Stochastic synthesis-degradation processes: first-passage properties and connections with resetting

TL;DR

This work develops a general framework for stochastic synthesis-degradation (SSD) processes and their first-passage properties by adapting resetting techniques. It derives exact renewal-like expressions for survival probabilities $Q^{(0)}(t)$ and $Q^{(1)}(t)$ in terms of the underlying non-degrading first-passage time distribution $P_0$ and the degraded survival $q_d$, leading to a closed-form MFPT $T_{b,d}$ and a Poissonian joint distribution $P(n,t)$. A small-$b,d$ expansion reveals a universal CV-criterion and a universal scaling for the critical synthesis rate $b_c(d)$, predicting when SSD speeds up or slows down search relative to a single non-degrading particle, with Brownian diffusion on a line and in an interval serving as illustrative cases. The results show that SSD can outperform single-particle search in bounded domains when $b$ exceeds a universal threshold and highlight the fundamental differences between SSD and resetting, including a nontrivial minimum in MFPT and a distinct short-rate expansion. Overall, the framework offers geometry-agnostic insights and practical guidance for optimizing reaction times in SSD-dominated transport and signaling in biology and related systems.

Abstract

Processes controlled by stochastic synthesis and degradation (SSD) are widespread in biology but their reaction kinetics are not well understood. Using methods borrowed from the theory of resetting processes, we determine the first-passage properties of a collection of independent particles that are synthesized and degraded at constant rates, and follow an arbitrary diffusive process in space. At equal synthesis and degradation rates, the mean reaction time with a target site can be minimized as in stochastic resetting, and a $CV$-criterion is derived. When the degradation rate is held fixed and the synthesis costs are taken into account, an optimal synthesis rate is obtained. In bounded domains, despite particle degradation, SSD improves the mean search time compared to a single non-degrading particle if the synthesis rate exceeds a critical value. The latter obeys a universal relation. We illustrate these findings with Brownian diffusion on the infinite line and in an interval.

Figures (3)

• Figure 1: a) In SSD, independent particles following an arbitrary stochastic motion are synthesized with rate $b$ at $x_0$ and degrade with rate $d$. b) In SR, a single particle is instantaneously reset to its starting position $x_0$ with rate $r$. Even when $b=d=r$, the two processes have different first-passage statistics, although they can exhibit identical densities.
• Figure 2: SSD Brownian particles with $D=1$ starting from $x_0=1$ on the semi-infinite line with an absorbing boundary at $x=0$. a) MFPT in Eq. (\ref{['T1gen']}) (blue line) and average number of synthesized particles until absorption (red line, right y-axis) with $b=d=r$. For comparison we depict the MFPT for stochastic resetting (yellow line) evans_diffusion_2011. The symbols represent Langevin dynamics simulations. b) Keeping degradation constant ($d=1$), total cost function as a function of the production rate $b$ for different values of the synthesis cost parameter $\lambda$.
• Figure 3: Brownian particles with $D=1$ in the interval $[0,1]$. a) Mean first exit time with $b=d=r$. The solid lines are obtained from Eq. (\ref{['T1gen']}), see the SM for details, and the symbols from Langevin dynamics simulations. The horizontal dotted lines represent the mean exit time $\langle T_0\rangle$ for a single non-degrading particle ($r=0$). b) Scaling of the critical synthesis rate vs. $d$ at fixed $x_0$. The symbols are numerical solutions of $T_{b,d}=\langle T_0\rangle$ with $T_{b,d}$ given by Eq. (\ref{['T1gen']}).