A bacterial regulatory protein is synthesised shortly after a cell is born
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Genetic circuits, regulation and biological information
Forces, Machines and Signals, Ch Numbers and noise in a small system
Problem
A bacterial regulatory protein is synthesised shortly after a cell is born, reaching a Poisson-distributed mean copy number of nˉ=7 molecules per cell within the first few minutes.
((a)) Starting from the defining property of a Poisson-distributed count, σn2=nˉ, derive a general expression for the relative fluctuation σn/nˉ in terms of nˉ alone, and evaluate it for this protein just after birth.
((b)) By the time this cell next divides, the same protein has accumulated to N=320 copies. Each copy is then sorted independently into one or the other daughter cell with probability 21, uncorrelated with any other copy. Starting from the variance of a binomial count with p=21, derive the general result ⟨(N1−N2)2⟩=N for the mean-squared difference between the two daughters' copy numbers, and evaluate the root-mean-square difference expected between the daughters of this cell.