The interplay of intrinsic and extrinsic bounded noises in biomolecular networks

Abstract

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a biomolecular network. The influence of intrinsic and extrinsic noises on biomolecular networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: (i) the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, (ii) a model of enzymatic futile cycle and (iii) a genetic toggle switch. In (ii) and (iii) we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possible functional role of bounded noises.

Figure 1. Noise-free Enzyme-Substrate-Product system.

Product formation (averages of simulations, plotted with dotted standard deviation) for both exact and approximated Michaelis-Menten kinetics. We have set and ; the initial configuration is in A and in B.

Figure 2. Stochastically perturbed Enzyme-Substrate-Product system.

Product formation (averages of simulations, plotted with dotted standard deviation) for both exact and approximated Michaelis-Menten kinetics. In A, B, C and D the initial configuration is , in all other panels is . Independent Sine-Wiener noises are present in all the reactions. For , in A, B, E and F, and in all other panels. Also, in A, C, E and G, and in all other panels.

Figure 3. Stochastically perturbed Enzyme-Substrate-Product system.

Product formation (averages of simulations, plotted with dotted standard deviation) for both exact and approximated Michaelis-Menten kinetics. In all panels the initial configuration is . Here single Sine-Wiener noises various intensities and autocorrelations are used. In A and , in B and , in C and , in D and , in E and , in F and , in G and , in H and , in I and , in J and , in K and and in L and . All other parameters are .

Figure 4. Stochastically perturbed Enzyme-Substrate-Product system.

Product formation (averages of simulations, plotted with dotted standard deviation) for both exact and approximated Michaelis-Menten kinetics in the fast time-scale . In all panels the initial configuration is . Here a single Sine-Wiener noise affects complex formation. In A and , in B and , in C and , in D and .

Figure 5. Stochastic models of futile cycles.

Single run and averages of simulations for substrate of the futile cycle models. In panel A the noise-free futile cycle and in panel D the extended noise-free model including the additional species . In bottom plots the cycle affected by bounded Sine-Wiener noise with: in B and , in C and , in E and , in F and . The initial configuration is always ; the kinetic parameters are , , , and (noise-free and the bounded noise case), and , and .

Figure 6. Stochastic models of futile cycles.

Empirical probability density function for at after simulations for the futile cycle models with the parameter configurations considered in Figure 5. In panel A the noise-free cycle, in B the cycle affected by sine-Wiener noise with and , in C the noise-free modified cycle including the additional species . In bottom panels the cycle affected by sine-Wiener noise with: in D and , in E and and in F and .

Figure 7. Periodically perturbed toggle switch.

In the top panels a single run for Zhdanov model (33) with (A) and (C). In bottom plots averages of simulations are shown with (B) and (D). In all cases , , , and and the initial configuration is . The noise realization is plotted for the single runs.

Figure 8. Periodically perturbed toggle switch.

Empirical probability density function at various times, after simulations for Zhdanov model with the parameter configurations considered in Figure 7. In A and , in B and , in C and , in D and , in E and and in F and .

Figure 9. Periodically perturbed toggle switch.

Empirical probability density function for plotted against time, i.e. the probability of being in any reachable state for . Lighter gradient denotes higher probability values. We used data collected with simulations of model (33) where and two perturbation intensities are used, in A and in B. In the -axis the species amount is represented, in the -axis the time (in minutes) is given.

Figure 10. Stochastically perturbed toggle switch.

In top plots, single runs for Zhdanov model with Sine-Wiener bounded noise: in A and in C. In bottom panels the averages of simulations: in C and in D. In all cases , , , and and the initial configuration is . We remark that noise parameters are equivalent to the perturbation of Figure 7; noise realization is plotted for the single runs.

Figure 11. Stochastically perturbed toggle switch.

Empirical probability density function at , after simulations for Zhdanov model with Sine-Wiener noise. Parameters are as in Figure 10 and two perturbation intensities are used: in A and in B.

Figure 12. Stochastically perturbed toggle switch.

Empirical probability density function for plotted against time, i.e. the DCKE solution for in . Lighter gradient denotes higher probability values. We used data collected with simulations of Zhdanov model with Sine-Wiener noise where and two perturbation intensities are used: in A and in B. In the -axis the species concentration is represented, in the -axis minutes are given.

Figure 13. Stochastically perturbed toggle switch.

Empirical probability density function at for , after simulations for Zhdanov model with Sine-Wiener noise. In A , in B , in C and in D . In all cases and other parameters are as in Figure 10.

Figure 14. Stochastically perturbed toggle switch.

Empirical pro bability density function for plotted against time, i.e. the DCKE solution for in . Lighter gradient denotes higher probability values. We used data collected with simulations of Zhdanov model with Sine-Wiener noise. In A , in B , in C and in D . In all cases the noise intensity is . In the -axis the species concentration is represented, in the -axis minutes are given.