Group Assignment 1 MATH3041, T1 2015

Group 2

AUTHORS:

Mohit Rajput (z3467941) , Anthony Patterson (z3416603), Prosha Rahman (z3417078), Katherine Tay (z3384676) and Tristan Meyers (z332970)

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Table of Contents

Introduction ..................................................................................................................... 3 Model Selection ........................................................................................................... 6

Methods ........................................................................................................................ 12 Discrete Model .......................................................................................................... 12 Enhancement by Applying a Stochastic Forcing ........................................................ 20

Discussion ..................................................................................................................... 27

Conclusion .................................................................................................................... 28

References .................................................................................................................... 28

3

Introduction

Candida Albicans (C. albicans) is a diploid form of yeast that commonly exists within the

human body, mainly the abdominal region [1]. It can exist in three forms [1]:

1. Yeast cells (blastospores)

2. Pseudohyphal cells

3. True hyphal cells

Without having to pass an intermediate phase, it is able to switch stochastically between

an opaque phase and a white phase. These states have different characteristics,

strongly correlated by their environment; the white state can adhere to human epithelial

cells in the mouth and other cavities, while the opaque state is seen to colonise skin

easily [2]. Moreover, mice intravenously given C. albicans revealed that the dominant

virulent state is the white phenotype [2], whilst the opaque phase is chiefly noted to be

the mating phenotype [2].

The switch occurs in two particular alleles of C. albicans, mating type locus 𝛼 (MTL-𝛼)

and MTL-a [2]. The stochastic White-Opaque regulator 1 (Wor1) is responsible for this

drastic switch, giving rise to heterogeneity in the cell population. Each phase is easily

distinguishable by the distinct gene expression programmes and cell shapes, and is

stable for many cell divisions [3].

Opaque cell formation and conversion to stable opaque cells in alpha cells is induced by

the increased expression of Wor1 [3]. Once this switch has been prompted, the

presence of Wor1 increases the further expression of Wor1 [2], [3], thereby acting as a

positive feedback loop. In addition, the expression of Wor1 induces an additional, non-

direct positive feedback loop involving the regulator Wor2, which in turn increases the

expression of Wor1. Wor1 also suppresses the zinc cluster transcription factor (Czf1),

which is necessary for white cell stability [2]–[4]. Therefore, the opaque state is very

stable once induced. Meanwhile, the presence of Wor1 also inhibits the expression of

Efg1 which is necessary for cells to be in the white state. This process is visualised in

figure 1.

The switching can occur stochastically, and bistable nature of white-opaque switching

can be attributed by these positive and negative feedback regulations with other genes

[2]. These feedback regulations can simulate a suitable environment to assist the

progress of bistable phenotypes among cells within a population. In a positive feedback

system where the feedback loops cooperatively respond to an activator, the system can

display bistability [3]. Here, the positive feedback loop is stimulated by the existence of

a particular transcription factor in high concentrations. Conversely, bistability can also be

caused by two mutually repressing negative feedback loops. In this scenario, random

cells will have a higher concentration of one of the repressors, resulting in phase-specific

target gene expression. The strength of the feedback loop is proportional to the

concentration of active Wor1 in the opaque phase. This affects the stability of the

opaque phase [3].

4

C. albicans plays a crucial role as a model organism as it has been observed to be one

of the most pathogenic and most frequent colonizing species [1]. Its pathogenicity partly

arises from its existence in a biofilm, giving it the ability to grow in a multicellular

community. The structure of the fungus also possesses two switching systems that

result in dramatic phenotypic changes. In C. albicans, Wor1 regulates in a binary “all-or-

none” [3] pattern and is highly expressed in opaque cells. Wor1 itself acts as a “master

regulator” [5] of white-opaque switching. This relatively condensed biological process is

therefore useful in making assumptions in mathematical models.

Figure 1: The feedback loops for the transcription factors involved in the white-opaque switch (reproduced from [2]). Arrows indicate an

increase in expression, whilst flat-head lines indicate a suppression of that transcription factor.

5

A model organism is a species that has been widely studied, usually because it is easy

to maintain and breed in a laboratory setting and has particular experimental advantages

[6]. Table 1 illustrates several model organisms and their common usage.

Table 1: Comparison of different model organisms [1], [6]

Yeast

Candida Albicans

Frog

Xenopus laevis

Fruitfly Drosophila

melanogaster

Mouse

Rattus norvegicus

Model Use Genetic model Experimental model Genetic model Genomic model

Advantages -Excellent genetics -Genome sequence complete -Genes can be easily cloned -Quick and easy to grow -Very powerful second site screening -Powerful molecular techniques -Cell cycle control similar to animals -Possess all basic eukaryotic cell organelles

-Accessibility of embryo (pond no shell) -Injection of RNA into identifiable blastomeres -Ectopic gene expression possible in early embryos, although manipulation of levels difficult -Excellent experimental embryology grafting induction preparations

-Excellent genetics -Genome sequence complete -Genes can be easily cloned -They are easily raised in the lab, with rapid generations, high fecundity, few chromosomes, and easily induced observable mutations. -Targeted gene disruption RNAi effective -Second site suppressor/enhancer screens -Powerful molecular techniques -Transposon tagging -SNP mapping -Targeted misexpression of genes in space and time -Mosaic analysis: determine where gene acts

-Similar brains to humans: homologous areas/cell types -Availability of material at all stages -Source of primary cells for culture -Developmental overview same as for all mammals -“Reverse” genetics: targeted gene knockouts by homologous recombination routine -Construction of chimeric embryos possible

Disadvantages -No distinct tissues -No genetics, although under development -Difficult to create transgenic animals

-Embryological manipulations difficult -Targeted gene disruption still difficult

-Embryonic manipulations difficult -Development and life cycle relatively slow (months) -Classic “forward” genetics difficult

Uses -Genetics and cell biology

-Developmental biology -Cell biology, -Toxicology neuroscience

-Genetics and developmental biology -Key aspects of how fertilized eggs develop into complex organisms. -Learning how genes and the environment interact to affect behaviour.

-Establish disease model by mimicking gene defects seen in humans. -Models used to test efficacy of new drugs. -Toxicology neuroscience -Source of primary cell cultures.

6

In a societal context, it is important to understand and model the mechanisms of C.

albicans. At a reasonable and healthy level, C. albicans makes up 20% of the total

bacteria in the stomach. However, it has been speculated that the modern diet with

increased carbohydrate intake can contribute to elevated levels of this yeast within the

human digestive system, exacerbating health impacts of immune-compromised patients

and causing yeast infections [2], [7]. In addition, C. albicans has been seen to be

resistant to antifungal drug treatments [8].

Model Selection One modeling approach used the dynamics of gene regulation by using a system of

differential equations describing the concentrations of the active and inactive forms of

proteins [4]. The system of equations is as follows:

𝑑𝐴

𝑑𝑡= 𝑘𝑎𝑃 − (𝑘−𝑎 + 𝑘𝑑)𝐴 (1)

𝑑𝑃

𝑑𝑡= 𝑘1 +

𝑘2(𝐾𝐴𝐻)

1 + 𝐾𝐴𝐻+ 𝑘−𝑎𝐴 − (𝑘𝑎 + 𝑘𝑑)𝑃 (2)

Here 𝐴, 𝑃 are the transcription factors, 𝐻 is the Hill’s coefficient and the parameters

𝑘−𝑎,𝑎,𝑑,1,2 are reaction constants. The initial conditions set were [𝑤ℎ𝑖𝑡𝑒, 𝑜𝑝𝑎𝑞𝑢𝑒] =

[100,100] to represent an initial cell count of 100 for both types of cells. The parameters

were standard values, set to 𝑘𝑎 = 5, 𝑘−𝑎 = 20, 𝑘1 = 0.01, 𝑘2 = 0.3 , 𝑘𝑑 = 0.08, 𝐾 = 5,

and 𝐻 = 2. Results are reproduced in figure 2.

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Figure 2: Dynamics of transcription factors with positive feedback based on a system of non-linear differential equations (see equations1,2)

8

As seen in the concentration graph in figure 2, the model did not reveal much about the

switching behaviour of the 2 cells over time, as there is an instant convergence and no

changing behaviour of the concentrations after 1 time step. Importantly, there is less

flexibility on the parameters used. Thus, it is harder to manipulate the model to allow for

different environmental factors unless something is known about how the reaction

factors 𝑘−𝑎,𝑎.𝑑.1.2 change with environmental conditions.

Another model that used a dyanmic relationship betwen three regulators was also

investigated. These transcription factors are Wor1, Wor2, and Efgz1 [9], where their

relationship can be visualised in figure 2. This system of equations is as follows:

𝑑[𝑊𝑂𝑅2]

𝑑𝑡= 𝑔1 +

𝑎1([𝑊𝑂𝑅1]𝑛)

(1 + [𝑊𝑂𝑅1]𝑛)− 𝑑1[𝑊𝑂𝑅2] (3)

𝑑[𝑊𝑂𝑅1]

𝑑𝑡= 𝑔2 +

𝑎2([𝑊𝑂𝑅2]𝑛)

(1 + [𝑊𝑂𝑅2]𝑛)+

𝑎4

1 + [𝐸𝐹𝐺1]𝑛− 𝑑2[𝑊𝑂𝑅1]

(4)

𝑑[𝐸𝐹𝐺1]

𝑑𝑡= 𝑔3 +

𝑎3

(1 + [𝑊𝑂𝑅1]𝑛)− 𝑑3[𝐸𝐹𝐺1]

(5)

Standard values for 𝑔1,2,3, 𝑎1,2,3 and 𝑑1,2,3 and initial conditions of [100,100,100] for all

three genes were used, and the results plotted in figure 3.

This genetic model was effective for displaying the behaviour that the 3 genes exibit with

each other in concentration as time increases. However, it does not allow for the same

level of flexibility that the discrete model in this report has.

The last model considered was the predator-prey model. This could parallel the

interaction with white and opaque cells by creating a positive feedback loop whereby

one cell type leads to an increase of switch rate as more cells are created.

Unfortunately, changing the signs of the coefficients of the original Lotka–Volterra

equation resulted in infinite gradients, and it was not viable to model a finite amount of

cells on a differential equation with infinite gradient.

The other option was to apply an unmodified version of the Lotka–Volterra equations.

However, an increase in predators ultimately leads to the decrease in the rate of change,

thus eventually depleting the number over time and not maintaining bistability. This

creates an infinite cycle between increased and decreased numbers of both white and

opaque cells, and it was not seen as a realistic interpretation of cell switching behaviour.

Figure 4 illustrates these trials.

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Figure 3: Using a dynamical system of equations to model feedback between three transcription factors. See equations 3-5.

10

Figure 4: Applying the predator-prey model to white-opaque switching.

11

Another approach considered the development of the discrete model. With this, all the

essential modifications to the environmental factors could be achieved by changes to the

parameters covered in the discrete model. A Markov Chain was used, seen as:

𝑃(1) = (𝑝𝑥 1 − 𝑝𝑥

1 − 𝑝𝑦 𝑝𝑦) (6)

where row 1 represents the white state, and row 2 represents the opaque state at time 0,

and column 1 and column 2 represents the respective states at time 1. This chain

denotes the probability that the state of a cell will be state x or y after 1 time period.

Multiplying this matrix 𝑛 times yields:

𝑃(𝑛) = (𝑝𝑥 1 − 𝑝𝑥

1 − 𝑝𝑦 𝑝𝑦)

𝑛

(7)

which is the probability the cell goes from state 𝑥, 𝑦 at time 0 to 𝑥, 𝑦 at time 𝑛. Taking the

limit as 𝑛 → ∞,

𝑃(𝑛) = (1/2 1/21/2 1/2

)𝑛

(8)

when 𝑝𝑥 = 𝑝𝑦 . This aids in understanding why in the control environment (when

probability of switch both ways were equal), both concentrations of white and opaque will

converge to 0.5., representing a stable system. Additionally, by focusing on the

concentrations of the white and opaque cells rather than the levels of Wor1 or Wor2, this

allows for a more generally approach to understand the switching process, which can

then be applied to other biological systems.

Thus, white-opaque switching process can be consider stochastic, occurring

approximately every 104 generations [9]. This results in a bistable relationship between

the two phases. However, the frequency at which the switches occur depend on

environmental conditions. This indicates that the activity of certain components of the

regulatory feedback loops can be changed by specific signals. More specifically, these

components are ones that ensure “epigenetic inheritance” (cellular trait variance not

caused by changes in DNA sequence) of the white and opaque phases.

Another factor that affects the rate of white-opaque switching is stimulation via UV

radiation. Low doses can speed up the switching 10 to 20 fold by inducing increased

levels of Wor1, leading to the producing of Wor2 and therefore a postie feedback loop is

established. Neutrophils and oxidants also similarly promote switching from white to

opaque. Finally, temperature, both low (4oC) and high (37oC) strongly stimulate opaque

cells to switch to white cells due to an increase in Wor1, with the latter causing a mass

conversion to the white cell phenotype [2].

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Methods

First, a control run of the model will be developed, then the control will be compared

with other models that clearly show the effects that external environmental factors have

on the regulation of the switching. These factors include increased and decreased

temperatures, increased ultraviolet (UV) radiation, increased CO2, and anaerobic

conditions (oxygen-free environment). An expansion upon this by the introduction of a

stochastic forcing will then be introduced.

Discrete Model

Initially, it is assumed a known proportion of white cells to opaque cells. In the following

evaluation of the model, all initial conditions start off with 100% white and 0% opaque.

There is an associated probability with the likelihood of a white cell switching to opaque,

or vice versa. The number of generations it takes for a cell to switch to take place is

between 103 and 104 generations [2]. Under the discrete time system modelling, it was

assumed that a fixed number of cells were being observed, which is why all future

graphs in this model will be displayed in proportions. This implies that if one every

thousand generations a switch takes place, then the probability of a cell switching is

1/1000.

Thus, the model it is assumed that 1/1000th of the currently existing white cells switch to

opaque in the next time step, and the reverse happens with the existing opaque cells.

Therefore, over the long term there is a convergence to a stable proportion value for

both the white cells and opaque, as seen in figure 5.

The control environment graph above also assumes that there is no positive or negative

feedback. Positive feedback is the behaviour of the switching cells (white to opaque)

activating the WOR1 gene which in turn encourages even more white cells to switch to

opaque [2]. This feedback mechanism will be considered separately later in the report. In

this example of the deterministic non-feedback model, 𝑁∗is the point where

𝑥(𝑛) = 𝑥(𝑛 − 1) and 𝑦(𝑛) = 𝑦(𝑛 − 1) for every 𝑛 ≥ 𝑁∗. (9)

Since the values will only converge to a constant as 𝑛 → ∞, a computational tolerance

value of 10−5 was used to closely match the visual behaviour of the model.

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Figure 5: Convergence to a stable population of white and opaque cells with a switch probability of 1/1000.

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The discrete model works under a fairly simplistic step-by-step approach which updates

the concentration proportion of the white and opaque concentrations over 𝑁 time

periods. The initial concentration of white ( 𝑥 ) is 100%, so 𝑥(0) = 1 , and the initial

concentration of opaque cells ( 𝑦 ) is 0% , so 𝑦(0) = 0 . There are also probabilities

associated with the model, so 𝑝𝑥 is the probability of white switching to opaque, and 𝑝𝑦

is the probability of the converse switch of any given cell. In the control environment, it

was assumed that the probability of the switch for both ways is 1/1000, so 𝑝𝑥 =1

1000 and

𝑝𝑦 =1

1000. This remained constant through all time steps under the constant probability

assumption. Next, the number of cells that switch from white to opaque from time step

(𝑛 − 1) to 𝑛 is denoted Δ𝑦(𝑛), while Δ𝑥(𝑛) is the equivalent notation for cells switching

from opaque to white. This results in the equalities:

Δ𝑥(𝑛) = 𝑝𝑦𝑦(𝑛 − 1),

Δ𝑦(𝑛) = 𝑝𝑥𝑥(𝑛 − 1). (10)

This process was then repeated for 𝑁 time steps to result in the convergence to a stable

concentration for the white and opaque cells.

The alteration to both 𝑝𝑥 and 𝑝𝑦 are next considered. UV radiation speeds up the

switching from both directions tenfold [2]. Thus, the probabilities were modified to shape

𝑝𝑥 = 0.01 and 𝑝𝑦 = 0.01 to reflect this stimulation. What is interesting to observe is that

the plots are identical to the control environment in shape, except the time that it takes to

a similar appearance to the control graph is reduced by tenfold. This is explained by the

fact that the increased 𝑝 values increase Δ𝑥 and Δ𝑦 tenfold, resulting in a faster

convergence, seen in figure 6.

15

.

Figure 6: Increasing the probability of switching with the influence of UV radiation.

16

Then considered was the case where one probability factor is increased relative to the

other. In the case of an environment with decreased temperatures (4oC), there was an

increase in the rate that opaque cells switch to white. In fact this is the case with both the

decreased and increased temperatures (37oC). For the sake of computational cost, a

factor of 3 was chosen to demonstrate the faster switching from the opaque to the white

phase. Here, 𝑝𝑥 is the same as in the control, and 𝑝𝑦 is increased by a factor of 3. As

seen in figure 7, there is a convergence towards a higher concentration of the white cell

concentration of 75% to the opaque concentration of 25%.

For the case of increased temperature, the conversion from opaque to white is much

stronger, resulting in “mass conversion” to the white state [2]. Here, a factor multiplier of

𝑝𝑦 was chosen to be 10 in order to demonstrate this conversion. In figure 8, the

convergent concentrations are 90.91% white and 9.09% opaque.

Now, anaerobic conditions can result in a bulk switch of cells to the opaque state [2].

This switch to the opaque phase is evidently more powerful than the switch to the white

phase with a temperature increase (37oC) with normal air conditions, as with both

increased temperature and anaerobic conditions, an opaque switch occurs. Therefore,

choosing a 𝑝𝑥multiplier of 15 seemed intuitive because it represents a more pronounced

effect (order of magnitude) than the switching from white to opaque has in anaerobic

conditions with increased temperature. Figure 9 shows the mass conversion in

concentration in anaerobic conditions at room temperature and then again at 37°C. The

convergent concentrations are 6.25% white and 93.75% opaque.

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Figure 7: Affect of temperature on the bistability of white-opaque switching

18

Figure 8: Temperature variation and its impact on the modeled white-opaque populations and their bistability

19

Figure 9: Increasing CO2 concentration at room temperature and at 37oC and its effect on the white-opaque switch.

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Enhancement by Applying a Stochastic Forcing

Now that the deterministic behaviour of the convergence of concentrations has been

modelled, a stochastic forcing is introduced. The parameters that will be modified are the

𝑝𝑥 and 𝑝𝑦, which were then made stochastic by pseudo-random number generation. The

new probabilities was modified with each time step by generating a random number

between 0 and 1 under a uniform distribution. The random number was then centred

around 𝑝𝑥 or 𝑝𝑦 by taking the inverse of the chosen cumulative distribution function

(CDF). In the model, the inverse CDF was chosen to be a normal distribution function

with mean and standard deviation of 𝑝𝑥/2 and 𝑝𝑦/2, respectively. This loose assumption

was made in order to introduce randomness into the model. Figure 10 shows the

probability distribution function (PDF) and CDF of 𝑝𝑥 that demonstrates the range of

values that 𝑝𝑥 and 𝑝𝑦 can take when set to the standard 1/1000. Note that the standard

deviations were set at the level shown above to be large enough to represent natural

variation in the switch rates, but not too large so that negative (non-practical) p-values

occur. As seen in figure 7, the probability is equivalent to 2 standard deviations away

from the mean. This was seen as negligible in the larger structure of the simulation

process.

It is interesting to note that with the first 4 cases, the switch rate graphs look similar to

white noise, which is created by generating thousands of numbers between 0 and 1 and

taking the inverse norm CDF and plotting it over a time period.

21

Figure 11-15 represents all modelled environmental factors with the stochastic forcing.

Figure 10: Probability distribution function (PDF) and cumulative distribution function (CDF) of the stochastic forcing.

22

Figure 11: Control environment with stochastic forcing

23

Figure 12: Stochastic forcing of white-opaque switching in a UV enhanced environment

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Figure 13: Effect on white-opaque concentrations in a low temperature environment and stochastic forcing.

25

Figure 14: Effect on concentration of white-opaque cells in a warm environment with stochastic forcing.

26

Figure 15: Anaerobic conditions at room temperature and 37oC and its affect on white-opaque populations.

27

Discussion

The stochastic model developed accurately reflects the bistability in white and opaque concentrations of cells in a media in regards to environmental impetus. Here, the transcription factors Wor1 and Wor2 are modeled by using the concentration of white and opaque cells as proxy. This model is empirical since it absorbs many of these conditions into a useful probability constant. This means it could be calibrated in a dose-response manner to UV exposure, temperature and oxygen concentrations in future experiments in order to increase the model skill. Moreover, the ease of use of the model means it can be applied to a variety of biological modeling situations requiring convergence to stable state(s). By increasing the value of either 𝑝𝑋 or 𝑝𝑦, this leads to a change in the rate of reaction,

therefore leading to a stable state sooner. The nature of the model means that many different states of stability could be achieved based on the selection of 𝑝𝑥, 𝑝𝑦, informed

by the nature of the reaction environment. Since an increase in temperature and UV exposure increase the rates of Wor1, this leads to a mass switch to opaque, which is consistent with the model. Additionally, the added element of stochasticity adds a level of randomness to the model. Comparing the control models runs, figure 5 shows an “idealised” curve, whilst figure 11 shows an inherient unpredicablity of the switching, which is consisent with experienmental observations [2-4]. However, when environmental forcing is then induced, the stochastic model still sucessfully achieves stable states. There are certain limitations of this empirical approach to modeling. One such consequence is the how stochasticity was handled. Here, it was chosen to be a uniform distribution. This stochastic forcing could be a function of the concentration of a transcription factor. For instance, during a strong environmental stimulus where Wor1 levels are increased and the positive feedback loop is triggered, the stochastic switching could tend to near negligible levels (once every 104 generations). Another possible limitation is the generality of the model. Whilst striving to develop an original model, many assumptions may have overlooked certain transcription processes. However, this could be accounted for by normalizing the probability parameters to a function based on the dynamics of transcription factors, such as those presented in equations 1-5. Thus, if the concentration of a transcription factor over time increases, the probability of switch, 𝑝𝑥 or 𝑝𝑦, would increase.

Therefore, although the model is general, the usefulness in modifying an all-encompasing parameter means that it could apply to a range of situations and adapt to future insight about the environmental behavior of C. albicans.

28

Conclusion C. albicans is a unique yeast organism in its ability to switch between two different “states” of white and opaque without any intermediate transition. This makes modeling the organism particularly attractive, as this almost binary existence alleviates the need to make assumptions. It was discovered that a simple stochastic model using a Markov Chain could be applied in order to model a bistable state between white and opaque cells induced by a presence of the receptor Wor1 by environemtal stimulus. Although there is room for improvement within the model parameters, the ease of modifiying parameter makes this model attractive for future use in modeling C. albicans as well as other biological systems that require a convergence towards stable states.

References

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