Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 13E / c

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- Solution c

Solution

At the deterministic feedback equilibrium use

λ_{2} m_{1} f (m_{2}) = β_{2} m_{2}

. The indicated elasticity is

H_{22} = 1 - \frac{\partial l o g f}{\partial l o g x _{2}} = 1 + h =: H .

The normalized drift becomes

A = (- β_{1} β_{2} 0 - β_{2} H)

; equal mean birth and death rates still give

D = diag (2 β_{1} / m_{1}, 2 β_{2} / m_{2})

. Solving the same covariance equation yields the feedback attenuation of gene-expression noise:

η_{22} ≃ \frac{1}{H m _{2}} + \frac{β _{2}}{H ( β _{1} + β _{2} H ) m _{1}} .

For

f (x_{2}) = k / x_{2}

,

h = 1, H = 2

, so

m_{2} ≃ λ_{2} k m_{1} / β_{2}, η_{22} ≃ \frac{1}{2 m _{2}} + \frac{β _{2}}{2 ( β _{1} + 2 β _{2} ) m _{1}} .

The inverse-feedback expression is interpreted near a positive equilibrium; a microscopic birth rate singular at zero needs a separate regularization there.

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