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Budding Yeast Clusters and the Turing Mechanism
Angelica Schwartz
Mentors: Anita Layton and Natasha Savage
Duke University
Abstract
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.
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.
Index of Abbreviations Used
The following table outlines the abbreviations used to represent each
solute involved in the modeled process:
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Abbreviation
Solute Name
RT
Activated cdc42
M
Bem1, E, and RT complex
Em
Bem1 and cdc42 complex (membranebound)
Ec
Bem1 and cdc42 complex (cytoplasmic)
RD
Inactivated cdc42
RDIm
Complex of RD*I (membrane-bound)
RDIc
Complex of RD*I (cytoplasmic)
I
RhoGDI protein
Eight Partial Differential Reaction-Diffusion Equations
𝜕𝑅𝑇
𝜕𝑡
= 𝑘2 𝐸𝑚 + 𝑘3 𝑀 ∙ 𝑅𝐷 − 𝑘−2 𝑅𝑇 − 𝑘4 𝐸𝑚 ∙ 𝑅𝑇 + 𝑘−4 𝑀 − 𝑘7 𝐸𝑐
∙ 𝑅𝑇 + 𝐷𝑚 𝛥𝑅𝑇
𝜕𝑀
= 𝑘4 𝐸𝑚 ∙ 𝑅𝑇 − 𝑘−4 𝑀 + 𝑘7 𝐸𝑐 ∙ 𝑅𝑇 + 𝐷𝑚 𝛥𝑀
𝜕𝑡
𝜕𝐸𝑚
= 𝑘1 𝐸𝑐 − 𝑘−1 𝐸𝑚 − 𝑘4 𝐸𝑚 ∙ 𝑅𝑇 + 𝑘−4 𝑀 + 𝐷𝑚 𝛥𝐸𝑚
𝜕𝑡
𝜕𝐸𝑐
= 𝜂 𝑘−1 𝐸𝑚 − 𝑘1 + 𝑘7 𝑅𝑇 𝐸𝑐 + 𝐷𝑐 𝛥𝐸𝑐
𝜕𝑡
𝜕𝑅𝐷
𝜕𝑡
= 𝑘−2 𝑅𝑇 − 𝑘2 𝐸𝑚 + 𝑘3 𝑀 ∙ 𝑅𝐷 + 𝑘−6 𝑅𝐷𝐼𝑚 − 𝑘6 𝐼 ∙ 𝑅𝐷
+ 𝐷𝑚 𝛥𝑅𝐷
𝜕𝑅𝐷𝐼𝑚
𝜕𝑡
= 𝑘6 𝐼 ∙ 𝑅𝐷 − 𝑘−6 𝑅𝐷𝐼𝑚 + 𝑘5 𝑅𝐷𝐼𝑐 − 𝑘−5 𝑅𝐷𝐼𝑚 + 𝐷𝑚 𝛥𝑅𝐷𝐼𝑚
𝜕𝑅𝐷𝐼𝑐
= 𝜂 𝑘−5 𝑅𝐷𝐼𝑚 − 𝑘5 𝑅𝐷𝐼𝑐 + 𝐷𝑐 𝛥𝑅𝐷𝐼𝑐
𝜕𝑡
𝜕𝐼
= 𝜂 𝑘−6 𝑅𝐷𝐼𝑚 − 𝑘6 𝐼 ∙ 𝑅𝐷 + 𝐷𝑐 𝛥𝐼
𝜕𝑡
Preliminary Mathematical Model: Single Solute
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.
Preliminary Mathematical Model: Implementation of Diffusion
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.
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.
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.
Linear System for
Diffusion Equation
Initial Conditions:100 points
A) Ec = .017
B) RDIc = 4.95
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.
Summary
1.
2.
3.
4.
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.
References
C) I = .05
D) RT @ point 50 = 17
1.
Goryachev, Andrew B., and Alexandra V. Pokhilko. “Dynamics of
Cdc42 network embodies a Turing-type mechanism of yeast cell
polarity.” (2008): Print.