Evolution of Cell Division
Nargess Khalilgharibi, CoMPLEX
Supervisors: Dr. Buzz Baum & Dr. Mark Miodownik
Total Word Count: 5236
Main Body: 4785 words
(Excluding abstract, figure captions, acknowledgment, references and appendices)
January 18, 2012
It is widely accepted that in order for a cell to maintain its average size through several divisions, a
coordination between the cell growth and cell division should exist. However, the details of this coordination is not well understood. In the recent years, several attempts have been made to propose a
generic model that explains the details of the process of cell growth and its coordination with cell division. Most of these models were developed using nonlinear dynamics and mathematical modelling.
Cellular automata have been used to study the growth of a population of cells. However, they have not
been used in studying the growth of an individual cell. In the following essay, I will introduce the previous mathematical and computational approaches to the problem. I will also propose a model, which
combines cellular automata with genetic algorithm in order to study the growth of an individual cell.
Contents
1
Introduction
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2
Cell Size Control: The Underlying Biology
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3
Mathematical Modelling and Simulation
3.1 Mathematical Modelling of Cell Size Control . . . . . . . . . . . .
3.2 Cellular Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 The Cellular Potts Model . . . . . . . . . . . . . . . . . . .
3.2.2 Cellular Automata Combined with the Genetic Algorithm
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Modelling Cell Growth Using Cellular Automata
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Results
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Discussion and Further Research
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Acknowledgement
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A First Appendix
12
B Second Appendix
B.1 The Code for the First Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 The Code for the Second Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
Introduction
The mechanics of cell division is considered more complicated than building ”a moon rocket or supercomputer” [3]. What makes it even more complicated is that cells should maintain their size over
generations. In order to achieve this, many cells display a coordination between their cell growth and
cell division [18, 19, 20]. Although multiple mechanisms have been suggested for the coordination of
cell growth and division [18], many aspects of this still remains a mystery [9, 13, 18, 19].
The cell cycle is constituted of discrete phases. Cell growth occurs in the first phase of the cycle called
interphase, which is divided into three stages: G1, S and G2. There are several ways for a cell to grow
its mass (e.g. endoreplication, accumulation of lipids) but one of the main ways is synthesis of biomass
and more specifically, synthesis of proteins [13]. Protein synthesis is regulated by nutrient levels in
the environment and growth factors [13, 18]. Growth factors are small peptides produced by other
cells and are required for activating intracellular signalling pathways. In the first period of the interphase, G1, the cell measures the nutrient concentrations and senses the presence of growth factors in
the environment. It has been suggested that these information is gathered and combined by a ”growthsignalling network”, whose main part is a protein called TOR (target of rapamycin) [13]. In fact, TOR
receives information from the environment and decides whether the cell should grow. If the condition
is not suitable for growth, the cell will enter a phase called G0. However, if the condition is suitable for
growth, TOR activates regulatory factors for protein synthesis. This results in an increase in cellular
mass, which continues until the cell reaches an appropriate size.
How does a cell realise that it has reached an appropriate size to divide? There is no certain answer
to this question, since the mechanisms which control cell size are not as well known and understood
as mechanisms controlling cell division [13]. In the following essay, I will introduce the major cell size
control mechanisms (Section 2) and will summarise the previous attempts to understand and model
them (Section 3). I will continue with explaining two different models I have developed for simulating
cell growth (Section 4). I will then present the results of my models (Section 5). The essay will end with
a discussion on my approach and suggestions for further research (Section 6).
2
Cell Size Control: The Underlying Biology
In proliferating cells, the size control is achieved by a coordination between the processes of cell growth
and cell division [18, 19, 20]. However, different cell types attain this coordination through different
mechanisms. Three major mechanisms have been proposed for this coordination [18].
In some cells, cell division is dependant on cell growth [13, 18, 19, 20]. This means that the cell cycle would only proceed to the division phase if it has reached a critical size in the growth stage. Some
experiments have revealed the existence of such a dependance of cell division on cell growth in yeast
(for example, see [19]) and some animal cells [18]. For example, Nurse [19] used a mutant type of the
fission yeast Schizosaccharomyces pombe to study the mechanism of cell size control and its coordination
with cell division. He proposed the existence of two checkpoints at the end of the G1 and G2 phases
of the interphase, before entering the stages of DNA synthesis (S phase) and nuclear division (M phase)
respectively. According to Nurse, these checkpoints would coordinate the processes of cell growth and
division and result in generation of cells of the same size.
In another group of cells, generally mammalian cells [9, 18], growth and division are regulated through
two independent mechanisms. In fact, activation through two unconnected, parallel signalling pathways makes growth and division correlated but not coupled to each other [18]. In these cells, maintenance of average cell size is reached by having constant rates of growth and division [9, 18]. This means
that even if a cell with an abnormal size is produced during the cell cycle by accident, this variation in
the cell size will be corrected after several generations. Thus, the average size of the cells will remain
unchanged [18].
A third mechanism, which is observed in many animal cell types, suggests the existence of a same
group of proteins (i.e. growth factors and mitogens) which activate both processes of growth and division. In fact, in these cells, growth and division are triggered through the same signalling pathways
[18].
1
3
3.1
Mathematical Modelling and Simulation
Mathematical Modelling of Cell Size Control
As explained earlier, coordination between cell growth and division can be achieved through different
mechanisms [18]. Thus, different models have been used to describe the growth of different types of
cells.
It has been suggested that an exponential growth rate can explain growth in cells which exhibit dependent growth and division processes (the first mechanism described above), such as the fission yeast
[9, 20]:
dV
=V
dt
⇒ V = ect
This size-dependant growth rate results in larger cells to grow faster and smaller cells to grow slower.
Thus, checkpoints are required to stop the large cells from excess growth and encourage the small cells
to grow to the appropriate size and hence, keep the average size of the cells constant.
Despite the strong experimental evidence for the existence of checkpoints in the fission yeast, a group
of scientists believe that checkpoints are not necessarily needed for size homeostasis [9]. More specifically, it has been suggested that cells which display a linear growth (the second mechanism described
above) do not need checkpoints to maintain their average size. For instance, Conlon and colleagues [9]
have shown that rat Schwann cells display a constant growth rate, equivalent to linear growth, which
is dependant on the concentration of growth factors in the surrounding environment. They have also
shown that this kind of growth rate would make the presence of the cell size checkpoints unnecessary,
meaning that the average cell size would be maintained in the absence of the checkpoints. They have
also claimed that other cells exist, especially other types of mammalian cells, which exhibit linear sizedependant growth.
Besides the still ongoing debate on whether cells grow linearly or exponentially, there have been some
attempts to find a generic model which can explain the cell size control in eukaryotic cells [20, 21, 22].
Mathematical modelling and nonlinear dynamics have been the tools for these studies. In all of these
models, size of the cell can be affected by growth rates and environmental noises and cell size checkpoints are considered as ”bifurcations in the dynamics of the cell cycle molecular network” [20], with
size being ”the main bifurcation parameter” [20]. In fact, these ”bifurcations in the vector field of the
kinetic equations” [22] are believed to be responsible for the ”irreversible transitions of the cell cycle”
[22]. Different models have proposed different local bifurcations to explain the dynamics of cell cycle.
A model by Tyson et al [22] proposes a saddle-node bifurcation and a saddle-node-loop bifurcation
for the checkpoints at the end of G1 and G2 phases, respectively. The stages proposed by their model
for the cell cycle were consistent with experimental observations. Furthermore, their model was successful in explaining the repeatability of the cell cycle. However, it failed in giving an explanation for
some other features of the cell cycle. For example, it could not explain the time it takes for the cells with
a size below average to grow to a critical size before entering the division (sizer phase) and the time it
takes for the cells with an appropriate size to complete division (timer phase) [21]. As proposed by Qu
and colleagues, another candidate for the checkpoints is Hopf bifurcation [21]. As mentioned by the
authors, their model was proposed for eukaryotes which represent either G1/S or G2/M transitions.
Thus, it is possible that a new dynamical model will be needed when the two modules are coupled to
form a complete cell cycle [21]. One model suggests that a Saddle-Node Infinite PERiod bifurcation
(SNIPER) triggered by an increase in cellular mass, results in cell size control [10]. However, as agreed
by the authors, when the growth rates are increased, the model predicts an excess growth of the cell
and thus, fails to explain the cell size control. A more recent universal model for cell size control in
eukaryotic cells, developed by Pfeuty and colleagues, seems to have overcome this issue by combining
the exponential cell growth with a ”two-variable cell cycle network dynamics” [20] .
Although these models have increased the insight into the dynamics of cell growth and division, the
”wiring diagrams” [20] that coordinate these processes still remain unknown. It has been suggested
that since there is a huge variation in the size of different eukaryotes, each species may have developed
its own way of coordinating these two processes and thus, a tremendous amount of work is still needed
to uncover these ”wiring diagrams” [20].
2
3.2
Cellular Simulation
Many of the attempts to simulate the cell cycle were done by considering the behaviour of a population of cells (for example, see [4]). However, the dynamics of cell growth and division has not yet been
studied by simulation of a single cell growth. Below, two excisting models for simulating cells are presented. None of these models were developed to study the processes underlying cell cycle. However,
they are important because both of them seem to be successful examples of growth from a single cell
and some aspects of the formation and growth of the cells can be observed in them.
3.2.1
The Cellular Potts Model
One successful simulation of the cell was done by cellular Potts model, developed by Glazier and Granier
in 1992 [12]. This model is an extension to the large-Q Potts model. Large-Q Potts model is a generalisation of the Ising model, a system of interacting spins arranged in a lattice. The large-Q Potts model
is constituted of a lattice. Each site of the lattice, σ(i, j), has a spin, whose value is an integer between 1
and N. The lattice sites with the same spin σ make a cell. Thus, the model is constituted of N cells. The
neighbouring sites interact with each other in such a way that bonds between sites with similar spins
have energy 0 and bonds between sites with mismatched spins have energy 1. The Hamiltonian of the
lattice is thus defined as:
H=
∑ 1 − δσ(i,j),σ(i0 ,j0 )
neighbours
At each time step, a site is chosen randomly and its state is changed from σ to σ0 with Monte Carlo
probability. This implies that in Potts model, the next state of each site is determined by the Hamiltonian. For temperatures above 0 K (which is the case for all the material in nature and specifically,
biological cells), if the energy difference between the two states is positive, the probability of transition
from spin σ to σ0 would be e
would be 1.
− k∆HT
B
and if the energy difference is negative, the probability of transition
The Potts model is widely used for studying ”diffusive grain growth driven by surface energy” [12]
and ”topological changes in cellular patterns in metals and soap froths” [12].
However, Graner and Glazier[12] argued that although in grain growth, the energy of the system can
be reduced by coarsening, this cannot be the case for biological cells. Thus, in these cells, the reduction
in the energy should be implemented by other means. To be more specific, in biological cells, the energy is reduced by the cell motion, which is caused by the differences in ”contact energies between the
cells of different types” [12]. Since in the Potts model, the Hamiltonian plays the most important role
in predicting the next state of the lattice, Graner and Glazier had to modify the Hamiltonian in order
to apply these changes. They added an ”elastic-area constraint” and ”a second quantum number, τ”
to the Hamiltonian . Three cell types were defined: light, dark and medium. Below is the modified
Hamiltonian of the cellular Potts model, proposed by Graner and Glazier:
H=
∑
neighbours
J(τ (σ(i, j)), τ (σ(i0 , j0 )))(1 − δσ(i,j),σ(i0 ,j0 ) ) + λ ∑[ a(σ) − Aτ (σ) ]2 θ ( Aτ (σ) )
σ
In the above equation, J is the surface energy between spins of type τ and τ 0 , λ is a Lagrange multiplier
which specifies the strength of the area constraint, a is the cell’s area and Aτ is the target area for the
cells of type τ [12].
Applying the above changes to the large-Q Potts model resulted in a successful simulation of biological cell sorting. Two stages of cell sorting were identified in the model: one rapid stage which was
dependant on the boundary and led to a ”uniform light-cell-medium layer” and a ”partial bulk cell
sorting”, and a slow stage which was independent of the boundary and led to complete cell sorting
[12]. Since the second stage was much slower that the first stage, it was suggested that it may explain
”why partial cell sorting is observed biologically” [12].
3.2.2
Cellular Automata Combined with the Genetic Algorithm
A cellular automaton is a spatially and temporally discrete system, consisting of a group of identical
cells arranged in a 1D, 2D or 3D lattice [14]. The state of each cell is chosen from a set of finite size.
Each cell interacts with the other cells in its neighbourhood (e.g. von Neumann neighbourhood, Moore
neighbourhood). This means that the next state of each cell will be determined by some transition rules
which take into account the state of neighbouring cells. The states of the cells in the lattice are usually
3
updated in parallel.
Since being introduced by John von Neumann in 1948, cellular automata have been used to study a
wide variety of fields, from biological and physical sciences to geology, mathematics and computer
sciences [14]. In biology, for example, different classes of CA (e.g. deterministic automata, lattice gas
models and solidification models) have been used for modelling different biological phenomena [11].
For example, they have been used for studying cancer growth [1], pattern formation [15] and woundhealing [5].
A model proposed by Basanta and colleagues [5, 7, 6] combines cellular automata with genetic algorithms to generate evolvable organisms which illustrate homeostasis and robustness in their shape.
A genetic algorithm is a subset of a large group of evolutionary algorithms, used to find solutions in
search and optimisation problems [8]. By using operators such as selection, crossover and mutation,
the GA mimics the natural process of evolution in finding the optimised solutions.
A genetic algorithm is constituted of several steps. First, a set of random solutions or genotypes are
created. These genotypes are then used as guidelines to grow the phenotype, whose fitness will be calculated using a problem-dependant fitness function. A selection algorithm will then be used to select
for the genomes whose phenotypes displayed high levels of fitness. The selected genomes form the
parent population and are subject to a round of crossover and mutation to form the next generation. Repeating this procedure for several generations will usually result in finding optimised solutions [16].
In their proposed model, Basanta and colleagues [5, 7, 6] created a deterministic three dimensional
cellular automata, whose growth was leaded by a linear genome. A set of 100 rules or genes, each
constituted of four integers, defined the genome [5]. The first three integers determined the condition
under which the rule should be applied. The action, which was either to grow, to move or to die, was
implemented in the fourth integer. Unlike many cellular automata in which only the state of the neighbouring cells is taken into account, Basanta and colleagues considered the states of the neighbouring
cells along with a cell division count and developmental time to decide whether to apply the action to
each cell or not [5]. For each cell, the conditions of expression of each of the 100 genes were examined.
For any of the genes, if the conditions were favourable, the action assigned to that gene would get a
vote. After going through all the genes, the action which got the highest vote would be applied to the
cell, as long as the number of votes for that action was higher than a threshold [5, 7, 6].
To select for organisms which displayed homeostasis and robustness, Basanta and colleagues used
genetic algorithm [5]. They allowed each phenotype or organism to grow for 50 steps and then used a
multi-objective fitness function to choose for organisms which showed a minimal change in their shape
between steps 50, 100 and 150. By the method mentioned above, Basanta and colleagues successfully
evolved a population of genomes which enabled a single cell to proliferate and form a large organism.
After reaching a particular size, the organism would enter a homeostatic phase, in which its shape
would not change any more.
4
Modelling Cell Growth Using Cellular Automata
The first attempt to model cell growth was to develop a cell which could grow continuously. A 2D cellular automata was chosen for this purpose.The initial cell was modelled as a square with two layers
of membrane around it. Cellular growth was assumed to be the result of the growth of the membrane.
Hence, a set of rules were developed to enable the cell grow on the edge. The rules were such that at
each step, the outer layer of the cell would grow, moving the second layer which was attached to the
cell interior outwards. The outward growth of the membrane would then lead to an increase in the
cell interior volume. A threshold for the perimeter to area ratio was defined. At each time step, this
ratio was measured and the rules were continued to be applied to the cell until the ratio reached the
threshold.
The cell which was modelled as mentioned above continued to grow until it reached a certain perimeter to area ratio, or more generally, a certain size. Then the cell entered a phase of homeostasis, where
its shape and volume did not change. For a more detailed description of the model see the first appendix.
4
It was found that although the cell started from a square shape with a sharp edge, it grew into a shape
with rounded corners (Figure 1). It is unclear why this happens but what is clear is that sharp corners
have been smoothed by just some local growth rules. Since sharp corners are not generally observed
in cells, this gives rise to the possibility that smooth edges in cells are actually the result of some local
growth rules.
Although this model successfully simulated the growth of a cell, it was unable to display the next
phase of the cell cycle, when the cell starts to divide. In addition, the rules applied to the cell to let it
grow were very specific. Therefore, it was highly probable that if modifications were made to them,
they would not be able to guide a proper cell growth. Thus, a second model was developed, whose
rule-set could be modified and put subject to evolution.
Figure 1: Stages of the cell growth, using the first model.
The new model also used a 2D cellular automata to simulate cell growth. However, unlike the first
model, the initial cell was a square with just one layer of membrane around it. Cellular growth was
again assumed to be the result of the growth of the membrane. A rule-set or ”genome” [5, 6, 7], comprising of a set of 100 rules [5], was used for growing the membrane. Similar to the model developed
by Basanta and colleagues [5], each rule was defined by three integers (although, in [5], four digit rules
were used to define the genome). The first two integers specified the conditions under which the rule
was true. Specifically, these integers looked at the number and position of the cells in the Moore neighbourhood. The third integer determined an action, to divide, to move to the interior, to move to the
outside, to be added to the interior of the cell or to block one of these actions. This resulted in a total of
1003200 distinct genomes for developing the cell.
An initial population of 100 random genomes was created. Each of these genomes were then used
to grow a cell, resulting in a total of 100 cells, each grown in a unique way. Genetic algorithm was used
to choose the genome which could produce the fittest cell [5, 7]. The fitness of each cell was measured
using a fitness function which took into account the perimeter to area ratio and connectivity of the cell
interior. A population of parent cells was selected using the tournament selection, a computationally
efficient ”fitness-proportionate” method of selection [17]. This population was then subject to a round
of crossover and mutation to produce the next population. For crossover, a pair of genomes where
randomly drawn from the parent population and one point was randomly chosen on them. The genes
before this point were then swapped with the probability of 0.7 between the two parent genomes. This
is known as one point crossover. After crossover, each gene was mutated with a probability of 0.001. A
more detailed description of this model can be found in the first appendix.
The method described above was repeated for 100 generation. The result was a population of genomes
which could grow a cell with a high level of fitness. The properties of the cells developed using the
final successful genomes were then analysed.
5
Results
The program was run thrice and 3 sets of 100 different genomes were selected. None of these genomes
displayed the actual steps of cell growth and division, at it happens in nature (Figure 2). This may be
due to the ruleset used in the model. To be more specific, the limited number of actions which were
used might have not been enough to generate the complete growth and division cycle. Moreover, con-
5
Figure 2: Variations in the shape of the cell during cell growth, guided by a selected evolved genome.
6
Variations in the shape of the cell during cell growth, guided by a selected evolved genome.
7
sidering the growth rates or developmental time as the conditions for performing an action may result
in genomes which could display a cellular growth more similar to the natural process. Furthermore,
improving a multi-variable fitness function which takes into account different parameters would lead
to a better choice of genomes.
Despite the fact that the model was unsuccessful to grow a cell and divide it into two cells of similar size and shape, the selected genomes shared some interesting common features.
One of the common properties of these genomes was that they all looked very similar, with minor
variations in some of the genes. This meant that although there exists a large number of possible
genomes, only a limited number of them, with a certain combination of rules and minor variations in
genes can end in a phenotypes with a high level of fitness.
Another common feature of the cells developed using the three sets of evolved genomes was that
the membrane seemed to be spread all over the cell interior (Figure 3). In fact, in a major group of
cells, the membrane divided the cell interior into several triangular parts. In another group of cells,
the cell interior was composed of a huge integrated part and a large number of small parts. The membrane surrounded the cell interior and filled the gaps between its parts. Other groups of cells also
existed which exhibited striped and checked patterns. It is well known that in a eukaryotic cell, the
cell interior is divided into various compartments by endomembrane system, which is constituted of
different membranes suspended in the cytoplasm. It is thought that this system may have evolved
during evolution [2]. In all the developed cells, the membrane displayed a similar structure which divided the cell interior into several parts. Moreover, this structure was emerged by evolving a random
genome under a specific selection criteria. Since the selection criteria looked for and chose the individuals which exhibited a larger area to perimeter ratio, the size of the cells displayed an unconstrained
growth. However, the structure of the interior of the cell did not change. Thus, it can be possible that
the endomembrane system is an emergent structure of the cell, evolving to keep its area to perimeter
ratio in a certain range.
Figure 4 shows the mean size of the cells developed by the genomes in the three sets, measured at each
step and plotted against time. It was observed that the size of the cell grew exponentially with time (In
the figures, the size was considered as the sum of the perimeter and the area of the cell. However, both
the perimeter and area of the cell exhibited a similar exponential growth). As mentioned earlier, this
suggests that checkpoints are not needed in the cell cycle, in order to keep the average size of the cells
constant [9].
A sierpinski-triangle-like structure of the interior of a group of the evolved cells was another interesting observation (Figure 3). This was very interesting as it showed that the simple local rules had
ended in the emergence of complex, self-similar structures in the cell, through evolution.
Although the features described above had emerged from carrying out the genetic algorithm on the
cellular automata, it was still possible that all, or at least some of these features could be achieved from
a randomly generated genome. To explore this possibility, a set of 100 random genomes was generated
and used to grow the cell. The fitness of the resulting cells were then calculated (Figure 5). It was
found that whilst the cells selected by the evolution had an average fitness of 2.134, the majority of the
cells grown by the random genomes exhibited an average fitness of 0.4022. Furthermore, just one of
the randomly generated genomes was able to grow cells which achieved a fitness score more than 0.9.
Therefore, it was concluded that the features could not be achieved in the absence of evolution.
6
Discussion and Further Research
In the last three decades of the twentieth century, a lot of advancements were made in understanding
the mechanisms underlying cell division. It was discovered that these mechanisms had been preserved
all the way through evolution [13]. However, the many aspects of cell growth and the role of evolution
in it have remained rather obscure.
Combining the cellular automata models, which have been proved to be useful for studying growth
mechanisms [1], with evolutionary algorithms enables us to investigate the role evolution plays in
forming and directing biological processes (for example, see [5]). However, they have not yet been
widely used in studying the process of cell growth [22] and up to this day, mathematical modelling is
8
Figure 3: Examples of the cells grown using the selected evolved genomes.
9
Figure 4: Average size of the cells generated by the selected genomes versus time. The curve, with the
equation size = 226.1e0.08272t and R-Square value of 0.9813, is an exponential fit to the data.
Figure 5: Histogram of the fitness of the cells generated by a population of (Left): evolved genomes
(Right): random genomes
Figure 6: The figure shows that the average fitness of the population increases during evolution.
10
the dominant approach to the field of cell growth (for example, see [9, 20, 21, 22]).
Although the model developed in this project was unsuccessful in illustrating the complete biological process of cell growth, the outcomes suggested that some features observed in eukaryotic cells may
be the result of selection of some certain properties (e.g. certain area to volume ratio) during evolution.
However, this conclusion was made on the basis of a limited number of observations and thus, further
work is required to explore the role of evolution in occurrence of these features. For example, having
discussed the issues of the model, one possible work is to consider more parameters (e.g. division
count and developmental time) in defining the rules and selection criteria of the model.
To conclude, the most important outcome of this project was to show that cellular automata models can
give us a deep understanding of the process of life and provide suggestions about the possible reasons
of appearance of some features in the structure and behaviour of cells. All in all, in the last decades of
the last century, a considerable amount of research has been done in the field of cell division, so many
aspects of the process are well known [13]. However, much remains to be explored about the process
of cell growth. Thus, this field is a very rich field for future work and cellular automata modelling is a
possible approach, able to give better insight into the field and the role of evolution in its underlying
processes.
7
Acknowledgement
I would like to thank my supervisors, Dr. Buzz Baum and Dr Mark Miodownik for their help and
support and for giving me the courage to use my imagination in creating my little in silico world of cells
and ruling over it.
11
A
First Appendix
The algorithm for the first model is as follows:
1. Repeat the following steps 100 times:
1.1 For each site on the cell membrane:
1.1.1 Calculate the number of vacant neighbouring sites.
1.1.2 Change at most three of the vacant neighbouring sites to membrane.
1.1.3 If the site is in contact with the inside of the cell, merge it with the cell interior.
1.2 If the ratio of the perimeter to the area of the resulting cell is larger than a given threshold, apply
the above changes. If not, keep the original cell.
The algorithm of the second model is as below:
1. Generate a random population of genomes. Each genome consists of 100 rules. Each rule is defined by 3 integers. The first two digits (integer numbers between 1 and 8) define the condition under
which the rule should be applied and the third digit (an integer number between 1 and 50) defines an
action.
2. Repeat for 100 generations:
2.1 For each genome, repeat for 30 time steps:
2.1.1 For the sites which are on the membrane of the cell:
2.1.1.1 Prepare a list of rules whose condition is satisfied.
2.1.1.2 Add to the counter of the action of the rules in the list above.
2.1.1.3 Apply the action with the highest counter.
2.1.2 Calculate the fitness of the cell grown by the genome. The fitness function is defined as
below:
fitness=area/perimeter+number of interior cells connected to other interior cells/number of membrane cells
2.3 Tournament Selection, a selection method with replacement:
2.3.1 Randomly, choose two genomes from the population.
2.3.2 Select the fittest genome with probability higher than the threshold (0.75) to be put in the
next generation. For this, generate a random number between 0 and 1. If this number is less than the
threshold, then the fittest genome would be put in the next generation. Otherwise, the less fit genome
would be put in the next generation [17].
2.4 Crossover [17, 16]
2.4.1 Divide the population into 50 pairs.
2.4.2 For each pair of genomes:
2.4.2.1 Randomly, choose a crossover point.
2.4.2.2 Swap the genes before the crossover point between the genomes with the crossover
probability of 0.7.
2.5 Mutation
For each genome, mutate the genes with the mutation probability of 0.001 [16].
B
Second Appendix
The codes for the two model are provided below. Both of the codes were written in MATLAB.
B.1
The Code for the First Model
%CREATE THE INITIAL CELL
n=131; %n should be odd
B = zeros(n,n);
B((n-5)/2:(n+3)/2,(n-5)/2:(n+3)/2)=1;
B((n-9)/2:(n-7)/2,(n-9)/2:(n+7)/2)=2;
B((n+5)/2:(n+7)/2,(n-9)/2:(n+7)/2)=2;
B((n-5)/2:(n+3)/2,(n-9)/2:(n-7)/2)=2;
B((n-5)/2:(n+3)/2,(n+5)/2:(n+7)/2)=2;
C=B;
imagesc(B);colormap(hsv(3))
%5x5
%5x5
%5x5
%5x5
%5x5
cell
cell
cell
cell
cell
12
%CELL GROWTH
for t=1:200
for i=2:n-1
for j=2:n-1
if (B(i,j)==2)
[neighbor_pos_i_0 neighbor_pos_j_0]=find...
(B(mod(i-2+n,n)+1:mod(i+n,n)+1,mod(j-2+n,n)+1:mod(j+n,n)+1)==0);
if(~isempty(neighbor_pos_i_0))
for(k=1:3)
pos=randi(length(neighbor_pos_i_0),1);
C(neighbor_pos_i_0(pos)+i-2,neighbor_pos_j_0(pos)+j-2)=2;
end
end
[neighbor_pos_i_1 neighbor_pos_j_1]=find...
(B(mod(i-2+n,n)+1:mod(i+n,n)+1,mod(j-2+n,n)+1:mod(j+n,n)+1)==1);
if(~isempty(neighbor_pos_i_1))
C(i,j)=1;
end
end
end
end
cell_paremeter=cell_count(C,n,2);
cell_area=cell_count(C,n,1);
if (cell_paremeter/cell_area)>0.05
B=C;
end
pause(0.1)
imagesc(B); colormap(hsv(3));
end
B.2
The Code for the Second Model
%DEFINE THE PROBABILITIES:
r_threshold=0.75;%parameter of Tournament Selection
p_c=0.7;%crossover probability
p_m=0.001;%mutation probability
genome_population=zeros(100,100,3);
average_fit=zeros(1,100);
fitness=zeros(1,100);
%CREATE THE INITIAL POPULATION OF GENOMES
for h=1:100
genome=zeros(100,3);
for i=1:100
for j=1:3
if j==1
genome(i,j)=randi(8,1);
elseif j==2
genome(i,j)=randi(8,1);
else
genome(i,j)=randi(50,1);
end
end
end
genome_population(h,:,:)=genome;
end
%CELL GROWTH
for no_generation=1:100;
for h=1:100
13
genome=genome_population(h,:,:);
%CREATE THE INITIAL CELL
n=101;%n should be odd
B = zeros(n,n);
B((n-5)/2:(n+3)/2,(n-5)/2:(n+3)/2)=1; %5x5 cell
B((n-7)/2,(n-7)/2:(n+5)/2)=2; %5x5 cell
B((n+5)/2,(n-7)/2:(n+5)/2)=2; %5x5 cell
B((n-5)/2:(n+3)/2,(n-7)/2)=2; %5x5 cell
B((n-5)/2:(n+3)/2,(n+5)/2)=2; %5x5 cell
C=B;
imagesc(B);colormap(hsv(3));
for t=1:30
for k=2:n-1
for l=2:n-1
if B(k,l)==2
next_action=zeros(1,50);
for i=1:100
next_action=next_act(B,k,l,n,genome,i,next_action );
end
if max(next_action)~=0
act=find(next_action,max(next_action));
if length(act)>1
act=act(randi(length(act)));
end
action=act;
C=Apply_action(C,k,l,n,action);
end
end
end
end
B=C;
pause(0.000001)
imagesc(B); colormap(hsv(3));
end
perimeter_count=0;
area_count=0;
surounded_cell_count=0;
for k=1:n
for l=1:n
if B(k,l)==1
area_count=area_count+1;
if neighbour_count(B,k,l,n)>6
surounded_cell_count=surounded_cell_count+1;
end
end
if B(k,l)==2
perimeter_count=perimeter_count+1;
end
end
end
fitness(1,h)=(area_count/perimeter_count)+...
(surounded_cell_count/area_count);
end
%TOURNAMENT SELECTION
avg_fitness=sum(fitness)/length(fitness);
average_fit(1,no_generation)=avg_fitness;
next_population=zeros(100,100,3);
for h=1:100
14
s1=randi(100,1);
s2=randi(100,1);
if fitness(1,s2)>fitness(1,s1)
s3=s1;
s1=s2;
s2=s3;
end
r=rand(1,1);
if r<r_threshold
next_population(h,:,:)=genome_population(s1,:,:);
else
next_population(h,:,:)=genome_population(s2,:,:);
end
end
%CROSSOVER AND MUTATION
for h=1:2:100
%CROSSOVER
c_point=randi([2,300]);
g1=next_population(h,:,:);
g2=next_population((h+1),:,:);
for i=1:c_point
g1(i)=g2(i);
end
for i=(c_point+1):100
g2(i)=g1(i);
end
%MUTATION
for i=1:100
for j=1:3
r_m=rand(1,1);
if r_m<p_m
if j==3
g1(1,i,j)=randi(50,1);
else
g1(1,i,j)=randi(8,1);
end
end
r_m=rand(1,1);
if r_m<p_m
if j==3
g2(1,i,j)=randi(50,1);
else
g2(1,i,j)=randi(8,1);
end
end
end
end
next_population(h,:,:)=g1;
next_population((h+1),:,:)=g2;
end
genome_population=next_population;
end
final_genome_population=next_population;
%FUNCTION FOR THE CALCULATION OF THE NEXT ACTION
function [ next_action ] = next_act( B,k,l,n,genome,i,next_action)
%This function determines the action which should be applied to the cell
%by calculating each action’s counter
no_neighbours=neighbour_count(B,k,l,n);
if genome(1,i,1)==1 && B(k-1,l-1)==1 && no_neighbours==genome(1,i,2)
15
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==2 && B(k,l-1)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==3 && B(k+1,l-1)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==4 && B(k-1,l+1)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==5 && B(k,l+1)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==6 && B(k+1,l+1)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==7 && B(k+1,l)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
if genome(1,i,1)==8 && B(k-1,l)==1 && no_neighbours==genome(1,i,2)
next_action(1,genome(1,i,3))=next_action(1,genome(1,i,3))+1;
x=1;
end
end
%FUNCTION FOR APPLYING THE NEXT ACTION
function [ C ] = Apply_action( C,k,l,n,action )
%This function applies the next action to each site of the cell.
if 1<=action<=8
index=(l-1)*n+k;
if 1<=action<=3
C(index-n+mod(action,3)-1)=2;
end
if 4<=action<=6
C(index+n+mod(action,3)-1)=2;
end
if 7<=action<=8
C(index+mod(action,2)*2-1)=2;
end
end
if 9<=action<=16
index=(l-1)*n+k;
if 9<=action<=11
C(index-n+mod(action,3)-1)=2;
C(index)=0;
end
if 12<=action<=14
C(index+n+mod(action,3)-1)=2;
C(index)=0;
end
if 15<=action<=16
C(index+mod(action,2)*2-1)=2;
C(index)=0;
16
end
end
if 17<=action<=24
index=(l-1)*n+k;
if 17<=action<=19
C(index-n+mod(action,3)-1)=2;
C(index)=1;
end
if 20<=action<=22
C(index+n+mod(action,3)-1)=2;
C(index)=1;
end
if 23<=action<=24
C(index+mod(action,2)*2-1)=2;
C(index)=1;
end
end
if action==25
C(index)=1;
end
end
%FUNCTION FOR CALCULATING THE NEIGHBOURING CELLS
function [ no_neighbours ] = neighbour_count( B,i,j,n )
%This function calculated the number of the neighbouring cells which belong
%to the cell interior.
no_neighbours=0;
for a=i-1:i+1
for b=j-1:j+1
if B(a,b)==1
no_neighbours=no_neighbours+1;
end
end
end
end
17
References
[1] T. Alarcon, H.M. Byrne, and P.K. Maini. A cellular automaton model for tumour growth in inhomogeneous environment. Journal of Theoretical Biology, 225:257–274, 2003.
[2] Bruce Alberts, Dennis Bray, Karen Hopkin, Alexander Johnson, Julian Lewis, Martin Raff, Keith
Roberts, and Peter Walter. Essential Cell Biology, chapter 15, pages 498–499. New York ; London :
Garland Science, 3 edition, 2010.
[3] Terence Allen and Graham Cowling. The Cell: A Very Short Introduction, chapter 4, page 60. Oxford:
Oxford University Press, 2011.
[4] Atilla Altinok, Didier Gonze, Francis Levi, and Albert Goldbeter. An automaton model for the
cell cycle. Interface Focus, 1:36–47, 2011.
[5] David Basanta, Mark Miodownik, and Buzz Baum. The evolution of robust developement and
homeostasis in artificial organisms robust development and homeostasis in artificial organisms.
PLoS Computational Biology Biol, 4(3):e1000030, 03 2008.
[6] David Basanta, Mark Miodownik, Peter J. Bentley, and Elizabeth A. Holm. Investigating the
evolvability of biologically inspired ca. 9th Conference on Artificial Life, 2004.
[7] David Basanta, Mark Miodownik, Elizabeth A. Holm, and Peter J. Bentley. Using genetic algorithms to evolve three-dimensional microstructures form two-dimensional micrographs. Metallurgical and Materials Transactions A, 36:1643–52, 2005.
[8] David Beasley, David R. Bull, and Ralph R. Martin. An overview of genetic algortihms: Part1,
fundamentals. University Computing, 15(2):58–69, 1993.
[9] Ian Conlon and Martin Raff. Differences in the way a mammalian cell and yeast cells coordinate
cell growth and cell-cycle progression. Journal of Biology, 2(1):7, 2003.
[10] Attila Csikasz-Nagy, Dorjsuren Battogtokh, Katherine C. Chen, Bela Novak, and John J. Tyson.
Analysis of a generic model of eukaryotic cell-cycle regulation. Biological Journal, 90:4361–4379,
2006.
[11] G. Bard Ermentrout and Leah Edelstein-Keshet. Cellular automata approaches to biological modelling. Journal of Theoretical Biology, 160:97–133, 1993.
[12] Francois Graner and James A. Glazier. Simulation of biological cell sorting using a twodimensional extended potts model. Physical Review Letters, 69(13):2013–17, 1992.
[13] David A Guertin and David M Sabatini. Cell size control. In Encyclopedia of Life Sciences, pages
1–10. John Wiley Sons, Ltd, 2005.
[14] Andrew Ilachinski. Cellular Automata: A Discrete Universe, chapter 1, pages 3–7. Singapore: World
Scientific, 2002.
[15] Julian F. Miller. Evolving developmental programs for adaption, morphogenesis, and self-repair.
Advances in Artificial life, 2801/2003:256–265, 2003.
[16] Melanie Mitchell. An Introduction to Genetic Algorithms, chapter 1, pages 10–12. Cambridge, Massachusetts ; London : MIT, 1996.
[17] Melanie Mitchell. An Introduction to Genetic Algorithms, chapter 5, pages 170–174. Cambridge,
Massachusetts ; London : MIT, 1996.
[18] David O Morgan. The Cell Cycle: Principles of Control, chapter 10. London : New Science Press in
association with Oxford University Press and Sinauer Associates, 2007.
[19] Paul Nurse. Genetic control of cell size at cell division in yeast. Nature, 256:547–551, 1975.
[20] Benjamin Pfeuty and Kunihiko Kaneko. Minimal requirements for robust cell size control in eukaryotic cells. Physical Biology, 4:194–204, 2007.
[21] Zhilin Qu, W. Robb MacLellan, and James N. Weiss. Dynamics of the cell cycle: checkpoints,
sizers and timers. Biological Journal, 85:3600–3611, 2003.
[22] John J. Tyson, Kathy Chen, and Bela Novak. Network dynamics and cell physiology. Nature
Reviews. Molecular Cell Biology, 2:908–916, 2001.
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