# Physics 434, 2015: Project 2 -- Multistability in gene expression

Consider a network of three genes, 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. For example, we may want to describe the deterministic model of this dynamics as ${\displaystyle {\frac {dx_{1}}{dt}}={\frac {A}{(1+(x_{2}/B_{1})^{2})(1+(x_{3}/B_{2})^{2})}}-rx_{1}}$ with ${\displaystyle B_{1}, and equations for ${\displaystyle x_{2},x_{3}}$ are given by cyclicly permuting indices in this equation. Alternatively we can write ${\displaystyle {\frac {dx_{1}}{dt}}={\frac {A}{1+(x_{2}/B_{1})^{2}+(x_{3}/B_{2})^{2}}}-rx_{1}}$, or even ${\displaystyle {\frac {dx_{1}}{dt}}={\frac {A_{1}}{1+(x_{2}/B_{1})^{2}}}+{\frac {A_{2}}{1+(x_{3}/B_{2})^{2}}}-rx_{1}}$. I would like you to explore all of these different formulations, or some others that you can think of that would realize the cyclic strong-weak repression system. Which of these models do you think can be implemented in a realistic biological scenario of multiple transcription factors binding to the DNA? How? Try to explore as many possibilities as possible.
Write a deterministic simulator for the system. Now transform this dynamics into a stochastic dynamics, and write a Gillespie simulator for this system. Choose some fixed value for ${\displaystyle A}$, and then explore different values of ${\displaystyle B_{i}}$ and ${\displaystyle r}$ in different kinetic schemes you have chosen to explore (maybe different members of your group should focus on different systems). Run the dynamics from different initial conditions and plot 3-d trajectories for these conditions, or build the phase portrait of the system (that is, figure out where the deterministic component of the vector ${\displaystyle d{\vec {r}}/dt=(dx_{1}/dt,dx_{2}/dt,dx_{3}/dt)}$ is pointing at different points in the phase space ${\displaystyle (x_{1},x_{2},x_{3})}$? How many stable steady states can you find for different value of the parameters? Are oscillations present? Sketch the phase diagram of this system, that is, the how many steady states, oscillations, etc., does this stochastic system have at different parameter values?