Noise control by autoregulation. (

*a*) The histogram (scale on left axis) to the right shows the unrepressed distribution, the one to the left the distribution when repression is turned on. The rate of protein production at any given protein number

*p* is given by a Hill repression function (dashed line, scale on right axis):

*k*_{R}/k=1/(1+[

*p/K*_{d}]”). Here,

*K*_{d} is the dissociation constant that specifies the threshold protein concentration at which the transcription rate is at half its maximum value.

*n* is the Hill coefficient and determines the steepness of the repression curve. For example, the cI repressor protein acts on the promoters P

_{R} and P

_{RM} of phage λ with a

*K*_{d} of about 50 and 1,000 nM, respectively (). Typical biological values for

*n* range from 1 (hyperbolic control) to over 30 (sharp switching). Note that the repression curve is very nearly linear in the region where it intersects the repressed histogram. (

*b*) Noise control by autoregulation: comparing analytic results (solid lines, Eq. ) with Monte Carlo simulations, as

*K*_{d} is varied (triangles), and

*n* is varied (circles). As in Fig.

*c*, the Fano factor (variance/mean) is plotted versus the mean. The protein half-life is fixed at 1 h, mRNA half-life at 2 min, and the burst size at 10; the unrepressed mean value is <

*p*>

_{unrep}* =* 1,200. Note that

*n* = 0 corresponds to a fixed transcription initiation rate that is half the base value, therefore giving a mean protein number of 600.

*K*_{d} is varied (triangles) from 100 to 2,000 in increments of 100, and then from 2,000 to 5,000 in increments of 1,000, with

*n* set to 2. (

*K*_{d} is given in molecule number; one molecule per cell corresponds to a concentration of ≈1 nanomolar.) The unrepressed (

*K*_{d} is infinite) limit is also shown.

*n* is varied (circles) from 0 to 20, with

*K*_{d} set at 800. The Monte Carlo simulations (symbols) are very well reproduced by the analytical values (solid lines) given by Eq. .

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