Rate enhancement of gated drift-diffusion process by optimal resetting

“Gating” in biochemistry typically refers to the transition between the open and closed states of an ion channel; the ions are allowed to flow through the channel only when it is open.¹ In a gated chemical reaction, the reactant molecules switch between a reactive and a non-reactive state; the collisions between the reactants must happen in their reactive states for a successful reaction. Therefore, gating is a signature of a constrained reaction/transport, be it an enzyme finding the correct substrate, a protein finding the target site along a DNA strand, or associating with a cleavage site on a peptide. Given their very generic nature, it is no wonder why gated processes showcase a myriad of applications spanning across fields from chemistry^2–5 and physics^6–12 to biology.^13–15 

Since the seminal works of Szabo et al.,^4,6,16 gated processes have gained considerable attention across a wide panorama of applications such as 1D gated continuous-time and discrete-space random search processes in confinement,⁸ intermittent switching dynamics of a protein undertaking facilitated diffusion on a DNA strand,¹⁷ 3D gated diffusive search processes with different diffusivities,⁹ and gated active particles,¹⁸ to name a few. Gopich and Szabo explored the possibility of multiple gated particles/targets in a model with intrinsic reversible binding.¹¹ On the mathematical side, Lawley and Keener established a connection between a radiative/reactive boundary and a gated boundary in diffusion controlled reactions.¹⁹ There has been a renewed interest in the field emanating from the works by Mercado-Vásquez and Boyer on a 1D gated diffusive process on the infinite line,²⁰ Scher and Reuveni on gated reactions on arbitrary networks including random walk models both in continuous²¹ and discrete time,²² and Kumar et al. on threshold crossing events of a gated process²³ and inference of first-passage times from the detection times of gated diffusive first-passage processes.²⁴ In a similar vein, in this work, we delve deeper into a gated diffusive process and, in particular, focus on design principles to improve the efficiency of a gated reaction [see Fig. 1].

For a gated diffusive process to be complete, a certain condition that mimics the open-gate-scenario, imposed either on the diffusing entity or on the target that it diffuses to, needs to be fulfilled. This additional restriction imposed due to gating certainly makes the process more selective, which is essential for the associated biochemical system to function properly. The cost of this selectivity, however, reflects on the completion time, which makes a gated diffusive process essentially slower than the corresponding ungated one. However, nature has its own way of curtailing such situations to allow effective reactions. For example, consider a chemical reaction such as in Fig. 1, where an unbinding or a resetting from a metastable state can lead to a facilitated reaction. The effects of such resetting events are proven to be crucial not only in such chemical reactions^25,26 but also in the backtracking of RNA polymerase²⁷ and the disassociation kinetics of RhoA in the membrane.²⁸ The motion of the reactants or the morphogens, which get produced constantly from a certain place inside the cell before degrading in time (due to their finite lifetime), can also be interpreted as stochastically restarted processes.²⁹ On the physics side, it has been established in the pioneering work of Evans and Majumdar³⁰ that stochastic resetting can be utilized as a powerful strategy to expedite the completion time of a 1D diffusive search process. This remarkable result led to many fascinating works where it was shown that indeed this intermittent resetting strategy can benefit the search processes conducted by diffusion controlled^31–40 and non-diffusive stochastic processes^41–44 (see here, Ref. 45, for an extensive review of the subject). Single particle experiments using optical traps have also paved the way for our understanding of resetting modulated search processes.^46,47 Despite these advances, there has been a persistent void in the understanding of resetting induced gated processes until recently, when resetting mechanisms have been used to reduce the average completion time of a gated diffusive process in 1D.^48,49

While resetting is a useful strategy to benefit the diffusive transport of ions/molecules in unbounded phase space, the same cannot be said for a drift-diffusive search process. There, resetting can be useful only if the rate of diffusive transport is higher compared to the rate of driven transport.^50,51,54,55 However, if the drift supersedes the transport of molecules across the channel, resetting can only hinder its completion, resulting in a longer transit time. This crucial interplay can then be understood in terms of the so-called Péclet number, which is a ratio between the diffusion- and drift-dominated microscopic timescales.^50,51 In fact, the role of resetting can be quantified within a universal set-up of first-passage under restart, which essentially teaches us that resetting will always be beneficial if the underlying process without resetting has a coefficient of variation (often known as the signal-to-noise ratio)—a statistical measure of dispersion in the random completion time, defined as the ratio of its standard deviation to its mean—greater than unity.^42,56,57 These observations naturally set the stage for understanding the role of resetting in a gated drift-diffusive search process which, to the best of our knowledge, has not been studied so far. In particular, the central objective of this work is to understand whether resetting can enhance the completion rate of a gated drift-diffusive search process and, if so, under what conditions. Unraveling the intricate roles of drift, diffusion, and gating, along with that of resetting, will be at the heart of this study.

To illustrate the set-up of the present work, let us consider a diffusive transport process confined to the positive semi-infinite space in the presence of a constant drift that is directed toward a target that randomly switches between an open and a closed state with constant rates. This is a general scenario mimicked by, e.g., a chemical reaction (see Fig. 1), where the collisions between reactants (CR₁ and R₂) lead to the formation of product only when at least one of the reactants is in an activated state (when R₂ exists as $R2⋆$⁠) and not otherwise. Since resetting is an integral part of any consecutive chemical reaction that has a reversible first-step (when C binds to, or unbinds from, R₁), the reaction scheme shown in Fig. 1 can be modeled as gated drift-diffusion with resetting. In this paper, we study the completion time statistics for this general setup. In particular, we perform a comprehensive analysis to understand the role of optimal resetting in enhancing the rate of such transport and construct a complete phase diagram that distinguishes between the phases where (i) resetting accelerates gated drift-diffusion beyond the original (without resetting) ungated process, (ii) resetting still proves itself to be beneficial by improving the rate of gated drift-diffusion but not beyond the original ungated process, and (iii) resetting cannot expedite gated drift-diffusion.

The rest of the paper is organized as follows. In Sec. II, we describe the model and follow a Fokker–Planck approach to write down the governing equations for a gated drift-diffusion process that is subject to resetting. In the same Section, we solve those equations to calculate the mean completion time, denoted $⟨TrG⟩$⁠. We calculate the optimal resetting rate r^⋆ that minimizes $⟨TrG⟩$ and thereby investigate the conditions for resetting to expedite the process completion in Sec. III. There, various limits of the system parameters are also examined in detail. We calculate the maximal speedup that can be achieved, within our set-up, by the optimal resetting rate in Sec. IV. In Sec. V, we construct a full phase diagram that characterizes all the possible regimes underpinning the role of resetting in the process’s completion. We conclude with a brief summary and outlook of our work in Sec. VI. Some of the detailed derivations have been moved to  Appendixes A–D for brevity.

We start by casting the problem of gated chemical reactions (⁠$CR1+R2⋆→C+P$⁠, introduced in Fig. 1) as gated drift-diffusion. A convenient way to do so is to map the reaction coordinate associated with the reactant $CR1+R2⋆$ onto the starting position (x₀ > 0; see Fig. 2) of a particle that undergoes Brownian motion. Similarly, the reaction coordinates for the products C + P can be mapped onto the position of a target (placed at the origin, see Fig. 2). The chemical potential drive, which governs the reaction to its completion, can then be translated to a constant bias λ that the particle experiences while it diffuses to the target with a diffusion coefficient D. Note that λ is considered to be positive when it acts toward the target and negative when it acts away from the target. Such an effective one-dimensional projection of the energy landscape of an enzyme or protein conformation is a well-adapted approach in the literature (see Refs. 58–60 for more details). Nonetheless, there are a few key assumptions that we make during the mapping. First, we take into account the well-established fact that the chemical master equations for the discrete states can be coarse-grained into the Fokker–Planck equations for the reaction coordinates in the continuous space using a system size expansion.^61,62 However, this conversion generically renders the drift and diffusion terms to be spatially dependent. In other words, the reaction coordinate associated with the catalysis process $CR1+R2⋆→C+P$ is expected to diffuse in an arbitrary energy landscape, where the product state C + P is usually denoted by the global minimum. However, to simplify the problem, we replace the general potential with a linear one, which leads to a constant drift velocity λ toward (or away from) the gated target. This is the second assumption that lies behind our analysis. Albeit these simplified approximations, the major advantage here is the elegant analytical tractability of the model, from which one can also unveil the crucial interplay between the gated boundary and the resetting/unbinding mechanism.

To model gating, the target is considered to randomly switch between a reactive state (σ = 1 state in Fig. 2 that resembles $R2⋆$ in Fig. 1) and a non-reactive state (σ = 0 state in Fig. 2 that resembles R₂ in Fig. 1). The conversion from the non-reactive to the reactive state happens with a constant rate α > 0, and that from the reactive to the non-reactive state happens with a constant rate β > 0 (Fig. 2). Therefore, the reactive occupancy of the target, i.e., the probability of finding the target in its reactive state is p_r := α/(α + β). Similarly, the non-reactive occupancy, or the probability that the target is in its non-reactive state is p_nr:= (1 − p_r) = β/(α + β). In analogy to the chemical reaction, the process ends only when the particle hits the target in its reactive state. When it hits the target in its non-reactive state, it simply gets reflected and continues to diffuse.

As mentioned in Sec. I, the unbinding of the catalyst C from R₁ can be interpreted as resetting at x₀. For simplicity, here we consider Poissonian resetting with a constant rate r. In the gated drift-diffusion scenario, this means that the particle is taken back to x₀ after stochastic intervals of time, taken from an exponential waiting time distribution with a mean r⁻¹. Note that here we assume that the resetting is instantaneous and that once the particle is reset at x₀, it immediately starts moving again. In other words, we neglect any refractory period, i.e., idle time after each resetting event before drift-diffusion resumes, considering that the time-scale of waiting at x₀ after reset (which maps to the time required for C to bind R₁ in Fig. 1) is much smaller compared to that of either drift-diffusion or resetting. We also assume that the intermittent dynamics of the target are independent of resetting, similar to Ref. 48.

In Fig. 3, we plot $⟨TrG⟩$ as a function of the resetting rate r for different values of the drift velocity λ, where the reactive occupancy of the target, p_r, is kept constant. It is evident from Fig. 3 that when λ > 0, i.e., when the drift acts toward the target, the average MFPT shows a non-monotonic variation with r for lower values of λ, indicating that the introduction of resetting expedites first-passage when the process is diffusion-controlled. In contrast, $⟨TrG⟩$ increases monotonically with r for sufficiently higher values of λ, meaning that the introduction of resetting delays the first-passage when the process is drift-controlled. This trend can be explained in the following way. When diffusion dominates over drift, the particle tends to diffuse away from the target. In such cases, resetting can effectively truncate those long trajectories, which reduces the overall first-passage time. In contrast, when drift dominates over diffusion, the particle tends to execute a directed motion toward the target (λ > 0); resetting can only hinder such transport resulting in a longer completion time. Note that resetting can accelerate the first-passage even when the dynamics are drift-controlled if the drive is away from the target (λ < 0).

Summarizing, we see that either for pure diffusion or when the drift acts away from the target (λ ≤ 0), the average MFPT consistently shows a non-monotonic variation with r. This means that the introduction of resetting always accelerates the first-passage for λ ≤ 0. Therefore, a hallmark of a resetting transition is apparent for λ > 0, while no such transition is expected for λ ≤ 0. Indeed, similar to ungated diffusive processes under resetting,^30,31 the introduction of resetting was shown to be always beneficial for pure diffusive gated processes (λ = 0), and no such transition was observed there.⁴⁸ Here, we focus on the scenario when the drift acts toward the target and then reveal the physical conditions under which a resetting strategy turns out to be beneficial. In passing, we will also briefly discuss a situation where the system is confined between two boundaries (one additional reflecting boundary apart from the stochastically gated target). In analogy to a chemical reaction, the reflecting boundary is often used to mimic high activation energy barriers in reaction coordinates. We refer to  Appendix B for this discussion, where we investigate the role of resetting in detail.

The resetting transition is traditionally captured by the optimal resetting rate (ORR, denoted r^⋆), defined as the rate of resetting that minimizes the average MFPT. Figure 3 reveals that for pure diffusion (λ = 0), the optimal resetting rate has a certain positive value (denoted $r⋆=r0⋆$⁠, not marked in Fig. 3). With an increase in the drift toward the target, ORR gradually decreases to finally become zero at a critical value of λ (denoted λ_c, not shown in Fig. 3), which marks the point of resetting transition. If we continue to increase λ beyond λ_c, ORR remains zero. Note that when the drift acts away from the target (λ < 0), r^⋆ increases as that drift becomes stronger. The optimal resetting rate r^⋆ thus acts as an order parameter, in the same spirit as in classical phase transitions, to explore the resetting transition. Since ORR minimizes $⟨TrG⟩$⁠, it can be calculated from the relation $d⟨TrG⟩/dr|r=r⋆=0$⁠, which leads to a complicated transcendental equation that cannot be solved analytically. Solving the same numerically, we calculate r^⋆, the optimal resetting rate for λ > 0. In a similar manner, the optimal resetting rate for pure diffusion (for λ = 0, denoted $r0⋆$⁠) is also obtained in order to calculate the scaled ORR, $r⋆/r0⋆$⁠.

In Fig. 4, we plot the scaled ORR with respect to λ for different values of the reactive occupancy p_r (we keep α constant and tune p_r by solely changing β). The non-zero values of the scaled ORR for λ < λ_c in Fig. 4 show that resetting accelerates the first-passage in that regime. In stark contrast, for λ ≥ λ_c, the scaled ORR is zero, which shows that resetting cannot accelerate the first-passage in that regime. For the ungated (p_r = 1) process, λ_c = 1 for our choice of parameters, which is in exact agreement with earlier literature.⁵⁰ It is evident from Fig. 4 that λ_c decreases when p_r is decreased below unity, which means that resetting expedites first-passage time up to a critical drift (which is essentially smaller than λ_c for the ungated process) when gating is introduced to the system.

In order to better understand how the critical value of λ changes with the reactive occupancy of the target, we next numerically calculate λ_c as a function of p_r. In Fig. 5, we construct a phase diagram spanned by the reactive occupancy p_r and the drift velocity λ, where λ_c acts as the separatrix that divides the entire phase space into two parts: one where resetting expedites first-passage (the white regime) and the other where it cannot (the gray regime). We can recover the core results observed in Fig. 4 from Fig. 5 in the following way. For each value of p_r ∈ [0, 1] (encountered by moving horizontally through Fig. 5), the dynamics is diffusion-controlled and resetting is beneficial if λ < λ_c, whereas the dynamics is drift-controlled and resetting turns out to be non-beneficial if λ ≥ λ_c. It is clear from Fig. 5 as well that with an increase in reactive occupancy, λ_c increases to finally become unity for the ungated process (p_r = 1). Figure 5 is, therefore, a complete yet compact representation of the condition for resetting to expedite gated drift-diffusion. Next, we discuss an alternative approach that can successfully generate this phase diagram by carefully exploring the resetting criterion in the limit r → 0.

It is evident from Eq. (8) that in the limit r → 0, resetting is expected to expedite the completion of the process, i.e., $⟨TrG⟩<⟨TG⟩$⁠, when f < 0. This is a sufficient condition (though it may not be necessary) for resetting to be useful. When f > 0, however, resetting is expected to delay the completion of the process, i.e., $⟨TrG⟩>⟨TG⟩$⁠. Therefore, the condition f = 0 should divide the entire phase space created by (λ, p_r) into two parts: one where resetting is beneficial (f < 0) and the other where it is not (f > 0). Indeed, when we plot f = 0 following Eq. (10) in the same phase diagram presented in Fig. 5, it exactly overlaps the existing separatrix plotted earlier by calculating the critical drift λ_c as a function of p_r. Therefore, the condition f < 0 marks the phase where the introduction of resetting accelerates transport to the target (the white regime), while f > 0 marks the phase where the introduction of resetting delays the same (the gray regime), as expected. Revisiting Eq. (10), we note that in the limit of p_r → 1 (for β → 0), the first term of Eq. (10) vanishes. Then, f < 0 boils down to Pe < 1 (meaning Pe = 1 is the resetting transition point), the condition where resetting accelerates ungated drift-diffusion, as obtained in earlier works.^50,51,55

The phase diagram in Fig. 5 is generated only for a fixed value for α. Next, we perform a similar exercise for other values of α and display the results in Fig. 6. It is evident from Fig. 6 that the separatrix, i.e., the phase boundary that separates the two phases—the “resetting-beneficial” phase at the left and the “resetting-detrimental” phase at the right—varies with α. In fact, for larger values of α, the separatrix divides the phases in such a way that the “resetting-beneficial” phase becomes considerably smaller and the “resetting-detrimental” phase occupies most of the phase space, implying that the effect of resetting becomes rather constrained in that limit. Next, we examine two important limits of the gating rates, viz., α → 0 and α, β → ∞.

The appearance of an ORR in the limit α → 0, i.e., when the target is poorly reactive (the so-called cryptic regime^20,21), is counter-intuitive in contrast to the case of α = 0 (a purely reflective boundary). It can be argued that a diffusing particle usually makes several encounters (also aided by drift) with the target regardless of the target’s state. In the current context, although most of the time the particle may remain unsuccessful in finding the target in a reactive state, it can still get absorbed whenever there is an element of chance for the target to become reactive. More details about such cryptic targets and their natural appearances in chemical, biological, and ecological systems can be found in Refs. 2, 20, 52, and 53.

So far, we have performed a detailed analysis to understand the effect of resetting on the dynamics of gated drift-diffusion. In Sec. IV, we turn our attention to the maximal speed-up that can be gained by resetting the process at an optimal rate.

To construct the complete phase diagram for the problem, we revisit Figs. 5 and 8 and recall that the entire phase space [spanned by (λ, p_r)] is divided into two parts, viz., $⟨Tr⋆G⟩<⟨TG⟩$ (i.e., where optimal resetting expedites transport) and $⟨Tr⋆G⟩≥⟨TG⟩$ (i.e., where optimal resetting cannot expedite transport), and the transition between these two phases takes place at a critical point λ = λ_c. The study of the maximal speedup of the gated process with resetting compared to the ungated process without resetting in Sec. IV suggests that we can further divide the former phase into two parts: one where the rate of transport for the gated process with optimal resetting is higher than that of the original ungated process, i.e., $⟨Tr⋆G⟩<⟨T⟩$⁠, and the other where it is not, i.e., $⟨Tr⋆G⟩≥⟨T⟩$⁠. The transition between these two phases happens at $λ=λc0≤λc$⁠. Since gating essentially makes drift-diffusion slower in the absence of resetting, i.e., $T<TG$ [see Eq. (9)], summarizing our observations, we construct a complete phase diagram for the present problem (displayed in Fig. 10), which consists of three distinct phases: (i) phase I: when optimal resetting makes the gated process faster than the original ungated (and hence the original gated) process, given by $Tr⋆G<T<TG$⁠, (ii) phase II: when optimal resetting makes the gated process faster than the original gated process but not the original ungated process, given by $T≤Tr⋆G<TG$⁠, and (iii) phase III: when resetting cannot make the gated process faster than the original gated (and hence the original ungated) process, given by $T<TG≤Tr⋆G$⁠. The transition points $λc0$ create the separatrix between phases I and II, whereas the transition points λ_c create that between phases II and III. Since $λc0=λc$ for p_r = 1, in Fig. 10, we find phases I and II to merge together in the absence of gating. This diagram in the phase space of two important parameters of the system, namely the reactive occupancy and the bias, allows us to delineate the exact nature of resetting in the process of completion. In other words, we can gain maximal benefits from a precise and a priori knowledge of the parameter space.

In this work, we performed an in-depth analysis of the completion time statistics of drift-diffusive transport to a stochastically gated target in the presence of Poissonian resetting. In particular, we strategically explored the conditions where resetting can enhance the rate of such transport, as has been shown for ungated processes where resetting stabilizes the non-equilibrium motion^{30,45,64–66} by removing the detrimental long trajectories that result in a slower transport rate. Projecting the general problem of gated drift-diffusion with resetting to a gated chemical reaction initiated by a catalyst (as discussed in Fig. 1), the main results of the present work can be interpreted as follows.

We observed that the rate of product formation depends on an interesting interplay between the chemical potential drive that governs the reaction (λ), the probability (p_r) that the gated reactant stays in its activated state (when R₂ exists as $R2⋆$⁠), and the rate of unbinding (or resetting, with rate r) of the catalyst C from CR₁. When the chemical potential drive toward the product (λ > 0) is low/moderate such that the reaction is diffusion-controlled, the rate of reaction is maximized for an optimal unbinding rate (r^⋆) of the catalyst. In contrast, when the drive λ is strong, i.e., the reaction is drift-controlled, the unbinding of the catalyst C from CR₁ decreases the rate of reaction. A transition is thus observed at a critical drive, λ_c, which grows with p_r and becomes maximal for p_r = 1, i.e., for the ungated reaction. Strikingly enough, we observed that for $λ<λc0$ (when $λc0≤λc$ is another critical value of λ that increases with p_r and attains a maximum $λc0=λc$ at p_r = 1), optimal unbinding with a rate r^⋆ can make the reaction even $>10$ times faster compared to the ungated reaction in the limit r → 0 (i.e., where the binding step is almost irreversible).

These observations lead to a complete phase diagram based on the qualitative and quantitative effect of optimal unbinding (resetting) on a gated chemical reaction (modeled by drift-diffusion to a gated target) that consists of three distinct phases. Recalling that⁶⁰ the rate of product formation ∝(the mean completion time of reaction)⁻¹, these three phases are identified through the following conditions: (i) where $⟨Tr⋆G⟩<⟨T⟩<⟨TG⟩$⁠, i.e., where the rate of gated chemical reaction is enhanced by optimal unbinding of catalyst beyond that of ungated/gated reactions when the binding is almost irreversible [r → 0], (ii) where $⟨T⟩≤⟨Tr⋆G⟩<⟨TG⟩$⁠, i.e., when optimal unbinding improves the rate of gated reaction but not beyond the ungated reaction in the limit r → 0, and (iii) where $⟨T⟩<⟨TG⟩≤⟨Tr⋆G⟩$⁠, i.e., when unbinding fails to make the gated reaction faster than either of the gated/ungated reactions for almost irreversible binding.

The model considered here generally applies to gated drift-diffusion under the influence of resetting. The major advantage, besides its analytical tractability, is that one can also gain deep insights about the intricate trade-offs between gating and resetting mechanisms, both of which are essential components of chemical reaction networks. A generalization of this simple model to a generic space-dependent diffusion process in an arbitrary energy landscape in the presence of gated targets would be a potential research avenue. A detailed numerical analysis to this end would be a worthwhile pursuit. Notably, such theoretical models can capture physical situations arising in experiments that study, e.g., completion time statistics of protein folding by gated fluorescence quenching. There, the protein is tagged/labeled by a fluorophore that reversibly binds the protein (unbinding is similar to resetting) to impart fluorescence properties.⁶⁷ Once the tagged protein folds to its native state, the quencher selectively reacts with the active site of that folded protein in its fluorescent state, provided that site is in its open (exposed) conformation. If the active site of the folded protein remains in a closed (hidden) conformation, the quencher fails to react with it, which implicates gating. A successful reaction thus occurs only in the exposed conformation, which leads to subsequent quenching of fluorescence, thereby marking the completion of the folding process.⁶⁸ We believe that our work can shed light on understanding and harnessing the various gating and resetting protocols inherent to such systems.

A.P. gratefully acknowledges research support from the DST-SERB Start-up Research Grant No. SRG/2022/000080 and the Department of Atomic Energy, Government of India. D.M. acknowledges SERB (Project No. ECR/2018/002830/CS), DST, Government of India for financial support, and IIT Tirupati for the new faculty seed grant. S.R. acknowledges the Elizabeth Gardner Fellowship by the School of Physics and Astronomy, University of Edinburgh, and the INSPIRE Faculty research grant by DST, Government of India, executed at IIT Tirupati. The numerical calculations reported in this work were carried out on the Nandadevi cluster, which is maintained and supported by IMSc’s High-Performance Computing Center. We acknowledge the anonymous Reviewers for their insightful remarks.

The authors have no conflicts to disclose.

Arup Biswas: Data curation (lead); Formal analysis (equal); Methodology (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Arnab Pal: Formal analysis (equal); Investigation (equal); Methodology (lead); Supervision (equal); Writing – review & editing (equal). Debasish Mondal: Formal analysis (supporting); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Somrita Ray: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Supervision (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study are available within the article, its appendixes, and in Ref. 69.