Fluctuations in first passage times and utility of resetting protocol in biochemical systems with two-state toggling

TL;DR

The paper investigates first-passage times in biochemical systems that toggle between two states under stochastic dynamics, examining how fluctuations (quantified by $CV^2$) behave as the bias toward a target is varied and how resetting protocols can expedite attainment of the target. It uses Gillespie-based simulations to compute FPTs for three models (extinction in auto-catalytic population dynamics, membrane detachment under fluctuating force, and mRNA threshold crossing under promoter switching) and derives analytical conditions for when an optimal resetting rate vanishes, replacing the traditional $CV^2=1$ criterion with two state-dependent criteria. The main findings are that FPT fluctuations are non-monotonic with respect to bias, resetting is beneficial only in re-entrant regions of the parameter space, and the analytical boundaries (Eq. e1 and Eq. e2) agree with simulations, indicating a general property of two-state toggling systems. These insights extend the scope of resetting theory beyond diffusive transport to discrete chemical kinetics and may guide experimental design in biological and biophysical contexts.

Abstract

Interesting theoretical problems of target search or threshold crossing, formally known as {\it first passage}, often arise in both diffusive transport problems as well as problems of chemical reaction kinetics. We study three systems following different chemical kinetics, and are special as they {\it toggle between two states}: (i) a population dynamics of cells with auto-catalytic birth and intermittent toxic chemical-induced forced death, (ii) a bond cluster model representing membrane adhesion to extracellular matrix under a fluctuating load, and (iii) a model of gene transcription with a regulated promoter switching between active and inactive states. Each of these systems has a target state to attain, which defines a first passage problem -- namely, population becoming extinct, complete membrane detachment, or mRNA count crossing a threshold. We study the fluctuations in first passage time and show that it is interestingly {\it non-monotonic} in all these cases, with increasing strength of bias towards the target. We also study suitable {\it stochastic resetting} protocols to expedite first passage for these systems, and show that there is a re-entrant transition of the efficacy of this protocol in all the three cases, as a function of the bias. The exact analytical condition for these transitions predicted in earlier literature is verified here through simulations.

Figures (6)

• Figure 1: A model of population dynamics. (Left panel) Schematic of the first passage problem showing the initial state (at $t=0$), a state at intermediate time, and the final state (at $t=\tau$). (Right panel) The kinetics in the two states $"+"$ and $"-"$ with the autocatalytic birth, death, and state toggling are shown with their corresponding rates.
• Figure 2: (a) $CV^2$ as a function of $k_d$ shows an U-shape for three different $k_{off}$ indicated with labels and colors. Here $k_{on}=0.51$, $n_0=5$, and $k_b=0.3$. Inset: mean FPT $\langle T_r\rangle^-$ decreases monotonically with $k_d$ (for the three different $k_{off}$ according to the colors). The points A, B, and C marked on the $k_+$ axis (for $k_{off}=0.05$) correspond to high, low, and high $CV^2$, respectively. (b) The solid black line obtained using Eq. \ref{['e2']} demarcate the boundary between the regions $r_{\ast}^- \ne 0$ and $r_{\ast}^- = 0$. The mean FPT $\langle T_r\rangle^-$ vs $r$ at the three points A, B, and C (at $k_{off}=0.05$) are shown in separate boxes. The empty black circles with error bars represent the points where $r_{\ast}^-$ vanishes, obtained through Gillespie simulations.
• Figure 3: Membrane adhesion and bond cluster model. (Left panel) Schematic of the first passage problem showing the initial partially attached membrane (at $t=0$), the limiting case of fully attached membrane, and the final state of fully detached membrane at $t=\tau$. (Right panel) The kinetics in the two states $"+"$ (with nonzero force) and $"-"$ (with zero force) are shown with the corresponding attachment, detachment, and state toggling rates.
• Figure 4: (a) $CV^2$ as a function of $k_+$ shows an U-shape for three different $k_{off}$ indicated with labels and colors. Here $k_{on}=20$, $F=3.5$, $N_0=3$, $\overline{N}=10$, $k_0=0.2$, $\gamma=1$,and $f_d=1$. Inset: mean FPT $\langle T_r\rangle^-$ decreases monotonically with $k_+$ (for the three different $k_{off}$ according to the colors). The points A, B, and C marked on the $k_+$ axis (for $k_{off}=4$) correspond to high, low, and high $CV^2$, respectively. (b) The solid black line obtained using Eq. \ref{['e2']} demarcate the boundary between the regions $r_{\ast}^- \ne 0$ and $r_{\ast}^- = 0$. The mean FPT $\langle T_r\rangle^-$ vs $r$ at the three points A, B, and C (at $k_{off}=4$) are shown in separate boxes. The empty black circles with error bars represent the points where $r_{\ast}^-$ vanishes, obtained through Gillespie simulations.
• Figure 5: Model of mRNA transcription. (Left panel) Schematic of the first passage problem showing the initial mRNA count (at $t=0$), the limiting case of zero transcript, and the final state where the mRNA count has reached the threshold $X$ (at $t=\tau$). (Right panel) The kinetics in the transcriptionally active $"+"$ state, inactive $"-"$ state, and toggling between two with suitable rates is depicted.