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Exact results for noise power spectra in linear biochemical reaction networks
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Image of FIG. 1.
FIG. 1.

Noise power spectra for the gene expression models in Eq. (15) [dashed line, Eq. (18)] and Eq. (19) [solid line, Eq. (26)]. Parameters for the explicit mRNA model of Eq. (19) are , , , and . These correspond to cro in a recent model of phage lambda (Ref. 17). Parameters for the implicit mRNA model of Eq. (15) are , , . These are chosen to match the point statistics, and , and initial value, , for the explicit model. Note that it is impossible to match the full power spectrum since the dependence on is different. The presence of two relaxation times in the explicit model compared to a single relaxation time in the implicit model is clearly seen.

Image of FIG. 2.
FIG. 2.

Noise power spectrum from Eq. (29) for the final product species of the gene expression model in Eq. (27), showing the effect of varying the rate of the post-translational modification step . Parameters are , , and (see Fig. 1). Results are shown for (upper solid line), (dashed lines), and (lower solid line). The high-frequency behavior is dominated by the intrinsic noise in and is independent of . As is reduced, the crossover to the universal high-frequency behavior moves to lower frequencies, resulting in an overall reduction in the noise strength (the area under the curve).

Image of FIG. 3.
FIG. 3.

Correlation function for the model in Eq. (48) with and varying with (it is not necessary to specify any of the other reaction rates). The lines are for , 1.0, and (dashed line). In the limit , the correlation function decays monoexponentially, corresponding to the coarse-grained scheme in Eq. (57) with , but starts from . The difference between this starting point and (for any finite value of ) is the sum rule deficit described in the main text.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Exact results for noise power spectra in linear biochemical reaction networks