Budding Yeast Clusters and the Turing Mechanism Angelica Schwartz

1. Budding Yeast Clusters and the Turing Mechanism Angelica Schwartz Mentors: Anita Layton and Natasha SavageDuke University Abstract Eight Partial Differential Reaction-Diffusion Equations Preliminary Mathematical Model: Single Solute One of the most important aspects of a cell is its ability to stay functionally symmetric. Cell division is a symmetry-breaking event. A particular example of this is when a yeast cell exhibits cellular polarity during the process of a yeast bud formation. A round cluster of activated cdc42, a cell division protein, forms spontaneously on the cell membrane at the site of the impending bud. Only one single cluster will form, but the location is not necessarily unique. Using models based on the work of A. Turing, who modeled the “spontaneous emergence of cellular polarity” (Goryachev, et. al), we modeled the emergence of the activated cdc42 cluster on the membrane of a yeast cell. All of the following simulations and graphs were created using the Java language in the jGrasp environment and / or MatLab and its graphing utilities. The next part of the Preliminary Mathematical Models combines the reaction and diffusion parts of the one solute equation and yields the following results. These graphs show the results if the reaction constant k is -1 and the diffusion constant D is 1. Introduction Cdc42 is a cell division control protein involved in regulating the cell cycle. The particular type of cell used in the following models was a yeast cell. In the yeast cell’s division process, there are two phases. The second phase, which is not discussed in this paper, is the actual separation of the bud as a distinct cell from the original. The first phase is the formation of a round cluster of activated cdc42 on the membrane. This phase is necessary for the emergence of a singular location for the bud to form on. This process can be modeled using eight partial differential equations of eight different solutes, each involving both a reaction and diffusion part. As shown above, the peak is first lowered because the amount of activated cdc42 being consumed in the other reactions is larger that the cdc42 that is being produced. However, after a certain point, the eight reactions start yielding activated cdc42, and then the cluster grows and broadens from diffusion. As shown in the example above, the width increases as the cluster is growing. Preliminary Mathematical Model: Implementation of Diffusion Depending upon these constant values, the graphs will look different. For example, if the reaction rate is larger in magnitude, all the concentrations might decay to zero before they can diffuse. Conversely, if the diffusion constant is larger in magnitude and the reaction rate is not as fast, then the diffusion might occur before the concentrations can decay to zero. The Preliminary Mathematical Model that I worked with was a single partial differential equation of one single solute, C, containing both a reaction part and a diffusion part. The reaction part is simply an exponential decay, simple to solve. The diffusion part is a little more complicated. The diffusion can be estimated using the second derivative. In this case, we used the numerical approximation of the second derivative in order to work with discrete data points. Index of Abbreviations Used Summary The following table outlines the abbreviations used to represent each solute involved in the modeled process: The diffusion part of the partial differential equations can be modeled using the second derivative. Then the numerical approximation is used to acquire discrete data points. The linear system Ax = b is used to solve the diffusion part of the equations for membrane-bound solutes. The activated cdc42 is at first used up by the reactions and then grows as the other reactions produce activated cdc42. A singular cluster of activated cdc42 is formed on the membrane of the yeast cell before the bud forms during cell division. The location of this cluster is not necessarily unique, but only one single cluster will emerge. Even starting with more than one peak of cdc42 will result in merging into one cluster. Mathematical Model: Simulation of cdc42 Cluster Formation One adjustment made on these models versus the simpler model is the difference of the diffusion part in the membrane-bound and cytoplasmic solutes. For the membrane-bound solutes, the diffusion constant was estimated to be about .0025. The linear system must be solved at each time step. However, for the cytoplasmic solutes, we can assume that the diffusion constant is equal to infinity because everything is being mixed around almost instantly. Therefore in this case, after the reaction part of the equations is implemented, I calculated the average of the new concentrations and set all the points equal to that concentration. This saved memory and time on the execution of the program. Initial Conditions:100 points Ec = .017 C) I = .05 RDIc= 4.95 D) RT @ point 50 = 17 References Linear System for Diffusion Equation Goryachev, Andrew B., and Alexandra V. Pokhilko. “Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity.” (2008): Print.