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What can we learn when fitting a simple telegraph model to a complex gene expression model?

Fig 7

Robustness of results with respect to cooperative regulation and extrinsic noise.

A: Positive and negative feedback models with cooperative regulation. Feedback is mediated by cooperative binding of two protein copies to the gene. B: For each cooperative feedback model, the HD between the simulated distribution and its telegraph model approximation is shown as a function of the mean expression level for 625 parameter sets. The simulated distribution is well captured by the telegraph model, manifested by HD <0.065. C: Under cooperative regulation and fast gene switching, the deterministic rate equation for the positive feedback model is given by . It may have two stable fixed points (and an unstable fixed point) and thus gives rise to deterministic bistability. The intersections of y = ρ(λ + μx²)/(λ + γ + μx²) (blue curve) and y = dx (red dashed curve) give the locations of the three fixed points (green circles). For a positive feedback loop with deterministic bistability, the effective telegraph model still accurately captures the resulting bimodal distribution. The parameters of the positive feedback model are chosen as ρ = 50, d = 1, λ = 2, γ = 160, μ = 0.5. The effective parameters are estimated to be . D: For each cooperative feedback model, the (absolute values of) relative errors of the three effective parameters , , and are computed under 625 parameter sets, along with their sample means, sample variances, and the sample frequencies of relative errors being greater than 0.2. E: Under cooperative regulation, the positive feedback model still has smaller time-dependent mean curve compared to the effective telegraph model, while the negative feedback model still has larger time-dependent mean curve. F: For each complex model and each parameter set, the simulated distributions are fitted to the telegraph model under four noise levels (0%, 5%, 10%, and 50%). The relative errors of the three effective parameters are computed for all parameter sets, along with the three statistics of relative errors (same as in D). The three statistics of are shown for the negative feedback model, and the three statistics of are shown for the other three complex models.

doi: https://doi.org/10.1371/journal.pcbi.1012118.g007