Fluctuations in reactive networks subject to extrinsic noise studied in the framework of the chemical Langevin equation

Theoretical and experimental studies have shown that the fluctuations of in vivo systems break the fluctuation-dissipation theorem. One can thus ask what information is contained in the correlation functions of protein concentrations and how they relate to the response of the reactive network to a perturbation. Answers to these questions are of prime importance to extract meaningful parameters from the in vivo fluorescence correlation spectroscopy data. In this paper we study the fluctuations of the concentration of a reactive species involved in a cyclic network that is in a nonequilibrium steady state perturbed by a noisy force, taking into account both the breaking of detailed balance and extrinsic noises. Using a generic model for the network and the extrinsic noise, we derive a chemical Langevin equation that describes the dynamics of the system, we determine the expressions of the correlation functions of the concentrations, and we estimate the deviation of the fluctuation-dissipation theorem and the range of parameters in which an effective temperature can be defined.

Figure 1

Cyclic three-state Michaelis-Menten mechanism maintained out of equilibrium. The enzyme E catalyzes the transformation of the substrate S in the product P. The concentrations of both S and P are controlled by mechanisms external to the catalytic cycle and $s\left(t\right)$ follows the dynamics of an Ornstein-Uhlenbeck process governed by Eq. (1). This chemical system can be modeled by a set of pseudo-isomerizations, one of which $\left(\text{E}\to \text{ES}\right)$ is associated with a time-dependent rate constant $k_a\left(t\right)$ given in Eq. (5).

Figure 2

Correlation function in the presence of extrinsic noise. (a) $\langle N_E\left(0\right)N_E\left(t\right)\rangle$ for a system maintained in a NESS by a constant driving force and a fast fluctuating driving force $\left({\tau }_s\ll \tau \right)$. The analytic expression given in Eq. (25) (red line) is compared to numerical results (+) for the set of parameters in dimensionless units ($N=1000,s_m=2,{\tau }_r=0.25,{\tau }_s={10}^{-4},{\sigma }_s=0.2$). The correlation function without extrinsic noise given in Eq. (24) is represented in blue. (b) $\langle N_E\left(0\right)N_E\left(t\right)\rangle$ for a system maintained in a NESS by a constant driving force and by a fast fluctuating driving force $({\tau }_s\ll \tau )$. The analytic expression given in Eq. (26) (red line) is compared to numerical results (+) for the set of parameters ($N=12000,s_m=1.2,{\tau }_r=0.3125,{\tau }_s=0.72,{\sigma }_s=0.06$). The correlation function without extrinsic noise in Eq. (24) is represented in blue.

Figure 3

Effective temperature in the presence of an extrinsic noise. We plot the ratios $\frac{T_{eff\left(t\right)}}{T_{eff,f}}$ and $\frac{T_{eff}\left(t\right)}{T_{eff,s}}$ as a function of $t$ for the chemical system characterized by the parameter set $N=12000,{\tau }_r=0.3125,s_m=1.2$ and submitted to the noise ${\tau }_s=0.01,{\sigma }_s=0.12 (△)$ and ${\tau }_s=0.1,{\sigma }_s=0.12 (△)$. The expressions of $T_{eff}\left(t\right),T_{eff,f}$, and $T_{eff,s}$ are given in Eqs. (34, 35, 36).

Figure 4

Probability distribution $P\left(n_r\right)$ of the reaction number $n_r$ of $\text{E}\to \text{ES}$ during a time $\tau$. (a) Fast Ornstein-Uhlenbeck process: probability distribution for the number of reactions of $\mathrm{E}\to \text{ES}$ during the time $\tau$ obtained numerically $(+)$ and theoretically (red line). The analytic expression is given in Eq. (A5). Both theoretical and numerical curves are obtained for $N=1000000,k=1,\tau =0.005,s_m=2,\sigma =800$, and ${\tau }_s=0.0001$ in dimensionless units. (b) Slow Ornstein-Uhlenbeck process: probability distribution for the number of reactions of $\mathrm{E}\to \text{ES}$ during the time $\tau$ obtained numerically $(+)$ and theoretically (red line). The analytic expression is given in Eq. (A8). Both theoretical and numerical curves are obtained for $N=1000000,k=1,\tau =0.001,s_m=2,\sigma =0.08$, and ${\tau }_s=0.01$ in dimensionless units.