## LAB 2

### Switching - the key to life?

In this lab, we will explore a variety of switches that occur in biology. What do I mean by a switch? Biologically, a system has a switch if a transient input to the system can move it from one state to a different one and a different stimulus can move it back. For example in the biology of a cell, there are many switches that turn on certain biochemical reactions so that the given cell can produce the proteins that are necessary. Mathematically, a switch can be represented by a differential equation:

x' = f(x,p)

where p is a parameter or input. Suppose that when p=0 the function has three roots, x1 < x2 < x3. We assume that x1,x3 are stable roots and x2 is unstable. If initial conditions are started below x2 then the system will tend to the lower root x1 whereas if the initial conditions are greater than x2 the solution will tend to x3 . The key point is that there are two distinct stable states. If there is a way to transiently kick x above the threshold , x2 then x will stay at the higher state until something kicks it down again. The system now has a "memory" and can hold onto that memory long after the stimulus or kick goes away.