Tuesday, October 21, 2014
The "Life Expectancy" of a Molecule
Schrödinger's "time of expectation" in What is Life? could use a little clarification. It is the expected amount of time for some change or transition to take place in a molecule at a given temperature. Suppose the transition in question is the time it takes to inactivate a virus. Using probability theory we can assign a probability q to the chance that the molecule will make the transition in a given time interval Δt and a probability p = 1 - q that it does not. The transition state diagram is shown in the figure below.
The molecule starts in the upper left state and it moves one step to the down or right depending on whether on not it makes the transition. The expected number of intervals that the molecule will survive is determined as follows from the transition matrix.
The powers of the transition matrix can be determined by doing repeated multiplications and noting that the right column doesn't change and that the top left component is just pk. To get the last component we use the fact that the sum of each column is 1 for a probability matrix. The expected value of k is just the sum of k times the probability that the molecule has survived k times. The sum can be shown to be equal to p/q2. The expected time of survival is the time interval multiplied by the expected number of survivals. We can check this result by comparing the two functions above for the expectation.
The result is not a linear function of the probability p since q = 1 - p. For very low rates of survival however the survival time is approximately a linear function of the probability of surviving an interval of time.
We are now in a position to modify Schrödinger's formula for the time of expectation. First we need to rewrite p as an exponential function of time by noting k = t/Δt to get the decay rate λ. This gives the same curve that we got for the finite steps. Substituting for p and q gives the correction for the time of expectation.