## 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.