# Biophysics Problem -- Summation Issues

## Homework Statement

The average number of mRNAs in the cell at any time t is <m>(t) = Σ m * p(t). Sum over all the differential equations derived in a) in order to obtain a differential equation for <m>(t)

## Homework Equations

So the differential equation I got in a) was dp/dt = (-kp * Pm) - (m * km * Pm) + (kp * Pm-1) + ((m+1) * km * Pm+1)

That would make d<m>/dt = Σ m * dPm/dt = Σ m * ((-kp * Pm) - (m * km * Pm) + (kp * Pm-1) + ((m+1) * km * Pm+1))

What I need is d<m>/dt without any p(t)'s left on the right hand side

## The Attempt at a Solution

I first separated the kp and km terms:

d<m>/dt = Σ m * kp * (-Pm + Pm-1) + Σ m * km * ((-m * Pm) + ((m+1) * Pm+1))

Now I know I have to manipulate this using substitutions such as m' = m -1, m = (m-1) + 1, and shifting of indices, but I am completely unsure how to proceed. Any help would be greatly appreciated.
Sorry I forgot to mention, the summations are from m = 0 to infinity

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