Fig 1.
Model diagram of the feedforward network of the full model and the reduced model.
In the diagram of the full model (left), red arrows indicate fast reactions. In the diagram of the reduced model (right), which consists of slow species S2 and S4 and slow reactions, 〈S1〉 and 〈S1S3〉 represent conditional moments, 〈X1|X2, X4〉 and 〈X1X3|X2, X4〉, respectively. In both of the diagrams, degradation reactions are not shown, for simplicity.
Table 1.
Reactions and propensity functions in the feedforward network (Fig 1).
Fig 2.
EMB model provides much more accurate approximation of the original feedforward network model than AMB model.
(A-B) Trajectories of original full model with ϵ = 0.01 and the EMB model (A) and the AMB model (B). The lines and colored ranges indicate the mean of X4 and standard deviations of X4 from their mean, respectively. Histograms represent distributions of X4 at the steady state. Here, X1(0) = α1/β1, X2(0) = X3(0) = X4(0) = 0. Here, 104 stochastic simulations were performed. (C-D) Mean (C) and standard deviation (D) of stationary distribution (t = 8) simulated with the full model with varying ϵ = 0.1, 0.05, 0.01, the EMB model and the AMB model. (E-F) As β1 = α1 increases, the EMB model, but not the AMB model, predicts that the mean and the CV of S4 decrease, which is consistent with the simulations of the original full model.
Fig 3.
Model diagram of a transcriptional negative feedback loop with a fast feedfoward subnetwork and a slow dimerization.
In the diagram of the full model (top), red arrows indicate fast reactions. Note that the subnetwork consisting of the fast species E, Q, F, and R with fast reactions (red arrows) is feedforward. In the diagram of the reduced model (bottom), which solely consists of slow species and slow reactions, 〈R(R − 1)〉 represents a conditional moment, 〈XR(XR − 1)|XP〉. In both of the diagrams, all degradation reactions are not shown, for simplicity.
Table 2.
Reactions and propensity functions in the transcriptional negative feedback loop with a dimerization (Fig 3).
Fig 4.
The EMB model provides more accurate approximation of the transcriptional negative feedback loop model than AMB model.
(A-B) Trajectories of the full model with ϵ = 0.01 and the EMB model (A) and the AMB model (B). The lines and colored ranges represent E(XR: R) and E(XR: R) ± SD(XR: R)/2 of 104 stochastic simulations. Here Xi(0) = 0. (C-D) Mean (C) and standard deviation (D) of steady state distribution of the full model with varying ϵ = 0.1, 0.05, 0.01, the EMB model and the AMB model.
Fig 5.
Model diagram of a genetic oscillator with a fast reversible binding.
In the diagram of the full model (left), red arrows indicate the fast reversible binding and unbinding reactions. Note that the fast species R, DA and their complex form a complex balanced network. In the diagram of the reduced model (right), which consists solely of slow species and reactions, 〈DA〉 represents the conditional moment , where
. In both of the diagrams, degradation reactions are not shown, for simplicity.
Table 3.
Reactions and propensity functions in the genetic oscillator (Fig 5).
Fig 6.
Period distributions of full and reduced models.
(A) Fourier transforms of stochastic trajectories with about 104 cycles of the EMB model, the AMB model and the full model with ϵ = 0.1. (B) After partitioning the trajectory of 104 cycles into 2⋅103 trajectories so that each trajectory consists of about 5 cycles, the autocorrelation of each trajectory is calculated to estimate the period of each trajectory. The period distribution of 2⋅103 trajectories of the EMB model (above) better captures that of the full model with ϵ = 0.1 than that of the AMB model (bottom). (C-D) The mean and the standard deviation of period distributions of the EMB model, the AMB model, and the full model with ϵ = 10, 1, 0.1.
Fig 7.
Model diagram of a positive feedback loop with decoys.
(A) In the diagram of the full model, red arrows indicate the fast reversible binding and unbinding reactions between transcriptional factor (P) and either a regulatory promoter site (A) or one of N identical nonregulatory decoy binding sites (D). These two fast reversible bindings form a complex balanced network with species P, G0, GA, D, and P: D with conservations, and XD + XP: D = N. (B) The reduced model consists solely of a slow species, which is the total number of transcription factors T (
). 〈GA〉 represents stationary conditional moment,
. In both of the diagrams (A and B), degradation reactions are not shown, for simplicity. (C) The exact moment
, which is used for the EMB model and its approximation, which is used for the AMB model. When XT < N = 10, they show a discrepancy.
Table 4.
Reactions and propensity functions in the positive feedback loop with decoys (Fig 7A).
Fig 8.
Trajectories of the full model, the EMB model and the AMB model of the positive feedback loop with decoys.
(A-B) The lines and colored ranges represent E(XT) and E(XT) ± SD(XT)/2 of 104 stochastic simulations when αA = 10 and α0 = 4. Here Xi(0) = 0. (C-D) When αA = 20 and α0 = 8, so that overall XT is greater than the total number of decoy sites (i.e N = 10), the AMB model also becomes accurate.