Modeling the Fission Yeast Cell Cycle with Boolean Networks

Group 1: Boolean Networks Math 5445 May 14th, 2020

Authors: Corey Momsen Jessie Speert Mitchell Gross

Abstract The purpose of this paper is to see if the simplifications needed to convert an ODE system of the fission yeast cell cycle to a boolean network still allows for accurate depictions of the cellular dynamics. Two models were found, and they were the Davidich boolean network and the Humaidan boolean network. The Davidich network defines 12 primary nodes centered around the Start node, the Cdc2/Cdc13 node, and the Cdc2_Tyr15 node, with the nodes being the activations of proteins within the fission yeast during its cell cycle. The Humaidan network, on the other hand, expands the Davidich network by adding 4 phosphorylation nodes to be able to more accurately model the dynamics. Both models were then replicated on Matlab, and one full cell cycle was completed, with the completion of the cell cycle marked by reaching a steady state. Our replicated boolean networks were used to model the wild-type of fission yeast, find all steady states of each model, and see the dynamics of each model under mutations to the initial conditions as well as when nodes were eliminated altogether. Our results verified the results from the Davidich and Humaidan papers, supporting the conclusion that the boolean models are robust. Also, while there were some differences found between our Davidich model results and our Humaidan model results, both corresponded very well to each other. Our results led us to the overall conclusion that not only do boolean models perform comparably to ODEs despite making several assumptions for simplification purposes, but they also have the potential to be able to model more complex biological networks like human cells to be able to possibly better show how mutations can causes diseases and even cancer. However, a boolean model must be created of a type of human cell in the future to definitively show that they can accurately depict more complex cellular interactions.

Introduction: The cell-cycle for eukaryotic cells undergo four phases: G1, S, G2, M. Over this cycle, the cell grows to approximately twice its original size before splitting in a process called mitosis, whereby the cell splits into two identical cells. The cell-cycle is paramount to preservation of a multicellular organism as well as to sustain the species of single celled organisms. Errors in the cell-cycle in humans can lead to disorders, diseases, and, most importantly, cancer. Modeling the dynamics and evolution of simpler eukaryotic cells will allow for those models to then be expanded and adapted to more accurately reproduce the dynamics of the more complex cells in humans. Since eukaryotic cells share several fundamental similarities in the cell-cycle, the general process of each of the four phases are the same [5]. To know how the cell-cycle is modeled, the basics of the cell cycle must first be understood. The first phase, known as G1, contains the point where a eukaryotic cell irreversibly

2 commits to cell division. Whether the point of no return is at the start point (yeast cells) or the restriction point (animal cells), it serves as a barrier to the S phase and where the cell-cycle can be modeled as starting and eventually ending [5]. During the rest of the G1 phase, the cell grows, prepares for DNA replication, and evaluates the surroundings for whether it should undergo division [5]. The next and equally important phase is the S phase, and this is where DNA replication occurs. With the DNA in the shape of compact chromosomes, DNA is replicated to give a shape that resembles an “X.” Also, the cell continues to grow in preparation for cell division and the rate of synthesis of enzymes and proteins increases significantly. The next phase is G2, and it’s also known as the “gap” phase. This phase is the last chance for the cell to grow before division, and it is where the cell checks if all DNA is correctly replicated and if all proteins and enzymes have been produced in the correct amounts. Once everything is in order, the last phase is known as the M, or mitosis phase. This is where the DNA is split into identical pairs and two cells with identical nuclei are formed. With yeast having less protein and gene interaction, but still having the basic complexities of eukaryotic cell division, it accurately models that the yeast cell cycle is a crucial bridge to understanding human cell-cycle behavior. It should be emphasized that the boolean model starts and ends at the start point within the G1 cycle. A full, viable cell cycle is accomplished if all four phases were successfully completed and a steady state was reached. Since the start point is modeled in the network as “Start” and that node in the network is only “on” during the first time step of the network, the network will end at the start point in the G1 phase after one full cycle. Ordinary differential equations (ODEs) have traditionally been recognized as the most-comprehensive way to study, understand, and manipulate the system-dynamics of the model they represent. Still, the substitution of ODE-based models for boolean network models presents an advantage over the classic approach towards understanding system dynamics due to their ability to decrease the complexity that comes from unnecessary dynamic detail that may arise from the traditional approaches. It is often the case that ODE-based models encompassing dozens of parameters become a computational burden on both the mathematician and computational system that is processing the evolution of the model over time; in the interest of

3 saving time for the researchers, a boolean network model may be used instead to capture the overarching dynamics of the system. This becomes especially important when the model switches from the simpler fission yeast system to the complexity of the cellular interactions in the human body. Boolean network models can be especially helpful when focusing on the activation of genes and proteins in regulatory networks. Because the set of boolean values is binary (0 or 1), a gene-network that is modeled in this way will function as a set of states that differ based on which genes are active or inactive at each time step. For example, if each gene is represented as a node that holds a 0-value for “off” or silenced and a 1-value for “on” or being transcribed and translated, the value of each node can be combined into a bit-string that is unique for that state, and all of the bit-strings that can be achieved as the model is simulated over time would encompass the set of possible states the network may exist in. The study of fission yeasts using boolean networks is an ideal starting point since the fission yeast cell cycle is already thoroughly studied. Examples include when fission yeast was first discovered in 1893 by German-Scientist Paul Lindner as well as notable studies carried out in the mid-twentieth century by John Murdoch Mitchison and Urs Leupold [4]. With existing, complex ODE models that precisely define the system, the accuracy of the boolean networks can be measured. More importantly, it can be seen if the simplifications necessary to build the boolean model, such as assuming most interactions happen in one pre-defined time step and having genes and proteins be either on or off, can still lead to an accurate representation of a biological system.

The two boolean networks used to model fission yeasts that we examined rely on time-points that correspond to the degradation of central proteins. Defining an appropriate time step (dt) to traverse the effects of the simulation requires the consideration of potential impacts extremities may have on the system. For instance, a timescale that is too large has the potential to represent some changes to the system as having immediate effects when the event happens in a time that is much less than dt. Conversely, if the model is using a timescale that is too small, then the researcher must account for changes to the system that occurred over multiple time steps.

In the boolean network models related to fission yeasts, however, a time-basis that corresponds to the degradation of central proteins is sufficient to model the system. In the fission yeast cell cycle, the creation and degradation of central proteins happened in similar time frames, so all nodes within the two boolean systems analyzed had all interactions happen in one time step. In other words, no interactions in the two papers examined happened instantaneously nor did they happen over multiple time steps. In the first paper that was published in 2013 by German scientists Maria Davidich and Stefan Bornholdt, they wanted to compare their Boolean network model of mutated fission yeasts to an ODE model of yeast cell cycles [1]. In their Boolean network model, they used a set of very strict dynamical rules. They used the binary values 0 and 1 (sometimes referred to as OFF and ON) for the absence and presence of the modeled proteins crucial to the fission yeast cell cycle. The initial conditions were set to match the beginning dynamics of the start node as defined by previous studies of fission yeast as well as the ODE models. In their results, they found a majority of similarities in mutated fission yeasts between the two models. They also found out that a Boolean network model can be translated from an ODE model under precise limits and mathematics. In the second paper, published in 2018 by German scientists Dania Humaidan, Frank Breinig, and Volkhard Helms, they primarily focused more on the cell cycle of fission yeast as well as its cellular differentiation [2]. They conducted an experiment by using an Android application to represent the cell cycle of the fission yeast by using the same Boolean model as Maria Davidich and Stefan Bornholdt used, but extending the model by adding four phosphorylation nodes to try to better capture the dynamics in the fission yeast cell cycle. They found that for the wild-type (meaning the dominant cellular expression found in nature) of fission yeast, over the course of 11 time steps, there was only one difference in one time step separating this model and the Davidich boolean network model. To understand the motivation for the Humaidan boolean network, phosphorylations must first be understood. Phosphorylations are proteins that either activates or deactivates enzymes during the cell division cycle. It plays an urgent role in terms in the dynamics of the core oscillators (meaning the central proteins dictating cellular growth and division) [2]. This process

5 can occur in any of the four stages of the cell division cycle, and their role is expanded when looking at cellular division in humans compared to fission yeast. In their experiment, even though they found that their extended Boolean model was very similar with the original Boolean model by Davidich and Bornholdt in modeling the wild-type mutant, subsequent pathologies deviating from the wild-type were more significant. In addition to the extended Boolean model, four nodes were added: Sep1, Fkh2, Atf1, and Cdc10 [2]. In terms of the equation(s) that was used to represent the cycle cell of the fission yeasts is whether a given node is activated or deactivated in a time step. It was mentioned earlier that in the 2013 paper (Maria Davidich and Stefan Bornholdt), they assigned binary values [0,1] in terms of ON or OFF [1]. In this first equation, Si(t + 1) represents the node i in the next time step t + 1 in terms of any given state that is activated in the network at time t [1]. Sj are the certain proteins/genes that directly affect Si, and they are also binary values. The equation is expressed below:

Equation 1: Equation of the boolean networks. This equation was implemented in both papers and implemented in our models. 1 is for activation, 0 is for deactivation, and Si(t) means no change.

The activating and inhibiting interaction strengths are defined by the values of [1, -1], respectively [1]. aij = 1 is the activating interaction, aij =− 1 is the inhibiting interaction, and aij = 0 represents an absent interaction. Diagrams and explanations of the specific interactions that take place in each model will be given later in the paper. However, in both models, θ is the threshold. It is 0 for all nodes except for Cdc2/Cdc13, Cdc2_Tyr15, and Slp1. The default is 0 since an activation of a certain protein will cause it to stay active until it’s degraded or an inhibition takes place. For the exceptions, Cdc2/Cdc13 is self-activated in correlation of the node, which requires a threshold of θ =− 0.5. In other words, unless cyclin dependent kinases (CDKs, abbreviated here as Cdc2) and their corresponding cyclins (Cdc13) are inhibited, they are in an active state. Also, since Cdc2_Tyr15 is the phosphorylation between Cdc2 and Tyr15

6 and consistent interactions are required to maintain the phosphorylation, the threshold is set to be θ = 0.5. Next, Slp1 has a high activation barrier, so both models have θ = 1 to replicate this. Finally, “self-degradation” is a simplification where several proteins in the boolean network model have a consistent degradation rate within the cellular environment. This is modeled through having a self-inhibitory link on those proteins’ nodes, modeled through the expression aii =− 1 [1]. In the second equation, it is similar to the first one, but it is now reduced to two options with the self-degrading loop excluded.

Equation 2: Alternate equation of the boolean networks. Though self-degradation loops of certain proteins are eliminated while all other proteins are then modeled with self-activation loops, this is functionally identical to Equation 1. It is also the equation that more resembles what was taught in Math 5445 lecture.

Our group focused on the research related to two boolean network models that highlight predictions in the cell cycles of fission yeasts. First, we replicated the boolean network models with the same initial conditions, time scales, and node interactions listed in each of the two studies by using Matlab. The papers we analyzed are “Boolean Network Model Predicts Knockout Mutant Phenotypes of Fission Yeast” (2013) by Maria I. Davidich et al. and “Adding phosphorylation events to the core oscillator driving the cell cycle of fission yeast” (2018) by Dania Humaidan et al. Following the direct-implementation of the two boolean network models, we perturbed the initial conditions of each model as well as eliminating nodes all together and examining the differences between the models. We finally compared the results from our replicated models to the original papers to verify their results. Thus, our goal is to confirm the conclusions derived in both papers examined as well as comparing the improvements in the Humaidan’s model (2018) to the original Davidich’s model (2013) of fission yeast cell cycles. Knowing the importance of those phosphorylation nodes that were foregone in the Davidich boolean model will more precisely answer the open question on what the needed complexity in the boolean networks needs to be to accurately describe the pathologies/mutations that can occur. This is important since the ultimate goal of these networks

7 is to be able to accurately model the human cell dynamics, particularly mutations that result in diseases and/or cancer.

Previous Work Davidich 2013 In the paper, “Boolean Network Model Predicts Knockout Mutant Phenotypes of Fission Yeast” by Davidich and Bornholdt [1], the core dynamics of the cell cycle for fission yeasts are captured with a boolean network model. The model proposed represents the cell cycle in four phases: G1, S, G2, and M. The G1 phase of the cell relates to the growth phase of the fission yeast. The S phase is marked by DNA synthesis, a necessary precursor to cell divisions. The G2 phase is described as a “gap” between the other events of the cell cycle. Finally, the M phase refers to mitosis, the event that marks cell division from one parent cell into two daughter cells. Within each of the phases of the cell cycle in fission yeasts, important biochemical reactions govern the expression and concentration of proteins related to the yeast’s division. The entirety of the model is centered around interactions involving the M-promoting factor (denoted as Cdc2/Cdc13): Cdc2/Cdc13 interacts with Ste9, Slp1, and Rum1 (inhibiting proteins). The Cdc2 portion of the M-promoting factor is only active in complexes involving cyclins such as Cig1, Cig2, Puc1, or Cdc13; thus Cdc2 is only considered as part of one of these cyclin complexes. Although the level of Cdc2 is unchanging throughout the entirety of the model, the enzyme’s ability to exist in two different states is considered. Additionally, phosphorylation events surrounding Tyr15 have the potential to reduce the amount of active Cdc2 and are also captured by the model.

Figure 1: (a - left) Visual of the Davidich boolean model. Note that the Start, Cdc2/cdc13 and Cdc2_Tyr15 nodes are crucial for yeast cell cycle, and those all other nodes included are central for those nodes’ activation and inhibition. (b - right) Table of the meanings of the lines in 1a [2].

Threshold of Nodes that Activate Nodes that Inhibit Self-Degradation Node ( θ ) Node in Row Node in Row Node?

Start 0 None None Yes

Cig1 / Cdc2 0 Start None Yes

Cig2 / Cdc2 0 Start None Yes

Puc1 / Cdc2 0 Start None Yes

Cdc2 / Cdc13 -0.5 None Rum1, Ste9, Slp1 No

Ste9 0 PP Cdc2/Cdc13, Cig1/Cdc2, No Cig2/Cdc2, Puc1/Cdc2

Rum1 0 PP Cdc2/Cdc13, Cig1/Cdc2, No Cig2/Cdc2, Puc1/Cdc2

Slp1 1 Cdc2/Cdc13, None Yes Cdc2_Tyr15

Cdc2_Tyr15 0.5 Cdc25 Wee1 No

Wee1 0 PP Cdc2/Cdc13 No

Cdc25 0 Cdc2/Cdc13 PP No

PP 0 Slp1 None Yes

Table 1: Table of Rules from Figure 1. This table contains the defined rules used to replicate the model. Bolded squares indicate node interaction, where activation is +1 and inhibition and self degradation is -1 per node. Nodes can only affect other nodes if they are activated, and are 0 for all interactions if not activated. All interactions at a node are summed and compared to the threshold as in Equation 1 to determine whether a node is ON or OFF after each time step. Note that all node interactions occur over just 1 time step.

Figure 2: Figure of wild-type interaction of Davidich model using the rules in Table 1. Initial conditions defined by normal conditions at start point in fission yeast. Note that the steady state is identical to the initial conditions except that the starting node is OFF [2].

Figure 1 provides an excellent summary of the elements captured by the Davidich model, thus serving as a base-point for comparisons between different boolean network models for the cell cycle of fission yeasts. The self-activation of Cdc2/Cdc13, however, is not captured by the provided figure and is still worth noting: because Cdc2 always present and Cdc13 is constitutively expressed in the fission yeast, the Cdc2/Cdc13 complex stands as the only complex in the model to undergo positive feedback regulation. A threshold has also been added for Cdc2_Tyr15 and Slp1, both of which are only activated after key proteins have hit high-enough concentrations. These thresholds, biologically, represent safe-guards against premature cell division of the fission yeast, and thus are important to include in the model. All of these interactions are shown in Table 1.

As expected for boolean network models, each node in the network is assigned a binary value {0,1} depending on the concentration of the protein within a certain range, as shown in Figure 2. This assignment serves to simplify each node into a switch with two states (“ON” for present, or “OFF” for absent as defined by the concentration of protein above or below a threshold). Additionally, the interactions between each node are quantified by discrete values (+1 for activating and -1 for inhibiting) to change the value of applicable states. The nodes are then updated simultaneously throughout the progression of discrete time steps. The initial conditions for the model have also been chosen to mirror those in the biological system (only the starting node, Ste9, Rum1, and Wee1 are on). These initial parameters result in the correct succession of the cell cycle from G1 to S to G2 to M and back to G1 with the caveat of the starting node being switched to the OFF position (Figure 2). The G1 state with the starting node in the OFF position was found to be the steady-state of the system. When the model was run using each of the 4096 possible combinations of initial conditions, the researchers found that the model converged to one of fifteen different steady states. The model proposed by Davidich has high effectiveness when compared to similar ODE models that also represent the cell cycle of fission yeasts. In fact, Davidich’s model reproduced most of the major results from the ODE models in a fraction of the time, and an example of a result is shown in Figure 2. These results highlight the aforementioned benefits of boolean network models: boolean network models sacrifice small amounts of detail for runtime speed to produce a reasonable summary of the system’s dynamics. Additionally, the model proposed by Davidich is robust and provides comparable results to models that use continuous time ODEs. In our results section, we will compare the Davidich paper’s wild-type cell cycle in Figure 2 to our replicated Davidich model to verify the accuracy of our replication.

Humaidan 2018 The paper “Adding phosphorylation events to the core oscillator driving the cell cycle of fission yeast” by Humaidan, Breinig, and Helms [2], expands upon the model presented by Davidich five-years earlier by accounting for additional phosphorylation events that regulate the dynamics of the system. To encapture these within the model, additional rules were proposed to

11 govern the concentration of proteins as a result of these phosphorylation events. Four molecules, Sep1, Fkh2, Atf1, and Cdc10 participate in these new phosphorylation reactions to further act on the other molecules of the system. Because these molecules are only active within certain phases of the cell-cycle, additional consideration had to be taken to incorporate these timing elements into the model.

Figure 3: (a - left) Visual of the Humaidan boolean model. Note that the Start, Cdc2/cdc13 and Cdc2_Tyr15 nodes are crucial for yeast cell cycle, and those all other nodes included are central for those nodes’ activation and inhibition. Also, phosphorylation nodes have been added (b - right) Table of the meanings of the lines in 3a [2].

Threshold of Nodes that Activate Nodes that Inhibit Self-Degradation Node ( θ ) Node in Row Node in Row Node?

Start 0 None None Yes

Cig1 / Cdc2 0 Start None Yes

Cig2 / Cdc2 0 Start, Cdc10 None Yes

Puc1 / Cdc2 0 Start None Yes

Cdc2 / Cdc13 -0.5 Atf1 Rum1, Ste9, Slp1 No

Ste9 0 PP Cdc2/Cdc13, Cig1/Cdc2, No Cig2/Cdc2, Puc1/Cdc2

Rum1 0 PP, Sep1, Fkh2 Cdc2/Cdc13, Cig1/Cdc2, No Cig2/Cdc2, Puc1/Cdc2

Slp1 1 Cdc2/Cdc13, None Yes Cdc2_Tyr15

Cdc2_Tyr15 0.5 Cdc25 Wee1 No

Wee1 0 PP, Cdc10 Cdc2/Cdc13 No

Cdc25 0 Cdc2/Cdc13 PP No

PP 0 Slp1 None Yes

Sep1 0 Cdc2/Cdc13, Wee1 No Cdc2_Tyr15

Fkh2 0 Cdc2/Cdc13 Cig1/Cdc2, Cig2/Cdc2, No Puc1/Cdc2

Atf1 0 Cdc2/Cdc13, Wee1 No Cdc2/Tyr15

Cdc10 0 Cdc2/Cdc13 None No

Table 2: Table of Rules from Figure 3. This table has all of the same rules as in Table 1, so look at that description for a more complete explanation. 4 added phosphorylation nodes have been added, and their interactions also occur in 1 time step. Rules for phosphorylation nodes are so that Fkh2 activates during G1/S and M, Sep1 and Atf1 activate in G2/M, and Cdc10 is always active unless it is never initially activated by Cdc2/Cdc13.

Figure 4: Figure of wild-type interaction of Humadian model using the rules in Table 2, but only the primary 12 are nodes shown. Initial conditions are the same as in Figure 2. Furthermore, except for Rum1 in time step 8, Figure 2 and Figure 4 are identical [2].

As a result of adding the new rules to govern phosphorylation events excluded from the boolean network model proposed by Davidich in 2013, the concentration of active Cdc2 was found to oscillate during the cell-cycle; this is logical considering the model is representative of the repetitive process of cell division and cyclic processes will undergo repetitive patterns. Because Cdc2 is present in constant quantities, the molecule’s ability to form complexes allows it to serve as an important regulator in the cell cycle of fission yeasts (a conclusion that was also reached by Davidich in 2013). The addition of these phosphorylation events could have resulted in minimal changes to the results of the model, high-variance with regard to the dynamics of the pattern before reaching similar steady states, or the addition of previously forgotten steady states. Overall the phosphorylation events had minimal impacts on the resulting primary steady states for the system, but the dynamics of the concentrations in the fission yeast showed different patterns before reaching the conclusions drawn by Davidich in 2013 (Figure 4). The changing dynamics of the system highlight the additional detail that was included into the model by the rules proposed by Humaidan’s research team. Additionally, while the updated model does not uncover previously unfound steady states, the results match that of the Davidich paper to provide more support that these steady states are the fixed-points within the system. Finally, like in the Davidich model, the results in Figure 4 (extracted from the Humaidan paper) will be compared to our models wild-type cell cycle to verify the accuracy of the replication.

Results: To verify the results from the papers and to compare the models, both models were replicated in Matlab using the rules from Tables 1 and 2 as well as Equation 1. Each figure below is a visual representation of the boolean data over time. Though functionally identical to Figures 2 and 4, it’s easier to see trends over the time steps. Note that the black boxes within each figure represent when the corresponding node is activated, and white boxes are when the node is inactive. All interactions in Tables 1 and 2 occur in each time step, and time step 1 shows the initial conditions.

In other words, any given time step is only affected by the node activations immediately preceding that time step. The results from the Davidich and Humaidan models are sectioned separately and displayed, and the models were run until a steady state was found or an oscillation occured. A steady state has occured on a given time step x if the node activations at time step x equal the node activations at time step x+1, as seen in Figure 5 between time steps 10-16. Oscillations occur when a fixed set of different node activations occurs before repeating, as seen in Figure 10. Figures 5 and 11 show the wild-type cell cycles of each model. The only difference between the two is Rum1 in steps 7 and 8 since both reach steady states after 10 time steps with all other nodes identical (excluding the extra nodes in the Humaidan model). Furthermore, the steady states of each model were evaluated over all possible

12 16 initial conditions (2 possible initial conditions for the Davidich model for the 12 nodes and 2 possibilities for the Humaidan model). In Figure 12, while the model was run according to Table 2, only the 12 nodes found in the Davidich model are displayed for comparison purposes. The number of steady states found in the Davidich and Humaidan models are 15 and 13, respectively. Also, though occuring in different amounts, all steady states in Figure 12 are found in Figure 6 with the exception of the second, third, and fourth most frequent steady states of each model. Interestingly, each one occurs in an oscillation with every oscillation repeating after 3 times steps (which is why only three time steps for each steady state graph are shown). The oscillations found in Figures 6 and 12 will be explained in more detail later. Furthermore, the alterations of the model due to a change in the initial conditions were evaluated. Though Figures 7 and 8 have slight differences to Figures 13 and 14, Figures 7 and 13 (no Wee1 initially) both show that the cell is viable and Figures 8 and 14 (only Start node is initially active) both show that the cell is non-viable after the completion of the cell cycle. Finally, while Figures 9 and 15 both show similar results (Wee1 node is permanently inactivated) with a viable cell that would be smaller than normal, Figures 10 and 16 show significantly different results (Wee1, Ste9, and Rum1 nodes permanently inactivated). Though both models show that the cells are not viable, Figure 10 shows an oscillation while Figure 16 shows a steady state.

Davidich Model Results Wild Type

Figure 5: Shows the wild type progression over the cell cycle of the Davidich model, with initial conditions set to the initial conditions found at the beginning of the G1 phase in fission yeast. The model reaches steady state after 10 time steps, and the steady state is identical to the initial state except the start node is inactive.

Steady States

Figure 6: Figures 6a through 6c show all 15 steady states for Davidich, with 3 oscillating. While the primary steady state doesn’t oscillate, the 2nd, 3rd, and 4th most prevalent steady states do oscillate. Note that all oscillations repeat after 3 time steps.

Initial Condition Mutations

Figure 7: Mutation to the initial conditions where Wee1 is initially set to zero. Compared to the Davidich wild type progression (Figure 1), outside of Wee1 being inactive for the first 4 time steps, the two graphs are identical.

Figure 8: Shows the progression of the Davidich model if all initial conditions except for the start node are set to zero. While the steady state as well as the 5 time steps prior to reaching steady state are identical compared to Figure 5, the beginning is significantly different. Due to skipping G1 and S phase, the resulting cell is likely not viable.

Node Mutations

Figure 9: Davidich model if the cell’s Wee1 is permanently inactive. Outside of the inactive Wee1, the graph is identical to Figure 5. Note that the model cell will likely be smaller after cell division, but be viable long term.

Figure 10: Davidich model where Wee1, Ste9, and Rum1 are permanently inactive. An oscillation is reached as a steady state. While the oscillation bears resemblance to Figure 5 before the steady state, this cell is non-viable due to the mutations.

Humadian Model Results Wild Type

Figure 11: Wild type of the Humadian model, where the initial conditions are identical to the Davidich model, and the 4 added nodes are set to their corresponding activations at the beginning of the G1 phase in fission yeast. Steady state is reached after 11 time steps. Ignoring the extra nodes, this wild type progression is identical to the Davidich wild type progression in Figure 5 except that Rum1 is active in time steps 7 and 8.

Steady States

Figure 12: Figures 12a through 12c show 13 all steady states, with 3 oscillating. Note that the extra 4 nodes in the Humaidan model are not shown for an easier comparison to the Davidich model. While the primary steady state doesn’t oscillate, the 2nd, 3rd, and 4th steady states do oscillate, like Davidich. Note that all oscillations repeat after 3 time steps. Also, outside of the 2nd, 3rd, and 4th most prevalent steady states, the steady states of the Humaidan and the Davidich model are identical, though smaller steady states occur in significantly different degrees.

Initial Condition Mutations

Figure 13: Identical to the initial conditions in Figure 11 except Wee1 is initially inactive. Several differences can be found compared to Figure 11: Wee1 is inactive in the first 4 time steps, Sep1 as well as Atf1 are active for time step 5, and Rum1 is active for time step 6. However, both have the same steady state at time step 11, and are both viable.

Figure 14: Humaidan model if all nodes are initially inactive except for the starting node. While steady state as well as the 5 time steps prior are the same as in Figure 11, like in the Davidich model, the beginning is significantly different. This cell will likely not be viable as a result of essentially skipping most of G1 and S.

Node Mutations

Figure 15: Mutation simulation where Wee1 is permanently inactive. This becomes similar to Figure 13 except the steady state doesn’t include Wee1 and all four nodes are still active at steady state. This cell is viable, but it would likely not be as large as normal because the cell likely didn’t grow as much without Wee1.

Figure 16: Mutation simulation where Wee1, Ste9, and Rum1 are permanently inactive. As compared to Figure 11, the steady state and overall progression through the cell cycle is drastically different, and the result creates a non-viable cell since it skipped most of G1 and all of S as well as having too many nodes active at steady state.

Discussion: It can be concluded the Davidich model verifies the results drawn in the corresponding papers. Looking at Figure 5, it has identical results to Figure 2. This proves that the implementation of Table 1 correctly replicates the results from the Davidich paper. Also, looking at Figure 6, out of the 4,096 possible initial conditions, 3,220 gave the wild-type steady state (corresponding to 78.61% for 3220/4096). This matched previous ODE studies of fission yeast where it was found that the wild-type steady state was found the vast majority of the time, showing how robust the system is [1]. Figures 7-10, despite the oscillations in Figure 10, also correspond to the general dynamics of what previous ODE models found, as concluded in the Davidich paper [1]. Also, the Humaidan model verifies the results drawn in the Humadian paper with one exception. That exception can be seen by comparing Figure 4 to Figure 11, looking at the difference in time step 7 at Rum1. They should be identical, but that one difference in one step of Rum1 is enough to be suspicious if our model accurately depicts the model from the paper. To understand whose error the disparity is, it’s crucial to remember that, like the Davidich paper which this model is based off of, all node interactions occur within 1 step. This means that in Figure 11, time step 8 is only affected by the results in time step 7, and the same can be said about step 7 with step 6. Therefore, for Rum1, to be activated in step 8 but not step 7 like in the paper, a node must activate in step 7 (that wasn’t active in step 6) that positively affects Rum1. However, outside the Rum1 node, the only thing that changes in Figure 11 between step 6 and step 7 is that Slp1 becomes activated, but as seen in Figure 3, there is no direct connection between Slp1 and Rum1. Therefore, it can be concluded that either the paper has a minor error in its results or Table 2 misses a rule that they implemented but did not show in Figure 3. Either way, the conclusion that Humaidan that there results successfully expands the biological model of the yeast must be withheld until either the authors possible error is fixed or our model is fixed. However, despite the discrepancy, looking at the differences in Figures 6 and 12, the conclusion that phosphorylation can have major biological effects seems to be validated. The differences in the prevalence of steady states as well as a change in the 2nd, 3rd, and 4th most prevalent steady states shows that the mutations from the wild-type cell cycle can have

22 significantly different results between the models. Also, comparing Figure 10 to Figure 16 also shows that the added phosphorylation nodes can have significant effects on the fission yeast’s biological system. The oscillation seen in Figure 10 compared to the fixed point found in Figure 16 shows that some simplifications made in order to narrow the Davidich boolean model to only 12 nodes may cause slight errors during certain mutations. While there are differences between the models, what’s remarkable is how similar both models are to each other and past ODE models of the fission yeast cell cycle. Both have over 78% of the possible initial conditions ending up in the wild-type steady state (since the Humaidan model has 51,978 out of the possible 65,536 initial conditions ending up the wild-type steady state or 79.31% of the time). Also, both Figure 7 and Figure 8 compare well to Figures 13 and 14, with each comparable Figure showing the same cell viability and same general trend. Both models confirm most of the other model’s steady-states, and even Figure 10 and 16, which are the most different, are very similar before Figure 10 starts to oscillate. Furthermore, these results correspond to the overall trends seen in ODE models. All of these results verifies the most important conclusion in both papers, that boolean networks have the potential to accurately depict complex biological networks. While some slight modifications to the models might be needed for optimal performance, it should be noted that once the models were built, it took about 1-2 hours to get all results. This is a drastic improvement over the ODE models, where it can take hundreds of hours to get similar results, especially when trying to derive all steady states. This speaks volumes at the necessity of finding simpler networks, for not only are they more intuitive, but they allow for reasonable results for much more complicated networks when ODEs are struggling with even the simpler eukaryotic systems. Comparing the results to each other, Figure 5 and Figure 11 show the wild-type cell cycle for the Davidich and Humaidan models, respectively. The difference between them, Rum1 in steps 7 and 8, shows that improvements to boolean networks may have more significant effects on the minor dynamics to a steady state rather than steady states themselves or even major dynamics. Also, both show G1 in time step 2 (marked by the activation of Cig1/Cdc2, Cig2/Cdc2, and Puc1/Cdc2), G1/S at step 3 (marked by Wee1 activation), G2 at steps 4 and 5

(marked by activation Cdc2/Cdc13), G2/M at steps 6 and 7 (marked by the activation of Cdc2_Tyr15), and M (marked by the inactivation of Cdc2/Cdc13) [2]. Also, Figures 6 and 12 compare well to each other in regards to having similar steady states. While Davidich has 15 steady states and Humaidan has 13 steady states, most of those steady states correspond to each other. However, the 2nd, 3rd, and 4th steady states of each model oscillate, and the corresponding node activations are different. This is one slight deviance from the ODE models, for the total oscillations at steady state for the ODEs of fission yeast are few if any [2]. These oscillations likely resulted from the simplification that all nodes are either ON or OFF. In reality, there is a gradual change in activations over time. As a result, it might improve results if the models varied the time step. Choosing a new time-step to better differentiate between nodes interaction times could lead to better results. This would also reduce the simplifications of the model, and although the boolean models could become more complex, the results would likely be more accurate. Next, the initial conditions of each model were changed. Figure 7 was identical to the wild-type cell cycle in Figure 11 except that Wee1 was inactive for the first four time steps. Although the resulting cells would be smaller since Wee1 controls cell growth, the cells would still be viable [3]. Also, Figure 8 starts off with no initial conditions, Though most of G1 and S period is skipped as a result, the overall progression through the cell cycle is still similar to the wild-type. It should be noted that Figure 7 and Figure 8 both ended at the wild-type steady state of Figure 6. Similarly, Figures 13 and 14 both ended at the wild-type steady state found in Figure 11. Also, like the Davidich model, Figure 13 was very similar to the wild-type cell cycle, and Figure 14, though not viable due to little cell growth, still bears resemblance to the wild-type. This overall means that the models compare well to each other in minor mutations where only the initial conditions are changed. Finally, the models themselves were changed by inactivating nodes of the system to simulate possible mutations. In Figures 9 and 15, Wee1 was permanently inactivated. Both, like in the initial conditions, are very similar to their models’ wild-type progression, and both cells are viable (though they would be smaller due to lack of usual growth in G1 and S). Also, while the phosphorylation nodes in Figure 15 are all active in the steady state, the primary 12 nodes,

24 with the exception of Wee1, are at the wild-type steady state. Also, Figures 10 and 16 both result in non-viable cells due to no little to no growth during G1 and S, preventing their daughter cells from being viable. While Figure 10 shows an oscillation while Figure 16 doesn’t, it should be noted that the first 5 time steps of the primary 12 nodes are identical between the two. Despite the differences, the models compare well to each other, and shows that even with alterations with the intent of optimizing the boolean model, only slight improvements can be made since the system is very robust. This is a great proof-of-concept for showing that the boolean networks can be taken to the next level with more complicated systems. Much like verifying the results, a comparison between the networks also leads to the conclusion of the potential of boolean networks, for they could help bring new understanding to how cell dynamics. In conclusion, the fission yeast cell cycle was modeled using boolean networks. The motivation was to test whether the simplifications needed to go from ODE models to boolean models still allowed for the boolean models to give accurate results. For our boolean models, the Davidich and Humaidan models were found, with the Humaidian adding four phosphorylation nodes to try to improve the Davidich model. Both models are defined in Tables 1 and 2 respectively. We replicated each model and found the wild-type cell cycle, the total number of steady states and their prevalence, and made mutations to the initial conditions as well as the mutations to the models themselves. It was concluded that the papers’ conclusions were verified, and despite minor differences between the boolean models, that they compared reasonably well with each other. Both of these lend credence to the overall conclusion that boolean networks could be used for more complex biological systems. In the future, taking the first steps towards creating a boolean network of a specific human cell type would be needed to prove that they can definitively give accurate depictions of more complex cell dynamics.

Works Cited [1] Davidich, Maria I., and Bornholdt, Stefan. “Boolean Network Model Predicts Knockout Mutant Phenotypes of Fission Yeast.” PLoS ONE, vol. 8, no. 9, 2013, p. E71786. [2] Humaidan, Dania, et al. “Adding Phosphorylation Events to the Core Oscillator Driving the Cell Cycle of Fission Yeast.” PLoS ONE, vol. 13, no. 12, 2018, p. E0208515. [3] Kellogg, Douglas R. “Wee1-Dependent Mechanisms Required for Coordination of Cell Growth and Cell Division.” Journal of Cell Science, The Company of Biologists Ltd, 15 Dec. 2003, jcs.biologists.org/content/116/24/4883. [4] PeoplePill. “Paul Lindner: German Microbiologist (1861-1945) - Biography and Life.” PeoplePill, peoplepill.com/people/paul-lindner/. [5] “G1 And G2: What Happens in the Growth Phases of The Cell Cycle?” Albert Resources, 11 May 2020, www.albert.io/blog/g1-g2-phases-cell-cycle/.