Fig. 2: Holimaps for autoregulatory gene networks in steady-state conditions.
From: Holimap: an accurate and efficient method for solving stochastic gene network dynamics
a Stochastic model of an autoregulatory feedback loop, which includes bursty protein synthesis, protein decay, cooperative binding of protein to the gene, and unbinding of protein. b The LMA maps the nonlinear network to a linear one with effective parameter \({\hat{\sigma }}_{b}\). The high-order reactions G + hP ⇌ G* in the former are replaced by the first-order reactions G ⇌ G* in the latter. c The 2-HM maps the nonlinear network to a linear one with effective parameters \({\tilde{\sigma }}_{u}\) and \({\tilde{\sigma }}_{b}\). d The 4-HM maps the nonlinear network to a linear one with effective parameters \({\bar{\sigma }}_{u},{\bar{\sigma }}_{b},{\bar{\rho }}_{u}\), and \({\bar{\rho }}_{b}\). e Heat plot of the HD for the LMA as a function of the protein burst frequencies ρu and ρb. Here the HD for the LMA represents the Hellinger distance between the real steady-state protein distribution computed using FSP applied to the nonlinear system and the approximate protein distribution computed using the LMA. f Heat plots of the HDs for the LMA and Holimaps as functions of the unbinding rate σu and binding rate σb (normalized by the decay rate d) when ρb ≫ ρu. The red curves enclose the true bimodal region, i.e., the parameter region in which the protein number has a bimodal distribution, as predicted by FSP; the orange curves enclose the bimodal region predicted by the approximation method. The vertical white dashed line demarcates the region of σu ≥ d, where the linear network given by the LMA can never exhibit bimodality, from the region of σu < d where it can. g Comparison of the steady-state protein distributions computed using FSP, LMA, 2-HM, and 4-HM in different regimes of gene state switching. h The maximum HD as a function of the cooperativity h for the LMA and Holimaps. Here the maximum HD is computed when σu and σb vary over large ranges, while other parameters remain fixed. See Supplementary Note 1 for the technical details of this figure. Source data are provided as a Source Data file.
