Modeling genetic cascades, where the transcriptional rate of mRNA is assumed to be constant, and the transcriptional rates of mRNA are regulated only by protein for . The concentrations of proteins are denoted by for .
The trajectories of difference [Eq. (21)] for four different cases are plotted, where (a) , (b) , (c) , and (d) .
The sequence is plotted vs the cascade stage . It shows clearly how the expected output levels are dependent sensitively on the expected input levels, where (a) , , , and ; (b) , , , and ; (c) , , , and ; and (d) , , , and . For all the cases, we take .
The Monte Carlo simulation results are plotted, where the cascade length is 10, and the parameters are taken as , , and in (a), and , , and in (b). The -axis denotes the noise and -axis the cascade stage . In both (a) and (b), all proteins have the same expectation, i.e., for for different values. The algorithm of Monte Carlo simulation is from Gillespie (Ref. 22). The stochastic simulation results match well with the theoretical predictions (dotted, solid dotted-broken, and broken lines).
The noises vary as the function of input concentration and cascade length. The -axis denotes the cascade stage , -axis the input concentration , and -axis the noise. The parameters , , and are taken as , , and in (a), and , , and in (b).
The normalized covariance measures the statistical correlation between proteins and . Here, the normalized covariance is plotted vs the input concentration and scade stage , where the parameters , , and are taken as , , and in (a), and , , and in (b).
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