Past exam of the biological physics course of the University of Oxford / 2022 / 2 / b / Solution

The protein production rate is proportional to the current messenger RNA concentration, not an independently constant ${\beta }_P$: ${\beta }_P(t)=r{X}_R(t)$. Setting the two coupled rate equations to zero givesWith first-order removal, a messenger RNA molecule has mean lifetime $1/{\alpha }_R$. At biological translation rate $r$ per molecule, it produces an expected $r/{\alpha }_R=b$ proteins before its removal. Thus the translational burst size is a dimensionless mean number of offspring proteins per messenger RNA, not a concentration and not a fixed number produced in every event. When production and removal are independent Poisson events, mixing a Poisson number of biological translation events over an exponentially distributed lifetime makes the burst count geometrically distributed on the nonnegative integers. Its mean is $b$ and its variance is $b(1+b)$.