling in two dimension, and hence can not take in to account the full three dimensional migration
of cells. Jeon et al.[14] has also studied a similar problem, but they used a cell-cell
interaction derived from Lennard-Jones potentials. Three of such similar methods
along with many single cell based modeling in biology and medicinetechniques can
be found in [17].
Chapter 1. Introduction
1.3
6
Objectives
This study uses a three dimensional off lattice (composite) hybrid discrete-continuum
model in order to understand how variations of biomechanical properties of cells
(in particular elastic modulus), the viscosity of the extracellular environment and
adhesion strength of cells affects multicellular dynamics in healthy and cancerous cells, and hence tries to bridge the gap between microscopic and macroscopic
behavior of biological systems which are multi-scale by their nature. Particular
interest to us is the extent to which Brownian motion is important in the dynamics
of cells by varying viscosity of the environment. In this study mitosis and related
biological phenomena will not be considered.
1.4
Outline
In chapter 2, the details of mathematical and numerical techniques is reviewed as
is used in this study, which is then followed by results obtained and discussions
in chapter 3. In the last chapter conclusions along with future research directions
are provided. The attached simple C++ code in appendix A also gives results
reported in this thesis and more.
Chapter 2
Stochastic Motion of Cells
Stochastic differential equation models play a prominent role in a range of application areas, including biology, chemistry, epidemiology, mechanics, microelectronics, economics, finance and etc [18]. In this chapter, the mathematics of Brownian
motion of cells which is usually expressed by stochastic differential equation called
Langevin equation is given along with its numerical solution.
2.1
Stochastic Differential Equation
Brownian motion is the random motion of particles suspended in a fluid (a liquid or
a gas) resulting from their collision with the quick atoms or molecules in the gas or
liquid. In order to describe this motion mathematically, one can use deterministic
Newtonian mechanics where the interaction between each atoms and/or molecules
and the particles are taken in to account explicitly.
But short time steps needed to handle fast motions of atoms of the fluid and
very long runs needed to allow for evolution of slower modes (larger) particles
make such a study computationally too expensive. If the fast motions are not
of great interest themselves, a useful approximation can be made which reduces
the computational burden such a deterministic equation have. In this case the
solvent particles are omitted from the study and their effect on the solute are
represented by a combination of a random forces and frictional terms. This is
the essence of Langevin equation, a stochastic differential equation, which replaces
7
Chapter 2. Mathematics of Cell migration
8
Newtons equation of motion. To understand the role of inertia in Brownian motion,
Ornstein solved the Langevin equation,
m
dV(t)
= −ξV(t) + Fthermal (t)
dt
(2.1)
Where m is the inertial mass of Brownian particle. The force from the surrounding
medium on the Brownian particle is given by the combination of stokes friction
−ξV and random thermal force
Fthermal = (2KB T ξ)1/2 η(t)
with Gaussian white noise statistical properties
hη(t)i = 0
and
0
00
0
00
hηj (t )ηk (t )i = δj,k δ(t − t )
Equation 2.1 can also be written in the form
P
dV(t)
= −V + (2D)1/2 η
dt
Where D is diffusion coefficient. P '
m
ξ
(2.2)
is called the persistent time, and character-
izes the time for which a given velocity is remembered by the system. The persistent random motion that this equation describes is called the Ornstein-Uhlenbeck
process (OU-process). The solution for mean square displacement given by Ornstein, takes the form independently of Fürth [19],
t
hd(t)2 i = 2ndim D(t − P (1 − e− P ))
(2.3)
This formula also follows from other similar theories.
Fürth obtained this formula while studying motility of protozoa because his data
were not described purely Brownian motion,
Chapter 2. Mathematics of Cell migration
hd(t)2 i = 2ndim D
9
(2.4)
Where in Einstein theory
D=
kB T
ξ
(2.5)
ndim = 3 is the dimension of motion, T is the absolute temperature, and ξ = 6πηa
is stocks drag coefficient for a rigid sphere with radius ’a’ moving with constant
velocity through a fluid at rest and having dynamic viscosity η.
The first attempt to mathematical modeling of individual trajectory of a cell was
reported by Prizibram in his study on the self-propelled random motion of protozoa. By tracking trajectories of individual protozoa, Prizibram demonstrated
that the net displacement averages to zero while its square satisfies equation 2.4.
Prizibram found a value of D which was much larger and much more sensitive to
changes in temperature than Einstein relation states with his lower time resolution
data before Fürth.
The simplest dynamic stochastic differential equation which describes a single cell
migration is given by Langevin equation (Eq. 2.1), which was introduced to study
Brownian motion. Here it is rewritten with emphasis that the physical origin of
the terms involved are not exactly identical to purely thermal motion [20] as
m
dVi (t)
= −ξVi (t) + FR
i (t)
dt
(2.6)
Where FR
i (t) is a force cell experiences due to all random process, and is a force
with zero mean and delta- correlated autocorrelation functions (her it is due to
factors internal to the cell ) while the second term in the right hand side of the
above equation, −ξVi (t), is a drag force that represents all the actions that slows
down the motion of cells. ξ is drag coefficient, m is mass, t is time and V is
velocity of the cell.
In the presence of external cues, the dynamics of a cell is given by adding the
terms of all other forces acting on a cell, which then Eq. 2.6 takes the form,
m
dVi (t)
D
= −ξVi (t) + FR
i (t) + Fi (t)
dt
(2.7)
Chapter 2. Mathematics of Cell migration
10
Where FD
i (t) takes in to account all deterministic forces acting on a cell i, which in
our modeling contains cell-cell attraction and repulsion forces. All other forces, like
cell-ECM interactions, are taken care of by drag forces along with the stochastic
force as is discussed in section 1.2.2.
2.2
Forces Between Cells
Interaction between cells is a crucial mechanism. Cell adhesion has been found to
plays a major role in cancer metastasis and invasion which is indicated by recent
studies that has focused to drugs that alter the expression of cell adhesion proteins
in a tumor system as possible therapeutic treatment for cancer [21]. At present
there is no single universally accepted expression of cell-cell forces. For instance
three different forms have been compared for cell-cell adhesion and repulsion in
[21].
In this study we follow the approach used by Chignola and Milotti [22], which
is based on Hertz model. For sufficiently small cell membrane deformation, the
force between two cells are roughly proportional to kc |x|3/2 where kc is a parameter
related to the problem at hand and x is the relative deviation from equilibrium
position. From the observation that isolated cells in culture or suspensions often
have a spherical shape [7], it is assumed in this study that cells to have such a
shape through out simulation. After defining cells by experimentally validated
biomechanical properties like elastic modulus (E), Poisson’s ratio (v) and radius
(R), the magnitude of the repulsion force between two cells, the parameter kc and
x are given by, respectively;
F12 = kc |x|3/2
kc =
√
R1 R2 (R1 + R2 )
2
3 1−v1
(
4 E1
+
1−v22
)
E2
(2.8)
(2.9)
and
x=(
d − (R1 + R2 ) 3/2
)
R1 + R2
(2.10)
Chapter 2. Mathematics of Cell migration
11
Where d is center to center distance between two cells under consideration. Subscripts 1 and 2 indicates cell index, for instance R1 is the radius of cell 1 and so
on.
The attractive forces between cells when they separate takes similar form as repulsion force above, but is multiplied by a factor which takes in to account the
number of cell adhesion links, and is given by
Fadhesion
1
' [1 + tanh(
2
r
2
(x − xo))]F12
πσ 2
(2.11)
Where xo and σ are parameters that must be adjusted, and controls the strength
of adhesion.
From the observation that two cells can not overlap, a minimum distance is set
beyond which cells can not approach. Cells are viscoelastic and change their shape
during interaction, here we don’t take in to account explicitly shape deformation
but only phenomenologically by increasing a distance range of interaction to few
microns [22].
2.3
Numerical Solution of Langevin Equation
Langevin Equation (Eq.2.7) with expressions of deterministic forces as in section
2.2, is a stochastic differential equation which is difficult to solve analytically and
and one needs to turn attention to numerical methods in solving it. Deterministic equations, like for instance Newtonian dynamics as in Molecular Dynamics
simulation can be integrated according to a well-developed algorithms that share
commonly agreed-upon desirable properties. Stochastic differential equation (including Langevin equations) don’t have such a common agreed criteria [23] and
hence there are many numerical techiques to handle them, nine of which are compared by Benedict Leimkuhler and Charles Matthews [24].
In this study, the Langevin Equation is integrated using second order Langevin
integrator (a stochastic version of the Verlet algorithm) as is given in [25]
r(t +
∆t
1
) = r(t) + v(t)∆t
2
2
(2.12)
Chapter 2. Mathematics of Cell migration
v(t + ∆t) = (1 − 0.5α)v(t) + m−1 F(r(t +
r(t + ∆t) = r(t +
Where α =
ξ∆t
m
12
p
∆t
))∆t + 2m−1 KB T αR
2
∆t
1
) + v(t + ∆t)∆t
2
2
(2.13)
(2.14)
and R is Gaussian random number with zero mean and unit
variance.
Among many possible ways of checking the correctness of the numerical solution
of Langevin Equation is finding mean square displacement in the case of zero
force (i.e when there is no deterministic force acting on a cell) [23]. In this case,
the mean square displacement of OU-process at long time limit (t >>
m
)
ξ
should
satisfy
hd(t)2 i = 2ndim Dt
With D =
kB T
ξ
and in the other domain (i.e t <<
(2.15)
m
),
ξ
the mean square displace-
ment should be quadratic in time [26].
2.4
Description of The Simulation Program
The simulation program in the Appendix produces all the results reported in the
next Chapter. After defining parameters needed at the top of the code including what to simulate using parameter simindex (which is set to 1 for single cell
migration, 2 for cell pair dynamics, 3 for multicellular system in uniform viscous
environment and 4 for multicellular system in non uniform viscous extracellular environment) the main program calls the function INITIALIZATION which assigns
initial positions to cells over a sphere of radius ro in three layers, with each neighboring layers separated by the sum of their radius (each cells are assumed to have
the same radius). The function UPDATEA moves cells according to Eqn 2.12
followed by calculation of a force by CALC FORCE on this new move. UPDATEB
then moves cells by one time step as of Eqs. 2.13 and 2.14 (and also checks in which
layer a cell is during simulation, depending on the distance from the center of the
spheroid, and assigns the corresponding drag coefficient). A function NORMAL
Chapter 2. Mathematics of Cell migration
13
generates normally distributed random variables and is adapted from William et
al. [27]. Finally the main program calls function OUTPUTS every ioutput steps,
which contains all the data for conventional and three dimensional plots done, and
iterates all the above calls up to a desired simulation run (isteptot). All the data
reported are in SI units, but the code measure energy in units of reference energy
(kB T = 10−16 J) mass in units of of mass of the Cell (4 × 10−12 kg) and distances in
units of radius of the cell (10µm). During outputs, the code convert results back
to SI units, for instance distances are multiplied by 10−5 and time by to=0.002.
2.4.1
Visualizing Using Paraview
In addition to the usual plots done, for example to visualize how center to center
distances of cells evolves in time, I make three dimensional plots making use of
Paraview to see how collectively all cells are moving. This also allows to make
a comparison with experimental observation. Paraview supports over hundred of
file formats, hence this study uses one of the prefered formats of Paraview called
Visualization ToolKit (VTK) [28]. Function OUTPUTS in the code attached in
Appendix also gives a result of time evolution of all position co-ordinates in *.vtk
formats, which then are imported in Paraview to visualize cell migrations in three
dimension (including live animations).
Chapter 3
Results and discussions
This chapter gives results of the computational study as discussed in the previous
chapters. After checking the performance of the integrator in section 3.1 for single
cell motion in the absence of deterministic force on cells, section 3.2 describes the
result of simulation of two cells. Finally the discussion of a multicellular system
with 75 cells is discussed. Through out all the simulation Poisson’s ratio (v) is
kept constant to 0.5. The green curves in multiple dynamics plots represents the
outer layer of the cells which is kept at viscosity of 0.001 Pa.s distributed initially
over sphere while the red one corresponds the inner and is at viscosity of 1.0 Pa.s
initially.
3.1
Single cell dynamics
Figure 3.1. and 3.2. is a simulation output of a single cell with drag coefficients
of ξ = 2.0 × 10−15 N.s/m and ξ = 2.0 × 10−7 N.s/m to explore two of the extreme
domains of OU-process. The result of both curves indicates the theoretically
expected qualitative behavior that for short observation time the mean square
displacement should be quadratic in time, while for long observation time it should
be linear in time. This is in fact the case as can be seen from both curves. In
the case of long observation time, the theoretical value of diffusion constant gives
us a value of D = 3.0 × 10−9 , while the corresponding value in our simulation is
obtained by the the slope of the linear fit of the mean square displacement (the
slope of green curve in Figure 3.2), and is found to be 3.095 × 10−9 - which is in
14
Chapter 3. Mathematics of Cell migration
15
a good agreement with the simulation results averaged over small (75) number of
realizations.
Figure 3.1: Mean squared displacement (in m2 ) Vs time (in s) for ξ = 2.0 ×
10−15 N.s/m
Figure 3.2: Mean squared displacement (in m2 ) Vs time (in s) for ξ = 2.0 ×
10−7 N.s/m
Chapter 3. Mathematics of Cell migration
3.2
16
Cell Pair dynamics
Figure 3.3 shows the center to center distance between two cells in the case where
only a total of two cells are present in simulation. The biomechanical parameters of
the cell are taken in experimental range of tumor cells as in [7] with elastic modulus
of 400 Pa. The red curve in Figure 3.3 corresponds to cells living in water like
environment with viscosity (0.001 Pa.s) while the green one is obtained when cells
live in environment with viscosity of 1.0 Pa.s (which corresponds to hyaluronate
and collagen in extracellular environment). After initial small separation the cell
starts repelling each other until they reach the steady state (where the distance
is twice that of their radius, taken to be the same for both cells) where the net
deterministic force becomes zero. It is evident from the figure that Brownian
motion has a huge influence on the lower viscous environment in which the center
to center distance oscillates back and forth about their steady state value through
out simulation - a typical characteristics of random motion. We’ll consider in the
next section a case of multiple cells distributed over sphere, typical behavior of
tumor spheroids, and check whether this behavior remains true.
Figure 3.3: Center to center distance between two cells (in m) Vs time (in s)
for ξ = 2.0 × 10−7 N.s/m
Chapter 3. Mathematics of Cell migration
3.3
17
Multicellular Dynamics
Here we consider a total number of 75 cells distributed over a sphere in three
layer initially with each layer containing 25 cells. After fixing the first layer, two
scenarios are considered; one in which the viscosity of the environment is kept
uniform while in the second case I vary viscosity from the second to the third
(outer) layer of our spheroid. The initial configuration of these cell arrangement
is given in Fig. 3.4 and Fig. 3.5 . Both Figures are the same except that the
second encloses the cells with spherical points to emphasize that in deed they are
on sphere. The blue cells in the inner part of the sphere represents the cells that
are fixed.
Figure 3.4: Initial configuration of cells over sphere.
Figure 3.5: The same as in Fig. 3.4 but with a sphere drawn on it.
Chapter 3. Mathematics of Cell migration
3.3.1
18
When Young’s Modulus is 400 Pa
This value corresponds to tumor study used in [7]. Fig. 3.6 shows the center to
center distance of two cells tracked from each layer for the case of uniform viscous environment with 0.001 Pa.s. Both curves indicates the influence of random
motion. Fig. 3.7 is a result of uniform viscous environment at viscosity of 1.0
Pa.s, it shows that the Brownian motion is unnoticeable compared to the case of
uniform viscosity of 0.001 Pa.s as in Fig 3.6, which is an indication of small effect
the Brownian motion has when cells are in high viscous environment. Fig. 3.8
indicates X-axis displacement of those cells from the two layers as measured from
the center of the spheroid. The effect of Brownian motion is noticeable in both
curves except that the magnitude is less in the case of the inner layer (depicted
by red color). The magnitude decrease of Brownian motion in case of the inner
layer can be described by the effect of attraction and repulsion forces exerted by
layers from below and above it, unlike that of the outer layer where the other cells
exerting a force on them come from only the inner layers.
Figure 3.6: Center to center distance (in m) for uniform viscosity of 0.001
Pa.s
Fig. 3.9 shows the scenario in which the inner layer is held at high viscous environment (with viscosity of 1.0 Pa.s) while the outer cell layer is kept at small viscous
environment (with viscosity of 0.001 Pa.s). In this case the Brownian motion is
almost washed out from the inner layer, similar to pair cell dynamics, which is
Chapter 3. Mathematics of Cell migration
Figure 3.7: Center to center distance (in m) for uniform viscosity of 1.0 Pa.s
Figure 3.8: X-axis motion of two cells (in m) for uniform viscosity of 0.001
Pa.s
19
Chapter 3. Mathematics of Cell migration
20
an indication that as cell viscosity increases, the effect of Brownian motion decreases. The magnitude of cell center to center distance indicates that the outer
layered cells move far apart compared to the inner layer, which is expected given
the freedom the outer layer has.
Figure 3.9: Center to center distance(in m) for non uniform viscosity.
Figure 3.10: X-axis position for nonuniform viscos environment.
Figure 3.10 shows the x-axis position measured from the center of spheroids for
nonuniform viscous environment. It shows similar behavior as in center to center
distance that the effect of Brownian motion of cells in the outer layer is large
compared to the inner one. This curve also shows the case where the outer layer
Chapter 3. Mathematics of Cell migration
21
enter the inner one at around 1.4 s as is indicated by the switch from large influence
of Brownian motion to smaller one.
Fig. 3.11 shows a snapshot of the three dimensional simulation output corresponding to non uniform viscous environment as in above, and it shows that the
tendency of cells to form edges and accumulate there - a behavior of beginning
of cancer metastasis. While Fig. 3.12 shows the three dimensional out put of
uniform viscous environment with viscosity of 0.001 Pa.s. It shows that cells are
relatively distributed uniformly, almost covering the fixed cells colored by blue.
The difference between these two three dimensional simulation outputs indicates
that cells do respond to the change of their extra cellular environment.
Figure 3.11: Three dimensional visualization at the end of simulation for
nonuniform viscosity
3.3.2
When Young’s Modulus is 34 Pa
To see the effect of bomechanical parameter has on dynamics of cells, the elastic
modulus is changed to 34 Pa, which is a value in the range of fibroblasts [5]. The
result obtained for non uniform viscous environment is indicated in Fig. 3.13. It
shows that indeed change of biomechanical parameters do change cell dynamics
as can be seen from center to center distance of cells compared to their 400 Pa
counterpart discussed above. Specifically decreasing the Young’s modulus leads
in a different interplay between two interacting cells with magnitude as well as
Chapter 3. Mathematics of Cell migration
22
Figure 3.12: Three dimensional visualization at the end of simulation for
uniform viscosity of 0.001Pa.s
directions of forces are acting. The magnitude of center to center distance in the
outer layer case , for instance, is oscillating at a higher value than the 400 Pa
counter part (the relatively rigid one). A similar effect can be seen from Figure
3.14 which is the case of uniform viscous environment at viscosity of 0.001 Pa.s.
Another difference with their 400 Pa counter part is, the effect of Brownian motion
is magnified for the case of 34 Pa.
Figure 3.13: Center to center distance(in m) for non uniform viscosity.
Chapter 3. Mathematics of Cell migration
Figure 3.14:
3.3.3
23
Center to center distance (in m) for uniform viscosity of
0.001Pa.s.
Change of Adhesion Parameters
Keeping all the parameters fixed (with E=400 Pa and viscosity of 0.001 Pa.s) and
varying the adhesion parameter σ ( where xo is also kept fixed at 3.0) are found
to influence the migration of cells in such a way that decreasing the strength of
adhesion causes cells to migrate away (almost independently) from the mass of
cells that were initially together, which is a typical behavior of cancer metastasis.
Fig. 3.15
q and 3.16 corresponds to a value of sq=0.5 and 1.0 respectively ( where
sq = πσ2 2 ).
Chapter 3. Mathematics of Cell migration
Figure 3.15: Three dimensional output for sq=0.5
Figure 3.16: Three dimensional output for sq=1.0
24
Chapter 4
Conclusion and future directions
This study has focused to extent where the Brownian motion needs to be present in
the description of cell migration. Using three layers of cells distributed over sphere,
it is shown that the effect of Brownian term gets too small as we switch from low
to high viscous medium, and hence one can neglect the effect of Brownian motion
in dealing with high viscous environment. Biomechanial parameters has also been
found to affect the migration of cells. Specifically increasing a Young’s modulus of
the cells are found to decrease the center to center distance between Cells, which
is an expected behavior that the more the Young’s modulus of the material is,
the more rigid it become. Cell adhesion parameters are also found to influence
the migration of cells: as the adhesion strength decreases, at some point the cells
are observed to detach and move farther from the initial mass of cells-a typical
behavior observed in cancer metastasis suggesting that its therapy should also
focus on the adhesion behavior of cells in order to avoid malignant cells invading
other part of the body. Our model lacks biological processes like mitosis, explicit
representation of cellular deformation and related processes. These phenomena
will be taken care of in the next research along with long time simulation.
25
Appendix A
C++ Code
Here a simple C++ code that is used to generate all the results in the study
is given. Parameters are given with their usual symbols and the functions are
described briefly in chapter 2 section 2.4.
26
Appendix A . C++ code
27
Appendix A . C++ code
28
Appendix A . C++ code
29
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