Noise and critical phenomena in biochemical signaling cycles at small molecule numbers

Biochemical reaction networks in living cells usually involve reversible covalent modification of signaling molecules, such as protein phosphorylation. Under conditions of small molecule numbers, as is frequently the case in living cells, mass-action theory fails to describe the dynamics of such systems. Instead, the biochemical reactions must be treated as stochastic processes that intrinsically generate concentration fluctuations of the chemicals. We investigate the stochastic reaction kinetics of covalent modification cycles (CMCs) by analytical modeling and numerically exact Monte Carlo simulation of the temporally fluctuating concentration. Depending on the parameter regime, we find for the probability density of the concentration qualitatively distinct classes of distribution functions including power-law distributions with a fractional and tunable exponent. These findings challenge the traditional view of biochemical control networks as deterministic computational systems and suggest that CMCs in cells can function as versatile and tunable noise generators.

Figure 1

(Color online) Schematic of a CMC. Substrate $X_0$ is activated by enzyme $A$ into the modified form $X$ and deactivated by enzyme $D$. Each shaded submodule denotes an enzymatic conversion reaction (unbinding reactions not shown).

Figure 2

(Color online) Monte Carlo simulation of enzymatic conversion. Molecule numbers of the enzyme-substrate complex (solid) and of the activated product (dashed) in the case of only one enzyme molecule. Parameters: $b=u=c=1.0$, $e_t=1$. Vertical arrows denote conversion $\left(c\right)$, binding $\left(b\right)$, and unbinding $\left(u\right)$ processes. $\Delta t$ is the time interval between two successive conversion events.

Figure 3

(Color online) Waiting time distributions. Simulated PDF of the time intervals between successive conversion events. Parameters: $b=u=c=1.0$, $x_0=1=\text{const}\text{.}$ Case (a): Only one enzyme molecule. Case (b): Ten independent enzyme molecules. Inset shows the same data in a semilogarithmic plot.

Figure 4

(Color online) $X$ distributions in the [(b) and (c)] linear regime and in the [(d) and (e)] weakly saturated regime. Solid line (a) is the analytical solution to the linear case. Inset: Normalized autocorrelation function for linear case (symbols) with analytical solution (solid line). For parameters, see Table .

Figure 5

(Color online) Semilogarithmic plots of the simulated enzyme distributions in the [(b) and (c)] linear regime and in the [(d) and (e)] weakly saturated regime. For parameters, see Table .

Figure 6

(Color online) Power-law tails in the strongly saturation regime. Double-logarithmic plots of the $X$ distribution. In cases (a)–(e), the conversion rate $c$ has been gradually increased. For parameters, see Table .

Figure 7

(Color online) Collapse of exponential distributions at the critical point of parametric symmetry. Distributions of $X_0$ (dashed lines) and $X$ (solid lines). In cases (a)–(d), the conversion rate of the activating reaction only has been gradually increased. The left wing of (d) corresponds to Fig. 6c when plotted double logarithmically. For parameters, see Table .

Figure 8

(Color online) Tuning the CMC through the critical point by changing the enzyme concentration $a_t$. The simulated CMC has asymmetric rate constants, but becomes symmetric with respect to the effective coarse-grained parameters $k_m$ and $v_m$ for $a_t=50$. Parts (a)–(c) correspond to $a_t=45$, 50, and 55. Shown are double-logarithmic distributions of the activated (solid lines) and deactivated (dashed lines) substrates. For parameters, see text.