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 1 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. 4 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.