 Solution

ID: 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

β_{P}

β_{P} (t) = r X_{R} (t)

. Setting the two coupled rate equations to zero gives

X_{R, st} = \frac{β _{R}}{α _{R}}, X_{P, st} = \frac{r β _{R}}{α _{R} α _{P}} = \frac{b β _{R}}{α _{P}} .

With first-order removal, a messenger RNA molecule has mean lifetime

1/ α_{R}

r

per molecule, it produces an expected

r / α_{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)

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