# Physics 434, 2012: Homework 10

• Modify the dynamics of the system to include Langevin noise. Choose the values of ${\displaystyle B_{i}}$ such that the system is tri-stable, and the highest steady state has ${\displaystyle \geq 30}$ molecules in it, while the lowest has ${\displaystyle \sim 7-10}$. You may need to increase ${\displaystyle A}$ over the previously suggested value of 3.5 to make sure that the lowest state has at least these many molecules.
• Simulate the dynamics of the system for a long time. Does the system switch between the states? Is there a preferred order in the switching? (that is, does high ${\displaystyle x_{1}}$ preferentially come after high ${\displaystyle x_{2}}$ or ${\displaystyle x_{3}}$?)
3. Consider again an arrangement of signal ${\displaystyle s}$, response ${\displaystyle r}$, and a memory ${\displaystyle m}$, treated in a linear approximation, with ${\displaystyle k_{mm}\ll k_{rr}}$. Do not consider effects of noise. Arrange the memory and the response in a negative feedback circuit. There are two of these: (1) response activates memory, which in turn suppresses response, and (2) response suppresses memory, which activates the response. Are these circuits different or equivalent in the linear approximation? Can either of these circuits achieve perfect mean adaptation (zero gain at zero frequency)? Can an incoherent feedforward loop, analyzed before, achieve perfect adaptation? Does this agree with our statement in class that only integral feedback circuits should be able to achieve perfect adaptation? In your analysis of the gain of the system, you may see a tradeoff between low gain at zero frequency and high maximum gain. It might be worthwhile reading Ma et al, 2009, where this has been discussed in detail for nonlinear systems.