Random Walks in Random
Environments with Local
Dynamic Interactions
By
Christopher Matthew Baker
A thesis submitted to The University of Melbourne
for the degree of
Master of Science
Department of Mathematics and Statistics
October 2012
ii
© Christopher Matthew Baker, 2012.
Typeset in LATEX 2ε .
iii
A bright idea struck him. “I’ll write a little essay on it,”
he said.
Lewis Carroll
iv
Acknowledgements
I would like to thank my supervisors Professor Kerry Landman and Professor Barry
Hughes for their constant support. I would also like to thank Dr Federico Frascoli,
Bevan Cheeseman and Jack Hywood, who were always happy to give assistance and
advice. Finally I would like to thank the other masters students who helped make the
last two years a much more enjoyable experience.
v
vi
Acknowledgements
Abstract
Cancer cells move through an environment called the extracellular matrix (ECM). Cells
interact with the environment by producing matrix metalloproteinases (MMPs) which
are able to degrade the ECM to allow for greater cell movement. Here we develop a
model framework for such processes. The ECM is represented by a regular lattice with
lattice sites which can be in on of two states, allowed or blocked. Random walkers, which
represent the cells, interact locally with the lattice environment. When attempting to
move to a blocked site a walker has a fixed probability to change the state of the site to
allowed. Initially, the state of each site in the environment is randomly assigned with
some probability, and the state can only be changed through local interactions with
a walker. We look at one-dimensional random walks in random environments with
and without dynamic interactions. Equations for the mean square displacement are
derived, along with other properties of the walks. These quantities are also considered
in higher dimensions. Cancer cell migration is modelled using random walks with
dynamic interactions. Here, the focus is on how quickly cancer cells are able to escape
and invade other parts of the body.
vii
viii
Abstract
Contents
Acknowledgements
v
Abstract
vii
List of Figures
xi
1 Introduction
1
2 Eroding a Uniform Environment
7
2.1
Percolation in One Dimension . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Random Walk in a Static Environment . . . . . . . . . . . . . . . . . .
8
2.3
Erosion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3 Eroding a Random Environment
17
3.1
Size of Allowed and Blocked Regions . . . . . . . . . . . . . . . . . . .
18
3.2
Time Until a Reaching a Blocked Site . . . . . . . . . . . . . . . . . . .
18
3.3
Which Edge of the Cluster . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.4
Mean Square Displacement of the Walk . . . . . . . . . . . . . . . . . .
21
3.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4 Random Walks and Percolation in Three Dimensions
4.1
Percolation in Three Dimensions . . . . . . . . . . . . . . . . . . . . . .
ix
25
26
x
Contents
4.2
Ant in the Labyrinth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.3
Three-Dimensional Random Walks with Local Dynamic Interactions . .
30
4.4
The Eden Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5 Cancer Cell Migration
35
5.1
Cancer Models and Biological Motivation . . . . . . . . . . . . . . . . .
35
5.2
Model for Cell Migration . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.3
Single Cell Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
5.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
6 Conclusion
45
A Proof of the relationship between the radius of gyration and moment
of inertia
47
B Solution of the Linear Advection-Diffusion Equation
49
C Simulation of Random Walks with Local Dynamic Interactions
51
D Measuring Aspects of Three-Dimensional Clusters
55
Glossary
61
References
65
List of Figures
1.1
Depiction of a local dynamic interaction. . . . . . . . . . . . . . . . . .
2.1
Plot of the root mean square displacement of a random walk on a cluster,
showing the trend towards a constant. . . . . . . . . . . . . . . . . . .
2.2
6
11
Fit of the root mean square displacement of a one dimensional random
walk on a cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Plot of cluster size and α. . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.4
Plot of α for various p and ps . . . . . . . . . . . . . . . . . . . . . . . .
14
2.5
Plot of β for various p and ps . . . . . . . . . . . . . . . . . . . . . . . .
15
2.6
Plot of the cluster size over time in the erosion model . . . . . . . . . .
16
3.1
Probability distribution for a walker attempting to make a snip after n
steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.2
Probability for a walker to reach the left side of a cluster before the right. 22
3.3
Plot showing the diffusivity of a walker with varying p and ps . . . . . .
23
4.1
Plot of the divergence of the mean cluster size as p approaches pc . . . .
26
4.2
Plot comparing the radius of gyration and expected moment of inertia
of percolation clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.3
Fit of mean square displacement for the ant in the labyrinth. . . . . . .
31
4.4
Fit of mean square displacement for a three-dimensional random walk
with snippers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
32
xii
List of Figures
4.5
Comparison to Eden clusters. . . . . . . . . . . . . . . . . . . . . . . .
5.1
Schematic of the relative position of the epithelium to the extracellular
34
matrix, the basement membrane, and the endothelium. . . . . . . . . .
37
5.2
Cell movement rules - biased walk on a hexagonal and square lattices. .
38
5.3
Schematic of the cancer cell migration model . . . . . . . . . . . . . . .
39
5.4
Fit of pde to movement distribution . . . . . . . . . . . . . . . . . . . .
42
5.5
Simulation of cell invasion front. . . . . . . . . . . . . . . . . . . . . . .
43
5.6
Speed of cell invasion front for varying paramters. . . . . . . . . . . . .
44
1
Introduction
Biological tissue is made up of two main parts: the cells, and the extracellular matrix
(ECM). The ECM is a gel-like material which provides the structural support for cells,
as well as a pathway for nutrients and oxygen to reach individual cells [1]. Although the
ECM supports cells, it also inhibits movement. Some cells have the ability to produce
enzymes called matrix metalloproteinases (MMPs), which are able to break down the
ECM, and allow greater movement of the cell. MMPs allow cancer cells escaping a
tumour to spread to other parts of the body by making paths through the ECM. We
will model cancer cells moving through the ECM using lattice random walks.
The term random walk was first introduced by Karl Pearson, in a 1905 letter to
the English journal Nature [2]. He was looking for a solution to the problem where a
man starts at a point O, and steps n times, with each step in a random direction and
with equal length. He wanted to know the probability that the man was at a distance
1
2
Introduction
between r and r + δr from the starting point after n steps. The solution, given by Lord
Rayleigh in the following week’s edition of Nature [3], is
2 −r2 /n
e
r dr.
n
(1.1)
In this letter, Pearson introduced what is now known as an off-lattice random walk.
The random walk on a lattice was introduced by Pólya in 1919 [4]. A one-dimensional
random walk can be thought of as a person flipping a coin, taking a step left for each
head, and a step right for each tail. In two dimensions a random walk on a lattice
corresponds to a person wandering a network of streets, choosing at random the next
street to walk down at each intersection.
A well-known problem in random walks concerns the return probability [4]. Suppose
you are out at the pub with a friend who has had a little more to drink than they should
have, and who decides to go for a walk. Your friend, not being on top of his directions
walks randomly, and you want to know the probability your friend will return. It turns
out your friend is certain to return, as long as he is walking in either 1 or 2 dimensions.
A less frequently asked question is, how long do you have to wait for your friend to
return? Unfortunately, you could be waiting a very long time, as the average return
time is infinite.
Random walks have found many applications including in the study of particle
motion and polymer chains [5]. Our focus however will be the applications in the
field of biology. Mathematics has a huge range of applications in biology, from species
population dynamics and the spread of species and diseases, right down to wound
healing, cancer modelling, and even the workings of single cells [6–8].
One of the earliest models in mathematical biology was introduced by Malthus in
his 1798 work An Essay on the Principle of Population [6]. The model was simply that
the rate of change of a population is equal to the number of births minus the number
of deaths, both of which are a proportion of the current population. This leads to the
equation
dN
= bN − dN,
dt
(1.2)
3
with solution
N (t) = N0 e(b−d)t ,
(1.3)
where b and d are the birth and death rates, and N0 is the starting population. The
long-term behaviour of this model is either the decline of a species, or unbounded
growth. It does not allow for a steady solution. Verhulst [6] proposed an adjusted
model, where the environment had a carrying capacity, K, for which
dN
N
= rN (1 − ).
dt
K
(1.4)
This model, often called logistic growth, is much improved, and is possible to fit to real
world population data. It also has a discrete analog, called the logistic map [9], which
has become a famous equation in the study of dynamical systems.
Cellular automata models, also known as agent-based models’ have become a common tool for modelling biological systems, particularly cell movement [10–12]. Cellular
automata simulations are implemented on a lattice, where each lattice site is given a
state [13]. A set of rules is implemented, determining how the state of each lattice
site evolves in time. This often depends on the state of neighbouring sites. One of
the most famous cellular automata models is Conway’s Game of Life [14], which is a
deterministic system. From a specific starting configuration, the result after a certain
number of steps will always be the same.
These deterministic models differ from the cellular automata models used in cell
modelling [13], where the movement rules involve probabilities. If the model is of a
single cell moving randomly, it is equivalent to a random walk on a lattice. It has
been experimentally observed that cell movements over long time appear to be random
walks [15, 16]. Cellular automata models with many cells have also had success in
modelling systems of collective cell movement [17–19].
The simple exclusion process is an example of a model involving interacting cells.
Suppose n cells are placed on a lattice, and at each time step, n cells are chosen to
move randomly. This corresponds to each cell moving, and then being stationary for a
random amount of time before moving again [13]. When considering walks where there
are illegal moves, there are a number of choices for how the walker moves. The two
4
Introduction
standard choices are blind and myopic. We choose to use blind walkers, and is defined
as follows: If a walker is chosen to move, it moves to one of the nearest neighbour sites,
unless the destination is already occupied, in which case the move is aborted.
Partial differential equations can be derived from the movement rules for the average
occupation of sites. This is done by using discrete-time master equations, making a
mean field approximation, and taking the continuum limit. The equation for a single
blind walker turns out to be a simple diffusion equation [4]
∂C
= D0 ∇2 C,
∂t
(1.5)
where C is the concentration, and D0 is the single agent diffusivity. Equations can be
derived for a wide class of walks, and up to a mean-field approximation, the result is
always a non-linear diffusion equation of the form [20]
∂C
= D0 ∇ · (D(C)∇C),
∂t
(1.6)
where D0 is the single agent diffusivity and D(C) is the dimensionless diffusivity factor.
Another approach to get a differential equation describing the global behaviour of
random walkers from the movement rules is to use the Fokker-Planck equation [21]
∂C
∂2 1 2
∂
=
σ (x, t)C −
(µ(x, t)C) ,
(1.7)
2
∂t
∂x 2
∂x
where σ 2 is the variance of the length of a step, and µ is the mean displacement of a
step.
The models investigated here aim to capture the interaction of a single moving cell
with its surroundings. To simulate the ECM, each site on the lattice is given either
the state allowed or blocked. To simulate the action of MMPs, a walker attempting
to move onto a blocked site may change the state of the site, which makes the site
an allowed site. For convenience, we will often refer to this process as snipping, with
a probability ps . If the state of each site is initially determined randomly, with a
probability p of being allowed, and probability of 1 − p of being blocked, the initial
state of the environment corresponds to site percolation.
Percolation theory dates back to the 1950’s and was introduced by Simon Broadbent, who was working on gas masks [22]. There are two models of percolation, site
5
percolation and bond percolation, however we will only consider site percolation. In
site percolation, a site is open with probability p, and blocked with probability 1 − p.
Edges are then drawn between any adjacent open sites. Any group of sites which is
connected by edges is called a cluster. As p is altered, the structure of the clusters
changes [23]. One well-studied problem is the mean cluster size for some value of p.
The point where the mean cluster size becomes infinite is known as the critical point1 ,
and will be denoted pc .
A random walk on a percolation cluster can be interpreted in a similar way to a
random walk on a lattice. If a two-dimensional random walk on a lattice is thought
of as a person walking a network of streets, the intersections of streets correspond to
allowed sites, and the streets are edges. Blocked sites correspond to intersections that
cannot be visited. The effect of snipping can be considered as a walker having the
ability to clear an intersection. We will be considering models where the environment,
the state of each lattice site, is pre-determined, and can only be altered by the walkers
through dynamic interactions.
It is important to note that this differs from models of random walks in dynamic
percolation, such as the one considered in [25]. These models update the entire percolated environment every step. Such an update process can be done a number of ways,
including having the entire environment be randomised, and having the blocked sites
themselves undergo a random walk.
Random walks on percolation clusters have been studied previously [26, 27]. The
original motivation was to use random walks to study the structure of percolation
clusters near criticality [24]. These walks have been studied below, at, and above the
percolation threshold. We will only consider the model below the percolation threshold.
The action of MMPs clearing regions of the ECM is introduced into the model as
local dynamic interactions. A walker attempting to move to a nearest neighbour site
that is blocked is able to move onto the site, and permanently change the state of the
1
The percolation threshold can be defined to be the point at which there is a non-zero probability
of an infinite cluster, or the point where the mean cluster size diverges. However, it is known that
these points are the same for all lattices considered in this thesis [24].
6
Introduction
i-1
Ps
i-1
i
i+1
i
i+1
1-Ps
i-1
i
i+1
Figure 1.1: A walker, at position i, attempts to move to a i + 1 which is blocked. It is
successful with the probability ps , the walker makes the step, and i + 1 becomes an allowed
site. With probability 1 − ps the walker fails to clear this site i + 1, and the move is aborted.
site to allowed with probability ps . The walker is unsuccessful with probability 1 − ps ,
in which case the site remains blocked, and the walker stays on the original site. This
process is depicted in Fig. 1.1.
Chapter 2 discusses random walks in one dimension. We look at the mean square
displacement over time for walks on clusters of allowed sites, which are surrounded
by blocked sites, with and without local dynamic interactions. We also look at how
the clusters change over time due to the these interactions. In chapter 3 we discuss
one-dimensional walks with local dynamic interactions in environments where the state
all lattice sites is initially random; we derive probability distributions for each section
of the walk, and also look at the long term mean square displacement. Chapter 4
discusses random walks on percolation clusters in three dimensions. We again consider
cases with and without local dynamic interactions. Random walk models with local
dynamic interactions are used to study cancer cell migration in chapter 5.
2
Eroding a Uniform Environment
In this chapter we focus on properties of a single random walker in one dimension.
Many results for walks on an infinite domain are known, including the mean square
displacement, return probability and mean number of sites visited [4, 21]. Our focus
will be on the mean square displacement of walks on restricted domains. We will
look at a number of different situations, with the walker starting on a fixed randomly
generated percolation cluster in each case. We will first consider walks without local
dynamic interactions, and then introduce interactions. The interaction allows walkers
to slowly erode the environment, which increases the size of the allowed region.
7
8
2.1
Eroding a Uniform Environment
Percolation in One Dimension
The mean square displacement of a random walk on a cluster is limited by the size of
the cluster on which the walk takes place. As we are going to be considering walks
on percolation clusters, we need to know the mean size of a percolation cluster. The
following derivation is given by Hughes [24], and gives the probability of that there is
a cluster size m over the origin. As usual, each site on the lattice is given the state
allowed with probability p or blocked with probability 1 − p. The probability of there
being a cluster of size m at the origin is the probability of there being m consecutive
allowed sites, pm , times the probability of having blocked sites on either end, (1 − p)2 ,
multiplied by m, the number of possible places for the origin in the cluster. Hence the
desired result is
P(cluster size = m) = mpm (1 − p)2 .
(2.1)
In our model, the walker always starts on a cluster at the origin, so the origin is always
an allowed site. The only difference then is to change the probability of the origin
being allowed from p to 1, and we get
P(walker is on a cluster size m) = mpm−1 (1 − p)2 .
(2.2)
The average cluster size, hcp i, for a given p is calculated in the usual way, by multiplying
(2.2) by m, and summing over all m:
∞
X
∞
(1 − p)2 X 2 m
mp
p
m=0
m=0
"
#
∞
X
d
1+p
d
p
= (1 − p)2
pm =
.
dp dp m=0
1−p
hcp i =
m2 pm−1 (1 − p)2 =
(2.3)
The mean cluster size has a minimum of 1 for p = 0. As p → ∞, the mean cluster size
also tends to ∞.
2.2
Random Walk in a Static Environment
An original result for the mean square displacement is derived for a random walk
in a non-evolving random environment. The mean square displacement for a one
2.2 Random Walk in a Static Environment
9
dimensional random walk is [4]
hRn2 i = n.
(2.4)
In this situation, one would expect the mean square displacement of the walk to increase
like n, until the walker reaches the edge of a cluster. Once this happens, the mean
square displacement should tend towards a constant, which we denote by
2
lim hRn2 i = hR∞
i.
n→∞
(2.5)
To calculate this constant we need the distribution of starting and ending positions
for the walker. We place the walker randomly, so the distribution of starting positions
on a cluster is uniform. To find the distribution for ending positions we solve for the
steady state of the governing equation for a one-dimensional random walk;
pn+1 (l) =
1
[pn (l + 1) + pn (l − 1)] .
2
(2.6)
In one-dimensional random walks there can be an extra consideration. If the walker
always makes a step, then it will always be on an even site on an even step, and an
odd site on an odd step. However, in this model, when a walker attempts a step onto
a blocked site the move is aborted, and so we do not have parity considerations. A
steady state solution to (2.6) must not depend on n, so we write
lim pn (l) = u(l).
n→∞
(2.7)
Substituting into (2.6) we get
u(l) =
1
[u(l + 1) + u(l − 1)] .
2
(2.8)
Making the substitution
u(l) = Aαl
(2.9)
we arrive at the characteristic polynomial
α2 − 2α + 1 = 0
(2.10)
with solution α = 1. As there are repeated roots, the general solution is
u(l) = c1 + c2 l.
(2.11)
10
Eroding a Uniform Environment
Here we invoke left-right symmetry. Over many realisations there will be on average as
many walks that start from the left half of the cluster as the right half. The governing
equation (2.6) is not biased left or right, and so the steady state cannot be biased
toward either side of the cluster. This implies that c2 = 0. Since u(l) describes average
the occupation, so the sum over all lattice sites must add to 1. Hence we arrive at the
solution
1
,
(2.12)
m
where m is the number of sites in the cluster. Now that we know there is an equal
u(l) =
probability of the walker finishing at any site on the lattice, we can sum the mean
square displacement over all starting positions, and all ending positions. From a given
starting site, s, the mean square displacement at large n is
s−1
m−s
1 X 2
1 X 2
1
i +
j =
1 + 2m2 + m(3 − 6s) − 6s + 6s2 .
m i=1
m j=1
6
(2.13)
Now summing over all possible starting positions gives
m
m2 − 1
1 X1
1 + 2m2 + m(3 − 6s) − 6s + 6s2 =
,
m s=1 6
6
(2.14)
which is the long time mean square displacement for a random walk from a random
starting position on a cluster size m. From equation (2.3), the probability of the cluster
being size m is
mpm−1 (1 − p)2 .
(2.15)
2
To obtain hR∞
i, the probability of a cluster being size m is multiplied by the long term
mean square displacement for a walk on a size m cluster, and then summed over all
possible cluster sizes. This gives
2
hR∞
i
=
∞
X
mpm−1 (1 − p)2
m=1
m2 − 1
p
.
=
6
(1 − p)2
(2.16)
This analytic result is compared to simulation results in Fig 2.1. A very good match
between the long term root mean square displacement1 of a random walk, and the
square root of Eq. (2.16) is observed.
1
Although we have been discussing mean square displacement, it turns out root mean square
displacement is more natural to use for a one-dimensional random walk on a cluster.
11
Root Mean Square Displacement
2.2 Random Walk in a Static Environment
5
p=0.8
4.5
4
3.5
3
2.5
p=0.6
2
1.5
p=0.4
1
0.5
0
0
100
200
300
400
500
Steps
600
700
800
900
1000
Figure 2.1: Plot of root mean square displacement over 1,000 steps and 50,000 realisations. The solid lines are simulation results, while dashed lines are the square root of
Eq. (2.16). This shows the root mean square displacement tending towards the expected
long term value.
For an unrestricted random walk, the root mean square displacement increases at
the rate of the square root of the number of steps taken. A random walk on a cluster
behaves in the same way, until the edge of the cluster is reached, when the mean
square displacement saturates, and tends to the result of equation (2.14). On average,
the starting position of a walker is the center of the cluster, and so would be expected
√
to be able to reach the edge of a cluster with radius less than n in n steps. This
suggests that we can break up the equation for the root mean square displacement into
two parts, given we know the behavior before and after reaching a boundary. All that
is needed is the probability of a walker having hit the boundary.
The probability that the number of sites in a cluster is q or less can be calculated
by the following sum:
q
X
q
d X m
P(cp ≤ q) =
m(1 − p) p
= (1 − p)
p
dp
m=0
m=0
q+1
d
p
−
1
= (1 − p)2
= qpq+1 − (q + 1)pq + 1.
dp
p−1
2 m−1
2
(2.17)
Next we need to know what the mean square displacement will be, given the cluster
size is less than size q. This is done by modifying equation (2.16), to have an upper
12
Eroding a Uniform Environment
60
Root Mean Square Displacement
p=0.99
50
40
30
p=0.95
20
p=0.9
10
p=0.8
0
0
500
1000
1500
2000
2500
Steps
3000
3500
4000
4500
5000
Figure 2.2: Plot of the root mean square displacement of a walker on over 5,000 steps
and 50,000 realisations. The solid lines are plots of Eq. (2.19), with simulation data plotted
on top as black dashed lines.
terminal of q rather than ∞:
2
hR∞
|cp ≤ qi
q
X
m2 − 1
=
mpm−1 (1 − p)2
6
m=1
=
6p − pq [6p + {2 + p (3 − 6p + p2 )} q + 3(p − 1)2 q 2 + (1 − p)3 q 3 ]
.
6(p − 1)2
(2.18)
We approximate the root mean square displacement by multiplying the probability
√
that the walker has not reached a side of the cluster by n, and add the probability
that the walker has reached an edge of the cluster multiplied by the expected size of
the cluster:
p
hRn2 i ∼
q
√
√
√
√
2 |c ≤ 2 ni P(c ≤ 2 n).
n P(cp > 2 n) + hR∞
p
p
(2.19)
Eq. (2.19) fits simulation data extremely well, and this is shown in Fig. 2.2. For small
√
n, the root mean square displacement grows like the n, until the walker explores the
2
cluster, and then approaches hR∞
i.
13
2.3 Erosion Model
70
5
60
4.5
cluster size
50
4
40
α
3.5
30
3
20
2.5
10
0
1000
2000
n 3000
4000
5000
2
0
1000
2000
n
3000
4000
5000
Figure 2.3: Plot of the mean cluster size (left) and α (right) for the one-dimensional
erosion model, with p = 0.8 and ps = 0.8 over 5,000 steps and 100,000 realisations
2.3
Erosion Model
We develop a new model, which we call the erosion model. This model is a random
walk starting on a cluster, just as the previous model, but the environment now evolves
through local dynamic interactions with the walker. When a walker attempts to move
to a blocked site, it is able to move to and to change the state of the site from blocked
to allowed with probability ps . With probability 1 − ps the move is aborted. For
convenience we call this snipping. Initially, the walker is placed on a percolation cluster,
with all surrounding sites given the state blocked. Here we derive an original equation
for the changing size of the cluster.
As the walker is placed randomly, there is a 2/l chance of landing adjacent to the
edge, where l ≥ 2 is the current cluster size. The walker has a probability 1/2 of
moving towards the edge, and attempting a snip, which has the probability of success
of ps . Therefore, the change in the cluster size should be
dl
αps
=
,
dn
l
(2.20)
l2 = αps n + l02 ,
(2.21)
giving
where the value of l0 , the starting size of the cluster, is the mean cluster size from
Eq. (2.3). The prefactor, α could depend on p and ps and is found by simulation.
14
Eroding a Uniform Environment
2.7
2.6
2.5
α
p=0.9
2.4
2.3
2.2
p=0.8
2.1
2
0.1
p=0.1-0.7
0.2
0.3
0.4
0.5
ps
0.6
0.7
0.8
0.9
Figure 2.4: Plot of α for various p and ps from simulations of 10,000 steps and 15,000
realisations.
Eq 2.21 can be re-arranged to
α=
l2 − l02
.
ps n
(2.22)
The value of α does approach a limiting value for large n as seen for one p and ps
in Fig. 2.3, which is a good representative for all choices of p, ps . As seen in Fig. 2.4, α
is dependent on ps , and is only dependent on p for sufficiently large p. In formulating
Eq. (2.20) we assumed that the rate of change of the cluster size would be linear. If
remove this assumption, and replace αps with β, Eq. (2.20) becomes
β
dl
=
,
dn
l
(2.23)
l2 = βn + l02 .
(2.24)
with solution
This can be rearranged to get an equation for β:
β=
l2 − l02
.
n
(2.25)
As can be seen in Fig. 2.5, p has very little influence on β. A quadratic in ps does a
15
2.4 Conclusion
2.5
p=0.9
2
β
p=0.1 to 0.8
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
ps
0.6
0.7
0.8
0.9
Figure 2.5: Plot of β for various p and ps from simulations of 10,000 steps and 15,000
realisations.
good job of fitting the curves:
β = 0.00436 + 2ps + 0.547p2s .
Substituting (2.26) into Eq. (2.21) and taking the square root gives
s
2
1+p
2
l(n) = (0.00436 + 2ps + 0.547ps )n +
.
1−p
(2.26)
(2.27)
The accuracy of this is shown in Fig. 2.6. The fit is very good when ps is sufficiently
large.
2.4
Conclusion
This chapter covers results of percolation theory and random walks in one dimension.
A new equation for the root mean square displacement of a walk on a restricted domain
is derived. The environment is generated at random, and the walker does not alter it
during the walk. The equation fits very well with simulation data. The new erosion
model is introduced, which includes local dynamic interactions between the walker
and the environment. Throughout the walk, the walker is able to increase the size of
the allowed region by locally changing the state of sites from blocked to allowed. An
equation describing the growing cluster size over time is derived for this model.
16
Eroding a Uniform Environment
140
cluster size
120
p=0.1, ps=0.8
100
80
p=0.5, ps=0.3
60
40
p=0.9, ps=0.1
20
0
0
2000
4000
n
6000
8000
10000
Figure 2.6: Plot of the cluster size from simulations of 10,000 steps and 15,000 realisations. The dashed lines show the fit from equation (2.27).
3
Eroding a Random Environment
In this chapter we discuss properties of a one-dimensional random walk in an environment where the state of every site is initially random, and local dynamic interactions
are allowed. This is similar to the erosion model discussed in section 2.3, where each
lattice site is given one of two states, allowed or blocked. The key difference is that in
the erosion model, all sites not in the initial allowed cluster are blocked. We are now
considering and environment where there are many clusters of allowed sites, separated
by blocked sites. The walker is able to open up the blocked regions, and gain access to
new clusters of allowed sites. We find probability distributions for the expansion of the
cluster, the number of steps before interaction with the environment, and on which side
of the cluster interactions take place. We also look at the mean square displacement
of a random walk in the eroding environment.
17
18
3.1
Eroding a Random Environment
Size of Allowed and Blocked Regions
From Eq (2.2), we know the probability distribution for the initial cluster size is
P(walker is on a cluster size m) = mpm−1 (1 − p)2 .
(3.1)
Using the same logic used to derive this expression, the distribution for the length of
the blocked region at one of the edges of the inital cluster is
P(blocked region is size m) = (1 − p)m−1 p,
(3.2)
and the neighbouring cluster size is given by
P(neighbouring cluster is size m) = pm−1 (1 − p).
(3.3)
These two equations are related by replacing p with 1−p, and are valid when 0 < p < 1.
Both distributions are strictly decreasing in m for a fixed p. Expanding Eq. (3.2) we
get
(1 − p)m−1 p = p{1 − (m − 1)p + ...}.
(3.4)
If p 1/(m − 1), then it is essentially independent of m. This means that clusters
of varying sizes occur with very similar probability. As p increases, small clusters are
favoured. In the limit as p approaches 1, the probability that the blocked region is size
1 goes to 1, and the probability that the blocked region is larger than one goes to zero.
3.2
Time Until a Reaching a Blocked Site
The time until a walker reaches the boundary of a cluster and interacts with it is
equivalent to a random walk on a lattice with absorbing boundaries. In the absorbing
boundary walk, if a walker steps onto one of the boundaries it is removed from the
system. If we know the probability that a walker is alive at step n, that means that it
has never reached an edge. In our case, this means that the walker is yet to interact
with the boundary. The solution for the probability distribution for a random walk
with absorbing boundaries has been determined by Hughes [4] using the method of
19
3.3 Which Edge of the Cluster
images. The probability of the walker being at site s after n steps, in a walk starting
from the origin, with absorbing boundaries at −l and r is
X
1
πml
πm(s + l)
πm
n
†
sin
sin
.
Pn (s) =
cos
l+r
l+r
l+r
l+r
(3.5)
|m|<l+r
Using this result, we sums over all the sites on the lattice. This gives the probability
of not being absorbed after n steps:
r−1
X
s=−l+1
Pn† (s)
[2j + 1]π
cos
l+r
|2j+1|<l+r
[2j + 1]πl
[2j + 1]π
× sin
cot
.
l+r
2[l + r]
1
=
l+r
X
n
(3.6)
The difference between the probability of not having been absorbed at step n and
step n − 1 gives the probability that the walker is absorbed at step n,
r−1 n
o
X
†
†
Pn (s) − Pn−1 (s)
=
s=−l+1
[2j + 1]π
[2j + 1]π
cos
1 − cos
l+r
l+r
|2j+1|<l+r
[2j + 1]πl
[2j + 1]π
× sin
cot
.
(3.7)
l+r
2[l + r]
1
l+r
X
n−1
The details of these calculations are given by Hughes [4]. Fig. 3.1 gives an example of
how Eq. (3.7) behaves over time. The non-monotonic behaviour is because the distance
to the left edge is odd, and the distance to the right is even.
3.3
Which Edge of the Cluster
In some cases the choice of which side the cluster is reached first is implicit in time.
From the previous result, in a particular simulation, if the walker reaches an edge after
t steps, but the far boundary is more than t steps away, the walker must be situated
at the near boundary. As well as this, without hitting an edge, the walker cannot
switch from being on even sites on even steps to odd sites. Therefore, if one edge is an
odd number of steps away, and the other is even, then the side the walker reaches is
determined by the parity of t.
20
Eroding a Random Environment
0.12
0.10
0.08
0.06
0.04
0.02
5
10
15
20
25
Figure 3.1: The probability distribution (3.7) for the probability of a walker snipping
after n steps, with l = 3 and r = 4.
If the walker could reach either edge, in time t, and both edges are either an even
distance or both an odd distance from the initial position, the following argument can
be used to find the probability of the walker being at one edge over the other.
The size of the cluster is given by r, and without the loss of generality, we will
assume the left side of the cluster is at lattice site 1, and the right side is at site r. The
probability that a walker arrives at site 1 before reaching site r, from a starting site s
will be written as p(s). The boundary conditions to the problem are
p(1) = 1,
p(r) = 0.
(3.8)
From symmetry we also have
p(s) + p(r − s + 1) = 1.
(3.9)
The quantity p(s) can also be written in terms of neighbouring sites:
1
p(s) = [p(s − 1) + p(s + 1)].
2
(3.10)
3.4 Mean Square Displacement of the Walk
21
Now, we can use the boundary condition at s = 1 to solve for p(s) in terms of p(s + 1).
p(1) = 1,
1 1
+ p(3),
p(2) =
2 2
1
1
p(3) =
p(2) + p(4),
2
2
1 1
1
p(3) =
+ p(3) + p(4),
4 4
2
1 2
p(3) =
+ p(4).
3 3
(3.11)
By induction it is easy to establish that
p(s) =
1 s−1
+
p(s + 1).
s
s
(3.12)
This can be solved, using the boundary condition p(r) = 0 to get
p(s) =
r−s
r−1
(3.13)
As can be seen in Fig. 3.2, this fits extremely well with simulation data. If we
actually consider site 1 and site r as blocked sites, then p(s) gives the probability that
the next attempt at snipping is made at the left boundary.
3.4
Mean Square Displacement of the Walk
As we have discussed so far in this chapter, the behaviour of this walk is much more
complicated than the erosion model. However, we can still develop a heuristic argument
to describe the behaviour of the walk. As long as the snipping probability, ps , is greater
than zero, the walker is never trapped on a cluster, and will always be able to explore
new territory. Essentially, the walker is just being delayed, and so the mean square
displacement should be linear, as in an unbiased random walk. As we expect the walk
to be slower, due to having to clear blocked site, we include an unknown pre-factor:
hRn2 i = Dn.
(3.14)
The coefficient D is dependent on p and ps , and this is shown in Fig. 3.3. This data was
input to a Eureqa to search fora functional form of D(p, ps ). Eureqa is a program which
22
Eroding a Random Environment
1
4
Simulation data
p(s)
Probability of reaching the left side
before the right
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
4
0
2
4
6
8
10
12
Starting position
14
16
18
20
Figure 3.2: Plots of simulation data of random walks on finite lattices, and the percentage
of times when the walker reaches the left boundary first, dependant on the starting position.
Overlaid is the fit from equation (3.13). The plot is from a cluster with 40 sites.
searches for mathematical relationships between sets of input data [28]. However, it
was unable to find any simple or informative relation.This plot suggests that ps is the
most important parameter. While ps = 0.1, D remains small for all values of p. This
is contrasted to the case when p = 0.1. Here, increasing ps also greatly increases D.
While ps = 0.9, the value of p has very little influence on D.
3.5
Conclusion
We have discussed the probabilities involved with each stage of a one-dimensional random walk with local dynamic interactions. Every state in the environment is initially
assigned the state allowed or blocked. The walker starts on an allowed site, and erodes
the walls of its cluster. This opens up new regions of allowed sites. We have been able
to find exact probability distributions which describe each section of a random walk.
It could be possible to use these distributions to simulate a walk by drawing random
23
3.5 Conclusion
1
0.9
0.8
0.8
0.7
0.6
D
0.6
0.5
0.4
0.4
0.2
0.3
0.2
0
1
0.5
ps
0
1
0.8
0.6
0.4
0.2
0
0.1
p
Figure 3.3: Surface plot of D with varying p and ps , from simulations with 10,000 repeats
over 1,500 steps.
numbers to find when and where each event occurs. We also look at the mean square
displacement of the walk, and find it is linear in time, and dependent on p and ps .
24
Eroding a Random Environment
4
Random Walks and Percolation
in Three Dimensions
In three dimensions, exact percolation results become very hard to come by. The mean
cluster size is simple to calculate in one dimension, but in three dimensions we do
not even know the critical point exactly, and so we certainly don’t know cluster size
distributions. There has been much work estimating critical exponents and thresholds,
and less work on other aspects.
In 1976 de Gennes dubbed the model of a random walker moving on a percolation
cluster to be an ant in the labyrinth [24]. His idea was that the path of the ant would
give some insight into the structure of the cluster. An equation for the mean square
displacement for an ant in the labyrinth has been proposed, but there is some disagreement about the value of the parameters. We introduce local dynamic interactions
25
26
Random Walks and Percolation in Three Dimensions
70
Mean cluster size
60
50
40
30
20
10
0
0.05
0.1
0.15
p
0.2
0.25
0.3
0.35
Figure 4.1: Mean cluster size as a function of p. The black line is the equation(4.1), with
γ = 1.795
into the model, and we will look at how the mean square displacement evolves over
time. We will also compare how the resulting clusters relate to Eden clusters, which
are clusters arising from a different cluster growth algorithm.
4.1
Percolation in Three Dimensions
The most basic aspect to look at is the number of sites in a cluster. The average number
of sites in a cluster for a given p, the probability that a site is allowed, is denoted χ(p).
There is no exact expression for χ, but the asymptotic behaviour as p → pc is given by
Hughes [24] as
χ(p) ∼
= |p − pc |−γ .
(4.1)
The percolation threshold pc and critical exponent γ are not known exactly, but recent
estimates, for the simple cubic lattice considered here, give pc = 0.3116004 [29] and
γ ≈ 1.795 ± 0.005 [30]. Although this is only valid near pc , it works remarkably well for
0 < p < pc , and this is observed when plotted alongside simulation results in Fig. 4.1.
The constant pre-factor is obtained by insisting that χ(0) = 1, which gives
χ(p) ≈ pγc |p − pc |−γ
(4.2)
27
4.1 Percolation in Three Dimensions
Although we have a good approximation for χ(p), the size of a cluster gives us no
indication about its shape. The radius of gyration gives a measure of how spread out
a cluster is, but is actually a special case of the moment of inertia. The moment of
inertia describes how an object’s mass is spread out around a certain point rp . The
square moment of inertia is defined as
N
1 X
I =
(ri − rp )2 ,
N i=1
2
(4.3)
where ri is the position of the ith particle. The radius of gyration is determined by the
moment of inertia, measured around the centre of mass rcm , where
rcm
N
1 X
=
ri .
N i=1
(4.4)
The radius of gyration, Rg is defined by
Rg2
N
1 X
=
(ri − rcm )2 .
N i=1
(4.5)
This gives a good way to compare clusters of equal size to determine which is more
compact than another, but is not quite so useful for comparing clusters of varying
sizes. Hoshen [31] provides a measure for the compactness of a cluster, which takes
into account varying sizes. The cluster compactness, e, is calculated in terms of the
minimum and maximum possible radius of gyration of a cluster with a given number of
sites. The minimum radius of gyration is derived by looking at the radius of gyration
of a sphere, which gives
2
Rs,min
3
=
5
3s
4π
2/3
,
(4.6)
for a cluster with s sites. The maximum radius of gyration of a cluster occurs when all
s sites are in a line, which gives
2
Rs,max
=
s2
.
12
(4.7)
The coefficient of compactness is defined as [31]
2
(Rg2 )−1 − (Rs,max
)−1
e=
.
2
2
(Rs,min
)−1 − (Rs,max
)−1
(4.8)
28
Random Walks and Percolation in Three Dimensions
The coefficient of compactness varies between 0 for a linear cluster, and 1 for a spher2
is only valid for large clusters, as it
ical cluster. It is important to note that Rs,min
approximates the cluster as a sphere. Hoshen shows that it can be used accurately for
clusters with more than 40 sites.
4.2
Ant in the Labyrinth
The model of a random walk on a percolation cluster without local dynamic interactions
is known as the ant in the labyrinth . The walker is confined to the percolation cluster
that it started on, and it explores the cluster over time. Hence, the mean square
2
as each walker
displacement of the walk, Rn2 initially increases, and asymptotes to R∞
fully explores the cluster [24]. This has prompted a fit of the form [32]
w n
2
2
hRn i = R∞ − B exp −
.
θ(p)
(4.9)
Originally, Mitescu et al. [4] assumed w = 1. It was later found that w = 0.4 ± 0.1
gives a better fit, and there has been speculation that w = 2/5 in all dimensions [26].
The quantity θ is given by a critical exponent:
θ(p) ∼
= |pc − p|y .
(4.10)
2
Estimates for y are 2.8 ± .4, by Mitescu et al. [4] and 2.1 ± .3 by Vicsek [33]. The R∞
has also been given in terms of a critical exponent:
2 ∼
R∞
= |pc − p|κ ,
(4.11)
with the most recent numerical estimate given κ = 1.32 ± 0.13 [34]. However, I have
found in my own simulations, unlike equation (4.1) for the mean cluster size, this does
2
not work well away from pc . Instead, it is possible to relate R∞
to properties of the
cluster.
At large times, the probability of finding the walker at any individual site in the
cluster is uniform. Therefore
2
R∞
N
1 X
=
(ri − rs )2 ,
N i=1
(4.12)
29
4.2 Ant in the Labyrinth
25
Square Moment
of Inertia
20
15
Square Radius
of Gyration
10
5
0
0
0.05
0.1
0.15
p
0.2
0.25
0.3
0.35
Figure 4.2: The average square radius of gyration and moment of inertia of threedimensional clusters for varying p with 500,000 repeats. The moment of inertia is twice
the radius of gyration.
where rs is the starting position of the walk, and the sum is over all the possible sites
of the cluster, each given equal weight. This is actually the square moment of inertia,
measured around the start of the walk. This is just for a single possible starting point,
on one particular cluster. In each realisation of the ant in the labyrinth, the starting
2
point and cluster are random, so to describe the average behaviour of R∞
, we need the
expected value of the square moment of inertia. This involves another sum, over the
possible starting positions,
2
R∞

!2 
N
N
X
X
1
 ri − 1
{rj }  .
=
N i=1
N j=1
(4.13)
As hinted in Fig. (4.2), and as noted by Hughes [24], citing Mitescu and Roussenq [35],
and Roussenq [36], there is a relationship between hI 2 i and Rg2 , namely hI 2 i = 2Rg2 .
This is easy to see in one dimension. We already know the square radius of gyration
from equation (4.7), which is
Rg2 =
N2 − 1
.
12
(4.14)
30
Random Walks and Percolation in Three Dimensions
The expected moment of inertia is a double sum over all the squared distance between
all the possible starting and finishing points:
N
N
N2 − 1
1 X 1 X
(i − r)2 =
.
hI i =
N r=1 N i=1
6
2
(4.15)
Therefore, in one dimension at least,
2
R∞
= hI 2 i = 2Rg2 .
(4.16)
Now we can look for the same result in three dimensions. It turns out that the calculation can be followed through without making any assumption about the dimensionality,
or whether the particles are in a cluster or not. The proof of this is given in appendix
A. Again, there is no explicit equation for Rg2 in three dimensions. However, it is not
hard to calculate during simulations, and so for any set of simulations of the ant in the
2
labyrinth, R∞
is known exactly.
Equation (4.9) can be fitted to simulation data, which gives estimates for the critical
exponents w and y. The software package Eureqa is used to do this [28]. To find the
critical exponents we set the target equation to be
ω n
2
2
2
R∞ − hRn i = R∞ exp −
.
A(pc − p)y
(4.17)
where A, w and y are to be solved for. Eureqa produced A = 0.0582, w = 0.473 and y =
2.98. These all fall within previous estimates. Eq (4.17) is plotted with simulation data
in Fig. 4.3.
4.3
Three-Dimensional Random Walks with Local
Dynamic Interactions
We now introduce local dynamic interactions into the model of the ant in the labyrinth.
When a walker attempts to step onto a site which is blocked, it has probability ps to
successfully make the step, and change the state of the site to be allowed. The walker
fails, and the move is aborted with probability 1 − ps . We again refer to this process
4.3 Three-Dimensional Random Walks with Local Dynamic
Interactions
31
12
mean square displacement
10
8
6
4
2
0
0
200
400
600
800
1000
1200
1400
1600
steps
Figure 4.3: Plot of simulations of the mean square displacement of the ant in the
labyrinth, along with the fit from Eq. (4.17), with 50,000 realisations.
as snipping for convenience. The mean square displacement of an unrestricted random
walker in 3 dimensions is known [4] to be
hRn2 i = n,
(4.18)
and so an equation for the mean squared displacement on a cluster with snipping must
return to this when either p or ps are equal to zero. A reasonable first guess would be
to multiply n by the probability of making a successful step under the assumption that
the walker never returns to any site. This leads to
hRn2 i = (p + (1 − p)ps )n.
(4.19)
When this is compared to simulation results, it fares poorly, even for large p and ps .
However, we can again turn to Eureqa, which alters the equation to
hRn2 i = (p + (1 − p)ps )1.55 n.
(4.20)
32
Mean Square Displacement
Random Walks and Percolation in Three Dimensions
1500
p=0.7
ps=0.8
p=0.2
ps=0.8
1000
p=0.7
ps=0.2
500
0
200
400
600
800 1000 1200 1400 1600
Step
Figure 4.4: Shows the fit of equation (4.20) (black) for three parameter values with
simulation data (blue) from 10,000 realisations.
The parameter p + (1 − p)ps is the probability that the walker will be able to move onto
an unknown site. If the walker is moving onto a site which it has attempted to move
to before, this is inaccurate. For this equation to fit well, the parameters must be such
that the walker attempts to move to one site repeatedly very rarely. This corresponds
to either p or ps being large. The accuracy of this is shown in Fig (4.4).
4.4
The Eden Model
The Eden Model for cluster growth was first introduced by Murray Eden in 1961 [37,
38]. In this model, the cluster starts as a single lattice site, and then at each time
step, one site adjacent to the cluster is added at random. We can compare the types of
clusters generated by Eden growth to the clusters that are formed by a walker, eroding
the environment. If the snipping probability ps is small, then there is usually a large
amount of time between successive snipping events. At large time, the probability of
being on any cluster site becomes uniform. Therefore, one might expect that with
small ps , the cluster would be similar to an Eden cluster of the same size.
To make a comparison, we will use the moment of inertia of the clusters, along with
4.5 Conclusion
33
the coefficient of compactness (4.8). A comparison between the two types of clusters
for the square moment of inertia is plotted in Fig 4.5, for ps = 0.2. The coefficient of
compactness is not used for this ps as the clusters are not large enough for this to be
useful. The plot shows that the moment of inertia of the two types of cluster are very
similar as the cluster grows. The moment of inertia of the two types of cluster is quite
similar when ps < 0.4. With a higher snipping probability, it is much easier to obtain
data for larger clusters, which enables us to use the coefficient of compactness. For
ps = 0.6 the moment of inertia measured around the origin no longer matches, but the
radius of gyration is very similar. This suggests that with higher ps , the clusters can
still be similar, but are likely not to be centered around the origin. These clusters also
have a very similar coefficient of compactness to the Eden clusters, with e = 0.432.
4.5
Conclusion
The mean square displacement of a random walk on a percolation cluster, the ant in
the labyrinth model, has been fitted using a previously suggested form. The critical
exponents found here all fall within previous estimates. Introducing dynamic local
interactions into the model allows the walker to escape its initial cluster. We observe
through simulation that the mean square displacement is indeed linear, which depends
on both the percolation and probability of clearing blocked sites. The clusters produced
from a random walk with local dynamic interactions, starting from a single site with
every other site blocked, produces clusters which are similar in structure to those
produced from Eden growth.
34
Random Walks and Percolation in Three Dimensions
6
Square Moment
of Inertia
5
4
3
2
1
0
30
20
Cluster Size
10
0
200
400
600
800
1000
Step
1200
1400
1600
1800
2000
Figure 4.5: The solid line is the square moment of inertia of a cluster produced by a
random walker with snippers. The dashed lines is the same quantity for Eden clusters of the
same size. The parameters for the simulations were p = 0, ps = 0.2, with 5,000 realisations.
5
Cancer Cell Migration
We have discussed random walks in environments with and without local dynamic
interactions. We are now ready to apply this to a model of cancer cell migration.
Blocked sites represent regions of the extracellular matrix (ECM) which are too dense
for a cancer cell to move through. Cells moving through the environment are given a
probability to clear blocked areas, and this represents the action of matrix metalloproteinases (MMPs). We will be concerned with how quickly cells are able to escape the
tumour, and find vessels.
5.1
Cancer Models and Biological Motivation
Computer, or in silico, models of cancer growth have become widespread, and there are
many modelling approaches [11]. These range from cellular automata models, where
35
36
Cancer Cell Migration
each cell can either take up one lattice site, or many lattice sites, which allows for cell
deformation, to off-lattice models, which can handle a greater variety of cell and cluster
shapes [39].
Before discussing a model, we need to further investigate some of the biological
processes involved. The process of cells escaping the primary tumour, and forming a
new tumour in another nonadjacent part of the body, is called metastasis. This is the
process we aim to model.
We consider the most common class of cancers, carcinomas, which are cancers
derived from epithelial cells. Epithelial cells comprise the epithelium, which lines many
surfaces and cavities in the body. Breast, prostate, lung and skin cancer are examples
of carcinomas. Adjacent to the epithelium is the basement membrane, which is a thin
fibrous layer of tissue which marks the boundary between the epethelium and the
stroma.
The supportive structure of the ECM (extra celllular matrix) is made up of a variety
of fibrous proteins, collagen, elastin, fibronectin and laminin [1]. Collagen and elastin
are of the greatest importance to us, as these are the fibres that are cleared out to
allow greater cell movement. Laminin is mostly contained in the basement membrane
while fibronectin is important for cell adhesion to the ECM.
Collagen, in various forms, is found all around the body, and is the most important
protein in the makeup of bone, tendons and skin. Collagen is formed into long strands
called fibrils, which are then bundled into fibres [40]. Elastin also forms fibres, but
instead of having the high tensile strength of collagen, are elastic and so deform under
a force, but return the their original shape one the force is released.
For a cancer cell to invade the stroma, it must first undergo an epithelial-mesenchymal
transition (EMT). This process exchanges the e-cadherin proteins on the cell surface
with integrins, which are proteins capable of binding to the ECM [41, 42]. This is required because e-cadherin is involved in the strong binding between adjacent epithelial
cells, and therefore shedding these proteins allows cells to break off from the cancer.
The basement membrane must also be breached before cells are able to escape the
5.1 Cancer Models and Biological Motivation
Epithelial Cells
Basement
Membrane
Endothelium
37
Cancer Cells
Extracellular Matrix
Blood Vessel
Extracellular Matrix
Figure 5.1: Schematic of the relative position of the epithelium to the extracellular
matrix, the basement membrane, and the endothelium.
tumour.
Once leading cancer cells have invaded the stroma, they begin to migrate away from
the tumour. During migration, they break down fibres in the ECM, and leave chemoattractive pathways for other cells [42]. There are 25 types of matrix metalloproteinases,
many of which play a role in metastasis. However, we will focus on MMP-2, MMP-9
and MT1-MMP, also known as MMP-14.
MMP-9, along with most MMPs, is secreted by cells, that is, it is made by the cell
and then pushed out into the environment. In most situations, MMPs are secreted in
an inactive state, and need to be activated before they can begin proteolysis. MMP-9
is able to degrade a number of different proteins, but its major role is the breakdown
of elastin [41]. MMP-2 is able to break down fibronectin and vitronectin, which are
proteins that form bonds between collagen and elastin fibers. MT1-MMP belongs to
a special class of MMP’s called membrane tethered MMPs. These MMPs are not
secreted out into the ECM, but rather held on the surface of a cell. MT1-MMP has
many functions, but the relevant functions in this context is its able to degrade fibrillar
collagens, and also activate MMP-9 [41, 42].
The process of cancer cell movement through three-dimensional fibrillar ECM has
been described by Wolf and Friedl [43]. Pseudopodia, which are temporary projections
from a cell, of the leading edge invade and attach to the ECM, allowing MT1-MMP to
38
Cancer Cell Migration
Figure 5.2: The two lattice options in two dimensions. The current position of the cell is
marked by the dark blue, and the previous position of the cell is marked by light blue. The
hexagonal lattice (left) allows three movement options at each step, while the square lattice
(right) allows only two movement options at each step.
begin to degrade the matrix. The widest part of the cell is the nucleus, and so as the
cell moves forward, the space created by the leading edge needs to be widened. The cell
develops psuedopodia, which, with the help of MT1-MMP, create enough space for the
cell to move forward. The cell leaves a trail of MT1-MMP and MMP-2 which cause the
fragmented collagen to be re-aligned to be parallel to the cell, forming a micro track,
10-15µm in diameter, which matches with the size of a cell. This leaves a pathway for
future migration of cancer cells.
5.2
Model for Cell Migration
We choose to model cancer cell migration as an exclusion process, and insist that cells
are able to move past each other. Cells cannot move past each other in one dimension,
and we therefore choose to model this in two dimensions. The leading cancer cells
move through the ECM away from the tumour, and leave a track just large enough for
a cell to fit through. We choose a hexagonal lattice, which allows a cell to move in 3
different directions, which are all away from the cancer. This is opposed to a square
lattice, where there are only two directions a cell can move while always moving away
from the tumour. This difference between these lattices is depicted in Fig. 5.2.
39
5.2 Model for Cell Migration
Vessel
Tumour
Figure 5.3: Schematic of the model. The cancer cell, shown in purple, moves away from
the tumour, in three possible directions. We are interested in how long it will take to reach
the vessel.
We consider a cell leaving a tumour, and performing a directed random walk towards
a blood vessel. This is depicted in Fig 5.3. Initially, all the sites are blocked to represent
a dense ECM, requiring the cell to clear a path. In each time step, the cell attempts
to move, and does so with probability ps , also clearing the destination site. We are
interested in the time it takes the cell to move a distance x horizontally. Diagonal steps
only move half the distance to the right as a horizontal step, so they are labelled s for
short steps, while horizontal steps are l for a long step. A long step moves the cell two
spaces to the right.
To begin with, we can ask the question: What is the average time for a cell to move
a distance x to the right? We know that in each successful step, the cell moves one
space to the right with probability 2/3, or two spaces to the right with probability 1/3.
Therefore, in a successful step, a walker will move on average 4/3 to the right. The
walker makes a step with probability ps , and so the distance x moved on average after
t is
4
x = ps t.
3
(5.1)
40
Cancer Cell Migration
We can also determine the probability distribution for moving distance x after t
time steps. The first step is to find the probability distribution for moving distance x
after n moves. Firstly we need the number of short and long steps. This is given by
the following equations;
n = l + s,
x = 2l + s
(5.2)
l = x − n,
s = 2n − x.
(5.3)
which rearrange to
So the probability P (x, n) of the walker being at x after n steps is
2n−x x−n
n
2
1
P (x, n) =
.
x−n
3
3
(5.4)
Now we consider the probability, Q(n, t), of making n successful steps in t time steps;
t n
Q(n, t) =
p (1 − ps )t−n .
(5.5)
n s
The probability of being at position x after making n moves, in t time is P (x, n)Q(n, t).
To get the distribution for the probability of being at x after a certain amount of time,
n must be summed out;
P (X = x|t) =
x
X
P (x, n)Q(n, t)
(5.6)
n=d x2 e
2n−x x−n x
X
n
2
1
t n
=
ps (1 − ps )t−n
x
−
n
3
3
n
n=d x2 e
x x
d2e
t
t−d x2 e d 2 e
2d x2 e−x −d x2 e
3
(1 − ps )
ps
= 2
x
d 2 e x − d x2 e


x
x
1, d 2 e − t, d 2 e − x
−ps 
.
×3 F2 
;
1
− x + d x e, 1 − x + d x e 3(ps − 1)
2
2
2
2
(5.7)
(5.8)
2
The final line of working was done using Mathematica. The function 3 F2 is a generalised
hypergeometric function, which are defined as [44]


∞
X
a1 , ..., ap
(a1 )k ...(ap )k z k


F
;
z
=
.
p q
(b
1 )k ...(bq )k k!
b1 , ..., bq
k=0
(5.9)
41
5.2 Model for Cell Migration
As verification, we can calculate the expected distance travelled in a given time by
evaluating the following sum
2t
X
xP(X = x|t).
(5.10)
x=0
This can be computed numerically, and the result exactly matches equation (5.1).
We can also consider the problem from the continuum limit, using the Fokker-Planck
equation,
∂2
∂C
=
∂t
∂x2
1 2
σ (x, t)C
2
−
∂
(µ(x, t)C) ,
∂x
(5.11)
where σ 2 is the variance of any step, and µ is the mean displacement of a step [21, 45],
which, in our case, are
2ps 2ps
4ps
+
=
,
3
3
3
4ps 2ps
4ps
2
2
Var(x) = E(x ) − E(x) =
+
−
3
3
3
8ps
ps ,
= 2 1−
9
E(x) =
(5.12)
(5.13)
Therefore we have
∂C
=
∂t
8ps
∂ 2 C 4ps ∂C
1−
ps 2 −
.
9
∂x
3 ∂x
(5.14)
Equation (5.11) is a linear convection diffusion equation. The solution, with initial
condition

 1
C(x, 0) =
 0
for − 12 < x <
1
2
otherwise,
is given in Appendix B, and is
1
C(x, t) =
2
−Erf
−1 + 8tps /3 − 2x
p
4 (1 − 8ps /9) ps t
!
+ Erf
1 + 8tps /3 − 2x
p
4 (1 − 8ps /9) ps t
!!
.
(5.15)
Eq. (5.15) is plotted in Fig 5.4, along with Eq. (5.8) and simulation results. The
distribution found in Eq. (5.8) matches the simulation results exactly. The solution of
the partial differential equation (5.15) also does an excellent job, once t is large enough
so that there are no boundary effects.
42
Cancer Cell Migration
0.1
0.08
0.07
0.08
0.06
0.05
C
0.06
0.04
0.04
0.03
0.02
0.02
0.01
0
0
100
200
x
300
400
0
0
50
x
100
150
Figure 5.4: Simulation data, the distribution found in (5.8), and the solution of the partial
differential equation (B.11) are plotted for various ps and t. The left plot has ps = 0.5, and
t from 50 to 500 in increments of 50. The right plot has t = 100, while ps is varied from 0.1
to 1 in steps of 0.1. The slight mismatch for low ps is due to boundary effects.
5.3
Single Cell Source
Suppose that the basement membrane surrounding the cancer is pierced, and cells are
able to leave one at a time. This corresponds to us altering the model so that cells
enter the system. If there is space, we let a cell enter the system with probability r.
We will consider how the presence of many cells in the system alters how quickly the
cell front moves.
When there are many cells in the system we must decide the order in which they
move in a time step, and how they interact with each other. We again choose the cells
to be blind. A cell does not check its surroundings before attempting to move. If it
attempts to move to a site where there is a cell already present, the move is aborted.
The order of cell movement is known as random sequential ordering. If there are n
cells in the model, we generate a list, ~v of n random numbers. The v1 th walker moves
first, followed by the v2 th walker, all the way through to walker vn . This means that
sometimes walkers will move more or less than a single time in a time step.
The density of invading cells for various times with ps = 0.5 is plotted in Fig. 5.5.
There are two regions, a front of leading cells with high density, and a lower density
region behind. The height of the front region is determined by ps . With ps = 1, there
is no increase in density, while for ps = 0.1 the density at the front is very large. We
43
5.3 Single Cell Source
0.06
0.025
0.02
0.05
t increasing
t increasing
0.04
0.015
0.03
0.01
0.02
0.005
0
0.01
50
100 150 200 250 300 350 400 450 500
0
Cell Density
50
100 150 200 250 300 350 400 450 500
Cell Front
Figure 5.5: Cell density (left) and the front cell position (right) over time. The plots
have ps = 0.5, r = 0.1 at times 100, 200, 300, 400 and 500 with 10,000 repeats.
can also consider the position of the front cell. As seen in Fig. 5.5, the cell does seem
to be moving at a constant speed. We find the velocity by taking the mean of the front
cell and plotting it against time. The slope of the graph is the velocity. This produces
a straight line in each case, apart from when ps is small, where for small time, the slope
in non-linear.
The velocity of the front for various ps and rates is plotted in Fig. 5.6. The speed
of the front is seen to be almost completely independent of the rate at which cells are
introduced into the system. It would be reasonable to think that this could be due to
jamming in the system. However, this could only explain the behaviour for the case
when ps is small. For ps = 1, the reason cannot be jamming. Also, it is interesting
to note that increasing ps has diminishing returns. Doubling ps from 0.5 to 1 only
increases the front speed by about 25%. A possible explanation for this is that adding
cells to the system reduces the the chance that the front walker is chosen to move in a
given time step. If there are c cells in the system, the probability of a certain cell not
moving is
c
1
.
1−
c
This is monotonically increasing, and quickly approaches the limit 1/e ≈ 0.37.
(5.16)
44
Cancer Cell Migration
0.8
0.7
Front Speed
0.6
0.5
Increasing
ps
0.4
0.3
0.2
0.1
0.2
0.3
0.4
0.5
0.6
Rate
0.7
0.8
0.9
1
Figure 5.6: Speed of the cell front. This shows a large dependence on ps , and little
dependence on the rate. 1,000 realisations were simulated for each data point.
5.4
Conclusion
We have discussed in detail the processes involved in carcinoma cell migration. Cells
escaping tumours have been observed to only move away from the tumour. We chose
to model this as a random walk with local dynamic interactions on a hexagonal lattice,
with walkers only allowed to move towards the right.
We began with a model of a single cell moving from a hole in the basement membrane surrounding a tumour. We are interested in how long it will take for a cell to
reach a vessel. Hence, the focus is on how quickly the cell moves away from the tumour.
As we are interested in how far the cell moves from the tumour, we just measure the
x coordinate of the cell. We found the probability distribution the the position of the
walker, and we were able to use the Fokker-Planck equation to obtain a very good
approximation.
Cells were then allowed to escape the tumour at a constant rate. Here we found that
the rate at which the cells entered made very little difference to how quickly the cell
front moved. This suggests the snipping probability is the most important factor. An
extension to this would be to look for an equation that describes the cell concentration
across the whole domain, rather than just the position of the front cell.
6
Conclusion
The process of metastasis involves cancer spreading from part of the body, to another
non-adjacent part. To do this, cancer cells must navigate through the dense extracellular matrix (ECM). Tumours which do not metastasise are known as benign tumours.
Benign tumours are generally considered a much lower health risk than metastatic tumours. An understanding of the processes involved in metastasis could lead to improved
cancer treatment. To aid movement, the cells are able to produce matrix metalloproteinases (MMPs) which degrade the ECM and produce pathways for cancer cells to
use.
We model this as a lattice random walk in an environment where each site is given
one of two states, allowed or blocked. To simulate the action of MMPs, we allow local
dynamic interactions between the walker and the environment. A walker attempting
to move to a blocked site is given a probability to change the state of the site to from
45
46
Conclusion
blocked to allowed, and complete the step.
In one dimension we study a number of variations of this model. A new scaling
theory is obtained for the root mean square displacement of a walk in a randomly
generated environment without local interactions. The erosion model is introduced,
which allows for local dynamic interactions, and we find an equation describing the
size of the allowed cluster over time. We also consider a random walk in a random
environment with local dynamic interactions, and see how various parameters affect
the mean square displacement. Intermediate stages of this model are also investigated,
and probability distributions of each stage are derived.
The mean square displacement of a random walk on a percolation cluster in three
dimensions has been previously studied. We fit the form of the equation previously
proposed, and find parameters that fall in previously estimated ranges. The mean
square displacement becomes linear with the introduction of local dynamic interactions.
We were able to find an equation to model the mean square displacement. Clusters
formed from this model proved to be very similar in structure to those formed from
the Eden growth model.
We model cells escaping a tumour as a random walk with local dynamic interactions
on a hexagonal lattice. We set all sites in the environment to be initially blocked,
representing a dense ECM. We model a single cell moving through this environment,
away from the tumour. We find the probability distribution of the distance the cell
is from the tumour both exactly, and approximately using the Fokker-Plank equation,
which provides a very accurate estimate. We allow cells to enter the system at a
constant rate, and look at how far the leading cell is from the tumour. We find that
when there are many cells in the system, the speed of the front cell is most dependent
on how effective any cell is at clearing the ECM.
There are a number of ways we could extend this work. Target vessels could be
introduced for the cells to find. We would then determine which parameters which most
affect the probability of the cells finding a vessel. The model of cancer cell migration
discussed could be extended to a three-dimensional lattice, which may alter results.
Another extension would be to consider types of cancer other than carcinomas.
A
Proof of the relationship between the radius
of gyration and moment of inertia
Let rj be the vector denoting the position of the j th cluster site. The equations for the
expected square moment of inertia and the square radius of gyration become
"
#
N
N
X
X
1
1
hI 2 i =
(ri − rj )2 ,
N i=1 N j=1

!2 
N
N
X
X
1
 ri − 1
Rg2 =
{rj } 
N i=1
N j=1
= hr − hrii.
If Rg2 is subtracted from hI 2 i, we expect to get back Rg2 .
47
(A.1)
(A.2)
(A.3)
Proof of the relationship between the radius of gyration and moment
48
of inertia
hI 2 i − Rg2

#
"
!2 
N
N
N
N
X
X
X
X
1
1
1
 ri − 1
=
(ri − rj )2 −
{rj } 
N i=1 N j=1
N i=1
N j=1

!2 
N 
N
N

X
X
X
1
1
1
2
=
(ri − rj ) − ri −
{rj }

N i=1  N j=1
N j=1

!2 
N
N
N
N


1 X
1 X
1 X 1 X 2
ri − 2ri · rj + r2j − r2i + 2ri ·
{rj } −
{rj }
=

N i=1  N j=1
N j=1
N j=1
(
N
N
N
N
1 X
1 X 2
1 X 1 X 2
ri − 2
ri · rj +
rj − r2i +
=
N i=1 N j=1
N j=1
N j=1
!2 )
N
N
ri X
1 X
2 ·
{rj } −
{rj }
N j=1
N i=1

!2 
N
N
N 
N
N

X
X
X
X
X
ri
1
1
1
ri
2
2
2
r −2 ·
rj +
{rj } −
=
r − ri + 2 ·
{rj }

N i=1  i
N j=1
N j=1 j
N j=1
N j=1


!2
N 
N
N

X
X
X
1
1
1
2
=
r −
{rj }

N i=1  N j=1 j
N i=1
!2
N
N
1 X 2
1 X
=
r −
{rj }
N j=1 j
N j=1
= hr2 i − hri2
= hr − hrii
= Rg2
(A.4)
Hence
2
R∞
= hI 2 i = 2Rg2 .
(A.5)
B
Solution of the Linear Advection-Diffusion
Equation
The linear advection-diffusion equation,
∂C
∂ 2C
∂C
=D 2 −V
,
∂t
∂x
∂x
(B.1)
with the initial condition

 1
C(x, 0) =
 0
for − 21 < x <
1
2
otherwise,
is solved as follows. Substitute
C(x, t) = e−βx−αt w(x, t)
(B.2)
∂w
∂w
∂ 2w
= (α + β(V + Dβ))w − (v + 2Dβ)
+D 2.
∂t
∂x
∂x
(B.3)
into (B.1) to obtain
49
50
Solution of the Linear Advection-Diffusion Equation
Choosing α and β such that
α + β(V + Dβ) = 0
(B.4)
v + 2Dβ = 0
gives
V2
α=
4D
V
.
2D
(B.5)
C(x, t) = e− 4D + 2D w(x, t),
(B.6)
and β = −
Substituting (B.5) into (B.2) gives
tV 2
Vx
where w(x, t) is governed by the diffusion equation
∂w
∂ 2w
=D 2,
∂t
∂x
along with the new initial condition

 e− V2Dx
w(x, 0) =
 0
(B.7)
for − 12 < x <
1
2
otherwise.
The Green function solution to the heat equation is [46]
Z ∞
1
2
√
e−(x−ξ) /(4Dt) w(ξ, 0) dξ.
w(x, t) =
4πDt
−∞
Evaluating this integral using Mathematica gives the result
1 tV 2 − V x
−1 + 2tV − 2x
1 + 2tV − 2x
√
√
w(x, t) = e 4D 2D −Erf
+ Erf
,
2
4 Dt
4 Dt
(B.8)
(B.9)
where Erf is the error function, defined as
2
Erf(x) = √
π
Z
x
2
e−y dy.
(B.10)
0
Substituting this into equation (B.6) gives the final solution
1
−1 + 2tV − 2x
1 + 2tV − 2x
√
√
C(x, t) =
−Erf
+ Erf
.
2
4 Dt
4 Dt
(B.11)
C
Simulation of Random Walks with Local
Dynamic Interactions
Sample MATLAB code to simulate a one-dimensional random walk with local dynamic
interactions is included here. This framework is simple to extend to higher dimensions.
This does not work well for long walks in three dimensions, as it requires that the state
of every site in the environment is stored, which uses a large amount of memory. To
get around this, a list of all lattice sites that the walker has attempted to visit can
be stored. This has the advantage that the length of the walk is unconstrained, but
the disadvantage that the cluster size cannot be measured. This is because sites which
have not been seen by the walker are not generated. This code below was used in
chapters 2 and 3, and variations on it were used in chapters 4 and 5.
51
52
1
Simulation of Random Walks with Local Dynamic Interactions
function [output] = randomwalk 1d
2
3
%This function runs a random walk in one dimension in a random ...
environment.
4
%The walker has probability ps to clear a blocked site.
5
%The output is the
6
%average mean square displacement at each time step
7
8
n=150;
%number of steps
9
p=0.2;
%probability that a site is allowed
10
ps=0.2;
%probability to clear a blocked site
11
rep=1000;
%number of repeats
moves=[−1 1];
%possible steps
dist=zeros(n,rep);
%initialises array for the square distance at
12
13
14
15
%each step
16
17
cs=2n+1;
%estimate of the required environment size
18
st site=round(cs/2);
%sets the starting position to the center of
%the cluster
19
20
for r=1:rep
21
pos=st site;
%sets the walk to be at the starting position
22
env=ceil(rand(2*cs+1,1)−1+p);
23
env(pos)=1;
%generates the environment
%sets the starting position to be an ...
allowed site
24
25
for st=1:n
new pos=pos+moves(ceil(rand*2)); %chooses a random ...
direction for
%the walker to step
26
27
if env(new pos)==1
%checks if the new site is
%available
28
pos=new pos;
%walker moves if available
31
if rand<ps
%if blocked, the walker
32
% clears the site with probability ps
29
30
else
53
env(new pos)=1;
33
%the state of the site is
%changed to allowed
34
pos=new pos;
35
%the walker position is ...
updated
end
36
37
end
38
dist(st,r)=(pos−st site)ˆ2;
%the square displacement from
%the starting location is ...
39
recorded
end
40
41
end
42
output=mean(dist,2);
%the mean square displacement from the ...
starting
43
%location is output
54
Simulation of Random Walks with Local Dynamic Interactions
D
Measuring Aspects of Three-Dimensional
Clusters
Measuring the size of a cluster is very simple in one dimension, where the size of
the cluster is simply the length. In higher dimensions we need more sophisticated
algorithms to measure cluster size. A well known algorithm is the Hoshen-Kopelman
algorithm [47]. This algorithm scours every site on the lattice, and finds the size of
each cluster. However, as the walker always starts at the origin, we only need the size
of the cluster which includes the origin. We also require the position of every site in
the cluster to be able to calculate the moment of inertia.
I have developed my own algorithm to complete the task. The input required is
the lattice, where each site is either a one or a zero, the position of the origin, and a
matrix defining neighbouring sites. A list is formed of lattice sites which are in the
55
56
Measuring Aspects of Three-Dimensional Clusters
cluster, which initially consists of only the origin. All sites which are neighbours of the
origin are used to initialise a list of sites which needs to be checked. The state of each
of these sites is checked, and are either added to the list of sites in the cluster, or to a
list of sites which are not in the cluster. Any sites adjacent to any of the lattice sites
which were just found to be in the cluster are added to the list of sites to be checked.
If the site had been checked previously, it is not added to the list again. Once there
are no sites left to be checked, the algorithm has a list of lattice sites in the cluster.
This algorithm has the advantage that it does not waste time checking the state of
sites which are not adjacent to the cluster. This provides the largest advantage when
the cluster size is small compared to the environment. For measuring the evolving
cluster size due to a random walker eroding the environment, we introduce optional
inputs which are known cluster sites and sites which have been checked previously.
Allowed sites do not change back to blocked sites, and so we do not need to check
these sites again. The only adjacent site that can change is the newly cleared site,
which is added to the list of cluster sites which is input to the algorithm.
This algorithm was used in chapter 4 for finding cluster sites. Once a list of cluster
sites is obtained, calculating the size, radius of gyration and moment of inertia is
straightforward. When p is near pc cluster become very large, and is impractical to
generate large arrays of random numbers to measure cluster size. The algorithm given
below easily adapted to generate a cluster. The part of the code which checks to see if
a lattice site in the cluster can be changed to generate a random number to see if the
2
site is in the cluster. This is how the data to calculate hR∞
i in chapter 4 was obtained.
This simulation took a week to run, using parallel code on 12 cores. MATLAB code
for generating a list of cluster sites is given below.
1
function [cluster size,edge,checked,cluster] = ...
cluster size2(matrix,origin,varargin)
2
3
%Finds the size of the cluster around the origin. Assumes cubic matrix
4
%with equal side lengths. The origin is assumed to be in the cluster
57
5
6
%If edge=1 the cluster is fully in the environment, 0 otherwise
7
8
checked in=false;
9
cluster in=false;
10
11
%Here we check for optional inputs
12
%checked must be a list of lattice sites which do not need to be ...
checked
13
%cluster is a list of lattice sites known to be in the lattice
14
if nargin>0
15
num var args = length(varargin);
16
assert(mod(num var args,2)==0);
17
for m = 1:2:num var args
18
19
switch varargin{m}
20
case 'checked'
21
checked=varargin{m+1};
22
checked in=true;
case 'cluster'
23
24
cluster=varargin{m+1};
25
cluster in=true;
otherwise
26
27
error('unknown option')
28
varargin{m+1}
end
29
end
30
31
end
32
33
%adj is matrix defining what an adjacent site is
34
adj=[1,0,0;
35
−1,0,0;
36
0,1,0;
37
0,−1,0;
38
0,0,1;
39
0,0,−1];
58
Measuring Aspects of Three-Dimensional Clusters
40
n adj=size(adj,1);
%The number of adjacent sites to any lattice site
41
matrix size=size(matrix,1); %The size of the matrix − assumes a cube
42
43
pos=origin; %The defualt starting site to check is the origin
44
45
edge=1;
46
%Here we add the origin to the list of checked sites
47
if checked in==true
checked(end+1,:)=origin;
48
49
else
checked=origin;
50
51
end
52
%The origin is added to the list of cluster sites
53
if cluster in==true
cluster(end+1,:)=origin;
54
55
else
cluster=origin;
56
57
end
58
nc=0; %The number of cluster site adjacent to the origin. We check ...
each adjacent lattice site, and nc is incremented each time ...
there is a site in the cluster.
59
for a=1:n adj
60
new pos=pos+adj(a,:);
61
if max(new pos)≤ matrix size && min(new pos)>0
if ...
62
any(sum([abs(checked(:,1)−new pos(1)),abs(checked(:,2)−new pos(2))
,abs(checked(:,3)−new pos(3))],2)==0)==0
63
64
nc=nc+1;
65
check(nc,:)=new pos;
66
checked(nc+1,:)=new pos;
end
67
end
68
69
end
70
%If there are cluster sites apart from the origin we do the ...
following. The
71
%loop runs until there are no sites remaining to check.
59
72
if nc>0
73
while isempty(check)==0
74
pos=check(1,:);
75
if matrix(check(1,1),check(1,2),check(1,3))==1
76
cluster(end+1,:)=check(1,:);
77
for a=1:n adj
78
new pos=pos+adj(a,:);
79
if max(new pos)≤ matrix size && min(new pos)>0
if ...
80
any(sum([abs(checked(:,1)−new pos(1)),abs(checked(:,2)
−new pos(2)),abs(checked(:,3)−new pos(3))],2)==0)==0
81
82
check(end+1,:)=new pos;
83
checked(end+1,:)=new pos;
end
84
else
85
edge=0;
86
end
87
88
end
89
90
end
91
check=check(2:end,:);
end
92
93
end
94
cluster size=size(cluster,1);
60
Measuring Aspects of Three-Dimensional Clusters
Glossary
Definitions of biological terms used in this thesis are listed below.
Adhesion
The tendency of cells to attach to one another.
Basement membrane
A thin sheet of cells which underlies the epithelium and endothelium.
Carcinoma
A cancer originating from epithelial cells.
Chemotaxis
Directed motion facilitated by chemicals within the environment.
Collagen
A component of the ECM which makes up connective tissue
e-cadherin
An important signalling protein involved in binding cells together.
Elastin
A component of the ECM which allows tissue to return to its original shape after
deformation.
Epithelium
The layer of tissue made up of epithelial cells. This tissue lines most surfaces and
61
62
Glossary
cavities in the body, and also forms many glands.
Epithelial-mesenchymal transition (EMT)
The process of an epithelial cell taking on the properties of a mesenchymal cell,
which allows it to adhere and move through the mesenchyme.
Extracellular matrix (ECM)
The part of tissue which is not the cells. It provides structural support for cells,
and allows nutrients to pass through to the cells.
Fibril
A very fine strand.
Fibronectin
A component of the ECM which helps bind other components together.
Integren
Integrens allow attachment between cells and surrounding tissue.
Laminin
An important protein within the basement membrane
Matrix metalloproteinase (MMP)
A class of enzyme which is able to degrade the ECM. There are two classes of
MMPs, secreted and membrane tethered (MT-MMP). Secreted MMPs are made
by cells and then secreted into the cells’ surroundings. MT-MMPs are retained
on the outer surface of a cell once created, and are held there to degrade the
matrix in the vicinity of the cell.
Metastasis
The spread of cancer from one part of the body to another.
Mesenchyme
Loose connective tissue.
63
Proteolysis
Breakdown of protein into smaller molecules.
Pseudopodia
These are temporary projections of a cell into its surroundings.
64
Glossary
References
[1] D. L. Nelson and M. M. Cox. Lehninger Principles of Biochemistry (Worth Publishers, New York, 2000). 1, 36
[2] K. Pearson. The problem of the random walk. Nature 72(1865), 294 (1905). 1
[3] Lord Rayleigh. The problem of the random walk. Nature 72(1866), 318 (1905). 2
[4] B. D. Hughes. Random Walks and Random Environments, Volume 1: Random
Walks (Oxford University Press, Oxford New York, 1995). 2, 4, 7, 9, 18, 19, 28,
31
[5] M. N. Barber and B. W. Ninham. Random and Restricted Walks: Theory and
Applications (Gordon and Breach, Science Publisher, Inc., New York, 1970). 2
[6] J. D. Murray. Mathematical Biology: I. An Introduction, Third Edition (SpringerVerlag, Berlin, 2002). 2, 3
[7] E. Purcell. Life at low Reynolds number. American Journal of Physics 45(1), 3
(1977).
[8] A. J. Trewenack, K. A. Landman, and B. D. Bell. Dispersal and settling of translocated populations: a general study and a New Zealand amphibian case study. Journal of Mathematical Biology 55(4), 575 (2007). 2
[9] R. M. May. Simple mathematical models with very complicated dynamics. Nature
261, 459 (1976). 3
65
66
References
[10] H. Byrne and D. Drasdo. Individual-based and continuum models of growing cell
populations: a comparison. Journal of Mathematical Biology 58, 657 (2009). 3
[11] Y. Kam, K. Rejniak, and A. Anderson. Cellular modeling of cancer invasion:
Integration of in silico and in vitro approaches. Journal of Cellular Physiology
227(2), 431 (2012). 35
[12] M. Bezzi. Modeling biology by cellular automata. Ph.D. thesis, University of
Bologna (1999). 3
[13] A. Deutsch and S. Dormann. Cellular Automation Modelling of Biological Pattern
Formation: Characterization, Applications, and Analysis (Birkhäuser Boston,
New York, 2005). 3
[14] P. Sarkar. A brief history of cellular automata. ACM Comput. Surv. 32(1), 80
(2000). 3
[15] P. A. DiMilla, J. A. Quinn, S. M. Albelda, and D. Lauffenburger. Measurement
of individual cell migration parameters for human tissue cells. American Institute
of Chemical Engineers Journal 38(7), 1092 (1992). 3
[16] S. L. Goodman, G. Risse, and K. von der Mark. The e8 subfragment of laminin
promotes locomotion of myoblasts over extracellular matrix. The Journal of Cell
Biology 109(2), 799 (1989). 3
[17] M. J. Simpson, A. Merrifield, K. A. Landman, and B. D. Hughes. Simulating invasion with cellular automata: Connecting cell-scale and population-scale properties.
Phys. Rev. E 76, 021918 (2007). 3
[18] K. A. Landman, A. E. Fernando, D. Zhang, and D. F. Newgreen. Building stable
chains with motile agents: Insights into the morphology of enteric neural crest cell
migration. Journal of Theoretical Biology 276(1), 250 (2011).
[19] D. Bonchev, S. Thomas, A. Apte, and L. Kier. Cellular automata modelling of
biomolecular networks dynamics. SAR and QSAR in Environmental Research 21,
337 (2010). 3
References
67
[20] C. J. Pennington, B. D. Hughes, and K. A. Landman. Building macroscale models
from microscale probabilistic models: A general probabilistic approach for nonlinear
diffusion and multispecies phenomena. Physical Review E 84(4) (2011). 4
[21] J. Klafter and I. Sokolov. First Steps in Random Walks: From Tools to Applications (Oxford University Press, Oxford, 2011). 4, 7, 41
[22] J. M. Hammersley. Origins of percolation theory. Annals of the Israel Physical
Society 5, 47 (1983). 4
[23] R. Ewing and A. Hunt. Percolation Theory for Flow in Porus Media (Springer
Berlin, Heidelberg, 2009). 5
[24] B. D. Hughes. Random Walks and Random Environments, Volume 2: Random
Walks (Oxford University Press, Oxford New York, 1995). 5, 8, 25, 26, 28, 29
[25] R. Hilfer and R. Orbach. Continuous time random walk approach to dynamic
percolation. Chemical Physics 128(1), 275 (1988). 5
[26] R. B. Pandey, D. Stauffer, A. Margolina, and J. G. Zabolitzky. Diffusion on
random systems above, below, and at their percolation threshold in two and three
dimensions. Journal of Statistical Physics 34(3), 427 (1984). 5, 28
[27] C. D. Mitescu, H. Ottavi, and J. Roussenq. Diffusion on percolation lattices: The
labyrinthine ant. In Electrical Transport and Optical Properties of Inhomogeneous
Media, vol. 40 of American Institute of Physics Conference Series, pp. 377–381
(1978). 5
[28] M. Schmidt and H. Lipson. Distilling free-form natural laws from experimental
data. science 324(5923), 81 (2009). 22, 30
[29] J. Škvor and I. Nezbeda. Percolation threshold parameters of fluids. Physical
Review E 79(4), 041141 (2009). 26
68
References
[30] R. Ziff and G. Stell. Critical behaviour in three-dimensional percolation: is the
percolation threshold a lifshitz point? Tech. Rep. 4, University of Michigan Laboratory of Scientific Computing (1988). 26
[31] J. Hoshen. Percolation and cluster structure parameters: The radius of gyration.
Journal of Physics A: Mathematical and General 30(24), 8459 (1988). 27
[32] M. Sahimi and G. R. Jerauld. Random walks on percolation clusters at the percolation threshold. Journal of Physics C: Solid State Physics 16(29), 1043 (1983).
28
[33] T. Vicsek. Random walks on bond percolation clusters: ac hopping conductivity
below the threshold. Zeitschrift fr Physik B Condensed Matter 45(2), 153 (1981).
28
[34] E. Seifert and M. Suessenbach. Test of universality for percolative diffusion. Journal of Physics A: Mathematical and General 17(13), L703 (1984). 28
[35] C. Mitescu and J. Roussenq. Diffusion on percolation clusters. Annals of the Israel
Physical Society 5, 81 (1983). 29
[36] J. Roussenq. Effet de taille en percolation: une étude numérique. Ph.D. thesis,
Université de Provence (1980). 29
[37] M. Eden. A two-dimensional growth process. Proc. Fourth Berkeley Symp. on
Math. Statist. and Prob. 4, 223 (1961). 32
[38] M. Kolb, R. Botet, and R. Jullien. Scaling of kinetically growing clusters. Physical
Review Letters 51(13), 1123 (1983). 32
[39] K. A. Rejniak and A. R. A. Anderson. Hybrid models of tumor growth. Wiley
Interdisciplinary Reviews: Systems Biology and Medicine 3(1), 115 (2011). URL
http://dx.doi.org/10.1002/wsbm.102. 36
[40] B. Alberts. Essential Cell Biology: Second Edition (Garland Science, New York,
2004). 36
References
69
[41] S. E. Gill and W. C. Parks. Matrix metalloproteinases and their inhibitors in
turnover and degradation of extracellular matrix. Biology of Extracellular Matrix
2, 1 (2011). 36, 37
[42] E. I. Deryugina and J. P. Quigley. The role of matrix metalloproteinases in cellular
invasion and metastasis. Biology of Extracelullar Matrix 2, 145 (2011). 36, 37
[43] K. Wolf and P. Friedl. Mapping proteolytic cancer cell-extracellular matrix interfaces. Clinical and Experimental Metastasis 26, 289 (2009). 10.1007/s10585-0089190-2. 37
[44] R. Askey and A. O. Daalhuis. Digital library of mathematical functions (2012).
URL http://dlmf.nist.gov/. 40
[45] F. C. Klebaner. Introduction To Stochastic Calculus With Applications: Second
Edition (Imperial College Press, London, 2005). 41
[46] J. Ockendon, S. Howison, A. Lacey, and A. Movchan. Applied Partial Differential
Equations (Oxford University Press, Oxford, 1999). 50
[47] J. Hoshen and R. Kopelman. Percolation and cluster distribution. i. cluster multiple labeling technique and critical concentration algorithm. Physical Review B
14(8), 3438 (1976). 55