Graphs showing the numerical solution of Eqs. (2.1), (2.6), and (2.7) for the nonlinear function , where is the maximum of the transcription rate, is the threshold protein concentration at which the transcription rate is at half its maximum value, and the parameter is the Hill coefficient and determines the steepness of the repression curve. The time unit for the numerical solutions is in minutes. The protein half-life is , the mRNA half-life is the burst size is , , and [when (i.e., there is no feedback), the equilibrium of protein concentration is about 1200 corresponding to ]. The initial conditions are , , , , and . The solid, dot-and-dash, and dotted curves are the numerical solutions corresponding to , 5, and 20, respectively, which represents three different negative feedback strengths. (a), (b) The dynamics of mRNA and protein concentrations ( and ) are plotted, respectively. For both mRNA and protein, the concentration level at the equilibrium will decrease with increasing the strength of negative feedback. (c) The numerical solution is plotted. Note that at any time , the variance of the number of proteins is proportional to . Thus, the solutions corresponding to three different values reflect how negative feedback acts on the stochastic fluctuation of protein. (d) The noise ratio of protein for three different values is plotted vs time . For large time , the strong negative feedback will reduce effectively the relative noise strength, but when the system state is far from the stable equilibrium, strong negative feedback may lead a large noise ratio compared to weak negative feedback.
Plot showing the noise vs mean protein concentration with varying values of the Hill coefficient. The Hill coefficient is varied from 0 to 20 at times of 5, 10, 20, 60, and 150. The binding constant is fixed at 800 while the parameters , , , and are kept identical to Fig. 1. Boundary conditions have and . Results show that a simple relationship between the strength of negative feedback and reduction of noise does not exist in the transient domain.
Plot showing the noise vs mean protein concentration with varying values of the binding constant . The value is varied from 0 to 200 with a value of . All other conditions are identical with Fig. 2. The plot shows that the noise can be very sensitive to in the transient domain.
Plot showing the noise vs mean protein concentration with varying values of the Hill coefficient. All conditions are identical to Fig. 2, except the boundary conditions are changed so that and . This plot along with Fig. 2 shows that the complex dependence on the Hill coefficient occurs at very different boundary conditions.
Plot showing the noise vs mean protein concentration with varying values of the binding constant . Conditions are identical to Fig. 4 and differ from Fig. 3 by the boundary conditions. This plot shows that a complex dependence on occurs at these different boundary conditions.
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