Stochastic Resonant Signaling in Enzyme Cascades

We observe the phenomenon of stochastic resonant signaling in signal amplification enzyme cascades, where certain optimal reaction rates minimize the average threshold-crossing time. We develop a new analytical technique to obtain the mean first passage time, based on a novel decomposition of the master equation. Our analytical results are in good agreement with the exact numerical simulations. We demonstrate that resonant behavior may be a ubiquitous phenomenon in stochastic threshold crossing in cell signaling. The physical principles behind this phenomenon are elucidated.

Figure 1

(a) A 3-step cascade, where activation of $R^{*}$ in the upstream reaction, $R\to R^{*}$, results in subsequent activation of $A^{*}$ in the downstream reaction, $A+R^{*}\to A^{*}+R^{*}$. $g=2k$, $\mu =0.25$, $\lambda =1.5$, ${\mu }_2=0.1$, ${\lambda }_2=2.5$ with initial condition $(N_R,N_{R^{*}},N_A,N_{A^{*}},N_B,N_{B^{*}})=(100,0,100,0,100,0)$; (b)–(d) Log-log plot of the first passage time $T_f$ vs the reaction rate $k$. (b) Comparison of different computations for the starting 2-step cascade in (a): Gillespie simulation (○), matrix diagonalization of Eq. (1) (dotted line), first order approximation Eq. (4) (dashed line), and second order approximation (solid line), with the $A^{*}$ critical value $n_c=50$. (c) Gillespie simulation as in (b) with the second reaction replaced by $R^{*}+A\rightleftharpoons A{R}^{*}\to R^{*}+A^{*}$ and the critical value $n_c=35$. (d) For the full 3-step cascade with the $B^{*}$ critical value $n_c=75$.

Figure 2

(a) The PDF, $P(m,n)$, on the two-dimensional lattice is shown. Filled curves indicate the conditional probability distribution ${\tilde{P}}_m(n)$ of $A^{*}$, for realizations with a specific number of activated $R^{*}$ receptors, $m$. The cutoff is shown at $n=n_c$ and the marginal probability $Q_m=\sum _{n}P(m,n)$ is shown with thick bars on the vertical panel; (b) the $A^{*}$ distribution for different $R^{*}$ switching rates: the large rate (dotted line for any $m$), the small rate (dashed line for $m>\overline{m}$), and the intermediate rate (solid line for $m>\overline{m}$).