Physics 380, 2011: Homework 10

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Back to Physics 380, 2011: Information Processing in Biology.

1. In class we described bistable biochemical systems. One of the examples that we used was for a self-activating gene, which can act as a bistable, toggle switch (see articles by Gardner et al., 2000). In fact, bistability is a general example of multistability, which we have not yet described. In this problem, we will construct an example of a multistable system with three stable states.
   • Consider three genes in a network such that gene 1 strongly inhibits gene 2 and weakly inhibits gene 3; gene 2 strongly inhibits gene 3 and weakly gene 1; and gene 3 strongly inhibits gene 1 and weakly gene 2. Production should be of the Hill form, and degradation of mRNA/protein products should be linear. Do not resolve proteins from mRNA (that is, consider that a gene produces a protein directly, which later inhibits other genes). You may have very different activation shapes: multiplicative, competitive, etc. Very broad range of detailed activation laws will produce the same type of results.
   • Write down differential equations that would describe this dynamics.
   • Write a Matlab script that would solve the dynamics by the Euler method.
   • Run the dynamics from different initial conditions and plot 3-d trajectories for these conditions. How many stable steady states can you find? Can you pinpoint the saddle points and the unstable steady states this way?
   • Explore different parameters for this system. Is the number of stable steady states dependent on the form of the gene suppression and on its strength? Elaborate by example.
2. For Grad Students (Extra credit for undergrads): Can you imagine a realistic biochemical system with just two degrees of freedom that would still have three or more stable steady states? Build such a system and complete the same analysis of it as above.