The Trajectory Statistics of Biological Exploratory Dynamics

Abstract

Biological processes, from molecules diffusing to their regulatory destinations to whales pursuing food or mates, are widely exploratory. Such behaviors are effective when some verifiable functional state or outcome can be reached regardless of initial conditions. In these cases, systems repeatedly undergo distinct and abortive trajectories; the process only ends when the system finds “right” outcomes. This dominance of the terminal condition (and indifference to the initial state) provokes a very different perspective than the conventional “dynamical systems” framework that has been a centerpiece of the quantitative sciences for centuries. We hypothesize that many of these problems defy the initial-condition driven or gradients on landscapes so useful in problems ranging from mechanics to electrodynamics to chemical kinetics to mass and heat transport. We examine several mathematical frameworks that capture and unify key aspects of exploratory dynamics. One powerful way of thinking of such processes is the geometric distribution, where repeated failures are punctuated by a successful trajectory. We develop intuitions for how random walks with resets accelerate search processes. We highlight fresh and surprising behaviors of search under drift, cues, and checkpoints. Last, appreciating the probability of trajectories conditioned on satisfying macroscopic final outcomes reveals a language for the potency of variation sculpted by selection in their broadest forms. This can give the fictitious appearance of the future making itself known in the present, but we view this as conceptually similar to the way “fictitious forces” arise in non-inertial reference frames. These approaches stress unity, open questions, and applications across a range of biological phenomena.

I. INTRODUCTION

It is our very sad privilege to participate in this special issue dedicated to the life and work of Prof. Rudi Podgornik. Two of us (RP and JK) got our start in the fascinating world of biology by thinking about the packing of DNA in viruses and that led us to a wonderful encounter with Rudi Podgornik and Adrian Parsegian^{100,101,123,143}. We learned many things about the mechanics of DNA from Rudi. He was particularly proud of the fact that one of the earliest measurements of the DNA persistence length was done by a Slovenian scientist⁹⁷, announced in a paper in Nature several months before the paper of Watson and Crick. One of the striking features of Rudi’s work was its mathematical elegance and approachability. In this paper, we attempt to strike a similar tone in the context of a broad class of biological processes that can be thought of as examples of the widespread phenomena of exploratory dynamics. Indeed, we go further in the sense that we have actively imitated one of Rudi’s outstanding papers, his article entitled “On virus growth and form”¹⁴³.

In the nearly 350 years since the 1687 publication of Newton’s Principia, we have had an incredibly powerful and beautiful description of the temporal evolution of systems of all kinds^{5,16,64,87,94,124}. In most of these cases, if dissected carefully, we see a common paradigm described by the update rule

$$\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{f}\mathrm{f}(t+∆t)=\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{f}\mathrm{f}\left(t\right)+f(\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{f}\mathrm{f},t)∆t.$$ (1)

To find the value of “$\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{f}\mathrm{f}$” now, we take its value a short time ago and add to it some “update” that is computed using laws such as $F=ma$ or the law of mass action, for example. Differential equations like these describe planetary motion, the flow of heat, the conversion of atoms and molecules into different kinds of molecules, the time evolution of the electromagnetic field according to Maxwell’s equations, the changes in the sizes of populations and much, much more. And yet, we argue that there are huge swathes of biological phenomena that are not described in this way.

In their provocative 1997 book Cells, Embryos and Evolution, Gerhart and Kirschner provided an unusual vista on some of the unique processes of living organisms. Rather than an impressive compendium of the many great achievements of modern biology, each of their chapters expounds tentative principles that help explain the facts of the living world. One of those chapters introduces the notion of “the exploratory behavior of biological systems,” which paints a completely different picture of biological dynamics in which the trajectories of the system are a repeated series of trials which end upon achieving some target function. Following on the discovery of the dynamic instability of microtubules and its role in chromosome search and capture, they provided a plethora of examples from across scales of biological exploratory behavior, whether the foraging of animals to find food⁶, the development of the cells of the immune system¹²⁸, the active dynamics of microtubules as they search for chromosomes as a prerequisite to cell division⁷⁸, the plasticity of nervous connections⁶⁷ or the development of vasculature³². However, their exposition was verbal, not mathematical. Our goal in this paper dedicated to Rudi Podgornik is to examine to what extent the qualitative statement of the exploratory behavior of biological systems can be restated mathematically.

One of the most powerful facets of formulating our thinking in mathematical terms is that it reveals that phenomena or processes that were outwardly not the same, can fruitfully be viewed as the same thing. For example, the use of statistical mechanical models to treat allosteric molecules reveals the unity of ligand-gated ion channels, membrane receptors, tranion factors, hemoglobin and beyond^73,98. The aim of the present paper is to examine several different mathematical frameworks that describe a myriad of exploratory processes. What unites these cases is a common architecture shown schematically in Figure 1: some mechanism generates variation, that variation samples a space of possibilities, and only a small subset of outcomes is selectively stabilized. A central idea running through these ostensibly distinct thrusts is the distinction between micromanagement and verification. One way to build a system is to specify every step in advance (i.e. micromanagement) so that the desired outcome is guaranteed by construction. Biology often uses a different strategy: generate many possible trajectories, let them explore, and stabilize the rare successful outcome when it appears.

FIG. 1:

The variation and selection of exploratory dynamics. A cell or an organism needs to perform some function such as chromosome segregation. To that end, it generates a collection of possible microscopic trajectories (variation) and when some function such as attachment to the kinetochore is achieved, that trajectory is stabilized (selection).

Another more general question raised but not answered by our paper is the extent to which the analysis of these fascinatingly diverse biological processes might teach us much deeper lessons and principles about the physical description of the natural world. We find it a strange irony that in his famous book Chance and Necessity⁸², Jacques Monod came down strongly against the idea that living systems would teach us any new physical principles, noting:

“Biology occupies, among the sciences, a place that is at once marginal and central. Marginal in that the living world constitutes only an infinitesimal and very special part of the known universe, so that the study of living beings does not seem likely ever to reveal general laws applicable outside the biosphere. But if the ultimate ambition of all science is, as I believe, to elucidate the relation of man to the universe, then one must recognize that biology occupies a central place, since it is, of all the disciplines, the one that attempts to go most directly to the heart of the problems that must be resolved before one can even pose the question of human nature in terms other than metaphysical.”

Given Monod’s own enormous contributions to biology we find his position ironic. The lac operon is arguably one of the most apparently specific, contingent, bacteria-in-a-test-tube-with-lactose thing you could imagine. And what did it reveal? The logic of regulatory circuits, the distinction between structural and regulatory genes, the concept of the operon as a unit of coordinated expression, the molecular basis of induction and repression^47,80,81,84. So moved was Monod by his work on these topics that he named it the “second secret of life”¹²⁶.

Physics is full of such examples where the highly specific illuminates the very general. The specific heat of solids at low temperatures occupies a tiny slice of physical phenomena, yet Einstein and Debye’s treatment of it revealed deep insights into the quantum theory of matter^91,114. The photoelectric effect is a specialized laboratory observation about light impinging on surfaces that unlocked an entirely new picture of the nature of light and matter^91,112. Spectral lines from hydrogen helped establish the entire architecture of quantum mechanics and with it the periodic table and all of chemistry¹¹¹. In fact one could argue that life seems more likely to reveal deep principles precisely because it is so apparently improbable and so structured. A system that maintains itself far from equilibrium, replicates with variation, and evolves is not a sign that it is parochial. Rather, we hypothesize that it is a sign that there is more physical insight to be gained. And it is our hope that some item from the long list of exploratory processes highlighted above might give rise to deeper insights.

We are also amused and inspired by a second unexpected connection to another biological luminary, the eminent developmental biologist Conrad Waddington. Towards the very end of his life, he wrote a book entitled, “OR in World War 2 — Operational Research Against the U Boat,” in which he articulates the science underlying what we might call search with a purpose¹³¹. Waddington notes that, “There are many more places in which a similar approach would be valuable. It is one of the main purposes of this book, by expounding what operational research did in one particular field, to suggest what kinds of things it might be called upon to do in others.” Waddington’s perspective argues that this field facilitates “the general method of science employed to study any problem which may be of importance to an executive.” Our view is that biological exploratory dynamics calls for “executive action” in the sense of achieving some function such as the high fidelity copying of genetic information or the attachment of microtubules to chromosomes.

Despite biology’s many impressive “executive actions,” natural history reminds us that biology need not be “optimal.” This is true due to our poor human guesses about what actually matters to organisms over evolutionary time, or because inherited accidents also rule over living systems. Nonetheless, understanding how the success and timing of searches vary with natural parameters, including with respect to possible optima in minimal models, is revealing even without assuming that biological systems operate exactly at such optima. Characterizing the space of possible dynamics in simple models can identify when a particular strategy is beneficial, quantify the largest improvements that it can provide, and crystallize tradeoffs that could apply to a wide array of systems.

The remainder of the paper traces several different mathematical perspectives offered in Figure 2 with corresponding biological case studies illustrated in Figure 3 and is organized as follows. In section II, we set up one of the simplest and most revealing of mathematical structures for studying exploratory processes, namely, the geometric distribution. In this section, we will see that this distribution provides a broad umbrella for thinking about exploratory processes. Section A, “Quixotic exploration through random walks,” of the Supplemental Material reviews how the simplest exploratory motions, bare random walks, suffer from fundamental mathematical limitations in their speeds or probability of success. This basis motivates why biology amends random walks with surprising strategies such as resets to the start, ballistic drift, and chemical cues; these dynamical strategies are explored in section III. Within this framework, we also examine how a spatially localized chemical cue can guide the search, and identify a combination of cue strength and spatial extent that determines when resetting remains beneficial. Section IV studies how exploratory random walks change under resets to intermediate checkpoints rather than to the origin, and section V investigates parallel searches accomplished by multiple searchers. Such parallel exploration is common in biology; for example, multiple microtubules act in parallel during chromosome capture and segregation. More generally, completion may depend on the first, several, or all successful arrivals. Finally, in section VI, we consider the language of conditional probability as an insightful and unifying approach to a wide variety of search problems. These various approaches are summarized in Table I which will serve as a road map and reference point for the remainder of the paper.

FIG. 2:

The statistical physics of exploratory trajectories. (A) Repeated attempts for microtubules to reach chromosomes are described by the geometric distribution. $p_s$ is the probability of success on a given trial. This distribution applies much more broadly than to just the problem of chromosome search and capture. (B) The random walk with reset is a type of exploratory dynamics in which the seeker returns back to the origin after a certain number of steps known as the reset time. (C) Exploratory behavior can be codified by checkpoints in which the seeker no longer has to reset to the origin, but rather can reset only to some intermediate point. (D) Exploration through conditioning considers only the subset of trajectories that reach the target.

FIG. 3:

Gallery of biological examples illustrating the ways in which the different mathematical strategies are instructive. (A) Chromosome search and capture can be examined using the geometric distribution^43,58,138. (B) Random walks with resets are a convenient mathematical formalism for thinking about gradient formation in structures such as cilia and flagella. (C) The search of transcription factors for their target sites on DNA can be considered as a problem in conditional probability⁴⁰.

TABLE I:

Mathematical frameworks for exploratory dynamics.

framework	key equation	resulting behavior	exploratory dynamics story
geometric distribution	$P(k\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{s}\mathrm{n}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{d})={\left(1-p_s\right)}^{k-1}{p}_s$	$T=t_{\text{step}}/p_s$ (eqn. 8)	searcher (e.g. microtubule) explores at each timestep $t_{\text{step}}$ with success probability $p_s$ of reaching target (e.g. chromosome).
random walk with Poissonian reset (with or without drift)	$0=1+D\frac{d^{2}T}{d{x}^2}+v\frac{dT}{dx}-rT(x)+rT(0)$ (eqns. 43, 82)	$T=T_R\left(\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{x_T}{\lambda \left(D,v,T_R\right)}\right]-1\right)$ (eqns. 55, 93)	searchers diffusively search for target at $x_T$ with diffusion coefficient $D$ and drift $v$, resetting to origin on average every $T_R$ steps. An emergent length scale $\lambda$ governs mean time to success $T$.
random walk with Poissonian reset and spatially dependent drift due to cues	$0=1+D\frac{d^{2}T}{d{x}^2}+v(x)\frac{dT}{dx}-rT(x)+rT(0)$ (eqn. 108)	organized by $\mathrm{P}\mathrm{e}\equiv \frac{v{x}_T}{D}$ (see e.g. eqns. 116 and 118)	biochemical cues induce spatially-varying drift field; prudence of reset and cue admit surprising regime transitions in Péclet number (see especially Fig. 22)
random walk with intermediate checkpoints	$n$ independent subtrajectories: $T=\sum _{k=0}^n{T}_k$ (eqn. 133)	$T=\frac{n+1}{r}\left(e^{x_T/(n+1){\lambda }_D}-1\right)$ (see eqn. 134)	$n$ checkpoints preserve progress between a searcher’s starting location and a target at $x_T$, subject to a diffusive search length scale ${\lambda }_D=\sqrt{\frac{D}{r}}$.
first success among multiple independent searchers	$S^{\mathrm{t}\mathrm{o}\mathrm{t}}(t)=S(t{)}^N$	$T_{\mathrm{m}\mathrm{i}\mathrm{n}}\sim \frac{x_T^2}{4D\mathrm{l}\mathrm{n}N}$ for $N\gg 1$ (eqn. 155)	$N$ independent searchers, each with survival probability $S(t)$ of not yet finding the target by time $t$, collectively explore more efficiently, but only logarithmically, with the search time scaling as $1/\mathrm{l}\mathrm{n}N$.
$M$ successes among multiple independent searchers	$S_{M:N}(t)={\sum }_{\ell =0}^{M-1}\left(\genfrac{}{}{0pt}{}{N}{\ell }\right)[1-S(t){]}^{\ell }S(t{)}^{N-\ell }$ (eqn. 158)	$T_{M:N}\sim \frac{x_T^2}{4D\mathrm{l}\mathrm{n}(N/M)}$ for fixed $M\ll N;T_{N:N}\propto \mathrm{l}\mathrm{n}N$ for $M=N$ and $N\gg 1$ (eqns. 162, 167)	$N$ independent searchers explore in parallel; successful completion can require $M=$ one, several, or all searchers to reach the target.
conditioning search trajectories on final and initial states: Schrödinger bridges on graphs	$\hat{p}(\ell ,t)\equiv p(\ell \mathrm{a}\mathrm{t}t\mid (i\mathrm{a}\mathrm{t}t=0,j$ at $t=T))=\frac{(\mathrm{e}\mathrm{x}\mathrm{p}[t\mathbf{L}]{)}_{\ell i}(\mathrm{e}\mathrm{x}\mathrm{p}[(T-t)\mathbf{L}]{)}_{j\ell }}{(\mathrm{e}\mathrm{x}\mathrm{p}[T\mathbf{L}]{)}_{ji}}$	new marginal distributions $\stackrel{\mathbf{ˆ}}{\mathbf{p}}(t)$ and new, effective dynamics $\frac{d\stackrel{\mathbf{ˆ}}{\mathbf{p}}}{dt}=\stackrel{\mathbf{ˆ}}{\mathbf{L}}(t)\stackrel{\mathbf{ˆ}}{\mathbf{p}}(t)$	given transitions between states of a system at rates stored in a graph Laplacian matrix $\mathbf{L}$, conditioning trajectories to start at state $i$ and end in state $j$ at time $T$ induces effective dynamics over states at intermediate times, described by a new time-dependent $\stackrel{\mathbf{ˆ}}{\mathbf{L}}(t)$

II. EXPLORATION THROUGH REPEATED FAILURE

We begin by building the mathematical language of the statistical mechanics of exploratory trajectories by considering trajectories built up of “wrong” and “right” moves. This structure appears in many guises⁵⁸. In one biological system, an attempt may be a microtubule growth episode that either finds a kinetochore or collapses^{38,41,43,78,79,138}. In another, it may be an excursion of a transcription factor away from DNA followed by rebinding^{40,44,130,136,137}. The microscopic meaning of an attempt differs from system to system, but the statistical structure is the same. A series of wrong moves followed by a final right move.

Suppose that each attempt succeeds with probability $p_s$ and fails with probability $p_f=1-p_s$ ($f$ for failure and $s$ for success). If success occurs after $i$ failed attempts, then the probability of that trajectory is

$$p_i=\underset{i\text{factors}}{\underbrace{p_f{p}_f\cdots p_f}}{p}_s=p_f^i{p}_s.$$ (2)

Equivalently, we can write this more informatively as

$$p_i={\left(1-p_s\right)}^i{p}_s,$$ (3)

which is the geometric distribution¹⁰. This is the first foundational model for exploratory dynamics. It says that the central quantity is the probability of success per attempt. This fundamental equation has a structure that we will use over and over in thinking about the statistical mechanics of exploratory trajectories since it highlights the importance of the final state.

In particular, we ask how long does it take on average for the exploratory dynamics to terminate due to success in finding its target? The answer to that question is obtained by averaging over the time spent in all the possible trajectories and given by

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}\underset{\text{failures}}{\underbrace{(i{t}_{\text{step}}}}+\underset{\text{success}}{\underbrace{t_{\text{step}}}})p_i,$$ (4)

where we have introduced the quantity $t_{\text{step}}$ as the duration of each instance where the system makes an unsuccessful excursion (e.g. a microtubule polymerizes without finding the chromosome). The second term in the sum immediately yields $t_{\text{step}}$ since the distribution is normalized leaving us with

$$⟨t⟩=\sum _{i=0}^{\mathrm{\infty }}i{t}_{\text{step}}{p}_i+t_{\text{step}}.$$ (5)

Next, we need to evaluate

$$⟨t⟩=t_{\text{step}}\sum _{i=0}^{\mathrm{\infty }}i{p}_i+t_{\text{step}}=t_{\text{step}}{p}_s\sum _{i=0}^{\mathrm{\infty }}i{\left(1-p_s\right)}^i+t_{\text{step}},$$ (6)

which we recognize as having a term of the form

$$\sum _{i=0}^{\mathrm{\infty }}i{x}^i=x\frac{d}{dx}\sum _{i=1}^{\mathrm{\infty }}{x}^i=x\frac{d}{dx}\frac{1}{1-x}=\frac{x}{(1-x{)}^2}=\frac{1-p_s}{p_s},$$ (7)

where we have used the definition $x=1-p_s$. If we now invoke this result in the context of eqn. 6, we find the very intuitive result that

$$⟨t⟩=t_{\text{step}}\left(1+\frac{1-p_s}{p_s}\right)=\frac{t_{\text{step}}}{p_s}.$$ (8)

The intuition for this result is best served by concretely imagining that $p_s=1/M$ which means that 1 out of every $M$ trials is successful. This implies in turn that $⟨t⟩=M{t}_{\text{step}}$, stating that on average we need to carry out $M$ trials for the exploration to be a success when the probability of success on a given trajectory is $p_s=1/M$.

This example showcases the structure of a first model for many problems in exploratory dynamics that both we and others have described elsewhere^43,58,138 We will often characterize the exploratory process as a statistical mechanics of trajectories, with all such trajectories sharing the feature that they involve repeated “failures” until they succeed on the final trajectory. Interestingly, for the examples considered throughout the paper, we will see many different realizations of the space of trajectories, but as noted above, they are often of the form $p_i=\underset{\mathrm{i}}{\underbrace{p_f{p}_f\cdots p_f}}{p}_s$, and perhaps surprisingly, even when our mathematical analysis does not start with the geometric distribution, it will often end there. In an earlier paper, we showed a number of results in exploratory dynamics that can be described precisely in terms of the geometric distribution^43,58,138. As a result, we turn to the next class of mathematical approaches shown in Figure 2, random walks—and, a realization of them particularly practical in biology, namely random walks with resets.

III. EXPLORATION THROUGH RANDOM WALKS WITH RESET

A second foundational model is the random walk with reset^{13,14,27,28,110}, as shown in Figure 4. In this model, a “seeker” begins at the origin, $x=0$, and searches for a target at position $x_T$. The seeker walks randomly (e.g. diffuses), but at a given rate $r$ it is returned to the origin and begins again. As shown by Figure 5, the seeker suffers either deterministic resets at a fixed time $T_R=1/r$, or else resets occur stochastically at an average rate $r$. Figure 6 gives further examples of trajectories under deterministic resets at varying rates to visualize the arrivals to a target at an example location of $x_T=5$. This is a deliberately stripped-down model of repeated exploration. It is inspired by systems such as microtubule search for kinetochores in the process of spindle assembly. There, failed growth episodes are erased by catastrophes, and new attempts begin from a preferred location^43,78,138. Why should such resets or structural decorations on random walks be deployed by biological systems? In Sec. A of the Supplemental Material, we examine the properties of random walks without resets with an eye to why they might not suffice for some biological exploratory processes on generic mathematical grounds.

FIG. 4:

The random seeker with reset. Schematic of the steps available to the random seeker during a time step. During each discrete time step $∆t$, the walker can step to the right or left with probability $k∆t$ or reset to the origin with probability $r∆t$.

FIG. 5:

Random walks with resets. (A) Top: Deterministic resets in which after a fixed number of steps $T_R$, the walker resets to the origin. Bottom: This corresponds to resets performed with a periodic clock. (B) Random resets, where resets to the origin occur at a stochastic rate $r=1/T_R$, making the time between resets match $T_R$ only on average. Bottom: This protocol corresponds to a Poissonian clock instead of a perfectly periodic clock.

FIG. 6:

Time to reach a target in the presence of reset. Every $T_R$ steps, each walker deterministically returns to the origin, no matter its present proximity to the target. The resets are indicated by vertical black arrows. The three examples show representative trajectories reaching the target at position $x_T=5$, under deterministic reset times of (A) $T_R=100$, (B) $T_R=300$, and (C) $T_R=1000$ steps. The times at which each stochastic trajectory first reaches the target are marked by dots.

A. Simulating the Simple Random Walk With Reset

How do resets change the average time a seeker needs to reach a target? Direct numerical simulations give an immediate feel for this question. Figure 7 averages the required length of many simulated trajectories (such as those highlighted in Fig. 6) to arrive at a target located at $x_T=5$ for the first time, as a function of the reset time $T_R$, in $d=1$ dimension. Intriguingly, as the graph in Figure 7 makes clear, these resets can intensify the exploration process: prudently choosing a reset rate can minimize the average time to a target. This acceleration of typical searches is initially surprising, given that resets in fact abort many trajectories on the precipice of success. The graph in Figure 7 showing how success time varies with reset rate has rich qualitative and numerical structure, and now our goal is to understand the different parts of the graph.

FIG. 7:

Simulation of random walk with reset in one-dimension with a target at position $x_T$. The graph at left shows the average time to find the target, $T$, as a function of the number of steps until a hard reset to the origin, $T_R$. The inset shows a zoom of the same curve on a linear scale. Note sawtooth oscillations which reflect parity structure in the discrete counting search problem (not sampling artifacts); see also Sec. B, “First-passage properties of ordinary random walks,” of the Supplemental Material. The schematic on the right illustrates the possible positions of the seeker after consecutive steps, subject to reset after $T_R=7$ steps. The seekers shaded in green show the endpoints of trajectories that did not find the target and are hence sent back to the origin to begin their target search again.

Before proceeding with a quantitative analysis, it is useful to ask why the curve in Figure 7 should exhibit a minimum at all. If the reset time $T_R$ is too short, the searcher is repeatedly returned to the origin before it has a reasonable chance of reaching the target, and successful trajectories become rare. (If the time between resets $T_R$ is shorter than the target location (when both are measured as a number of steps), as in the red shaded region in Fig. 7, then the target is never reached.) On the other hand, if $T_R$ is very large, resets occur infrequently and the search approaches unconstrained diffusion. In that regime, the searcher can spend a long time on unproductive excursions far from the target, leading again to large search times; this is the original quixotic character discussed in Sec. A of the Supplemental Material. These two competing effects suggest that an optimal finite reset time should exist.

We can also anticipate the scale of this optimum using a simple diffusive argument. In a time $t$, a walker diffusing with diffusion coefficient $D$ typically explores a distance of order $\sqrt{Dt}$. Reaching a target located a distance $x_T$ from the origin therefore requires a characteristic time of order

$$T_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}\sim \frac{x_T^2}{D}\sim \frac{1}{k}\frac{x_T^2}{a^2},$$ (9)

where $k$ is the stepping rate and $a$ is the lattice spacing (as schematized in Fig. 4), and $D=k{a}^2$ is the emergent diffusion coefficient. Resetting on a timescale much shorter than $T_{\text{diff}}$ prevents the searcher from reaching the target, while resetting on a much longer timescale allows rare but costly excursions to dominate the search. Consequently, we expect the optimal reset time to be of the same order as the diffusive first-passage time,

$$T_R^{\mathrm{*}}\sim \frac{x_T^2}{D},$$ (10)

a prediction that will emerge naturally from the calculations that follow. (This is also the only timescale defineable by the physical parameters entering the problem, anticipating the optimal time will vary with this quantity.)

First, to appreciate the curve to the left of the optimal reset rate in Figure 7, can we understand the search time for the special case in which the reset time is equal to the distance of the target from the origin? Note that here we use dimensionless units of “steps” for both distance and time. For the case study considered here, the target is at $x_T=5$, thus any reset time shorter than $T_R=5$ can never find the target. If we allow 5 steps before reset, there is only one successful trajectory which is $RRRRR$ with probability 1/32. The seeker must make 5 successive steps to the right, each with probability 1/2. This means that out of the 2⁵ = 32 possible trajectories, 31 of them do not reach target and they are characterized by probability 31/32. Hence, if we want to sum over all the possible trajectories to find the average time to reach the target, we can write

$$⟨\#\text{steps}⟩=\sum _{n=1}^{\mathrm{\infty }}5n{\left(\frac{31}{32}\right)}^{n-1}\frac{1}{32},$$ (11)

where $n-1$ is once again the number of “failures” before success. The factor of $5n$ measures the fact that for each of the $n-1$ failed attempts, they were of length 5 before the reset and then the final trajectory is itself of length 5 since that one is of the form $RRRRR$. Using the geometric distribution as highlighted in Table I and described in detail in section II, we find that the average number of steps until hitting the target is 160, precisely what we see for the point on the vertical red line in Figure 7.

Let us now solve this problem for the more general case in which $x_T=n$ and we reset after $n$ steps. We assume that each step is either +1 or −1 with probability 1/2. In this special case there is a very simple observation. Since the walker has only $n$ steps available before reset, the largest position it can possibly attain is $x=n$. That can happen only if every one of the $n$ steps is a step to the right. Thus there is exactly one successful trajectory in each attempt,

$$(+1,+1,\dots ,+1),$$ (12)

and therefore the probability of success in a single attempt is

$$p_s={\left(\frac{1}{2}\right)}^n.$$ (13)

The probability of failure in a single attempt is hence

$$1-p_s=1-{\left(\frac{1}{2}\right)}^n.$$ (14)

Now let $i$ be the number of the attempt on which success first occurs. Then the probability that the first $i-1$ attempts fail and the $i$th attempt succeeds is

$$p(i)={\left[1-{\left(\frac{1}{2}\right)}^n\right]}^{i-1}{\left(\frac{1}{2}\right)}^n,i=\mathrm{1,2},3,\dots$$ (15)

This is once again a geometric distribution, specifically an application of eqn. 3, revealing that our work in the previous section sheds light on the problem of random seeker with resets as well. Since each attempt lasts exactly $i$ steps, the total number of steps taken before success on the $n$th attempt is

$$N=in.$$ (16)

Hence, the mean number of steps is given by,

$$⟨N⟩=\sum _{i=1}^{\mathrm{\infty }}in{\left[1-{\left(\frac{1}{2}\right)}^n\right]}^{i-1}{\left(\frac{1}{2}\right)}^n,$$ (17)

which is an application of eqn. 4. Using the standard identity for the mean of the geometric distribution, this becomes

$$⟨N⟩=n\frac{1}{(1/2{)}^n}=n{2}^n.$$ (18)

It is worth checking that this reproduces the case $n=5$. In that case we find

$$⟨N⟩=5\cdot 2^5=160,$$ (19)

as already determined above, via eqn. 5, and shown by the point circled in black in Figure 7.

Now that we have explained the numerical value of the time to find the target for the special case in which the reset time is equal to the distance of the target from the origin, we look to the opposite limit of the graph where the time to find the target is an increasing function of the reset time.

B. Approximate Description of The Simple Random Walk With Reset

An approximate approach to the seeker-with-reset problem that we find particularly appealing is to replace the explicit reset dynamics by random walk in a finite spatial domain. The central idea is simple. In the original problem, the seeker begins at the origin and repeatedly explores until reset returns it to the starting point. Rather than tracking the complicated history of many interrupted excursions, we ask whether there is a simpler description that captures the same physics at the level of scaling behavior. The approximation is to imagine that reset limits how far the seeker can typically wander and to replace that limitation by a reflecting boundary. This idea is justified when the target location is small compared to the resetting interval, $x_T\ll T_R$.

Hence, we keep the geometry of the original problem intact as shown in Figure 8. The seeker begins at the origin, the treasure remains at position $x=x_T$, and the effect of reset is represented by a reflecting wall placed a distance $L_R$ to the left of the origin. The resulting finite-box problem plays out on the interval

$$-L_R\le x\le x_T.$$ (20)

FIG. 8:

Replacing the random walker with reset by a walker in a finite box. The width of the box in the negative x-direction is set by the average number of steps $L_R$ that the seeker walks before being sent back to the origin through reset. A naive estimate for the characteristic exploration length under reset is $L_R=\sqrt{D{T}_R}$.

The point $x=x_T$ is absorbing because reaching the treasure terminates the search, while the boundary at $x=-L_R$ is reflecting because the seeker is not allowed to wander indefinitely far away from the target. A natural estimate for the scale of this confinement length comes from ordinary diffusion. A random walker diffuses a characteristic distance of order $\sqrt{D{T}_R}$ during the reset time $T_R$, where the diffusion coefficient $D\equiv \frac{a^2}{2\mathrm{\Delta }t}$ relates to the hopping rate $k$. The simplest guess would then take the effective distance of the reflective wall $L_R$ as $L_R=\sqrt{D{T}_R}$. This naive estimates already captures the essential scaling. However, more sophisticated treatments of this problem are tractable and interesting; these more precisely specify the appropriate lengthscale sampled by walks under reset. These analyses give numerical prefactors for the lengthscale $L_R$ that accords more closely with the dynamics of reset as we explore in Sec. C, “Analytics of random walks with deterministic reset,” of the Supplemental Material. Accordingly, we acknowledge such a dimensionless constant $C$ entering $L_R$ as,

$$L_R=C\sqrt{D{T}_R}.$$ (21)

In effect, we have replaced a difficult dynamical problem involving repeated resets by a much simpler first-passage problem in a finite domain.

Armed with this picture of motion in a domain effectively bounded by resets, how long on average will it take for a seeker to reach the treasure at $x_T$? To approach this question, which inquires about the mean first hitting time from the walker’s position to the target position $x_T$ under diffusive motion, we use the following gambit as illustrated in Figure 9. This strategy is well established⁸⁸, mechanically simple, but conceptually subtle. It will recur again and again throughout our studies of complex exploratory trajectories, rewarding some discussion now.

FIG. 9:

Origins of the local averaging property of a hitting time to a target. (A) The average time $T(x)$ for a seeker to hit a target starting from position $x$ is consistent with the analogous times $T(x-a)$ and $T(x+a)$ starting from neighboring positions. The purple shading of each site schematizes its local first hitting time; the self-consistency between adjacent sites means a site’s time (shading) smooths between those of its neighbors. The explicit mention of $k/(k+k)$ is to highlight that there is a branching ratio between two processes. In this case, those two processes have the same rate and hence the step probabilities are the same. (B) This self-consistency is a local average.

While we care about the time $T(x)$ a particular walker released at a particular position $x$ (say, the origin) will take to reach the target, we could easily ask the same question of any other starting position. For example, if we had started instead one site to the left at position $x-a$. The key idea is that once a walker arrives at a position, its propensity for where to step next, and crucially, the average time it takes to later wander anywhere else in the domain, does not depend at all on its past, and depends only on its current position. This property (the so-called Markov property of the walk) means there is a set of well-defined numbers $T(x-a)$, $T(x)$, $T(x+a)$, and so on, that specify the average times that a walker (released at $x-a,x$, $x+a$) will take to arrive at any target, no matter the paths they took to arrive at those starting points. Crucially, this requires that the time $T(x)$ to hit a target from $x$ must be internally consistent with those of its neighbors, $T(x-a)$ and $T(x+a)$, according to a very specific and natural pattern. Specifically, as shown in Fig. 9, the average time $T(x)$ to reach position $x_T$ satisfies a sort of local average with those of its neighbors, according to,

$$T(x)=\frac{1}{2k}+\frac{1}{2}T(x-a)+\frac{1}{2}T(x+a),$$ (22)

where the first constant term $\frac{1}{2k}\equiv \mathrm{\Delta }t$ represents the average time before a jump occurs to either neighboring position, where $k$ is the rate at which the walker steps right or left. This recursion relation eqn. 22 quantifies the intuition that if neighboring positions can quickly reach a target, and a walker hops between neighboring sites quickly, then the time to reach the target from the present position should also be rapid. The logic of this insightful procedure was developed by Kolmogorov^25,57 and such recurrence relations are often expansion of the form referred to as “backwards equations.” Another way to see the emergence of such equations is via explicit averages over trajectories; see Sec. D, “Average hitting times as explicit averages over trajectories,” of the Supplemental Material.

This recursion can be simplified by resorting to a Taylor expansion of the form

$$\begin{gathered} T(x)=\mathrm{\Delta }t+\frac{1}{2}\left(T(x)-a{T}^{\prime }(x)+\frac{a^2}{2}{T}^{\prime \prime }(x)\right) \\ +\frac{1}{2}\left(T(x)+a{T}^{\prime }(x)+\frac{a^2}{2}{T}^{\prime \prime }(x)\right) \end{gathered}$$ (23)

After performing the relevant cancellations we are left with

$$\frac{d^{2}T}{d{x}^2}=-\frac{1}{D}$$ (24)

where we define the diffusion coefficient $D=a^2/2\mathrm{\Delta }t$. The general solution of the differential equation is

$$T(x)=-\frac{x^2}{2D}+Ax+B.$$ (25)

To determine the coefficients $A$ and $B$ we need to impose the boundary conditions. Of course, if we start at $x_T$, we don’t have to make any steps and hence the boundary condition there is that the time to reach the target is zero, written mathematically as

$$T\left(x_T\right)=0.$$ (26)

For the boundary condition at $-L_R$, we note that reaching the point $-L_R$ means that the seeker has only one option which is to step back to the right. This condition is equivalent to

$${\left.\frac{dT}{dx}\right|}_{x=-L_R}=0.$$ (27)

To understand why this reflecting boundary condition materializes, consider the relation for the first hitting time at the boundary, where only one move is possible, enforcing,

$$T\left(-L_R\right)=\frac{1}{2k}+T\left(-L_R+a\right).$$ (28)

Taylor expanding the right-most term near $x=-L_R$ gives,

$$T\left(-L_R\right)=\frac{1}{2k}+T\left(-L_R\right)+{\left.\frac{dT}{dx}\right|}_{x=-L_R}a$$ (29)

Substituting $1/2k=a^2/2D$ yields

$${\left.\frac{dT}{dx}\right|}_{x=-L_R}=-\frac{a}{2D}.$$ (30)

In the continuum limit of relatively small lattice spacing $a/L_R\to 0$, this gives indeed,

$${\left.\frac{dT}{dx}\right|}_{x=-L_R}=0.$$ (31)

Imposing the reflecting condition gives

$$A=-\frac{L_R}{D},$$ (32)

and imposing the absorbing condition at the treasure gives

$$B=\frac{x_T^2}{2D}+\frac{L_R{x}_T}{D}.$$ (33)

As a result, the mean first-passage time becomes

$$T(x)=-\frac{x^2}{2D}-\frac{L_{R}x}{D}+\frac{x_T^2}{2D}+\frac{L_R{x}_T}{D}.$$ (34)

Since the seeker begins at the origin, the quantity of interest is simply

$$T(0)=\frac{x_T^2}{2D}+\frac{L_R{x}_T}{D}.$$ (35)

This expression has a simple interpretation. The first term is the ordinary diffusive contribution associated with reaching a target at distance $x_T$, while the second term reflects the penalty imposed by reset through the effective confinement length $L_R$. When the reset length is large compared with the target distance, so that $L_R\gg x_T$, the dominant contribution becomes

$$T(0)\approx \frac{L_R{x}_T}{D}.$$ (36)

Substituting the heuristic $L_R=C\sqrt{D{T}_R}$ of eqn. 21 gives,

$$T(0)\approx x_{T}C\sqrt{\frac{T_R}{D}},$$ (37)

showing that the search time grows as the square root of the reset time as shown in Figure 7.

How does the above compare to an exact analytical expression for the mean first passage time of a random searcher subject to deterministic reset? In the large reset time limit $T_R\to \mathrm{\infty }$, we find in Sec. E.2, “Equivalence of asymptotics of mean first passage times,” of the Supplemental Material that the mean first passage time is given by

$$T_{\text{exact}}(0)=x_T\frac{2}{\sqrt{\pi }}\sqrt{\frac{T_R}{D}}.$$ (38)

Comparing to eqn. 37 identifies the prefactor $C=\frac{2}{\sqrt{\pi }}$, resulting in the reflecting boundary being placed at

$$L_R=\frac{2}{\sqrt{\pi }}\sqrt{D{T}_R}.$$ (39)

Interestingly, this prefactor may be physically rationalized by understanding that the presiding length scale is not precisely the typical diffusive lengthscale $\sqrt{D{T}_R}$, but rather the expectation of the largest distance explored to the left $⟨{\mathrm{m}\mathrm{a}\mathrm{x}}_{0\le t\le T_R}x(t)⟩=(2/\sqrt{\pi })\sqrt{D{T}_R}$ between resets⁴⁵. Indeed, the confining box must be large enough to reproduce the full spatial region visited by the walker.

In addition, the result of eqn. 34 is also illuminating in the limit of no resets, namely $L_R\to \mathrm{\infty }$: the fact that $T(0)$ grows to infinity in this limit is another explicit affirmation of the fact that the mean time to reach a target in 1D with pure diffusion is infinite, as presaged by Sec. A of the Supplemental Material. The action of the reset is to introduce the finite lengthscale $L_R$ and makes this time finite.

Thus a seemingly complicated problem involving repeated interruption and restart admits a remarkably simple approximation in which reset merely confines the seeker to a finite region of exploration.

C. Exact Description of The Simple Random Walk With Reset

Having earned a qualitative feeling for the time for the seeker to find the target as a function of reset time, we resort to a full calculation to appreciate the structure of the whole curve shown in Figure 7. Thus far, we have spoken of deterministic resets as shown in Figure 5(A). However, as shown in Figure 5(B), a close and mathematically tractable alternative is to consider the case where the seeker suffers a stochastic reset with an average rate $r=1/T_R$ (instead of a deterministic periodic reset).

Once again, we appeal to the idea of making a recursive calculation that recognizes that the time to reach the target given the seeker is currently at position $x$ can be obtained by adding the times it would take had it been at adjacent positions. In particular, we can write

$$\begin{array}{l}T(x)=\frac{1}{2k+r}+\frac{k}{2k+r}T(x+a)+\frac{k}{2k+r}T(x-a)\\ +\frac{r}{2k+r}T(0),\end{array}$$ (40)

where the average rate of steps is by $2k+r$. The first term on the right, $1/(2k+r)$, is the length of a time step accomplishing any jump; it is the analog of the $1/2k$ term in the earlier more basic example of eqn. 22. The weights for each of the remaining terms are given by branching ratios. The two terms with factors $k/(2k+r)$ correspond to the seeker jumping left or right. Last, the term with factor $r/(2k+r)$ gives the branching probability of reset to the origin. In light of this recursive structure, we can rewrite our equation as,

$$\begin{gathered} T(x)(2k+r)=1+kT(x+a)+kT(x-a) \\ +rT(0) \end{gathered}$$ (41)

which invites us to perform Taylor expansions of $T(x+a)$ and $T(x-a)$ with the result

$$0=1-rT(x)+rT(0)+\frac{k{a}^2}{2}{T}^{\prime \prime }(x).$$ (42)

Hence, in the absence of drift, the differential equation for the mean first passage time is given by

$$0=1+D{T}^{\prime \prime }(x)-rT(x)+rT(0).$$ (43)

We can rewrite this as

$$0=D{T}^{\prime \prime }(x)-r\left(T(x)-T(0)-\frac{1}{r}\right).$$ (44)

This differential equation holds for all $x<x_T$, and must be supplemented with the absorbing boundary condition

$$T\left(x_T\right)=0,$$ (45)

which is the statement that the target is at $x=x_T$ as shown in Figure 7. We now introduce a transformed function

$$y(x)=T(x)-T(0)-\frac{1}{r}$$ (46)

which leads to the transformed differential equation

$$y^{\prime \prime }(x)-\frac{1}{{\lambda }_D^2}y(x)=0,$$ (47)

where we define ${\lambda }_D=\sqrt{D/r}$ as the length scale over which the seeker diffuses before the reset occurs. The general solution for this equation is

$$y(x)=A{e}^{-x/{\lambda }_D}+B{e}^{x/{\lambda }_D},$$ (48)

with $A$ and $B$ real constants. This gives us

$$T(x)=A{e}^{-x/{\lambda }_D}+B{e}^{x/{\lambda }_D}+T(0)+\frac{1}{r}.$$ (49)

Because of the resets, we know that even when trajectories wander to positions $x$ very far away, $x\to -\mathrm{\infty }$, the waiting time to reach the target $x_T$ will not be infinite. This is because searchers will eventually reset to the start, and then go from there. This yields the additional required boundary condition,

$$\underset{x\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}T(x)=\text{finite}.$$ (50)

This condition requires we take $A=0$. Next, we enforce the additional boundary condition $T\left(x_T\right)=0$, namely that the time to success is zero once the searcher has already reached the target. This condition applied to eq. 49 specifies the value of the remaining coefficient $B$ as,

$$B=-e^{-x_T/{\lambda }_D}\left(T(0)+\frac{1}{r}\right)$$ (51)

which results in the expression

$$T(x)=\left(T(0)+\frac{1}{r}\right)\left(1-e^{\left(x-x_T\right)/{\lambda }_D}\right).$$ (52)

When we start at the origin, this simplifies to

$$T(0)=T(0)\left(1-e^{-x_T/{\lambda }_D}\right)+\frac{1}{r}\left(1-e^{-x_T/{\lambda }_D}\right),$$ (53)

which can be simplified in turn to the form

$$T(0)=\frac{e^{x_T/{\lambda }_D}}{r}\left(1-e^{-x_T/{\lambda }_D}\right).$$ (54)

We can rewrite this as the extremely informative final expression, giving a theoretical prediction for the behavior of stochastic reset,

$$T(0)=\frac{1}{r}\left(e^{x_T/{\lambda }_D}-1\right).$$ (55)

How well does the stochastic model prediction of eqn. 55 compare to exact numerical simulations of the sort we set out to understand? Fig. 10 reports in green square markers the results of simulations with stochastic reset summarized by their average times to hit a target, analogous to the original deterministically-reset curve of interest in Fig. 7 (also reproduced for comparison as blue circles). The theoretical prediction of eqn. 55 for stochastic reset is shown as a black line (informed by the knowledge that $D=1/2$ in units of time and space steps in this discrete simulation, and recalling $1/r=T_R$). The theoretical prediction accords with the stochastic reset simulations, and also closely reproduces the major features of the related deterministic reset problem in blue. We note that the distinctions and comparative performances between deterministic resetting and stochastic resetting support an active arena of mathematical inquiry; for instance, investigations show that a deterministic resetting strategy can reach targets in smaller optimal times than stochastic resetting strategies¹⁸, but stochastic resetting strategies can show performance benefits for randomly-distributed targets²⁹.

FIG. 10:

Agreement of theoretical descriptions of random walk under reset with direct numerical simulations. The blue-edged circle markers are the same numerically simulated behaviors visualized in Fig. 7. The green-edged square markers show the analog of these direct simulations, but where the resets are executed stochastically with an average rate, instead of deterministically with the period of the reset. The gold straight line visualizes the large reset time asymptotic in the case of deterministic reset, $T\approx 2x_T\sqrt{2T_R/\pi }$. The dark green and dark purple lines visualize the theoretical expressions for stochastic resets (eqn. 56) and deterministic resets (derived in Sec. C.3, “Mean first passage time of random searcher,” of the Supplemental Material) respectively, showing that despite the technical difference in executions of these resets, this description describes the quantitative features of both reset scenarios with excellent agreement. (The red shaded region where $T_R<x_T$ delimits the physical region where resets are too fast to allow any successful encounters of the target.)

The next question we might ask in our qualitative attempt to understand the graph is how the position of the minimum depends upon the distance to the target. The distance to the target ($x_T$) and the diffusion constant $D$ combine to give the only time scale in the problem $x_T^2/D$. This implies that the minimum time to target must scale as $T_{\mathit{min}}\sim x_T^2/D$.

To formalize that result and estimate the prefactors, we consider the average search time as a function of the reset time $T_R=1/r$ reformulating eqn. 55 as

$$T\left(T_R\right)=T_R\left(e^{x_T/\sqrt{D{T}_R}}-1\right),$$ (56)

and seek the value of $T_R$ that minimizes it. Here, as mentioned earlier, we approximated the original problem shaped by a deterministic reset time using a closely related problem steered by stochastic resets with rate $r=1/T_R$. To that end, we define a dimensionless average search time, $\tilde{T}=TD/x_T^2$, and a dimensionless reset time $\widetilde{T_R}=T_{R}D/x_T^2$, which transforms eqn. 56 into

$$\tilde{T}\left(\widetilde{T_R}\right)=\widetilde{T_R}\left(e^{1/\sqrt{{\tilde{T}}_R}}-1\right).$$ (57)

Minimization leads to the condition

$$e^{1/\sqrt{{\tilde{T}}_R}}\left(1-\frac{1}{2\sqrt{{\tilde{T}}_r}}\right)=1.$$ (58)

This transcendental equation has a unique positive solution

$${\tilde{T}}_R^{\mathrm{*}}\approx 0.394,$$ (59)

corresponding to an optimal reset time in real units of,

$$T_R^{\mathrm{*}}\approx 0.394\frac{x_T^2}{D}.$$ (60)

The corresponding minimal value for the average search time is

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}\approx 1.54\frac{x_T^2}{D}.$$ (61)

This optimal reset policy is interesting. It does not correspond to perfunctory heuristics one might guess before performing the calculations we just discussed. To appreciate these real subtleties, it is revealing to return to the beginning and ask, what would be appealing guesses for the optimal rate of resets? Given just the specified physics of the problem (the target location $x_T$ and the statistics of raw search trajectories without reset), one immediate guess might be to choose the time $T_R$ between resets that matches the most common time that trajectories otherwise ordinarily first hit the target. More precisely, say we run the unrestricted search many times, as in Fig. 11(A), and keep track of the times ${\tau }_i$ that each trajectory $i$ first hits the target (shown by the set of red circles first hitting the red target line). This would yield a distribution of first arrival times, which could be summarized by an empirical histogram (shown by the grey histogram of Fig. 11(B)). In the continuum limit of (many) long searches, this can be described by a probability density function $f(t)$ shown as the red curve in Fig. 11(c)), which encodes the probability $f(t)dt$ of a searcher arriving at the target within a time interval of [$t,t+dt$]. Our initial naïve guess of taking the most common time that a searcher reaches the target for the reset corresponds to the mode of this first passage time probability distribution, which turns out to be $x_T^2/3$ (with $D=1$); see Sec. B.2, “Interpretations and continuum approximations,” of the Supplemental Material. As stressed by Fig. 12, this guess conspicuously disagrees with the actual optimal reset rate identified by our more careful calculations. Giving trajectories more time before reset than the modal arrival time $x_T^2/3$ is needed to minimize the time to arrive at the target.

FIG. 11:

The first hitting time probability density and survival probability from trajectories. (A) Many random trajectories (in black lines) wander until they first hit a target (here, shown at $x_T=5$, a location emphasized by the red horizontal line). The stochastic first arrival times ${\tau }_i$ of each trajectory $i$ are marked by red circles. Clearly, some first arrival times are much more common than others, as stressed by the gradation in density of these first arrivals. (B) The histogram of first arrival times to the target, shown as a grey empirical histogram for the example trajectories in (A), approach a continuum probability density $f(t)$ of first arriving near time $t$, shown in the red curve. Specifically, $f(t)dt$ gives the probability of first arriving at the target within an interval $dt$ of time $t$. (C) The total “survival probability” $S(t)$ that a searcher has not yet arrived at the target by time $t$, given by $S(t)={\int }_t^{\mathrm{\infty }}dtf(t)$, gives another handle on the statistics of search; such a survival probability is explored further in the context of Fig. 27. (The light vertical lines in all plots illustrate a particular time $t\approx T_R^{\mathrm{*}}$, for explicit visual reference, and comparison to Fig. 12.)

FIG. 12:

The optimal reset time revealed by eqn. 60, shown as the teal vertical line, in the context of the first-passage time density (of a random walk), shown as the purple curve. The most-common time to reach the target for the first time is given by the mode $x_T^2/3$ marked by the vertical grey line which notably is quite distinct from the (longer) optimal reset policy specified by eqn. 60. The integral of the first-passage time density up to an allotted time $t$ gives the net probability of the walk reaching the target by this time; integrating up to the optimal reset time (denoted by the purple shading) suggests a ≈14% probability of reaching the target in a bout between two resets according to the optimal policy.

Another tempting guess at the outset might be to choose some reset policy that nonetheless allows each search trajectory to have a decent chance of hitting the target before being compelled to reset. However, yet again, interestingly this expectation is not borne out. To appreciate this, it is instructive to consider the so-called “survival probability” $S(t)$ of a searcher not yet having succeeded at reaching the target by time $t$, as illustrated by Fig 11(C). This is a quantity worth appreciating as we will regularly refer to such behavior throughout calculations in this paper. As shown by Fig. 12, the actual optimal reset time remains surprisingly aggressive in pruning trajectories: the probability that the seeker reaches the target within the optimal reset time is only $S(T_R^{\mathrm{*}})={\int }_0^{T_R\mathrm{*}}dtf(t)\approx 0.14$, meaning that ≈ 86% of attempts are reset before finding the target. In general, these features would be obscure were it not for the understanding given by the exact calculation of eqn. 60. We are persuaded that these trajectory-level metrics are a powerful way to gauge the distinct character of microtrajectories, following on the idea of the potency of such trajectories we described elsewhere.

Eqn. 56 gives revealing limiting asymptotics that fortify intuition. First, consider the case of large reset time $T_R$: Taylor expanding eqn. 56 near $T_R\to \mathrm{\infty }$ gives $T\approx x_T\sqrt{\frac{T_R}{D}}+\cdots$ to first order, explicitly recovering the square-root dependence of the mean time to find the target, emphasized by the teal triangle in Fig. 10, that we reasoned about in the limiting physical scenario of the walker in a finite box in the preceding subsection and eqn. 37. Next, consider the opposite extreme of a small reset time $T_R$ in the vicinity of the target location itself, namely $T_R=x_T+\epsilon$ for small $\epsilon$. Substituting this and expanding near $\epsilon \to 0$ gives $T\approx x_T\left(\mathrm{e}\mathrm{x}\mathrm{p}\left[\sqrt{\frac{x_T}{D}}\right]-1\right)+\left(\mathrm{e}\mathrm{x}\mathrm{p}\left[\sqrt{\frac{x_T}{D}}\right]\left(1-\sqrt{\frac{x_T}{D}}\right)-1\right)\epsilon +\cdots$ where $x_T,T_R$, and $\epsilon$ are all measured in units of steps; the precise form is less important than the takeaway of a local linear behavior in the departure $\epsilon$ from the $T_R=x_T$ case. These behaviors are visible in Fig. 10 as eqn. 56 stitches together.

For additional concreteness and intuition, Sec. F.1, “Additional sweep of behavior of random walk under resets from simulation,” of the Supplemental Material, shows additional direct simulation sweeps reflecting how reset rates and target locations change the mean time to successfully reach targets, and optimal reset rates. Further interesting features visible in the behavior of Figure 7, including the sawtooth oscillations especially visible at small reset times (which are not sampling artifacts but reflect parity structure in the discrete search problem), are also ratifiable in light of lattice path counting, which we detail in Sec. B of the Supplemental Material. Together, all of these results provide a coherent explanatory framework for the rich competing behaviors shown in Figure 7.

D. Case Study: Intraflagellar Transport as a Random Walk with Reset

To what extent might these mathematical approaches shed light on biological phenomena? Reframing the problem in terms of concentration fields rather than the trajectory statistics of individual seekers provides a direct path to interesting biological case studies. One interesting example is offered by intraflagellar transport (IFT), directed motion of proteins in eukaryotic cilia driven by kinesin and dynein motors as indicated schematically in Figure 13(A). These structures are built up of about one thousand proteins of a few dozen types, which then assemble into a large protein complex with kinesin motors to form the anterograde IFT train¹²⁷, as depicted in Figure 13(B). Powered by ATP and the action of kinesin motors, this protein complex moves along ciliary microtubules toward the cilium tip at a speed of about $v\approx 1\mu \mathrm{m}/\mathrm{s}$. When it reaches the tip, the train disassembles and then reforms to make the retrograde IFT train, which contains a similar number of proteins and is transported by dynein motors toward the cell body at a similar speed. In addition to this directed transport by IFT trains, proteins in the lumen of the cilium also move diffusively, with a diffusion constant of about $D\approx 1{\mu \mathrm{m}}^2/s$. This means that in a typical cilium about $L\approx 5\mu \mathrm{m}$ in length, transport and diffusion across the length of the cilium take anywhere between a few and few tens of seconds. Though our focus on random walks with resets has thus far emphasized the individual walkers, here we examine how random walks with resets shapes not only individual search trajectories but achieves interesting concentration distributions⁸⁶ within cells that are crucial for exquisite and downstream functions for organisms.

FIG. 13:

Intraflagellar transport (IFT) spatially organizes cargo in cilia and flagella. (A) Cilia and flagella are ubiquitous structures in many different cell types. (B) Mechanisms of intraflagellar transport, where cargo interact with motorized complexes moving ballistically in the anterograde direction towards the ciliary or flagellar tip, and otherwise diffuse in the lumen. Retrograde transport chains also replenish complexes impoverished by cargo back to the cell body, visualized by the empty motors returning to the cytosolic body. (C) A one dimensional model of cargo transport, organized by two lanes, yields an effective resetting mechanism.

The motion of a protein in a cilium can be described by a two-lane model⁸⁶, as illustrated by Figure 13(C). (Specifically, this is true when anterograde transport dominates retrograde transport (such that the latter can be neglected). A two-lane model behaves qualitatively similarly to a fuller three-lane model acknowledging both anterograde and reterograde motion, as long as the net motion of cargo is towards the tip.)

Each lane in this two-lane model is described by a one dimensional lattice, with a lattice spacing $a$, like in Figure 13(C). In the transport lane the protein undergoes directed motion toward the tip of the cilium (the positive-$x$ direction) with a rate $v$. In the diffusion lane the protein hops left and right with rate $k$. The protein can switch between the two lanes, with rate $k_{\text{on}}$ describing the transition from the diffusion lane to the transport lane, and rate $k_{\text{off}}$ dictating transitions from the transport to the diffusion lane. In the limit when the time spent in the transport lane, which on average is $1/k_{\text{off}}$, is much larger than the time it takes to transport the protein to the tip of the cilium (a few seconds), the two lane model becomes a random walk with resets. In particular, every time a protein diffusing in the lumen of the cilium is captured by an anterograde IFT train, its fate is sealed and it is deposited at the tip before it has a chance to disassociate from the train, effectively sending the walker back to the origin. Direct experiments tracking the motion of a fluorescently-labeled sodium receptor protein cargo inside the cilium of a C. elegans nerve cell explicitly identified and measured this type of motion, with the protein either diffusing or being primarily transported by the anterograde IFT¹²⁷. The result is the accumulation of this protein at the tip of the cilium, which was shown to be required for sensitive sensing of salt in the environment. In a very different setting, similar accumulation of a depolymerizing kinesin, which also rides on the IFT trains, was observed in the flagella of Giardia²². Other examples of this sort of protein localization by intraflagellar transport have been reported, making the cilium tip into a hub for signaling, sensing, and regulation⁹⁰.

The spatial localization of proteins to the cilium tip observed in experiments can be described using the mathematics of diffusion with resets. Along the length of the cilium, the concentration at a location $x$ changes either by diffusion or by molecules being whisked away by the IFT at rate $k_{\text{on}}$. The tip enjoys an additional source term from the accumulation of delivered cargos, accumulated over possible intial arrival points all along the microtubule. These deliveries contribute a total positive arrival rate given by,

$$\text{additional}\text{source}\text{term}\text{at}\text{tip}x=0:k_{\text{on}}{\int }_0^{L}d{x}^{\prime }c\left(x^{\prime },t\right).$$ (62)

The integral accounts for the delivery of all the captured proteins along the length of the cilium, to the tip of the cilium at $x=0$. Taken together, these effects make the continuum equation for these dynamics adopt the form,

$$\frac{\partial c(x,t)}{\partial t}=\left\{\begin{array}{ll}D\frac{{\partial }^{2}c}{\partial x^2}-k_{on}c& \text{if}x\ne 0\text{(not}\text{at}\text{the}\text{tip),}\\ D\frac{{\partial }^{2}c}{\partial x^2}-k_{on}c+k_{\mathrm{o}\mathrm{n}}{\int }_0^{L}d{x}^{\prime }c\left(x^{\prime },t\right)& \text{if}x=0\text{(at}\text{the}\text{tip).}\end{array}\right.$$ (63)

The steady-state concentration is described by the balance between diffusion and resets (to the tip), given by

$$D\frac{{\partial }^{2}c}{\partial x^2}=k_{\mathrm{o}\mathrm{n}}c.$$ (64)

This equation has the general solution,

$$c(x)=A\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{x}{\sqrt{D/k_{\mathrm{o}\mathrm{n}}}}\right]+B\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{x}{\sqrt{D/k_{\mathrm{o}\mathrm{n}}}}\right]$$ (65)

$$\equiv A\mathrm{e}\mathrm{x}\mathrm{p}[x/\lambda ]+B\mathrm{e}\mathrm{x}\mathrm{p}[-x/\lambda ],$$ (66)

where we have defined the emergent length scale $\lambda \equiv \sqrt{D/k_{\mathrm{o}\mathrm{n}}}$, the same length scale we encountered when analyzing the first passage time for random walks with resets.

To find the constants $A$ and $B$ in this solution, we consider the boundary conditions set by the physics of the process. At the cytosolic end of the cilium ($x=L$), the ambient concentration of cargo matches that of the cytoplasm, $c_{\text{cyto}}$; this delivers the boundary condition $c(L)=c_{\text{cyto}}$. This fact, which expresses the reasonable assumption that the cytoplasm is not depleted of cargo, imposes the behavior,

$$c_{\text{cyto}}=c(L)=A\mathrm{e}\mathrm{x}\mathrm{p}[L/\lambda ]+B\mathrm{e}\mathrm{x}\mathrm{p}[-L/\lambda ].$$ (67)

In addition, the diffusive $\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}-{\left.D\frac{\partial c}{\partial x}\right|}_{x=0}$ experienced at the tip must balance the accumulated influx from resets along the length, $k_{\text{on}}{\int }_0^{L}d{x}^{\prime }c\left(x^{\prime },t\right)$. This gives the additional boundary condition,

$$-{\left.D\frac{\partial c}{\partial x}\right|}_{x=0}=k_{\text{on}}{\int }_0^{L}d{x}^{\prime }\left(A\mathrm{e}\mathrm{x}\mathrm{p}\left[x^{\prime }/\lambda \right]+B\mathrm{e}\mathrm{x}\mathrm{p}\left[-x^{\prime }/\lambda \right]\right),$$ (68)

or,

$$-D\left(\frac{A}{\lambda }-\frac{B}{\lambda }\right)=k_{\mathrm{o}\mathrm{n}}\lambda (A(\mathrm{e}\mathrm{x}\mathrm{p}[L/\lambda ]-1)-B(\mathrm{e}\mathrm{x}\mathrm{p}[-L/\lambda ]-1)).$$ (69)

Recalling that $k_{\mathrm{o}\mathrm{n}}=D/{\lambda }^2$, this yields,

$$-D\left(\frac{A}{\lambda }-\frac{B}{\lambda }\right)=\frac{D}{\lambda }(A(\mathrm{e}\mathrm{x}\mathrm{p}[L/\lambda ]-1)-B(\mathrm{e}\mathrm{x}\mathrm{p}[-L/\lambda ]-1)),$$ (70)

or,

$$B-A=A(\mathrm{e}\mathrm{x}\mathrm{p}[L/\lambda ]-1)-B(\mathrm{e}\mathrm{x}\mathrm{p}[-L/\lambda ]-1),$$ (71)

and subtracting $B-A$ from both sides finally gives,

$$A\mathrm{e}\mathrm{x}\mathrm{p}\left[L/\lambda \right]=B\mathrm{e}\mathrm{x}\mathrm{p}[-L/\lambda ].$$ (72)

This consequence of the second boundary condition eqn. 72 combines with the first cytosolic boundary condition eqn. 67 to require that

$$A\mathrm{e}\mathrm{x}\mathrm{p}[L/\lambda ]=B\mathrm{e}\mathrm{x}\mathrm{p}[-L/\lambda ]=\frac{c_{\text{cyto}}}{2},$$ (73)

so,

$$c(x)=\frac{c_{\text{cyto}}}{2}\left(\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{L-x}{\lambda }\right]+\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{L-x}{\lambda }\right]\right)$$ (74)

$$=c_{\text{cyto}}\cosh \left(\frac{L-x}{\lambda }\right).$$ (75)

This profile is perhaps of greatest interest in positions close to the ciliary tip, namely $L\gg x$ (in units of the presiding length scale $\lambda$). More specifically, when $L\gg \lambda$, so that $\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{L-x}{\lambda }\right]\to 0$ in the limit $\frac{\lambda }{L}\to 0$ at the tip $x=0$, we enjoy the approximation,

$$c^{\text{near}\text{tip}}(x)\approx \frac{c_{\text{cyto}}}{2}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{L-x}{\lambda }\right]=\frac{c_{\text{cyto}}}{2}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{L-x}{\sqrt{D/k_{\text{on}}}}\right].$$ (76)

Such an exponential concentration profile is illustrated in Fig. 14(A).

FIG. 14:

Gradients in cilia and flagella. (A) Intraflagellar transport in which the cargo can jump onto motors within the flagellum, thus serving as a reset to the flagellar tip. (B) Intraflagellar transport (IFT) in which the cargo can only be loaded at the cell body.

Does this quantitative picture accord with real gradients of cargo observed in cilia? To address this question, we first must estimate the on rate $k_{\text{on}}$ for cargo arriving to the cytoskeletal filaments where the transport (and thus resetting) occurs. To anticipate the scale of how quickly molecules arrive from the cytoplasm to the central cytoskeletal filament, we recall that the time that a molecule takes to transversely diffuse across the radius of the cilium is simply of order ${\tau }_{\text{diff}}\approx R^2/D$, implicating an on rate of order $k_{\text{on}}\approx 1/{\tau }_{\text{diff}}\approx D/R^2$. If we substitute this approximation to $k_{\text{on}}$ into the anticipated exponential length scale $\lambda \equiv \sqrt{D/k_{\mathrm{o}\mathrm{n}}}$, we find $\lambda \approx R$, namely, the length scale of the gradient is roughly equal to the width of the cilium. Strikingly, measurements of the exponential profile of the concentration of kinesin-13 cargo in the flagella of Giardia are consistent with that expectation⁷⁴. For each of the four pairs of flagella in this organism, the exponential profile gave the same $\lambda \approx 350$ nm length scale, which is consistent with the 200–300 nm range for the widths of microtubule-based flagella and cilia.

Intriguingly, this description linking the width of a cilium $R$ with the longitudinal gradient length scale $\lambda$ (via the arrival rate $k_{\text{on}}$ of cargo to central cytoskeletal filaments) anticipates a relationship potentially under cellular control, or else testable in a synthetic engineered setting. This description, albeit simplified, anticipates that the transverse geometry of a cilium or flagellum should couple tightly with the shape of the longitudinal gradient, with wider cilia or flagella eliciting shallower gradients (modulo other regulation or saturation effects)²¹. This intriguing speculation invites further inquiry both experimentally and theoretically.

Note that another mechanism under cellular control concerns where in space the cargo can load onto the machinery of the intraflagellar transport chain’s complexes. If cargo can load only at the base of the cilium (at the cell body), instead (as assumed earlier) occurring anywhere along the flagellum or cilium, the resulting dynamics and concentration gradient adopt a very different form. Specifically, except at the cell body at $x=L$ where loading occurs at a rate $k_{\text{on}}{c}_{\text{cyto}}$, the sink terms disappear throughout the domain, making the only way that concentration changes in time changes from the dynamics specified earlier in eqn. 63 to simply,

$$\frac{\partial c(x,t)}{\partial t}=D\frac{{\partial }^{2}c}{\partial x^2}.$$ (77)

The solution to this equation now has a simple linear profile in space,

$$c(x)=c_{\text{cyto}}\left(1+\frac{k_{\text{on}}}{D}(L-x)\right),$$ (78)

where this solution enforces the boundary conditions $-{\left.D\frac{dc}{dx}\right|}_{x=0}=k_{\text{on}}{c}_{\text{cyto}}$ and $c(L)=c_{\text{cyto}}$. In other words, merely by adjusting which spatial positions enjoy replenishment of cargo (just the base of the cilium, or all of the sites along the cilium), the concentration profile of cargo over space can be tuned to change from an exponential behavior to a linear gradient, as shown by Fig. 14(B). In fact, distinct examples of such a linear profile have been observed for kinesin-2 motors in experimental works²⁰.

In summary, the calculations we just performed show how resets and diffusion conspire to set the typical concentration profiles of molecules along cilia and flagella, in tunable ways. Such concentration profiles reflect the probability of encountering molecules at any point in space, no matter whether they have visited that point in space before. This analysis gives a natural informative complement to the separate mathematics in the other sections of this paper that attend to first arrivals of searchers like molecules, organelle structures, cells, or organisms to specified targets of interest.

E. Random Walk With Resets and Drift

We now generalize the calculation of random walk with reset to include a bias, or drift, in the direction of the target. The physical motivation comes from the fact that in many biological contexts the seeker is not an unbiased random walker but moves preferentially toward its target, whether because of a chemical gradient, a mechanical bias, explicit flow, motor-assisted transport, or an asymmetry in the dynamics of growth and shrinkage. We model this by allowing the rate of stepping toward the target to differ from the rate of stepping away, as shown in Fig. 15. Specifically, we assign rate $k_1=k+\mathrm{\Delta }k$ to steps in the positive direction (toward the target) and rate $k_2=k-\mathrm{\Delta }k$ to steps in the negative direction (away from the target), where $∆k\ge 0$ measures the degree of bias. The reset rate $r$ remains unchanged.

FIG. 15:

Schematic of the steps available to the random seeker during a time step when there is a bias in one direction. The top graph posits a linear concentration gradient of some cue that leads to a constant drift velocity as shown in the lower graph.

The recursive equation for the mean first-passage time $T(x)$ now takes the form

$$T(x)(2k+r)=1+k_{1}T(x+a)+k_{2}T(x-a)+rT(0),$$ (79)

where as before the factor $2k+r$ is the total rate of leaving position $x$, and the three terms on the right involving $T(x)$ account for stepping right, stepping left, and resetting to the origin, respectively. Substituting $k_1=k+\mathrm{\Delta }k$ and $k_2=k-\mathrm{\Delta }k$ gives

$$T(x)(2k+r)=1+(k+\mathrm{\Delta }k)T(x+a)+(k-\mathrm{\Delta }k)T(x-a)+rT(0).$$ (80)

We now perform a Taylor expansion of $T(x\pm a)$ to second order in the step size $a$, leaving

$$rT(x)=1+2k\cdot \frac{a^2}{2}{T}^{\prime \prime }(x)+2\mathrm{\Delta }k\cdot a{T}^{\prime }(x)+rT(0).$$ (81)

Rearranging and identifying the diffusion coefficient $D=k{a}^2$ and the drift velocity $v=2\mathrm{\Delta }k\cdot a$ in the continuum limit, we arrive at the ordinary differential equation

$$0=1+D{T}^{\prime \prime }(x)+v{T}^{\prime }(x)-rT(x)+rT(0),$$ (82)

which holds for all $x<x_T$, supplemented by the absorbing boundary condition $T\left(x_T\right)=0$. This is the natural generalization of eqn. 43: the only new term is $v{T}^{\prime }(x)$, which encodes the drift toward the target. When $v=0$ we recover the unbiased result exactly.

To solve eqn. 82, we again define a helper function given by,

$$y(x)=T(x)-T(0)-\frac{1}{r},$$ (83)

which transforms eqn. 82 into,

$$D{y}^{\prime \prime }(x)+v{y}^{\prime }(x)-ry(x)=0.$$ (84)

This is a second-order linear ODE with constant coefficients. We seek solutions of the form $y(x)=e^{\sigma x}$, which leads to the characteristic equation

$$D{\sigma }^2+v\sigma -r=0.$$ (85)

The two roots are

$${\sigma }_{\pm }=\frac{-v\pm \sqrt{v^2+4Dr}}{2D}.$$ (86)

Note that ${\sigma }_{+}>0$ and ${\sigma }_{-}<0$ for all positive values of $v,D$, and $r$. The general solution is hence

$$y(x)=A{e}^{{\sigma }_{-}x}+B{e}^{{\sigma }_{+}x},$$ (87)

which results in

$$T(x)=A{e}^{{\sigma }_{-}x}+B{e}^{{\sigma }_{+}x}+T(0)+\frac{1}{r}.$$ (88)

As in the unbiased case, we require the mean first-passage time to remain finite as $x\to -\mathrm{\infty }$. This is again because resets will eventually return even the most eccentrically waywards seeker back to the origin and give another chance at success. Since ${\sigma }_{-}<0$, the term $e^{{\sigma }_{-}x}$ grows without bound as $x\to -\mathrm{\infty }$, whereas $e^{{\sigma }_{+}x}\to 0$ because ${\sigma }_{+}>0$. As a result, this condition imposes $A=0$ to guarantee that $T(x)$ is finite. This leaves

$$T(x)=B{e}^{{\sigma }_{+}x}+T(0)+\frac{1}{r}.$$ (89)

Imposing the absorbing boundary condition $T\left(x_T\right)=0$ (that the time to reach the target is zero once the seeker is already there) gives

$$B=-e^{-{\sigma }_{+}{x}_T}\left(T(0)+\frac{1}{r}\right),$$ (90)

and hence

$$T(x)=\left(T(0)+\frac{1}{r}\right)\left(1-e^{{\sigma }_{+}\left(x-x_T\right)}\right).$$ (91)

Setting $x=0$ and solving for $T(0)$ yields

$$T(0)=\frac{1-e^{-{\sigma }_{+}{x}_T}}{r{e}^{-{\sigma }_{+}{x}_T}}=\frac{1}{r}\left(e^{{\sigma }_{+}{x}_T}-1\right).$$ (92)

We can write this compactly as

$$T(0)=\frac{1}{r}\left(e^{x_T/\lambda }-1\right).$$ (93)

where the characteristic length scale is

$$\lambda =\frac{1}{{\sigma }_{+}}=\frac{2D}{\sqrt{v^2+4Dr}-v}.$$ (94)

Here, it is worth noting that we have taken $v>0$ throughout, corresponding to a productive drift towards the target. If instead the drift points away from the target, $v<0$, the same algebraic calculation applies with $v=-\left|v\right|$. In this case the characteristic length becomes

$${\lambda }_{-}=\frac{2D}{\sqrt{v^2+4Dr}+\left|v\right|},$$ (95)

and the mean first-passage time retains the form

$$T(0)=\frac{1}{r}\left(e^{x_T/{\lambda }_{-}}-1\right).$$ (96)

Unlike the case of drift toward the target, however, the mean first-passage time diverges in the no-reset limit $r\to 0$, since a walker drifting away from the target has a finite probability of never reaching it.

The behavior of the first-passage time eqn. 93 (and its length scale eqn. 94) is the central result. The mean first-passage time with drift and reset has exactly the same exponential form as the unbiased (diffusion with reset) case seen in eqn. 56, but with a modified length scale $\lambda$ that depends jointly on the drift velocity $v$, the diffusion coefficient $D$, and the reset rate $r$. Fig. 16 summarizes and emphasizes the commonality of an exponential form of the mean time to reach a target, with different effective length scales, across disparate search dynamics.

FIG. 16:

The typical time to reach a target adopts a common mathematical form for a surprising variety of search dynamics. (A) A simple random walk with reset shows a mean time to find a target given by eqn. 56 and is shown by the red curve. The additional complexity of a drift towards the target is given by eqn. 93 and shown by the (faster) teal curve. Both curves show that the time to reach a target can be minimized by using some finite reset rate. These curves display the evocative case of a target located at $x_T=5$, diffusion coefficient $D=1$, and velocity $v=0.05$. (B) Searches by coarse-grained drift alone (as considered in Ref.⁵⁸’s eqn. 27, taking the catastrophe rate indicated there as $k_c\equiv 1/T_R$ and recognizing $d$ as $x_T$) shows a similar algebraic form of the time to reach the target. Here, without diffusive motion, however, no finite optimal reset rate minimizes the average time to arrive at the target. These disparate cases can be understood by recognizing that the mean time to reach the target always displays an exponential dependence on the target location $x_T$, only with respect to different effective length scales $\lambda \left(T_R\right)$ for each search dynamic. The appearance of a finite optimal reset rate depends on how aggressively this effective length scale changes with the reset interval $T_R$.

Interestingly, as shown in Fig. 16(B), a quite separate “drift-only” search behavior analyzed by Ref.⁵⁸’s eqn. 27 in the context of collapsing microtubule catastrophes also shows a similar algebraic form. The microscopic dynamics are nevertheless distinct: whereas the drift direction is fixed in the present model, in the microtubule problem the direction of motion after a reset can be either toward or away from the target. Despite these differences, the resulting mean search time can be written as

$$T\left(T_R\right)=T_R\left(e^{\frac{x_T}{v{T}_R}}-1\right)$$ (97)

$$=T_R\left(e^{x_T/\lambda \left(T_R\right)}-1\right),$$ (98)

where $T_R=1/r$ and $\lambda \left(T_R\right)\equiv v{T}_R$. Because this effective length scale grows linearly with $T_R$, the mean search time decreases monotonically with $T_R$, and the optimum is therefore the no-reset limit, $T_R\to \infty$. Thus, the same mathematical form can arise from quite distinct search dynamics, with the dependence of the effective length scale on resetting determining the resulting search behavior.

In addition to the diffusive reset length ${\lambda }_D=\sqrt{\frac{D}{r}}$ defined above, it is useful to introduce the ballistic reset length

$${\lambda }_B=\frac{v}{r},$$ (99)

corresponding to the distance traveled by drift during the mean time interval between resets. The effective length scale for the case in which both diffusion and drift are in play can then be written as

$$\lambda =\frac{{\lambda }_B+\sqrt{{\lambda }_B^2+4{\lambda }_D^2}}{2}.$$ (100)

Several limits are instructive. When the drift vanishes, $v=0$, we have ${\lambda }_B=0$ and recover the unbiased result exactly,

$$\lambda ={\lambda }_D=\sqrt{\frac{D}{r}}.$$ (101)

More generally, in the diffusion-dominated regime ${\lambda }_B\ll {\lambda }_D$, or equivalently $v\ll \sqrt{Dr}$, the effective length scale reduces to the diffusive lengthscale,

$$\lambda \approx {\lambda }_D.$$ (102)

In the opposite limit, when drift dominates over diffusion, ${\lambda }_B\gg {\lambda }_D$, or equivalently $v\gg \sqrt{Dr}$, we write

$$\lambda =\frac{{\lambda }_B}{2}\left(1+\sqrt{1+4\frac{{\lambda }_D^2}{{\lambda }_B^2}}\right).$$ (103)

Expanding the square root for ${\lambda }_D/{\lambda }_B\ll 1$ gives

$$\lambda \approx {\lambda }_B+\frac{{\lambda }_D^2}{{\lambda }_B}\approx {\lambda }_B.$$ (104)

In this ballistic regime, the mean first-passage time becomes

$$T(0)\approx \frac{1}{r}\left(e^{x_T/{\lambda }_B}-1\right).$$ (105)

Here the relevant length scales are the distance traveled between resets, ${\lambda }_B=v/r$ and the distance to the target $x_T$.

In our present context, whether or not a finite reset rate can improve the search can be expressed in terms of this Péclet number (see Sec. G, “Dependence of the mean first-passage time on the reset rate $r$ in 1D with diffusion and drift,” of the Supplemental Material) through the condition

$$\mathrm{P}\mathrm{e}<2.$$ (106)

This condition has been derived previously for diffusion with drift and stochastic resetting¹⁰⁶.

This condition exposes a central takeaway: resetting is therefore beneficial only when diffusion is sufficiently important relative to drift. In this regime, trajectories can wander away from the target, and resets provide a useful way to restart the search from the origin. When $\mathrm{P}\mathrm{e}>2$, drift carries the seeker toward the target efficiently enough that resets mostly interrupt productive trajectories. The optimal reset rate then tends to zero, and the best strategy is not to reset.

We can understand where this condition comes from by looking near the no-reset limit. In the limit of very frequent resets, $T_R\to 0$, the walker is returned to the origin before it has time to reach the target, and the mean first-passage time diverges. In the opposite limit of no resetting, $T_R\to \mathrm{\infty }$, the mean first-passage time approaches the ballistic time $x_T/v$. Expanding the search time via eqn. 97 about this no-reset limit gives,

$$\tilde{T}=\frac{1}{\mathrm{P}\mathrm{e}}+\frac{1}{{\tilde{T}}_R}\left(\frac{\mathrm{P}\mathrm{e}-2}{2{\mathrm{P}\mathrm{e}}^3}\right)+\cdots ,$$ (107)

where ${\tilde{T}}_R=T_{R}D/x_T^2$. The sign of the first correction determines whether introducing a small amount of resetting improves the search. For $\mathrm{P}\mathrm{e}<2$ this correction is negative, so resetting initially lowers the mean first-passage time; since the search time diverges again as $T_R\to 0$, a finite optimum must occur. For $\mathrm{P}\mathrm{e}>2$, introducing resets instead increases the search time, and the optimum remains the no-reset limit. This core behavior is justified more completely and rigorously in Sec. G of the Supplemental Material, but the essence is already well captured by this expansion.

F. Random Seeker with an Exponential Cue

The random walk models considered so far treat the explorer as having, at most, a compass: a constant drift can point it toward the target, but gives no indication of how close the target is. Many biological seekers have access to richer information, responding to cues that grow stronger or weaker as they move through space. The next generalization is to allow the drift to vary with position.

For example, as hinted in Fig. 17, endothelial tip cells respond to the growth factor VEGF during angiogenic sprouting. The exquisite accomplishment of such a cue-guided search process is to position almost all cells in vertebrate bodies within a hundred microns of some capillary² (ch. 23, p. 1448),⁵⁴. Further examples abound. Axons are guided by the extracellular protein netrin, and bacteria such as E. coli bias their run-and-tumble motion in chemical gradients^7,32,55. These examples should not be identified with a single microscopic sensing rule. They motivate a family of models indicated schematically in Figure 18 in which cue detection is summarized by an effective drift field $v(x)$. This class of models is closely related to the literature on stochastic resetting in external potentials, where spatially varying drift fields are due to effective forces or guiding landscapes^1,107.

FIG. 17:

Many biological settings steer exploratory motion using cues of biochemicals, such as the exquisitely sensitive development of new blood vessels and capillaries across the circulatory system that respond to gradients of oxygen and associated signaling molecules such as vascular endothelial growth factor (VEGF).

FIG. 18:

Schematic of the steps available to the random seeker in the presence of a cue. A concentration gradient of an underlying cue field, such as the growth factor VEGF during angiogenic sprouting or the extracellular protein netrin guiding axon growth, results in a gradient in the effective drift velocity experienced by the random seeker.

Here, we assume that the cue acts by changing the relative rates of stepping toward or away from the target. Generalizing the continuum-limit derivation of eqn. 82 to a position-dependent bias $∆k(x)$ (and hence position-dependent drift velocity $v(x)$) gives

$$0=1+D{T}^{\prime \prime }(x)+v(x)T^{\prime }(x)-rT(x)+rT(0),$$ (108)

where $D=k{a}^2$, and $v(x)=2a\mathrm{\Delta }k(x)$. This is the “backward equation” for the first hitting time $T$ derived using the same scheme outlined in Figure 9. Because the two stepping rates are written as $k\pm ∆k(x)$, their sum remains fixed at $2k$. The cue therefore changes the local directional bias, and hence the local drift velocity, while leaving the diffusion coefficient unchanged.

As a concrete example, consider an exponential cue profile

$$c(x)=c_0{e}^{-\left(x_T-x\right)/{\ell }_c},$$ (109)

where $c_0$ is the concentration at the target and ${\ell }_c$ is the cue decay length. This profile is defined for all $x<x_T$. If the drift velocity is proportional to the local gradient of the cue, namely,

$$v\left(x\right)=\chi \frac{dc}{dx},$$ (110)

where $\chi$ is a mobility that characterizes the relation between the gradient and the drift speed, then

$$v(x)=\frac{\chi c_0}{{\ell }_c}{e}^{-\left(x_T-x\right)/{\ell }_c}$$ (111)

$$\equiv v_{\mathrm{*}}{e}^{-\left(x_T-x\right)/{\ell }_c},$$ (112)

where $v_{\mathrm{*}}=\chi c_0/{\ell }_c$ is the drift speed at the target. Larger values of the constant mobility $\chi$ encode stronger “sensing” of the underlying cue field by the random seeker. (In general, of course, the mobility $\chi$ can adopt more complex functional forms; the sweep of physics describing these couplings of velocity to cue is the majestic field of chemotaxis. Here, we sweep all of these complexities into a single effective and constant mobility.)

Armed with this expression for the velocity over space, the eqn. 108 for the mean first-passage time becomes,

$$0=1+D{T}^{\prime \prime }(x)+v_{\mathrm{*}}{e}^{-\left(x_T-x\right)/{\ell }_c}{T}^{\prime }(x)-rT(x)+rT(0).$$ (113)

The absorbing boundary condition is $T\left(x_T\right)=0$; the solution’s behavior is further specified by requiring $T(x)$ to remain finite as $x\to -\mathrm{\infty }$. (Namely, no matter what $x$ the seeker begins at, it can arrive at the target in a finite amount of time.) Indeed, when $x$ becomes a highly negative number, the exponential cue becomes negligible, and the seeker no longer experiences the drift. Hence, eqn. 108 reduces to the pure diffusion equation with resets, eqn. 43, corresponding to the case $v(x)=0$. In this far-field limit, the mean first-passage time must therefore recover the exponential behavior with respect to the position obtained previously in eqn. 55.

Eqn. 113 states how a full set of five physical parameters shape the search process under such a cue, as highlighted by Fig. 19(A). To make progress despite this profusion of parameters, it is useful to nondimensionalize positions by $x_T$ and time by $x_T^2/D$. This leads us to define the dimensionless variables

$$\tilde{x}=\frac{x}{x_T},\tilde{T}=\frac{DT}{x_T^2},$$ (114)

and the three dimensionless parameters

$$\tilde{\ell }=\frac{{\ell }_c}{x_T},\mathrm{P}\mathrm{e}=\frac{v_{\mathrm{*}}{x}_T}{D},{\tilde{T}}_R=\frac{D}{r{x}_T^2}.$$ (115)

FIG. 19:

Physical parameters and their nondimensionalized representations for search with cue. (A) Five physical parameters in the raw description of the cue problem, entering eqn. 113. (B) Three non-dimensional parameters economize and organize the study of the search problem.

Thus, the five physical parameters collapse into three dimensionless control parameters, $\tilde{\ell },\mathrm{P}\mathrm{e}$, and ${\tilde{T}}_R$, as illustrated in Fig. 19(B). The parameter $\tilde{\ell }={\ell }_c/x_T$ measures the range of the cue relative to the target distance: small $\tilde{\ell }$ corresponds to a localized cue, while large $\tilde{\ell }$ corresponds to a broad cue. The Péclet number $\mathrm{P}\mathrm{e}=v_{\mathrm{*}}{x}_T/D$ measures the strength of drift relative to diffusion: small Pe corresponds to diffusion-dominated motion, while large Pe corresponds to drift-dominated motion. The dimensionless reset interval ${\tilde{T}}_R=D/\left(r{x}_T^2\right)$ compares the mean time between resets, $1/r$, to the diffusive time $x_T^2/D$ needed to explore the target distance: small ${\tilde{T}}_R$ corresponds to frequent resetting, while large ${\tilde{T}}_R$ corresponds to rare resetting.

Using these dimensionless variables, the earlier equation on the mean first passage time, eqn. 113, can be rewritten in the cleaner and more expressive form,

$$0=1+{\tilde{T}}^{\prime \prime }(\tilde{x})+\mathrm{P}\mathrm{e}{e}^{-(1-\tilde{x})/\tilde{\ell }}{\tilde{T}}^{\prime }(\tilde{x})-\frac{1}{{\tilde{T}}_R}\tilde{T}(\tilde{x})+\frac{1}{{\tilde{T}}_R}\tilde{T}(0),$$ (116)

as derived in Sec. H.1, “Nondimensionalization of the exponential-cue backward equation,” of the Supplemental Material.

Even after nondimensionalization, the mean first-passage time depends on three parameters: $\mathrm{P}\mathrm{e},\tilde{\ell }$, and ${\tilde{T}}_R$. For each given cue, we ask which reset time gives the fastest search. We denote this optimal reset time by ${\tilde{T}}_{R,\text{cue}}^{\mathrm{*}}$ and the corresponding minimum search time by ${\tilde{T}}_{\text{cue}}^{\mathrm{*}}$. These values reflect interesting aspects of exploratory decisions important to organisms. For instance, Fig. 19 schematizes how a bee searching for an aromatic flower decides how frequently to reset its position back to the hive. Remarkably, bees follow odor, visual, and social (waggle dance) cues on lengthscales of centimeters to tens of meters^{15,24,113,120,122}. In the language of the present mathematics, how quickly they can arrive at flower targets is subject to behavioral decisions captured by ${\tilde{T}}_{R,\text{cue}}^{\mathrm{*}}$ and ${\tilde{T}}_{\text{cue}}^{\mathrm{*}}$ under different types of cue. Ultimately, asking these questions reflects sensitivity to the two remaining parameters Pe and $\tilde{\ell }$.

As we show below, two distinct dimensionless combinations organize cue-guided searches. One controls the optimized search time, while the other controls the resetting strategy. These two knobs are illustrated schematically in Fig. 21.

FIG. 21:

Two dimensionless quantities control cue-guided search. (A) ${\mathrm{P}\mathrm{e}}_{\text{cue},0}=v(0){\ell }_c/D$ controls the resetting strategy: small values give blind-like resetting, while large values make resetting rare. (B) ${\mathrm{\Theta }}_{\text{drift}}={\tilde{t}}_{\text{drift}}/{\tilde{T}}_{\text{blind}}^{\mathrm{*}}$ controls the optimized search time: small values are drift dominated, while large values remain close to the blind-search limit.

How different are the statistics and optimal timing of cue-guided search compared to blind searches without cues? Two natural limits guide the comparison. For a negligible cue, we recover blind search, for which the dimensionless optimal reset time and optimal search time are fixed constants. As derived from eqn. 57, the smallest time to arrive at a target without any cues occurs at

$${\tilde{T}}_{R,\text{blind}}^{\mathrm{*}}\simeq 0.394,{\tilde{T}}_{\text{blind}}^{\mathrm{*}}\simeq 1.54.$$ (117)

These values are useful constant reference points to understand the performance of cue-guided searches. We therefore normalize by these blind-search values. We compare the two problems through the ratios

$$\frac{{\tilde{T}}_{\mathrm{c}\mathrm{u}\mathrm{e}}^{\mathrm{*}}}{{\tilde{T}}_{\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{d}}^{\mathrm{*}}},\frac{{\tilde{T}}_{R,\text{cue}}^{\mathrm{*}}}{{\tilde{T}}_{R,\text{blind}}^{\mathrm{*}}}.$$ (118)

Both ratios approach one when the cue becomes negligible. The meanings of these metrics of performance are visualized in Fig. 20(B).

FIG. 20:

Biological importance of search strategy and resulting arrival time for search under cue. (A) A schematic example of a biological search where a decision about the optimal reset parameters for a search matters is a bee executing a one dimensional search for a flower target, but subject to deciding the optimal rate at which it returns home to a hive to rest. Bees can use long-range cues, including odors, to seek and orient towards flowers¹⁵. (B) Two features of the resulting search behavior, namely the optimal reset interval ${\tilde{T}}_R^{\mathrm{*}}$ that minimizes the average time to encounter the target, and the resulting minimum time ${\tilde{T}}^{\mathrm{*}}$ itself, are meaningful metrics of performance, and soon analyzed in Fig. 22.

Next, to appreciate the behavior of this larger physical problem, we evaluate the solution of eqn. 116 for the mean first-passage time (see Sec. H.2, “Solving the backward equation for the mean first-passage time,” of the Supplemental Material). Figure 22 shows how these two quantities vary across the ($\mathrm{P}\mathrm{e},\tilde{\ell }$) plane. When the cue is weak, both ratios remain close to one and the optimal strategy approaches that of blind search. As the cue becomes stronger and extends farther from the target, the search becomes increasingly drift dominated. The optimal search time decreases, while the optimal reset time increases. In this regime, resetting becomes less useful: once the cue reliably guides the searcher toward the target, a reset only sends it back to a less favorable position. For any finite cue strength, the optimal reset time remains finite, but in the strong-cue regime it becomes very large, corresponding to an effectively rare-reset search, as shown in Sec. H.3.b, “Large ${\tilde{T}}_R$ for fixed $\tilde{\ell }$ and $\mathrm{P}\mathrm{e}$,” of the Supplemental Material.

FIG. 22:

Effect of an exponential cue on optimized search. The searcher experiences the drift $v(x)=v_{\mathrm{*}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\left(x_T-x\right)/{\ell }_c\right]$, with $\tilde{\ell }={\ell }_c/x_T$ and $\mathrm{P}\mathrm{e}=v_{\mathrm{*}}{x}_T/D$. (A) Optimal reset time with the cue, normalized by the blind-search optimum, ${\tilde{T}}_{R,\text{cue}}^{\mathrm{*}}/{\tilde{T}}_{R,\text{blind}}^{\mathrm{*}}$. For visualization, values of $\frac{{\tilde{T}}_{R,\text{cue}}^{\mathrm{*}}}{{\tilde{T}}_{R,\text{blind}}^{\mathrm{*}}}$ exceeding 10⁴ are clipped to the same fixed dark brown color, to accommodate the divergence of the optimal reset time in the strong-cue limit. The black curve marks the informative boundary ${\mathrm{P}\mathrm{e}}_{\text{cue},0}=1$. (B) Optimized mean first-passage time, normalized by the blind-search optimum, ${\tilde{T}}_{\text{cue}}^{\mathrm{*}}/{\tilde{T}}_{\text{blind}}^{\mathrm{*}}$. The black curve marks the informative boundary ${\mathrm{\Theta }}_{\text{drift}}=1$.

In this rare-reset regime, the optimized search time approaches the deterministic time needed to follow the cue from the starting point to the target,

$${\tilde{t}}_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}={\int }_0^1\frac{d\tilde{x}}{\tilde{v}(\tilde{x})}={\int }_0^1\frac{d\tilde{x}}{\mathrm{P}\mathrm{e}{e}^{-(1-\tilde{x})/\tilde{\ell }}}=\frac{\tilde{\ell }\left(e^{1/\tilde{\ell }}-1\right)}{\mathrm{P}\mathrm{e}}.$$ (119)

This asymptotic result is derived in Sec. H.3.a, “Large Pe for fixed $\tilde{\ell }$ and ${\tilde{T}}_R$,” of the Supplemental Material, where we also show that the optimized search time approaches this drift-time limit.

To understand what controls the optimal search time itself, we compare the deterministic drift time with the optimal blind-search time and define the normalized drift time

$${\mathrm{\Theta }}_{\text{drift}}=\frac{{\tilde{t}}_{\text{drift}}}{{\tilde{T}}_{\text{blind}}^{\mathrm{*}}}=\frac{\tilde{\ell }\left(e^{1/\tilde{\ell }}-1\right)}{\mathrm{P}\mathrm{e}{\tilde{T}}_{\text{blind}}^{\mathrm{*}}},$$ (120)

where ${\tilde{T}}_{\text{blind}}^{\mathrm{*}}\simeq 1.54$ is the dimensionless optimal blind-search time defined above. When ${\mathrm{\Theta }}_{\text{drift}}\gg 1$, the optimal search time remains close to the blind-search limit. When ${\mathrm{\Theta }}_{\text{drift}}\ll 1$, the optimal search time approaches the deterministic drift time. Replotting the search-time ratio in the (${\mathrm{\Theta }}_{\text{drift}},\tilde{\ell }$) plane produces nearly vertical contours, showing that much of its dependence on $\mathrm{P}\mathrm{e}$ and $\tilde{\ell }$ collapses onto ${\mathrm{\Theta }}_{\text{drift}}$, as shown in Sec. H.3.a of the Supplemental Material.

A different combination of parameters controls the resetting strategy. The relevant question is whether, at the starting position, drift is strong enough to carry the searcher through one cue length before diffusion moves it away. A central result of our analysis is that this behavior is organized by the Péclet number evaluated at the scale of the cue,

$${\mathrm{P}\mathrm{e}}_{\text{cue},0}=\frac{v(0){\ell }_c}{D}=\mathrm{P}\mathrm{e}\tilde{\ell }{e}^{-1/\tilde{\ell }}.$$ (121)

This single combination of cue strength and cue length controls the transition from blind-like resetting to a regime in which resetting becomes increasingly rare.

When ${\mathrm{P}\mathrm{e}}_{\text{cue},0}\ll 1$, diffusion dominates over one cue length at the starting point, and the optimal reset time remains close to that of blind search. When ${\mathrm{P}\mathrm{e}}_{\text{cue},0}\gg 1$, the cue already provides reliable directional motion from the starting position, so resetting becomes rare and the optimal reset time becomes very large. For broad cues, $\tilde{\ell }\gtrsim 1$, curves of constant ${\mathrm{P}\mathrm{e}}_{\text{cue},0}$ approximately align with the transition in Fig. 22(A), showing that the reset-time dependence on $\mathrm{P}\mathrm{e}$ and $\tilde{\ell }$ largely collapses onto ${\mathrm{P}\mathrm{e}}_{\text{cue},0}$. The localized-cue regime discussed below is the main exception.

Thus, the cue presents two distinct knobs. The normalized drift time ${\mathrm{\Theta }}_{\text{drift}}$ controls the optimal time to reach the target, whereas ${\mathrm{P}\mathrm{e}}_{\text{cue},0}$ controls the optimal resetting strategy. Importantly, these are independent dimensionless coordinates: the transition in search time and the transition in reset frequency need not occur together. These two regimes are also visible directly in Fig. 22. In panel (A), ${\mathrm{P}\mathrm{e}}_{\mathrm{c}\mathrm{u}\mathrm{e},0}\ll 1$ corresponds to blind-like resetting, whereas ${\mathrm{P}\mathrm{e}}_{\text{cue},0}\gtrsim 1$ marks the onset of increasingly rare resetting. Similarly, in panel (B), ${\mathrm{\Theta }}_{\text{drift}}\gg 1$ corresponds to blind-like search times, while ${\mathrm{\Theta }}_{\text{drift}}\ll 1$ corresponds to the drift-dominated regime.

Finally, the purple region marked in Fig. 22(A) reveals a less intuitive regime: for sufficiently localized cues, the optimal reset time can be smaller than in blind search. In this regime, resetting gives the searcher repeated opportunities to reach and exploit the localized cue. Thus, while broad and strong cues suppress resetting, localized cues can instead enhance its usefulness. The onset of this regime and the corresponding threshold cue length are analyzed in Sec. H.4, “Existence of a minimum for the optimal reset time,” of the Supplemental Material.

IV. RANDOM WALK WITH RESETS TO CHECKPOINTS

The reset models considered in the previous section share a common feature: every failed attempt returns the seeker to the same starting point after reset. In contrast, however, many natural exploratory processes consolidate partial progress. Progress that has already been consolidated is not lost, and only the most recent exploratory episode must be repeated as indicated schematically in Figs. 23 and 24. These are the biological analog of a mountain summit expedition in which one stops for the night at a series of refuges: if the weather forces a retreat the following morning, one returns to the last refuge rather than all the way to the base camp. The mathematics of such a climb is the subject of this section.

FIG. 23:

Random walk with resets to checkpoints. (A) Resets globally to the beginning of an exploratory journey. (B) Partial resets to the closest checkpoints (shown as orange camp tents), thus saving intermediate progress along a journey.

FIG. 24:

The seeker that resets to checkpoints (shown as orange camp tents). Once the seeker arrives at the intermediate treasure at checkpoint 1, it commences a new search to arrive at checkpoint 2. The process repeats until arriving at the giant pot of gold at the target.

Biological implementations of such checkpoints reflect rich and potentially-useful behaviors; Ravasio and colleagues give a creative contemporary synthesis¹⁰⁴. Other examples include a microtubule that has laid down a GTP island mid-filament, which does not collapse all the way to the centrosome when catastrophe strikes; instead, it retreats only to the most recent island. An axon that has already reached a waypoint does not retract to the cell body when its growth cone stalls. Instead, it resets only to the last stable branch point. A B cell that has already achieved an intermediate affinity does not lose all prior progress when a hypermutation fails: it reverts only to the last positively selected clone.

We will show that intermediate checkpoints reduce the mean first-passage time from a quantity that grows exponentially in the target distance to one that grows only linearly, and that the optimal checkpoint spacing is simply the characteristic exploration length ${\lambda }_D$ that appeared in the single-reset problem. These results are exact, and they follow from a single key observation: once a checkpoint has been reached, the future problem is statistically identical to a fresh single-segment problem starting from the origin. The segments are therefore independent, and everything follows from that independence. Previewing our approach, we consider the cases shown in Fig. 25 in turn.

FIG. 25:

Different implementations of checkpoint architectures. (A) Global reset back to the origin without any intermediate checkpoints. (B) A single intermediate checkpoint receiving partial resets once the seeker has progressed past it, instead of being subject to resets all the way back to the origin. (C) Multiple, uniformly-spaced checkpoints between the start and the target location. (D) The limit of many checkpoints.

A. The Single Intermediate Checkpoint

We begin with the simplest case that illustrates the principle: one intermediate checkpoint placed at the midpoint of the journey as shown in Fig. 25(B). The target is at position $x_T$ and the checkpoint is at $x_1=x_T/2$. The seeker starts at the origin. While the seeker is between the origin and $x_1$, any reset returns it to the origin. When the seeker reaches $x_1$ for the first time, that position is consolidated and becomes the new reset point. The seeker then attempts to reach $x_T$, with any subsequent reset returning it to $x_1$ rather than to the origin. We work with diffusion coefficient $D$ and reset rate $r$, and we recall from the previous section that the mean first-passage time for a single segment of length $L$ with resets to the start of that segment is

$$⟨T{⟩}_L=\frac{1}{r}\left(e^{L/{\lambda }_D}-1\right),{\lambda }_D=\sqrt{\frac{D}{r}}.$$ (122)

As noted above, the key observation is that the two exploratory segments are independent. Once the seeker has reached $x_1$, the future evolution depends only on the distance to the next target, $x_T-x_1=x_T/2$, and not on how long it took to reach $x_1$ or how many resets occurred along the way. This is simply the Markov property: the seeker starts fresh from $x_1$. Hence, the total first-passage time is the sum of two independent first passage times and is given by,

$$T_{\text{total}}=T_0+T_1,$$ (123)

where $T_0$ is the time to cross the first segment from 0 to $x_1=x_T/2$ (with resets to 0), and $T_1$ is the time to cross the second segment from $x_1$ to $x_T$ with resets to $x_1$. Since both segments have the same length $L=x_T/2$, both $T_0$ and $T_1$ have the same statistical distribution, and their means are equal. The mean total time is thus given by

$$⟨T_{\text{one}\text{checkpoint}}⟩=⟨T_0⟩+⟨T_1⟩=\frac{2}{r}\left(e^{x_T/2{\lambda }_D}-1\right).$$ (124)

Compare this to the mean first-passage time without any checkpoint, which from eqn. 122 with $L=x_T$ is

$$⟨T_{\text{no}\text{checkpoint}}⟩=\frac{1}{r}\left(e^{x_T/{\lambda }_D}-1\right).$$ (125)

The ratio of these two times is

$$\frac{⟨T_{\text{one}\text{checkpoint}}⟩}{⟨T_{\text{no}\text{checkpoint}}⟩}=\frac{2\left(e^{x_T/2{\lambda }_D}-1\right)}{e^{x_T/{\lambda }_D}-1}.$$ (126)

To understand what this ratio means, we introduce the dimensionless distance $\mathrm{\Lambda }=x_T/{\lambda }_D$ permitting us to write

$$\frac{⟨T_{\text{one}\text{checkpoint}}⟩}{⟨T_{\text{no}\text{checkpoint}}⟩}=\frac{2\left(e^{\mathrm{\Lambda }/2}-1\right)}{e^{\mathrm{\Lambda }}-1}.$$ (127)

For large $\mathrm{\Lambda }$, the denominator grows as $e^{\mathrm{\Lambda }}$ and the numerator grows only as $2e^{\mathrm{\Lambda }/2}$, so the ratio decays as

$$\frac{⟨T_{\text{one}\text{checkpoint}}⟩}{⟨T_{\text{no}\text{checkpoint}}⟩}\approx 2e^{-\mathrm{\Lambda }/2}=2e^{-x_T/2{\lambda }_D},x_T\gg {\lambda }_D.$$ (128)

One intermediate checkpoint reduces the mean first-passage time by a factor that is exponentially small in $x_T/2{\lambda }_D$. This is often the biologically relevant regime, since the distance to the target can be much larger than the typical distance explored between reset events. For a target that is ten exploration lengths away, a single checkpoint at the midpoint reduces the mean search time by a factor of roughly $2e^{-5}\approx 0.013$, a reduction of almost two orders of magnitude. In the opposite limit, $x_T\ll {\lambda }_D$, checkpoints still provide a speedup, but the exponential behavior of eqn. 128 no longer applies. Instead, the improvement becomes weaker and scales linearly with $x_T/{\lambda }_D$, as shown in Sec. I, “The Near-Target Limit for a Single Checkpoint,” of the Supplemental Material.

Because $T_0$ and $T_1$ are independent, the variance of the total time is simply the sum of the variances of the two segment times given by

$$\mathrm{V}\mathrm{a}\mathrm{r}\left[T_{\text{one}\text{checkpoint}}\right]=\mathrm{V}\mathrm{a}\mathrm{r}\left[T_0\right]+\mathrm{V}\mathrm{a}\mathrm{r}\left[T_1\right]=2\mathrm{V}\mathrm{a}\mathrm{r}\left[T_k\right],$$ (129)

where $\mathrm{V}\mathrm{a}\mathrm{r}\left[T_k\right]$ is the variance of a single segment of length $x_T/2$. To measure noise per unit signal, we calculate the coefficient of variation, given by the ratio of the standard deviation to the mean is given by

$$\frac{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}\left[T_{\text{one}\text{checkpoint}}\right]}}{⟨T_{\text{one}\text{checkpoint}}⟩}=\frac{\sqrt{2\mathrm{V}\mathrm{a}\mathrm{r}\left[T_k\right]}}{2⟨T_k⟩}$$ (130)

$$=\frac{1}{\sqrt{2}}\frac{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}\left[T_k\right]}}{⟨T_k⟩}.$$ (131)

The coefficient of variation of the full journey is reduced by a factor of $1/\sqrt{2}$ relative to that of a single segment. The checkpoint not only reduces the mean search time exponentially but also makes the arrival time more reproducible, which might matter biologically if we hypothesize that developmental and cell-division processes operate on tight schedules.

B. The General Case: $n$ Uniform Checkpoints

We now generalize to $n$ uniformly spaced checkpoints, dividing the full distance $x_T$ into $n+1$ segments of equal length

$$L=\frac{x_T}{n+1},$$ (132)

as shown in Fig. 25(C). The checkpoints are at positions $c_k=kL$ for $k=0,1,\dots ,n$, with $c_0=0$ the origin and $c_{n+1}=x_T$ the target. When the seeker is in the interval $\left(c_k,c_{k+1}\right)$, any reset returns it to $c_k$. Upon reaching $c_{k+1}$, that position is consolidated and becomes the new reset point.

By the same Markov argument as before, the time to traverse each segment is independent of all other segments. The total first-passage time is

$$T_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\sum _{k=0}^n{T}_k,$$ (133)

where each $T_k$ has the same distribution as the first-passage time for a segment of length $L$ with resets to the start of that segment. Taking the mean and using eqn. 122. we find

$$⟨T_n⟩=(n+1)\cdot \frac{1}{r}\left(e^{L/{\lambda }_D}-1\right)=\frac{n+1}{r}\left(e^{x_T/(n+1){\lambda }_D}-1\right).$$ (134)

The no-checkpoint result is recovered by setting $n=0$, giving $⟨T_0⟩=r^{-1}\left(e^{x_T/{\lambda }_D}-1\right)$ as expected. The variance is equally simple and given by

$$\mathrm{V}\mathrm{a}\mathrm{r}\left[T_n\right]=(n+1)\mathrm{V}\mathrm{a}\mathrm{r}\left[T_k\right],$$ (135)

meaning the coefficient of variation (the mean divided by the standard deviation) scales as $1/\sqrt{n+1}$, decreasing with the number of checkpoints. Furthermore, since the total first-passage time is the sum of $n+1$ independent segment times, the central limit theorem implies that, for large $n$, the distribution of $T_n$ approaches a Gaussian. Thus, increasing the number of checkpoints not only decreases the mean search time but also makes the completion time progressively more reproducible.

C. The Limit of Many Checkpoints

It is instructive to ask what happens as $n\gg x_T/{\lambda }_D$ with $x_T$ fixed. As the segment length $L=x_T/(n+1)\to 0$, we expand the exponential in eqn. 134 to first order,

$$e^{L/{\lambda }_D}=1+\frac{L}{{\lambda }_D}+O\left(\frac{L^2}{{\ell }^{\mathrm{*}2}}\right),$$ (136)

and substitute to obtain

$$⟨T_n⟩\approx \frac{n+1}{r}\cdot \frac{x_T}{(n+1){\lambda }_D}=\frac{x_T}{r{\lambda }_D}=\frac{x_T}{\sqrt{Dr}}.$$ (137)

In this limit the mean first-passage time grows only linearly in $x_T$, in stark contrast to the exponential growth of the no-checkpoint result. Dense checkpoints completely eliminate the exponential cost of long-distance search, replacing it with a cost proportional to the distance divided by the geometric mean of the diffusion coefficient and the reset rate.

D. The Optimal Checkpoint Spacing

Given fixed $x_T,D$, and $r$, how many checkpoints minimize the mean first-passage time? We treat $n+1$ as a continuous variable $m$ and minimize $⟨T⟩=m{r}^{-1}\left(e^{x_T/m{\lambda }_D}-1\right)$ with respect to $m$. Differentiating and setting to zero gives the condition

$$e^{x_T/m\mathrm{*}{\lambda }_D}\left(1-\frac{x_T}{m\mathrm{*}{\lambda }_D}\right)=1,$$ (138)

which must be solved numerically in general. In the biologically relevant limit $x_T\gg {\lambda }_D$, the solution satisfies

$$\frac{x_T}{m\mathrm{*}}\approx {\lambda }_D,$$ (139)

so the optimal segment length is approximately ${\lambda }_D$. The optimal strategy places one checkpoint per characteristic exploration length. Fewer checkpoints and the exponential cost of crossing long segments dominates. More checkpoints and the overhead of repeatedly consolidating short segments dominates. The optimal spacing is set entirely by the physics of the reset process.

E. Summary of Scaling Results

Table II summarizes these key results for the mean first-passage time as a function of the number of checkpoints, writing $\mathrm{\Lambda }\equiv x_T/{\lambda }_D$ for the target’s distance in units of the diffusive exploration distance.

TABLE II:

Summary of scaling results for the mean first-passage time to a target located at $x_T$ as a function of the number of checkpoints $n$, where $\mathrm{\Lambda }\equiv \frac{x_T}{D}$.

	case	$⟨T⟩$	scaling with $x_T$
	no checkpoints	$r^{-1}\left(e^{\mathrm{\Lambda }}-1\right)$	exponential
	one checkpoint at midpoint	$2r^{-1}\left(e^{\mathrm{\Lambda }/2}-1\right)$	sub-exponential
	n uniform checkpoints	$(n+1)r^{-1}\left(e^{\mathrm{\Lambda }/(n+1)}-1\right)$	sub-exponential
	dense checkpoints $(n\gg \mathrm{\Lambda })$	$x_T/\sqrt{Dr}$	linear

The progression from exponential to linear scaling as the number of checkpoints increases is the main message of this section. The biological implication is direct: a system that can consolidate partial progress is exponentially more efficient at long-distance search than one that must start over completely after every failure. The checkpoint spacing that achieves this efficiency is not a fine-tuned parameter: it is simply the characteristic exploration length ${\lambda }_D=\sqrt{D/r}$, which is already determined by the diffusion and reset properties of the system.

A biological example that is conceptually similar to the random seeker with intermediate checkpoints is the translocation of polymers across membranes. This mechanism is often described as a translocation ratchet^68,85,96,118. Concretely, the problem of DNA viruses ejecting their DNA into bacterial hosts is one in which tens of thousands of basepairs have to get from within the viral capsid to the cellular interior. The checkpoint idea is relevant here because the binding of proteins such as RNA polymerase to the translocated viral genome can inhibit backward motion of the DNA, thereby making later stages of translocation less likely to undo earlier progress^30,42,46,144. More generally, similar ideas arise whenever a multistep process passes through relatively stable intermediates that make reverse transitions less likely.

V. SEVERAL SEARCHERS AND SEVERAL TARGETS

1. Parallel search with $N$ independent agents

Many biological search processes do not rely on a single seeker. Instead, multiple agents explore the same environment in parallel. Examples pervade biology, as hinted at in Fig. 26; these include many microtubules searching for chromosomes⁴¹, many transcription factors searching out their binding sites on DNA¹¹⁹, or many cellular protrusions exploring the extracellular environment¹¹. In such situations, the relevant timescale is no longer the first-passage time of a single searcher, but rather the time at which the first member of the population reaches the target.

FIG. 26:

Many biological functions are organized by many searchers pursuing many targets. (A) The canonical example of dynamically-unstable microtubules exploring space while setting up the kinetochore is a manifestly parallel search problem. (B) Multiple filopodia in the growth cones of neurons extend and contract in various directions during synaptogenesis, ultimately underlying the exquisite specificity of neuronal connections. (C) At a much more macroscopic scale, the first root (of many searching) to encounter water or nutrients often sets the destiny of plants. (D) Slime molds such as the charismatic Physarum polycephalum use many individual extensions of their plasmodia to perform parallel searches for food in their environments.

Consider $N$ independent searchers evolving under identical dynamics and starting from the same initial condition. Denoting by ${\tau }_i$ the first-passage time of the $i$th searcher to a target located at $x_T$, we define the earliest-arrival time

$${\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{m}\mathrm{i}\mathrm{n}\left({\tau }_1,\dots ,{\tau }_N\right).$$ (140)

This quantity describes the time of first successful encounter among the population of searchers. We first study the earliest arrival. Then, in §V2, we generalize this construction to appreciate the timing of at least multiple arrivals (in particular, the $M$-th arrival, with the last arrival recovered by setting $M=N$).

We first introduce the survival probability $S(t)$ of a single searcher, namely the probability that a single searcher has not yet reached the target by time $t$ starting at the origin, or

$$S(t)=\mathbb{P}\left({\tau }_i>t\right),$$ (141)

where $\mathbb{P}$ denotes probability. This $S(t)$ is illustrated in Fig. 27(A)–(B). The corresponding survival probability for a single searcher is shown in Fig. 28(A), illustrated for the case of a constant drift random walk under reset in one dimension. The event ${\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}>t$ means that none of the $N$ searchers has reached the target by time $t$. Since each searcher survives independently with probability $S(t)$, the probability that all $N$ searchers survive up to time $t$ is

$$\mathbb{P}\left({\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}>t\right)=S(t{)}^N.$$ (142)

FIG. 27:

Survival probabilities quantify the likelihood that a specified success criterion has not yet been met by time $t$. (A) A representative single-searcher trajectory that has not yet reached the target by time $t$, corresponding to the event $\tau >t$. Over an ensemble of trajectories, the fraction satisfying this condition is the survival probability $S(t)=\mathbb{P}(\tau >t)$. (B) The area under a survival-probability curve gives the corresponding mean arrival time, $⟨T⟩$. (C) A representative realization of $N$ independent searchers in which none has reached the target by time $t$. The probability of this event is the collective survival probability $S^{\text{tot}}(t)=\mathbb{P}\left({\tau }_{\text{min}}>t\right)=S(t{)}^N$. (D) A representative realization in which fewer than $M$ of the $N$ searchers have reached the target by time $t$. The probability of this event defines the $M$th-arrival survival probability $S_{M:N}(t)$.

FIG. 28:

Survival probabilities and arrival statistics for the 1D drift–diffusion–reset model. (A) Single-searcher survival probability $S(t)$, giving the probability an individual searcher has not yet reached the target by time $t$. The mean time to reach the target is indicated by the black vertical line. (B) The collective survival probability $S(t{)}^N$ for $N=10$ independent searchers, showing a reduction in the mean time for at least one of the $N$ searchers to reach the target. (C) The survival probability $S_{M:N}(t)$ for $N=10$ independent searchers with $M=3$ required successes, with the mean $M$th-arrival time indicated by the black vertical line. Parameters used in these simulations are $=0.5{\mu \mathrm{m}}^2/\mathrm{m}\mathrm{i}\mathrm{n},v=1\mu \mathrm{m}/\mathrm{m}\mathrm{i}\mathrm{n},r=1{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$, and $x_T=2\mu \mathrm{m}$.

This collective survival probability is illustrated schematically in Fig. 27(B), and the corresponding survival curve for $N=10$ independent searchers is shown in Fig. 28(B).

The mean earliest-arrival time is

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}=⟨{\tau }_{\mathrm{m}\mathrm{i}\mathrm{n}}⟩={\int }_0^{\mathrm{\infty }}S(t{)}^{N}dt,$$ (143)

as derived in Sec. J.1, “Mean earliest-arrival time from the collective survival probability,” of the Supplemental Material and illustrated geometrically in Fig. 27(C). This reduces the parallel-search problem to determining the survival probability $S(t)$ of a single searcher. These relations are standard results for the extreme first-passage time of independent searchers^59,132,142.

To determine the survival probability of a searcher initialized at the origin, we introduce the more general position-dependent survival probability $S(x,t)$, defined as the probability that a searcher initialized at position $x$ at time $t=0$ has not yet reached the target at $x_T$ by time $t$. The single-searcher survival probability introduced above is then

$$S(t)=S(0,t).$$ (144)

To determine $S(x,t)$, we use the same backward-equation approach introduced previously for the mean first-passage time, now applying the local averaging argument to survival probabilities rather than times. The derivation is given in Sec. J.2 of the Supplemental Material. For diffusion with constant drift $v$ and Poissonian resetting to the origin at rate $r$, this gives⁹²

$$\frac{\partial S(x,t)}{\partial t}=v\frac{\partial S(x,t)}{\partial x}+D\frac{{\partial }^{2}S(x,t)}{\partial x^2}-rS(x,t)+rS(0,t),$$ (145)

with the absorbing boundary condition

$$S\left(x_T,t\right)=0,$$ (146)

and initial condition

$$S(x,0)=1,x<x_T.$$ (147)

We now turn to the large-$N$ asymptotic behavior. When many searchers explore in parallel under diffusion and drift, what does the earliest-arrival time approach as $N\to \infty$? Once this asymptotic limit is established, can we determine a threshold number of searchers beyond which the system is effectively in this large-$N$ regime?

For large $N$, the first successful searcher arrives increasingly early. To understand which part of the single-searcher dynamics controls this limit, let

$$F(t)=1-S(t)$$ (148)

denote the probability that a single searcher has reached the target by time $t$. At sufficiently early times, $F(t)\ll 1$. Among $N$ independent searchers, the expected number that have reached the target by time $t$ is $NF(t)$. If this number is much smaller than one, it is unlikely that any searcher has yet succeeded; if it is much larger than one, at least one successful trajectory is overwhelmingly likely. Intuitively, the first arrival therefore occurs when the expected number of successful searchers becomes of order one,

$$NF(t)=O(1).$$ (149)

A more formal derivation of this criterion is given in Sec. J.3, “Characteristic earliest-arrival time and relation to the mean,” of the Supplemental Material.

This observation is useful because the large-$N$ problem is therefore controlled by exceptionally fast single-searcher trajectories. In this short-time regime, the first-passage probability has the asymptotic form

$$F(t)\sim \frac{2\sqrt{Dt}}{x_T\sqrt{\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{x_T^2}{4Dt}+\frac{v{x}_T}{2D}\right],t\to 0^{+},$$ (150)

as derived in Sec. J.4, “Short-time and large-$N$ asymptotics of the earliest-arrival time,” of the Supplemental Material. The full order-statistics calculation, including the relation between a characteristic first-arrival time and the mean $T_{\text{min}}$, is also given there.

Using this short-time form, the mean earliest-arrival time of a single searcher among $N$ admits the large-$N$ expansion (in increasing powers of $1/\mathrm{l}\mathrm{n}N$) of

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}=T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(0)}+T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(1)}+T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(2)}+\cdots ,$$ (151)

with

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(0)}=\frac{x_T^2}{4D\mathrm{l}\mathrm{n}N},$$ (152)

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(1)}=\frac{x_T^2}{8D}\frac{\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{n}N}{(\mathrm{l}\mathrm{n}N{)}^2},$$ (153)

and

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(2)}=\frac{x_T^2}{4D(\mathrm{l}\mathrm{n}N{)}^2}\left[\frac{1}{2}\mathrm{l}\mathrm{n}\pi -{\gamma }_E-\frac{v{x}_T}{2D}\right].$$ (154)

The leading behavior is therefore

$$T_{\mathrm{m}\mathrm{i}\mathrm{n}}\sim \frac{x_T^2}{4D\mathrm{l}\mathrm{n}N},N\to \mathrm{\infty }.$$ (155)

This inverse-logarithmic scaling is a classical result of extreme first-passage statistics^{59,132,141,142}. Parallel search therefore accelerates the first encounter only logarithmically with the number of searchers. Because $T_{\text{min}}\propto 1/\mathrm{l}\mathrm{n}N$, halving $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$ requires increasing the population from $N$ to $N^2$.

The structure of this expansion is illustrated in Fig. 29 The leading term $T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(0)}$ is independent of drift⁵⁹. The first correction, $T_{\text{min}}^{(1)}$, is also drift independent and decays only logarithmically with $N$, producing a slow approach to the leading large-$N$ behavior. Drift first appears in $T_{\text{min}}^{(2)}$. Consequently, although all fixed drift velocities ultimately approach the same leading $1/\mathrm{l}\mathrm{n}N$ scaling, their earliest-arrival times can remain substantially different over a broad range of finite $N$.

FIG. 29:

Mean earliest-arrival time as a function of the number of parallel searchers $N$ for diffusion without drift ($v=0$) and with drift toward the target ($v=10$). Solid curves show the exact mean $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$, while the corresponding corrected asymptotic curves include $T_{\text{min}}^{(0)}+T_{\text{min}}^{(1)}+T_{\text{min}}^{(2)}$. The common leading-order result $T_{\mathrm{m}\mathrm{i}\mathrm{n}}^{(0)}=x_T^2/(4D\mathrm{l}\mathrm{n}N)$, which applies for any drift, is shown separately as a brown dashed curve.

The physical origin of this universal behavior can be understood from eqn. 150. Because the first arrival occurs when $NF(t)=O(1)$ according to the heuristic eqn. 149, increasing $N$ selects progressively earlier arrival times. At these short times, the diffusive factor $\mathrm{e}\mathrm{x}\mathrm{p}\left[-x_T^2/(4Dt)\right]$ dominates the first-passage probability. Accordingly, the requirement $NF(t)\approx N\mathrm{e}\mathrm{x}\mathrm{p}\left[-x_T^2/(4Dt)\right]=$ constant suggests that among a sufficiently large population, one searcher realizes the exceptionally rare and fast diffusive trajectory required to reach the target, giving rise to the universal $1/\mathrm{l}\mathrm{n}N$ behavior in eqn. 155.

Drift affects these rare trajectories differently. Its contribution to the short-time first-passage probability, $\mathrm{e}\mathrm{x}\mathrm{p}\left[v{x}_T/(2D)\right]$, remains finite as $t\to 0$. Drift therefore changes the probability of observing a fast target-reaching trajectory, but does not change the dominant short-time scaling that determines $T_{\text{min}}^{(0)}$. This distinction is visible in Fig. 29; for finite $N$, drift can substantially accelerate the first arrival, while at sufficiently large $N$ rare diffusive events dominate the leading behavior. Importantly, this asymptotic regime may require extremely large populations, so drift can remain quantitatively important over the range of searcher numbers relevant to biological systems.

Given the parameters of a particular search process, we can now ask how many parallel searchers are “enough.” A complementary version of this question has been considered by asking how the optimal reset rate changes with the number of searchers⁹. Here, instead, we introduce a criterion based on the timescale of the winning trajectory: how many searchers are required for the first successful trajectory to occur before resetting typically has time to act? Since the mean reset time is $1/r$, we define a characteristic population $N_c(\epsilon )$ by requiring the first-arrival time to be a small fraction $\epsilon$ of this timescale. Using the large-$N$ condition $NF(t)=O(1)$ at $t=\epsilon /r$ gives

$$N_c(\epsilon )\sim \sqrt{\frac{\pi \rho }{\epsilon }}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{\rho }{\epsilon }-\frac{v{x}_T}{2D}\right],\rho =\frac{r{x}_T^2}{4D},$$ (156)

with the derivation given in Sec. J.5, “Approach to the reset-independent large-$N$ regime,” of the Supplemental Material This characteristic population size is illustrated in Fig. 30.

FIG. 30:

Approach to the reset-independent regime as the number of parallel searchers increases. The mean earliest-arrival time $T_{\mathrm{m}\mathrm{i}\mathrm{n}}$ decreases with $N$, and the characteristic population size $N_c$ is defined by $T_{\mathrm{m}\mathrm{i}\mathrm{n}}=\epsilon /r$. The numerical value of $N_c$ is compared with the large-$N$ asymptotic prediction from eqn. 156.

For $N<N_c$, resetting can still influence the winning trajectory, whereas for $N>N_c$ the first arrival typically occurs before a reset has time to act. The figure also compares the characteristic population size obtained from the full numerical calculation with the large-$N$ asymptotic prediction in eqn. 156.

A second notion of “enough” arises when each additional searcher carries a cost⁷⁵. We consider an effective cost

$$𝒞(N)=T_{\mathrm{m}\mathrm{i}\mathrm{n}}(N)+\gamma N\simeq \frac{x_T^2}{4D\mathrm{l}\mathrm{n}N}+\gamma N,$$ (157)

where $\gamma$ is the cost associated with each additional searcher, measured in units of time, and could represent, for example, the time required to create or deploy a searcher. More generally, the linear cost $\gamma N$ could be replaced by a population-dependent cost $G(N)$. For the linear case considered here, the diminishing gain from parallelization competes with the cost of additional searchers, selecting a finite optimal population $N\mathrm{*}$, whose derivation is given in Sec. J.6, “Optimal number of searchers in the presence of a resource cost,” of the Supplemental Material.

The comparison between $N\mathrm{*}$ and $N_c$ then distinguishes two regimes. If $N\mathrm{*}<N_c$, adding searchers becomes too costly before the system reaches the reset-independent regime, and resetting still influences the winning trajectory at the resource optimum. If $N\mathrm{*}>N_c$, the system can afford enough searchers that the first arrival typically outruns resetting before the resource optimum is reached.

2. Multiple searchers and multiple required successes

So far, we have focused on the first success among $N$ parallel searchers. We now consider a different completion criterion in which a single target must be reached by multiple searchers before the process is considered complete. This differs from searching for multiple distinct targets, although both involve multiple successful search events.

One suggestive biological setting is kinetochore capture during cell division. Multiple dynamic microtubules can search for and attach to the same kinetochore, with the number of microtubule attachments supported by a single kinetochore varying substantially across organisms: a budding-yeast kinetochore binds a single microtubule, fission-yeast kinetochores bind a few, whereas mammalian kinetochores typically bind on the order of tens of microtubules^53,66,134. These attachment numbers do not imply that cell division waits for a prescribed $M$th attachment before proceeding. Rather, they show that a single biological target can receive multiple successful exploratory arrivals, motivating the more general question of how the time to the $M$th success depends on the number of parallel searchers.

Figure 31 illustrates more generally how biological processes can distinguish between “OR”-like logic, in which a single successful encounter is sufficient, and threshold or “AND”-like logic, in which several successful encounters are required. Search processes with stochastic resetting have also been extended to situations in which a searcher can encounter several possible targets¹⁴. Motivated by this broader logic, we generalize the earliest-arrival problem by asking for the time at which $M$ of the $N$ searchers have reached the target.

FIG. 31:

Multiple targets for exploratory dynamics can sculpt biological activity. (A) Some biological trajectories terminate and unlock downstream activity when the first of any searchers encounters a target, effectively implementing “OR” logic among the search trajectories. For instance, an egg can be fertilized by any of arriving sperm and the first arrival among a group is often salient. A sea urchin egg being fertilized is a particularly direct example of such a dynamic. (In reality, fertilizations do not occur deterministically once sperm encounter eggs but rather at some finite probability estimable by measurements and modeling^61,62, but high fertilization rates motivate the abstraction depicted here.) (B) Other biological contexts do not permit just one encountered target to elicit downstream consequences. Instead, they can require at least several targets to be met before terminating and prompting downstream activity, effectively implementing “AND” logic. For instance, the division of the mitotic spindle is under tight error regulation to effectively execute division only when exactly twenty three chromosomes are captured³⁹, an example of aspired AND logic.

Denoting by ${\tau }_i$ the first-passage time of the $i$th searcher, we define ${\tau }_{M:N}$ as the $M$th smallest value among (${\tau }_1,\dots ,{\tau }_N$). Equivalently, ${\tau }_{M:N}$ is the first time at which $M$ distinct searchers have successfully reached the target.

By time $t$, a single searcher has arrived with probability $1-S(t)$ and remains unsuccessful with probability $S(t)$. Among the $N$ independent searchers, the number of successful arrivals by time $t$ is binomially distributed, with “success” corresponding to arrival by time $t$ and success probability $1-S(t)$. For the $M$th arrival not to have occurred yet, there may have been no successful arrivals, exactly one successful arrival, exactly two, and so on, up to at most $M-1$ successful arrivals. Summing over these possibilities gives the survival probability of the $M$th arrival^132,142,

$$S_{M:N}(t)=\mathbb{P}\left({\tau }_{M:N}>t\right)=\sum _{i=0}^{M-1}\left(\genfrac{}{}{0pt}{}{N}{i}\right)[1-S(t){]}^{i}S(t{)}^{N-i}.$$ (158)

Figure 27(D) illustrates this construction: the $M$th arrival has not yet occurred at time $t$ whenever fewer than $M$ searchers have reached the target.

The corresponding mean completion time is

$$T_{M:N}=⟨{\tau }_{M:N}⟩={\int }_0^{\mathrm{\infty }}{S}_{M:N}\left(t\right)dt.$$ (159)

Figure 28(C) shows the corresponding $M$th-arrival survival probability and mean completion time for $N=10$ independent searchers with $M=3$ required successes.

The two limiting cases are particularly informative. For $M=1$, only one searcher needs to succeed and we recover the earliest-arrival problem, $T_{1:N}=T_{\mathrm{m}\mathrm{i}\mathrm{n}}$. At the opposite extreme, $M=N$ requires every searcher to succeed, so completion is set by the slowest member of the population. In this case,

$$\mathbb{P}\left({\tau }_{N:N}>t\right)=1-[1-S(t){]}^N,$$ (160)

since the last arrival has not yet occurred whenever at least one searcher remains unsuccessful.

We first consider the regime $N\gg M$, with $M$ fixed. By time $t$, the expected number of successful searchers is $NF(t)$, where $F(t)=1-S(t)$. If $NF(t)\ll M$, obtaining $M$ successes is unlikely, whereas once $NF(t)\gg M$, the required number of arrivals has typically been exceeded. The $M$th arrival therefore occures in the regime

$$NF(t)=O(M),$$ (161)

or equivalently $F(t)=O(M/N)$. For $N\gg M$, this remains in the short-time tail of the single-searcher first-passage distribution.

Using the short-time form of $F(t)$ derived above, the mean $M$th-arrival time obeys

$$T_{M:N}\sim \frac{x_T^2}{4D\mathrm{l}\mathrm{n}(N/M)},N\to \mathrm{\infty }\text{with}\text{fixed}M.$$ (162)

As shown in Fig. 32, this large-$N$ prediction captures the behavior of the mean completion time for fixed $M$, while also highlighting the contrasting behavior of the last-arrival case $M=N$. A derivation of eqn. 162 is given in Sec. K.1, “Large-$N$ behavior of the $M$th arrival,” of the Supplemental Material. The first-arrival result is recovered by setting $M=1$, and this behavior is consistent with the general large-$N$ theory of fixed first-passage order statistics^59,142. More generally, requiring more successful searchers increases the completion time, since the process must wait for progressively later arrivals among the $N$ searchers; equivalently, the benefit of parallel search is reduced from $N$ effective searchers to roughly $N/M$.

FIG. 32:

Mean $M$th-arrival time $T_{M:N}$ as a function of the number of independent searchers $N$ for $M=1,M=10$, and $M=N$. Solid curves show the numerical results, while dashed curves show the corresponding leading large-$N$ asymptotic predictions.

We next ask how many parallel searchers are needed for $M$ successful arrivals to typically occur before resetting becomes important. The corresponding characteristic number of searchers is

$$N_c(M,\epsilon )\sim M{N}_c(\epsilon )=M\sqrt{\frac{\pi \rho }{\epsilon }}\mathrm{e}\mathrm{x}\mathrm{p}\left[\frac{\rho }{\epsilon }-\frac{v{x}_T}{2D}\right],\rho =\frac{r{x}_T^2}{4D},$$ (163)

where $\epsilon =t/r$ is again the time measured in units of the reset time. The number of searchers required to outrun resetting therefore increases linearly with $M$. This scaling follows naturally from the same effective parallelization that controls the $M$th-arrival time: requiring $M$ successes effectively reduces the number of parallel searchers from $N$ to $N/M$. Maintaining the same short-time search advantage therefore requires increasing $N$ by a factor of $M$. We derive this result in Sec. K.2, “Approach to the reset-independent regime for the $M$th arrival,” of the Supplemental Material.

The opposite limit, $M=N$, corresponds to requiring every searcher to succeed before the process is complete. Closely related last-arrival logic arises in ovarian aging, where menopause timing is controlled by the last small fraction of primordial follicles to leave the ovarian reserve⁶⁰. The probability that all $N$ searchers have reached the target by time $t$ is $[1-S(t){]}^N$. At the late times relevant for the final arrival, $S(t)\ll 1$, so

$$[1-S(t){]}^N\simeq e^{-NS(t)}.$$ (164)

If $NS(t)\gg 1$, it is therefore very unlikely that every searcher has succeeded, whereas if $NS(t)\ll 1$, essentially all searchers have reached the target. The final arrival consequently occurs in the regime $NS(t)=O(1)$, or equivalently $S(t)=O(1/N)$. For large $N$, the last arrival is therefore controlled by the long-time tail of $S(t)$.

For the drift–diffusion process with Poisson resetting considered here, this long-time behavior can be understood intuitively in terms of repeated attempts. Each reset returns the searcher to its initial position and begins a new attempt to reach the target. Let $q$ denote the probability that a single attempt reaches the target before the next reset. The probability of failing in $n$ successive attempts is then

$$(1-q{)}^n=\mathrm{e}\mathrm{x}\mathrm{p}[n\mathrm{l}\mathrm{n}(1-q)].$$ (165)

Since resets occur at a constant rate $r$, the typical time between resets is $1/r$, so the number of attempts grows proportionally with the duration of a long trajectory, $n=O(rt)$. The probability of remaining unsuccessful therefore decreases exponentially with time. This repeated-attempt argument suggests the long-time form

$$S(t)\sim C{e}^{-\alpha t},t\to \mathrm{\infty },$$ (166)

where $\alpha >0$ is the long-time decay rate⁹². The precise value of $\alpha$ depends on the distribution of the attempt durations and is derived more formally in Sec. K.3, “Long-time survival and the last arrival,” of the Supplemental Material. Combining this exponential decay with the condition $NS(t)=O(1)$ gives $NC{e}^{-\alpha t}=O(1)$, and hence the leading large-$N$ behavior of the mean last-arrival time⁶⁰,

$$T_{N:N}\sim \frac{\mathrm{l}\mathrm{n}N}{\alpha },N\gg 1.$$ (167)

Unlike the fixed-$M$ case, adding more searchers now increases the completion time, because it also increases the chance that one searcher will take unusually long to reach the target.

These different regimes are illustrated in Figure 32, which shows the mean $M$th-arrival time as a function of $N$ for the first arrival ($M=1$), a fixed intermediate number of required successes ($M=10$), and the last arrival ($M=N$). For fixed $M$, increasing the number of searchers decreases the completion time, whereas requiring all $N$ searchers to succeed produces the opposite behavior. The dashed curves show the corresponding leading large-$N$ asymptotic predictions.

This contrast highlights the difference between processes controlled by the fastest successful trajectories and those that must wait for the slowest member of the population.

VI. EXPLORATION THROUGH CONDITIONING

So far in our journey through exploratory dynamics in sections §I-§V, we contemplated trajectories united only by the condition that they eventually successfully terminate at a target state (no matter their particular approaches or delays in arriving there). Yet biology may impose and experience broader flavors of conditioning than mere eventual success. For instance, trajectories may only be relevant to an organism if they terminate by (or at) an allotted time, set by physiological, environmental, or reproductive demands.

Accordingly, to appreciate how these wider senses of conditioning sculpt biological exploration, our final foray into mathematical representations of exploratory dynamics focuses on revealing incarnations of the powerful idea of conditional probability^4,40,49. Though our discussion will focus primarily on the easily visualizable case of one-dimensional diffusion, we note that many biological processes can be described as random walks between the nodes of graphs with the coin flips of a honest coin used to describe the simplest random walk replaced by rolls of a multisided and dishonest die that allow the transition rates between different nodes to be unequal.

At the heart of this viewpoint is Bayes’ theorem. In words, it states that the probability of the present state, given a desired future outcome, is obtained by reweighting the probability of the present state according to how likely that future outcome is from the present state. Symbolically, we write this as

$$\begin{array}{r}P(\text{present}\text{state}\mid \text{future}\text{outcome})=\\ \frac{P(\text{future}\text{outcome}\mid \text{present}\text{state})}{P(\text{future}\text{outcome})}P(\text{present}\text{state}).\end{array}$$ (168)

A more compact way to write the same idea is

$$P\left(X_t\mid 𝒪\right)=\frac{P\left(𝒪\mid X_t\right)}{P(𝒪)}P\left(X_t\right),$$ (169)

where $X_t$ denotes the current state of the system and $𝒪$ denotes a specified future outcome. The significance of this equation is that it provides a recipe for constructing dynamical laws. The primary example we will use to illustrate those ideas is what is sometimes called the Schrödinger or Brownian bridge^19,116. Rather than merely analyzing trajectories after they have succeeded, Bayes’ theorem tells us how a desired future outcome reweights the present state of the system. The Brownian bridge provides the canonical example. Ordinary diffusion is transformed into a new stochastic process whose trajectories are generated according to the condition that they arrive at a specified endpoint. Our goal is to explore how this same logic can be extended beyond diffusion to more general exploratory processes.

One of the very interesting facets of the problems we describe below is that in almost all of the cases, we will construct our conditional probabilities using Bayes-like ratios of constrained microscopic trajectories. Many of the building blocks for our analysis were written down succinctly and clearly long ago in a famed 1943 article by S. Chandrasekhar entitled “Stochastic Problems in Physics and Astronomy”¹⁷. In pages 3–7 of that extremely impressive tome, Chandrasekhar demonstrates how to solve the generic random walk, the problem with both reflecting and absorbing boundary conditions as well as the first-passage properties needed to figure out the probability of arriving at position $x$ at time $t$ for the first time. He carries out these analyses both in discrete and continuous language. In the pages that follow, we try to develop a concise but explicit language for describing these various conditional probabilities in both the discrete and continuous frameworks and then assembling these pieces using Bayes-like ratios.

The goal of this part of the paper is to illustrate how to construct and analyze such conditioned ensembles of trajectories, and to see how the conditioning process induces a new dynamics. In particular, we will study the statistics of diffusive paths conditioned on future outcomes, such as arriving at a prescribed position at a fixed final time, or reaching a target for the first time at that final time⁸. These constrained trajectory ensembles let us ask how apparently purposeful dynamics can emerge from purely diffusive motion once successful outcomes are imposed as constraints. Throughout, we consider a one-dimensional Brownian particle with diffusion coefficient $D$, starting at position $x_i$ at time $t_i$.

A. Free diffusion

The elementary building block for all of the conditioned processes considered below is free Brownian diffusion shown in Figure 33(A). An intuitive way to think about this process⁴ is that over a small time interval $∆t$, the displacement $∆x$ is drawn from a Gaussian distribution with mean zero and variance $2D∆t$,

$$∆x\sim 𝒩(0,2D∆t).$$ (170)

FIG. 33:

Conditioning on the trajectories of the random seeker. (A) Representative trajectories for a random seeker with no constraints. (B) Representative trajectories for a random seeker that leaves position $x_i$ at time $t_i$ and arrives at the target at position $x_T$ at time $t_f$. (C) Representative trajectories for a random seeker that leaves position $x_i$ at time $t_i$ and arrives at the target at position $x_T$ at time $t_f$ but with the added constraint that it never arrives at $x_T$ at any time before time $t_f$.

As a result, the transition probability density for a particle diffusing freely from position $x$ at time $t$ to position $y$ at time $t+∆t$ is the Gaussian propagator^31,133

$$p_0(y,t+\mathrm{\Delta }t\mid x,t)=\frac{1}{\sqrt{4\pi D\mathrm{\Delta }t}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{(y-x{)}^2}{4D\mathrm{\Delta }t}\right].$$ (171)

Here and throughout, $p_0$ denotes the free-diffusion probability density and we will often use the $p_0$ shorthand rather than writing the whole formula. In particular, the probability density of finding the particle at position $x$ at time $t$, given that it started at $x_i$ at time $t_i$, is

$$p_0\left(x,t\mid x_i,t_i\right)=\frac{1}{\sqrt{4\pi D\left(t-t_i\right)}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{\left(x-x_i\right)}^2}{4D\left(t-t_i\right)}\right].$$ (172)

Another important quantity associated with free diffusion is the first-passage-time density^108,109. Consider a particle starting at position $x<x_T$ at time $t$ and diffusing freely until it reaches the target position $x_T$ for the first time. We introduce the notation $f_{\mathrm{F}\mathrm{P}}\left(\tau \mid x,t;x_T\right)$ for the probability density that this first arrival occurs after an elapsed time $\tau$. For free one-dimensional diffusion, this density is

$$f_{\mathrm{F}\mathrm{P}}\left(\tau \mid x,t;x_T\right)=\frac{x_T-x}{\sqrt{4\pi D{\tau }^3}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{\left(x_T-x\right)}^2}{4D\tau }\right].$$ (173)

This quantity gives the probability density that the particle first reaches the target position $x_T$ at time $t+\tau$, given that it was at position $x$ at time $t$. The corresponding survival probability is obtained by integrating over all the seekers who have arrived at the target before time $t$ as

$$S_0\left(x,t\right)=1-{\int }_0^t{f}_{\mathrm{F}\mathrm{P}}\left(x,\tau \right)d\tau .$$ (174)

For one-dimensional Brownian motion, this integral can be evaluated explicitly¹⁰⁸ as

$$S_0(x,t)=\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{x_T-x}{\sqrt{4Dt}}\right).$$ (175)

B. Brownian bridge

We next consider the Brownian bridge, the simplest example of a diffusive trajectory conditioned on its endpoint. This idea goes back at least to Schrödinger’s formulation of conditioned diffusions, now often called the Schrödinger bridge problem^19,116, and is treated systematically in references on Brownian motion and stochastic processes¹². In the Brownian bridge considered here, trajectories are conditioned only to arrive at the target position $x_f$ at a prescribed final time $t_f$, as seen in Figure 33(B). Importantly, trajectories are still allowed to cross $x_f$ before the final time.

We introduce the notation

$$p_{\mathrm{b}\mathrm{b}}\left(x,t\mid x_i,t_i;x_T,t_f\right)$$ (176)

for the conditional probability density of being at position $x$ at time $t$, given that the trajectory starts at position $x_i$ at time $t_i$ and ends at position $x_T$ at time $t_f$. For $t_i<t<t_f$, Bayes’ theorem and the Markov property give

$$p_{\mathrm{b}\mathrm{b}}\left(x,t\mid x_i,t_i;x_T,t_f\right)=\frac{p_0\left(x,t\mid x_i,t_i\right)p_0\left(x_T,t_f\mid x,t\right)}{p_0\left(x_T,t_f\mid x_i,t_i\right)}.$$ (177)

The numerator counts all trajectories that start at position $x_i$ at time $t_i$, pass through position $x$ at the intermediate time $t$, and subsequently arrive at the final position $x_f$ at time $t_f$. The denominator counts all trajectories that start at $x_i$ at time $t_i$ and arrive at $x_f$ at time $t_f$. The conditional probability density therefore corresponds to the ratio between these two trajectory ensembles. Note that our use of the word “count” is imprecise because there are an infinite number of such trajectories. Substituting the Gaussian diffusion kernels and simplifying yields a Gaussian conditional distribution,

$$p_{\mathrm{b}\mathrm{b}}\left(x,t\mid x_i,t_i;x_f,t_f\right)=\frac{1}{\sqrt{2\pi \mathrm{V}\mathrm{a}\mathrm{r}[x(t)]}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{(x-⟨x(t)⟩{)}^2}{2\mathrm{V}\mathrm{a}\mathrm{r}[x(t)]}\right],$$ (178)

with mean trajectory

$$⟨x\left(t\right)⟩=x_i+\frac{t-t_i}{t_f-t_i}\left(x_T-x_i\right),$$ (179)

and variance

$$\mathrm{V}\mathrm{a}\mathrm{r}[x(t)]=2D\frac{\left(t-t_i\right)\left(t_f-t\right)}{t_f-t_i}.$$ (180)

The variance vanishes at both endpoints because the trajectory is constrained to begin at $x_i$ at time $t_i$ and end at $x_T$ at time $t_f$.

The same construction can be carried out locally in time to obtain the one-step transition kernel for the Brownian bridge,

$$p_{\mathrm{b}\mathrm{b}}\left(y,t+\mathrm{\Delta }t\mid x,t;x_T,t_f\right)=p_0(y,t+\mathrm{\Delta }t\mid x,t)\frac{p_0\left(x_T,t_f\mid y,t+\mathrm{\Delta }t\right)}{p_0\left(x_T,t_f\mid x,t\right)}.$$ (181)

This one-step transition was used to generate the trajectories shown in Figure 33(B).

C. First-passage Brownian bridge

We now introduce a more restrictive conditioned ensemble. Rather than merely requiring the trajectory to arrive at the target position $x_T$ at time $t_f$, as in the Brownian bridge, we require that $t_f$ be the first time at which the target is reached, as illustrated in Figure 33(C). Trajectories that reach $x_T$ at any earlier time are therefore excluded.

We denote the corresponding conditional probability density by

$$p_{\mathrm{F}\mathrm{P}\mathrm{b}\mathrm{b}}\left(x,t\mid x_i,t_i;x_T,t_f\right),$$ (182)

which gives the probability density of finding the trajectory at position $x$ at an intermediate time $t$, given that it starts at ($x_i,t_i$) and first reaches $x_T$ at time $t_f$^8,99.

The construction again follows the Bayes-like logic introduced above. For a trajectory to pass through position $x$ at time $t$, it must satisfy two requirements. First, it must travel from $\left(x_i,t_i\right)$ to $(x,t)$ without having reached the target at any earlier time. Second, starting from $x$ at time $t$, it must first reach $x_T$ after exactly the remaining time $t_f-t$.

The first requirement is described by the absorbing-boundary propagator

$$p_{\mathrm{a}\mathrm{b}\mathrm{s}}\left(x,t\mid x_i,t_i;x_T\right),$$ (183)

which gives the probability density of finding the particle at $x<x_T$ at time $t$, given that it started at $x_i<x_T$ at time $t_i$ and has remained below the absorbing boundary at $x_T$ at all earlier times. For free one-dimensional diffusion, the method of images gives

$$p_{\mathrm{a}\mathrm{b}\mathrm{s}}\left(x,t\mid x_i,t_i;x_T\right)=p_0\left(x,t\mid x_i,t_i\right)-p_0\left(x,t\mid 2x_T-x_i,t_i\right),$$ (184)

for $x<x_T$, where $p_0$ is the free-diffusion propagator introduced above.

The second requirement is described by the first-passage-time density $f_{\mathrm{F}\mathrm{P}}$ introduced in the preceding subsection. The conditional probability density is therefore obtained by counting trajectories that satisfy both requirements and dividing by the total probability density of first passage from the initial to final state and given by

$$p_{\mathrm{F}\mathrm{P}\mathrm{b}\mathrm{b}}\left(x,t\mid x_i,t_i;x_T,t_f\right)=\frac{p_{\mathrm{a}\mathrm{b}\mathrm{s}}\left(x,t\mid x_i,t_i;x_T\right)f_{\mathrm{F}\mathrm{P}}\left(t_f-t\mid x,t;x_T\right)}{f_{\mathrm{F}\mathrm{P}}\left(t_f-t_i\mid x_i,t_i;x_T\right)},$$ (185)

for $x<x_T$. The numerator represents trajectories that begin at $\left(x_i,t_i\right)$, survive below the target up to time $t$, pass through $x$ at that time, and subsequently first reach $x_T$ exactly at $t_f$. The denominator represents all trajectories beginning at $\left(x_i,t_i\right)$ that first reach $x_T$ at $t_f$. For the remainder of this subsection, we set $x_i=0$ and $t_i=0$ for readability and introduce the remaining distance to the target,

$$z=x_T-x.$$ (186)

It is also convenient to define

$$\mu (t)=\frac{x_T\left(t_f-t\right)}{t_f},\sigma (t{)}^2=\frac{2Dt\left(t_f-t\right)}{t_f}.$$ (187)

Substituting the absorbing propagator and the first-passage densities into eqn. 185 gives the compact expression

$$p_{\mathrm{F}\mathrm{P}\mathrm{b}\mathrm{b}}\left(x_T-z,t\mid 0,0;x_T,t_f\right)=\frac{z}{\mu (t)\sqrt{2\pi \sigma (t{)}^2}}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{[z-\mu (t){]}^2}{2\sigma (t{)}^2}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{[z+\mu (t){]}^2}{2\sigma (t{)}^2}\right)\right].$$ (188)

The derivation of eqn. 188 and the moments quoted below are given in Sec. L, “Derivation of the first-passage Brownian bridge statistics,” of the Supplemental Material.

The conditioned density can be used to determine both the average trajectory and the fluctuations around it. The mean position is

$$\begin{array}{c}⟨x(t)⟩=x_T-\left(\mu (t)+\frac{\sigma (t{)}^2}{\mu (t)}\right)\mathrm{e}\mathrm{r}\mathrm{f}\left(\frac{\mu (t)}{\sqrt{2}\sigma (t)}\right)\\ -\sqrt{\frac{2}{\pi }}\sigma \left(t\right)\exp \left[-\frac{\mu (t{)}^2}{2\sigma (t{)}^2}\right],\end{array}$$ (189)

while the variance is

$$\mathrm{V}\mathrm{a}\mathrm{r}[x(t)]=\mu (t{)}^2+3\sigma (t{)}^2-{\left(x_T-⟨x(t)⟩\right)}^2.$$ (190)

The mean and standard deviation predicted by these expressions are shown together with representative conditioned trajectories in Figure 33(C).

Whether the average first-passage bridge moves monotonically toward the target depends on the competition between the distance to the target, $x_T$, and the characteristic distance explored by diffusion over the allotted time $t_f$ to reach the target. When

$$\sqrt{2D{t}_f}<x_T,$$ (191)

the average first-passage bridge moves monotonically toward the target. In this regime, reaching $x_T$ by time $t_f$ already requires an unusually large displacement toward the target, while arriving much earlier is unlikely.

In contrast, when

$$\sqrt{2D{t}_f}>x_T,$$ (192)

the mean trajectory becomes non-monotonic: it initially moves away from the target before turning back toward $x_T$. In this regime, reaching the target too early is likely enough that the first-passage constraint selects against trajectories that approach $x_T$ too quickly.

In the strongly non-monotonic regime, $\sqrt{2D{t}_f}\gg x_T$, the maximal displacement of the average trajectory away from the target is itself set by the diffusive length scale. If $t_{\mathrm{*}}$ denotes the time at which the mean trajectory is farthest from $x_T$, then

$$x_T-⟨x\left(t_{\mathrm{*}}\right)⟩\simeq 2\sqrt{\frac{D{t}_f}{\pi }}.$$ (193)

The derivation of the transition between the monotonic and non-monotonic regimes and of eqn. 193 is given in Sec. L of the Supplemental Material.

The contrast between the three rows of Figure 33 summarizes how knowledge of a future outcome reshapes the dynamics. For unconstrained diffusion, there is no preferred direction of motion and the distribution simply broadens with time. Conditioning only on the final position produces a Brownian bridge whose mean trajectory is directed toward the prescribed endpoint. Conditioning instead on the first arrival at that endpoint imposes an additional requirement: the trajectory must both arrive at $x_T$ at time $t_f$ and avoid arriving there at any earlier time. As for the Brownian bridge, this conditioned ensemble can be expressed locally as a new stochastic dynamics. Applying the same construction over a short time interval $\Delta t$ gives the transition kernel

$$p_{\mathrm{F}\mathrm{P}\mathrm{b}\mathrm{b}}\left(y,t+\mathrm{\Delta }t\mid x,t;x_T,t_f\right)=p_{\mathrm{a}\mathrm{b}\mathrm{s}}\left(y,t+\mathrm{\Delta }t\mid x,t;x_T\right)\frac{f_{\mathrm{F}\mathrm{P}}\left(t_f-t-\mathrm{\Delta }t\mid y,t+\mathrm{\Delta }t;x_T\right)}{f_{\mathrm{F}\mathrm{P}}\left(t_f-t\mid x,t;x_T\right)},y<x_T.$$ (194)

The absorbing propagator appears here rather than the free propagator because the first-passage condition must be enforced during the short interval from $t$ to $t+∆t$ itself. This local transition kernel provides a direct prescription for generating first-passage bridge trajectories and was used to generate the sampled paths shown in Figure 33(C).

The Brownian bridge and first-passage Brownian bridge illustrate how conditioning on a prescribed future outcome can induce new effective dynamics through the reweighting of microscopic trajectories. This viewpoint also provides a complementary interpretation of the resetting processes considered earlier. A search with resetting can be viewed as a sequence of unsuccessful excursions followed by a final successful trajectory. Conditioning allows us to isolate this last trajectory and ask what paths are typical given that success occurs at a prescribed time. In this sense, resetting organizes exploration through repeated attempts, whereas bridges describe the statistics of the successful attempt itself. To our knowledge, the corresponding first-passage Brownian bridge with stochastic resetting has not previously been characterized. In Sec. M, “First-passage Brownian bridges with stochastic resetting,” of the Supplemental Material, we construct this process explicitly. We then decompose conditioned trajectories into failed excursions followed by a final successful excursion and derive analytical results for the number of resets in the conditioned ensemble.

D. A Conditioned Bridge on a Graph

The Brownian and first-passage bridges considered above offer two ways of thinking about conditioning: retrospectively, as selecting a subset of trajectories, and dynamically, as reweighting the local transitions that generate them. We now ask what this same construction looks like beyond diffusion in continuous space.

As noted at the beginning of this section, “diffusion” can be interpreted so much more broadly than the simple coin flips of traditional Brownian motion or the continuum version as embodied in Fick’s law. Indeed, as we argue here, many of the most fundamental biochemical networks can be thought of as random walks on the nodes of graphs^37,77, with transition rates that depend upon which node of the graph the random walker occupies.

To illustrate how conditioning induces new dynamics on biochemical networks, we adopt the highly simplified three-nodedgraph shown in Figure 34. We consider four instructive versions of dynamics on this graph, culminating in what we will call a “graph bridge,” conditioned to begin at node 1 at $t=0$ and arrive at node 3 at a prescribed final time $t_f$. Case 1: Unbiased diffusion on the three-noded graph. As shown in Figure 34(A), we first consider the triangular graph in which every allowed transition has the same rate $k$, with arrows in both directions between every pair of nodes. For all of our examples, we will consider the initial condition of releasing the graph walker at node 1, corresponding to the initial condition

$$\mathbf{p}(0)=\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right),$$ (195)

associated with the master equation

$$\frac{d}{dt}\left(\begin{array}{l}{p}_1\\ p_2\\ p_3\end{array}\right)=k\left(\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right)\left(\begin{array}{l}{p}_1\\ p_2\\ p_3\end{array}\right).$$ (196)

FIG. 34:

Conditioning on the trajectories of the triangular graph. Throughout, we set $k=1$, so that time is measured in units of $1/k$. (A) Relaxation to equilibrium on the three-noded graph with all rate constants equal. (B) Broken detailed balance on the three-noded graph. (C) The three-noded graph with an absorbing boundary condition at node 3. (D) The three-noded graph for a random seeker that leaves node 1 at time $t_i=0$ and arrives at the target at node 3 at time $t_f=3$.

We can solve this set of coupled equations directly by elimination. As shown in Figure 34(A), the solution is given by

$$p_1(t)=\frac{1}{3}+\frac{2}{3}{e}^{-3kt},$$ (197)

and

$$p_2(t)=p_3(t)=\frac{1}{3}-\frac{1}{3}{e}^{-3kt}.$$ (198)

This is diffusion on a graph. Probability starts localized at node 1 and then spreads until all three nodes have equal occupancy of 1/3. Interestingly, when we turn to conditioning in Case 4, we will find these solutions serve as the propagators for solving the conditioned problem.

Case 2: Broken detailed balance on the three-noded graph. In earlier work, we have shown how broken detailed balance greatly enriches the behaviors available to biochemical networks⁷². The simplest implementation of broken detailed balance is to imagine that one of the edges is driven as shown in Figure 34(B).

To be concrete, let the transition $1\to 2$ have rate $\epsilon k$, while all other rates remain equal to $k$. In this case, the master equation becomes

$$\frac{d}{dt}\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right)=k\left(\begin{array}{ccc}-(\epsilon +1)& 1& 1\\ \epsilon & -2& 1\\ 1& 1& -2\end{array}\right)\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right).$$ (199)

Using the fact that $p_1+p_2=1-p_3$, we can solve for $p_3(t)$ immediately yielding

$$p_3(t)=\frac{1}{3}\left(1-e^{-3kt}\right).$$ (200)

Next, we can solve for $p_1(t)$ using the same trick by noting that $p_2+p_3=1-p_1$ which yields

$$\frac{d{p}_1}{dt}=k\left[1-(2+\epsilon )p_1\right],$$ (201)

whose steady state value is $1/(2+\epsilon )$ and whose rate of approach to steady state is $(2+\epsilon )k$, so that

$$p_1(t)=\frac{1}{2+\epsilon }+\frac{1+\epsilon }{2+\epsilon }{e}^{-(2+\epsilon )kt}.$$ (202)

The last probability $p_2$ follows from conservation, $p_2=1-p_1-p_3$, resulting in

$$p_2(t)=\frac{1+2\epsilon }{3(2+\epsilon )}+\frac{1}{3}{e}^{-3kt}-\frac{1+\epsilon }{2+\epsilon }{e}^{-(2+\epsilon )kt}.$$ (203)

The system now forgets its initial condition at two distinct rates, the old $3k$ carried by the bystander at node 3 and a new relaxation rate $(2+\epsilon )k$ carried by the altered edge. The steady state it reaches is no longer uniform and given by

$$p_1^{\mathrm{s}\mathrm{s}}=\frac{1}{\epsilon +2},$$ (204)

$$p_2^{\mathrm{s}\mathrm{s}}=\frac{2\epsilon +1}{3(\epsilon +2)},$$ (205)

and

$$p_3^{\mathrm{s}\mathrm{s}}=\frac{\epsilon +2}{3(\epsilon +2)}=\frac{1}{3}.$$ (206)

When $\epsilon =1$, all three probabilities are 1/3, as in the unbiased case. But when $\epsilon \ne 1$, the system has a nonequilibrium steady state with a net circulating flux. For example, the net flux from node 1 to node 2 is

$$J_{12}=\epsilon k{p}_1^{\mathrm{s}\mathrm{s}}-k{p}_2^{\mathrm{s}\mathrm{s}}=\frac{k(\epsilon -1)}{3(\epsilon +2)}.$$ (207)

The same net flux circulates around the other edges. Thus the system reaches a time-independent distribution, but it is not an equilibrium distribution. The random walkers have a circulation.

Case 3: Absorbing boundary condition on the three-noded triangle. As shown in Figure 34(C), in this case, the fact that node 3 is absorbing is captured through the fact that there are no outgoing arrows from this node. This assumption results in a master equation of the form

$$\frac{d}{dt}\left(\begin{array}{l}{p}_1\\ p_2\\ p_3\end{array}\right)=k\left(\begin{array}{ccc}-2& 1& 0\\ 1& -2& 0\\ 1& 1& 0\end{array}\right)\left(\begin{array}{l}{p}_1\\ p_2\\ p_3\end{array}\right).$$ (208)

As usual, we have the initial condition $p_1(0)=1$ implying that the solution is

$$p_1(t)=\frac{1}{2}{e}^{-kt}+\frac{1}{2}{e}^{-3kt},$$ (209)

$$p_2(t)=\frac{1}{2}{e}^{-kt}-\frac{1}{2}{e}^{-3kt},$$ (210)

and

$$p_3(t)=1-e^{-kt}.$$ (211)

with the long-time limit given by

$$p_1(t)\to 0,p_2(t)\to 0,p_3(t)\to 1$$ (212)

as shown in Figure 34(C).

Case 4: The conditioned bridge on the three-noded graph. In the final and most important case we neither let the walker diffuse freely on the graph, nor do we subject it to absorbing boundary conditions at node 3. Instead, we condition on arrival at a particular place (node 3) at a particular time ($t_f$), as shown in Figure 34(D). We ask what dynamics the trajectories that respect this constraint must follow.

We invoke Bayes’ theorem to determine the reweighted rates, exactly as we did for the Brownian bridge. Imagine the walker sits at node $i$ at time $t$ and consider its next step to a neighboring node $j$. With no constraint, that step is taken at the bare rate $k$. However, we keep only those histories that end at node 3 at time $t_f$, and a step to $j$ is consistent with that demand in proportion to the chance of completing the trip once we are at $j$. That chance is captured by the graph propagator $P_{ij}(\tau )$, the probability that a walker initially at node $i$ is found at node $j$ a time $\tau$ later under the unbiased dynamics of Case 1. It is the exact counterpart of the diffusion propagator $p_0\left(x_f,t_f\mid x,t\right)$, with positions replaced by nodes on a graph, and describes propagation under the unconditioned, unbiased dynamics of Case 1. The chance of completing the trip from $j$ is then $P_{j3}\left(t_f-t\right)$, the propagator carrying node $j$ to the target over the time that remains. Bayes’ theorem tells us that the conditioned probability of the step $i\to j$ is its bare probability, multiplied by the propagator to the target from $j$ and divided by the propagator to the target from $i$, so that the bare rate is reweighted to

$$k_{ij}^{\mathrm{*}}(t)=k\frac{P_{j3}\left(t_f-t\right)}{P_{i3}\left(t_f-t\right)}.$$ (213)

The numerator measures the chance of ultimately reaching the target after taking the step to node $j$, while the denominator measures the chance of reaching the target before the step is taken. We can formalize this by writing

$$\frac{P_{33}\left(t_f-t\right)}{P_{13}\left(t_f-t\right)}=\frac{\text{prospects}\text{for}\text{eventual}\text{success}\text{after}\text{the}\text{move}}{\text{prospects}\text{for}\text{eventual}\text{success}\text{before}\text{the}\text{move}}$$ (214)

which measures the factor by which the move $1\to 3$ improves the walker’s prospects. Conditioning on the destination turns the constant rate $k$ into a time-dependent rate determined by the propagators to the target.

These propagators are not a new calculation; they are a reinterpretation of Case 1’s solution. Recall how that solution was obtained. We released the walker at node 1, that is, we solved the master equation with all of the probability placed on node 1 at $t=0$, and we found the probability $p_j(t)$ of locating the walker at node $j$ a time $t$ later. But a probability computed under the certainty that the walker began at node 1 is already a conditional probability, the chance of being at node $j$ given a start at node 1, and that is exactly the transition probability $P_{1j}(t)$. The occupation probabilities of a walk released from a definite node simply are the propagators out of that node. The Case 1 results

$$p_1(t)=P_{11}(t)=\frac{1}{3}+\frac{2}{3}{e}^{-3kt},$$ (215)

and

$$p_2(t)=p_3(t)=P_{12}(t)=P_{13}(t)=\frac{1}{3}-\frac{1}{3}{e}^{-3kt},$$ (216)

are therefore already the propagators carrying node 1 to nodes 1, 2, and 3, with no further work.

Two features of the triangle let this single calculation supply every propagator the reweighting needs. First, the unbiased rates do not change in time, so the propagator depends only on how much time has elapsed, and we may read it over the remaining time $\tau =t_f-t$. Second, under the unbiased dynamics the three nodes are interchangeable, every node equivalent and every edge equal, so the propagator can depend only on whether its start and end are the same node or two different ones, never on their labels; releasing the walker instead from node 2 or node 3 would return these very same two curves. The whole propagator is thus built from just two numbers, a same-node value $\frac{1}{3}+\frac{2}{3}{e}^{-3k\tau }$ and a different-node value $\frac{1}{3}-\frac{1}{3}{e}^{-3k\tau }$. The bridge now singles out node 3 as the target, and this is the only place the three nodes cease to stand on an equal footing. The propagator to the target from the target itself is the same-node value,

$$P_{33}\left(t_f-t\right)=\frac{1}{3}+\frac{2}{3}{e}^{-3k\left(t_f-t\right)},$$ (217)

while the propagators to the target from the two non-target nodes are the different-node value,

$$P_{13}\left(t_f-t\right)=P_{23}\left(t_f-t\right)=\frac{1}{3}-\frac{1}{3}{e}^{-3k\left(t_f-t\right)}.$$ (218)

It is the demand that the walk end at node 3, and not any asymmetry of the dynamics, that sets node 3 apart from nodes 1 and 2.

Introducing the shorthand $u=e^{-3k\left(t_f-t\right)}$ for the remaining-time factor, the reweighted rates follow at once. The edge joining nodes 1 and 2 is left untouched, since those two nodes carry the same propagator to the target and their ratio is unity, so that

$$k_{12}^{\mathrm{*}}=k_{21}^{\mathrm{*}}=k.$$ (219)

The two rates into the target are accelerated to

$$\lambda (t)\equiv k_{13}^{\mathrm{*}}(t)=k_{23}^{\mathrm{*}}(t)=k\frac{P_{33}\left(t_f-t\right)}{P_{13}\left(t_f-t\right)}=k\frac{1+2u}{1-u},$$ (220)

and the two rates out of the target are suppressed to

$$\mu (t)\equiv k_{31}^{\mathrm{*}}(t)=k_{32}^{\mathrm{*}}(t)=k\frac{P_{13}\left(t_f-t\right)}{P_{33}\left(t_f-t\right)}=k\frac{1-u}{1+2u}.$$ (221)

As $t\to t_f$ the remaining time vanishes, $u\to 1$, the rate into the target diverges, and the rate out of it falls to zero. The walker is driven onto node 3 and held there, precisely as the bridge demands. When $u\ll 1$, however, the terms proportional to $u$ become negligible, and both $\lambda (t)$ and $\mu (t)$ approach the unconditioned rate $k$. Figure 35 illustrates these induced rates for bridges of different durations: for long bridges, the rates remain close to their unconditioned value $k$ for much of the trajectory before changing sharply as the prescribed final time approaches.

FIG. 35:

Conditioned transition rates on the three-node graph for trajectories conditioned to start at node 1 at $t=0$ and arrive at node 3 at a prescribed final time $t_f$. Panels show three conditioning times, $t_f=0.1,1$, and 10, with $k=1$, so that time is measured in units of $1/k$ and transition rates in units of $k$. The rate $\lambda (t)=k_{13}^{\mathrm{*}}(t)=k_{23}^{\mathrm{*}}(t)$ into the target node is shown in blue, the rate $\mu (t)=k_{31}^{\mathrm{*}}(t)=k_{32}^{\mathrm{*}}(t)$ out of the target node in purple, and the unchanged rate $k_{12}^{\mathrm{*}}(t)=k_{21}^{\mathrm{*}}(t)=k$ in orange.

With these reweighted rates the master equation becomes

$$\frac{d}{dt}\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right)=\left(\begin{array}{ccc}-(k+\lambda )& k& \mu \\ k& -(k+\lambda )& \mu \\ \lambda & \lambda & -2\mu \end{array}\right)\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right),$$ (222)

with the time-dependent rates $\lambda (t)$ and $\mu (t)$ given above and the initial condition $\mathbf{p}(0)=(1,0,0)$.

Although the rates depend on time, the same Bayesian logic that fixed them delivers the solution as a product of propagators. The probability of finding the conditioned walker at node $i$ at time $t$ is the propagator that carries it from the start to node $i$, times the propagator that carries it from node $i$ on to the target, divided by the propagator for the whole trip,

$$p_i(t)=\frac{P_{1i}(t)P_{i3}\left(t_f-t\right)}{P_{13}\left(t_f\right)}.$$ (223)

It is striking that this solution carries no trace of the reweighted rates $\lambda$ and $\mu$, being assembled entirely from the unbiased Case 1 propagators; the altered rates and this product of bare propagators are two faces of the same conditioning, the one describing it step by step and the other all at once.

We also note that this expression is the exact graph analogue of the Brownian bridge formula

$$p_{\mathrm{b}\mathrm{b}}(x,t)=\frac{p_0\left(x,t\mid x_i,0\right)p_0\left(x_f,t_f\mid x,t\right)}{p_0\left(x_f,t_f\mid x_i,0\right)}.$$ (224)

In both cases the conditioned probability is obtained by multiplying a forward propagator from the initial condition by a backward propagator to the target and dividing by the propagator for the entire journey. The forward propagator is the Case 1 result released from node 1, namely $P_{11}(t)=\frac{1}{3}+\frac{2}{3}{e}^{-3kt}$ and $P_{12}(t)=P_{13}(t)=\frac{1}{3}-\frac{1}{3}{e}^{-3kt}$. Introducing the elapsed-time and remaining-time factors

$$w=e^{-3kt},u=e^{-3k\left(t_f-t\right)},U=e^{-3k{t}_f}=wu,$$ (225)

the three occupation probabilities are

$$p_1(t)=\frac{(1+2w)(1-u)}{3(1-U)},$$ (226)

$$p_2(t)=\frac{(1-w)(1-u)}{3(1-U)},$$ (227)

and

$$p_3(t)=\frac{(1-w)(1+2u)}{3(1-U)}.$$ (228)

Equivalently, we can return from the compact variables $w,u$, and $U$ to the original time variables $t$ and $t_f$. This gives the conditioned occupation probabilities directly as sums of elementary time-dependent terms,

$$p_1(t)=\frac{1-2e^{-3k{t}_f}}{3\left(1-e^{-3k{t}_f}\right)}+\frac{2}{3\left(1-e^{-3k{t}_f}\right)}{e}^{-3kt}-\frac{1}{3\left(1-e^{-3k{t}_f}\right)}{e}^{-3k\left(t_f-t\right)},$$ (229)

$$p_2(t)=\frac{1+e^{-3k{t}_f}}{3\left(1-e^{-3k{t}_f}\right)}-\frac{1}{3\left(1-e^{-3k{t}_f}\right)}{e}^{-3kt}-\frac{1}{3\left(1-e^{-3k{t}_f}\right)}{e}^{-3k\left(t_f-t\right)},$$ (230)

$$p_3\left(t\right)=\frac{1-2e^{-3k{t}_f}}{3\left(1-e^{-3k{t}_f}\right)}-\frac{1}{3\left(1-e^{-3k{t}_f}\right)}{e}^{-3kt}+\frac{2}{3\left(1-e^{-3k{t}_f}\right)}{e}^{-3k\left(t_f-t\right)}.$$ (231)

This form separates the contribution of the two endpoint constraints. Each probability contains a constant term, a term proportional to $e^{-3kt}$, and a term proportional to $e^{-3k\left(t_f-t\right)}$. The factor $e^{-3kt}$ is largest near the initial time and decays as the system relaxes away from the initial condition. By contrast, $e^{-3k\left(t_f-t\right)}$ increases as $t$ approaches $t_f$, reflecting the growing influence of the final-time conditioning. Between these two boundary layers in time (for large $t_f$), the constant terms give the intermediate-time plateau values of the conditioned probabilities.

As shown in Figure 36, these occupation probabilities satisfy both ends of the bridge. At $t=0$ we have $w=1$, so that $p_1=1$ and $p_2=p_3=0$, implying that the walker is released at node 1. At $t=t_f$ we have $u=1$, so that $p_3=1$ and $p_1=p_2=0$, implying in turn that the walker is delivered with certainty to node 3. Between these endpoints the probability drains out of node 1, passes in part through the bystander node 2, and gathers at node 3, shepherded the whole way by the reweighted rates that the conditioning has induced. Although we have constructed this bridge using the unconditioned, unbiased dynamics of Case 1, nothing in the conditioning procedure requires this particular choice. The same construction can be applied to other underlying dynamics, including driven nonequilibrium dynamics such as those of Case 2. We now make this generalization explicit.

FIG. 36:

Occupancy probabilities for trajectories on the three-node graph conditioned to start at node 1 at $t=0$ and arrive at node 3 at a prescribed final time $t_f$. Rows correspond to nodes 1, 2, and 3, while columns show three conditioning times, $t_f=0.1,1$, and 10, with time measured in units of $1/k$ (equivalently, $k=1$). Solid curves denote the conditional occupancy probabilities, and dashed red lines indicate the corresponding steady-state occupancies $p_i^{\text{ss}}$ of the unconditioned process. Gray bands show simulated occupancy probabilities from conditioned trajectories.

E. General Markovian description, conditioned on starting and ending states at specified times

The triangular graph is a special example of a continuous time, discrete state Markov chain with $N=3$ nodes. How do the renormalized dynamics exposed by conditioning on arrival to a final state, or more demandingly doing so at a final time, lift to more general graphs? To address this question, consider a general Markov chain with states $i$ between 1 and $N$, where the rate at which state $i$ jumps to state $j$ is $k_{ij}$. Let the probability of occupying these states at a time $t$ be stored in a vector $\mathbf{p}(t)={\left[p_1(t),\dots ,p_N(t)\right]}^{\mathrm{\top }}$, and populate the transition rates into a Laplacian matrix $\mathbf{L}$, whose $(j,i)$th entry stores the transition rate $k_{ij}$ from state $i$ to $j$, and whose diagonal entries give the total exit rate from each state, namely $L_{ii}=-\sum _{j\ne i}{k}_{ij}$.

Then these probabilities evolve over time as,

$$\frac{d\mathbf{p}}{dt}=\mathbf{L}\mathbf{p}(t).$$ (232)

The brevity of eqn. 232 belies the variety and complexity of the dynamics admitting such a description; such systems are sometimes called kinetic schemes or chemical master equations. Technically, the general solutions to such dynamics like eqn. 232 (starting from an initial condition $\mathbf{p}(t=0)={\mathbf{p}}_0$) can be written compactly as a solution involving a matrix exponential, namely,

$$\mathbf{p}(t)=\mathrm{e}\mathrm{x}\mathrm{p}[t\mathbf{L}]{\mathbf{p}}_0$$ (233)

$$\equiv \left(\mathbf{I}+t\mathbf{L}+\frac{1}{2}t\mathbf{L}\mathbf{L}+\cdots \frac{1}{m!}{t}^m{\mathbf{L}}^m+\cdots \right){\mathbf{p}}_0,$$ (234)

where the second line uses the power series expansion representation of the matrix exponential; the $\mathbf{I}$ is the $N\times N$ identity matrix; and ${\mathbf{L}}^m$ represents the product of $m$ factors of $\mathbf{L}$. We hastily stress that in our view, the ability to write dynamics in the symbolic form of eqn. 234 does not yet represent true intellectual progress, up to a few very coarse advantages of mechanical manipulation. Rather, to the contrary, this representation should not obscure that the true conceptual, mathematical, and biological functionality and flexibility of a physical system obeying eqn. 232 still remains essentially uninterrogated by eqn. 234; this equation is essentially just a notational repackaging of the original problem. And indeed, in our view, new tools for thinking about these facilities are urgently needed. For instance, we have previously explored some useful tools for conceptually understanding the flexibility of such systems in or out of equilibrium using practical techniques from graph theory⁷², which replace arduous linear algebraic calculation with drawing of key diagrams. Yet we strongly argue that much more creative thinking on these axes is critically needed.

This said, at least one virtue of eqn. 233 is that this representation furnishes a very concrete input-output map, or propagator, from an initial condition ${\mathbf{p}}_0$ to a final state $\mathbf{p}(t)$. That is, the term $\mathrm{e}\mathrm{x}\mathrm{p}[\mathbf{L}t]$ represents a matrix transformation specifying the transition probability of occupying a state $i$ at time $t=0$ and ending at any other state $j$ at any desired time $t$. Specifically, eqn. 233 says that the probability $p_j(t)$ of being in state $j$ at time $t$ is given exactly by the $j$th element of $\mathrm{e}\mathrm{x}\mathrm{p}[\mathbf{L}t]{\mathbf{p}}_0$, computed as $\sum _j(\mathrm{e}\mathrm{x}\mathrm{p}[\mathbf{L}t]{)}_{ji}{\left({\mathbf{p}}_0\right)}_i$. So, if the system begins at state $i$ at time 0 (namely with the initial condition ${\mathbf{p}}_0=[\underset{i}{\underbrace{0,\dots ,1}}\dots ,0{]}^{\mathrm{\top }}$), then this matrix multiplication shows that the output probability of being in some different state $j$ is exactly reported by $(\mathrm{e}\mathrm{x}\mathrm{p}[\mathbf{L}t]{)}_{ji}$. In other words,

$$p(\text{state}j\text{at}\text{time}\mathrm{t}\mid \text{state}i\text{at}t=0)=(\mathrm{e}\mathrm{x}\mathrm{p}[\mathbf{L}t]{)}_{ji}.$$ (235)

As shown in Fig. 37, since the matrix exponential gives a general notation for the transition probabilities between specified starting and ending times, we may use it to express the probability of being at some intermediate state $\ell$ at time $t$, given that the system began at state $i$ at time $t=0$, and ends at state $j$ at time $t=T$. In particular, $(\mathrm{e}\mathrm{x}\mathrm{p}[t\mathbf{L}]{)}_{\ell i}$ expresses the probability of starting at state $i$ and arriving at an intermediate state $\ell$ at time $t$. In the remaining time ($T-t$) before trajectories must terminate at state $j$ at final time $T$, the probability that trajectories can do so while beginning anew at $\ell$ is $(\mathrm{e}\mathrm{x}\mathrm{p}[(T-t)\mathbf{L}]{)}_{j\ell }$ (the fact that we need not consult any prior history (for instance from $i$) exploits the Markov property of trajectories). Last, we must normalize this against the net probability that any path (regardless of whether $\ell$ is visited at the proper time) takes a time $T$ from $i$ to $j$, which is $(\mathrm{e}\mathrm{x}\mathrm{p}[T\mathbf{L}]{)}_{ji}$. These components allow us to write the probability of occupying any state $\ell$ in the system, given the enforced conditioning, as

$$p(\ell \text{at}t\mid i\text{at}t=0,j\text{at}t=T)$$

$$=\frac{p(\ell \text{at}t\mid i\text{at}t=0)p(j\text{at}T\mid \ell \text{at}t)}{p(j\text{at}t=T\mid i\text{at}t=0)}$$ (236)

$$=\frac{(\mathrm{e}\mathrm{x}\mathrm{p}[t\mathbf{L}]{)}_{\ell i}(\mathrm{e}\mathrm{x}\mathrm{p}[(T-t)\mathbf{L}]{)}_{j\ell }}{(\mathrm{e}\mathrm{x}\mathrm{p}[T\mathbf{L}]{)}_{ji}}.$$ (237)

FIG. 37:

Probability $p(\ell ,t)$ of visiting an intermediate state $\ell$ at time $t$, once trajectories are conditioned on beginning at state $i$ at time $t=0$ and ending at state $j$ at time $T$.

This expression gives the generalizations of eqns. 226–228 for generic state space graphs.

How do these conditional probability distributions tend to behave over time, compared to their unconditional counterparts? The behavior of eqns. 226–228 explain how central aspects of the interesting anatomy seen in the earlier section’s three-nodeded graph example in Fig. 36 may be expected to be very general. In particular, Fig. 36 hints that these bridge conditional probabilities “procrastinate” in satisfying their endpoint constraints, if they can afford to via a sufficiently long conditioning time $T$. This procrastination is most visible for the case of $T=10$ (the right-most column of plots) in Fig. 36: sufficiently long bridges appear to reach an intermediate steady-state that matches the unconditioned distribution ${\mathbf{p}}^{\text{ss}}$ over states, bookended with transient boundary layers that lead from the start state to this intermediate steady-state, and then from the intermediate steady-state to the final conditioned state. To understand this general behavior, consider the literal algebraic contrast of the conditional dynamics $\hat{p}(\ell ,t)$ of being in state $\ell$ at time $t$ conditioned on both start and end states, as specified by eqn. 237 (as shown in Fig. 37) with the unconditional dynamics of trajectories merely starting at state $i$ at $t=0$: the latter unconditional proability of being in state $\ell$ at time $t$ is merely $p(\ell ,t)=\mathrm{e}\mathrm{x}\mathrm{p}(t\mathbf{L}{)}_{i\ell }$. Interestingly, notice that the conditional dynamics $\hat{p}(\ell ,t)$ depart from the unconditional dynamics $p(\ell ,t)$ only by a ratiometric factor, namely,

$$\frac{\hat{p}(\ell ,t)}{p(\ell ,t)}=\frac{(\mathrm{e}\mathrm{x}\mathrm{p}[(T-t)\mathbf{L}]{)}_{j\ell }}{(\mathrm{e}\mathrm{x}\mathrm{p}[T\mathbf{L}]{)}_{ji}}.$$ (238)

This ratiometric factor exposes three epochs in how a bridge probability changes in time. Early in time, highlighted as the orange small $t$ trace in Fig. 38, the ratiometric factor of eqn. 238 approaches unity. This happens because both the numerator and denominator of eqn. 238 are well approximated by the steady-state probability of the terminal state $j$: that is, regardless of whether the system starts at an intermediate state $\ell$ or the initial state $i$, the probability of being in state $j$ given by the matrix propagator should approach the steady-state value of the chain, $p^{\text{ss}}(j)$. As a result, the conditional probability $\hat{p}(\ell ,t)$ closely mimics the unconditional relaxation from the starting state $i$ to the relevant intermediate state $\ell$. What time $t$ is sufficiently early depends on the rate at which the original unconditional Markov chain of states relaxes to its equilibrium; this characteristic relaxation timescale $\tau$ is captured by the second smallest-size eigenvalue of the Laplacian transition rate matrix $\mathbf{L}$.

FIG. 38:

Typical anatomy of a long bridge involves dwelling at the unconditional steady-state distribution for intermediate times bookended by transitions before and after resembling other unconditional transients.

At intermediate times $t\gg \tau$ but $(T-t)\gg \tau$, the conditional probability still approximates the unconditional probability, $\hat{p}(\ell ,t)\approx p(\ell ,t)$; but now sufficient time has passed (compared to the relaxation time) that the unconditional system has plateaued to near its steady-state value $p(\ell ,\infty )$, highlighted as the flat teal region in Fig. 38. At late times, namely $t\gg \tau$, $(T-t)<\tau$, a boundary layer traced schematically in blue in Fig. 38 manifests, in order to satisfy the endpoint constraint at last. Thus the coarse anatomy seen in the traces of the particular triangular cyclic graph of Fig. 36 are in fact generic features in relevant conditioning time limits.

More generally, these expressions of eqns. 226–228 describing conditional distributions can be readily generalized in two important senses. First, one might pursue trajectories that only satisfy some softer, or more macroscopic, constraint than deterministically starting and terminating in specified states. Specifically, let the microscopic state of a system $x(t)$ specify some biological activity of interest $g[x(t)]$, where the functional bracket reminds us of a potential dependence on the whole trajectory; for instance, $g[x(t)]\equiv {\alpha }_i{x}_i(t)$ could encode the biochemical production or signaling rate of state $i$ given by a proportionality ${\alpha }_i$ at a specific time $t$; or $g[x(t)]\equiv {\int }_0^{T}dt{\alpha }_i{x}_i(t)$ would encode the accumulated such activity over the whole trajectory of length $T$. One could focus only on those trajectories that match a stipulated value $⟨g⟩$ of this activity $g$ on average. Whether this constraint is demanded at a particular time (like the start or end), or aggregated over the whole trajectory length (making this a functional path constraint), this conditioning induces a new probability distribution over satisfactory trajectories (that departs from the unconditional distribution over paths). In fact, there are many possible probability distributions over such surviving trajectories. One logical principle to choose a particular one is to demand the distribution over surviving trajectories that satisfies the constraint but is otherwise as close as possible to the original unconditional probability distribution over paths $P[x(t)]$, for some informationally prudent notion of “close.” Said mathematically, the new probability distribution $\hat{P}[x(t)]$ (over full trajectories $x(t)$) should be the one that has the smallest Kullback-Leibler divergence (relative entropy) with respect to the original unconditional probability distribution while still satisfying the constraints, namely $\hat{P}[x(t)]\equiv {\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{\hat{P}\text{matching}⟨g⟩}\mathrm{K}\mathrm{L}(P[x(t)]\mid \hat{P}[x(t)])$. This general prescription is a maximum (relative) caliber principle (with respect to the original unconditional probability distribution), along the foundational ideas of Jaynes⁵¹ and many subsequent practitioners^{23,34,35,102,103,139}. This recipe clearly accommodates the fixed deterministic endpoint constraint discussed earlier as a special case, for instance considering all trajectories starting at $i$ with $g[x(t)]\equiv \left(x_j(T)=1\right)$; it also smoothly specifies a recipe for a more gently-constrained probability distribution over paths that are found in state $j$ with some desired softer probability $\alpha \le 1$.

A second generalization obtainable from the same prescription considers paths that start from some distribution ${\mathbf{p}}_0$ and end at some distribution ${\mathbf{p}}_T$ over states, instead of deterministically starting from particular (singleton) start and end states. This distributional problem was originally posed and analyzed by Schrödinger¹¹⁷ and the ensuing mathematical forms are called Schrödinger bridges; Léonard provides a superb, mathematically rigorous discussion of these ideas⁷¹. In summary, all of these examples show the key idea that a new distribution over acceptable trajectories is induced by start and endpoint constraints, while being as close as possible to an original unconditioned distribution over paths (in an informational sense).

Conditioning in this sense does not only induce intermediate probability distributions over states of the system; it also formally imposes effective new transition rates among the states to comply with the starting and ending conditions. In brief, though a full description of these dynamics is beyond the scope of the present paper, one may see that conditioning induces a new probability distribution $\stackrel{\mathbf{ˆ}}{\mathbf{p}}(t)$ with a time-dependent set of effective transition rates $\mathbf{L}(t)$ such that $\frac{d\stackrel{\mathbf{ˆ}}{\mathbf{p}}}{dt}=\stackrel{\mathbf{ˆ}}{\mathbf{L}}(t)\stackrel{\mathbf{ˆ}}{\mathbf{p}}$. In particular, such renormalized transition rates adopt the form of ${\hat{L}}_{\ell i}(t)=L_{\ell i}\frac{(\mathrm{e}\mathrm{x}\mathrm{p}[(T-t)\mathbf{L}]{)}_{j\ell }}{(\mathrm{e}\mathrm{x}\mathrm{p}[(T-t)\mathbf{L}]{)}_{ji}}$ The precise mechanics here are much less important than the concrete enunciation of the more general point: conditioning effectively accomplishes new forces atop the unconditional dynamics.

Taken together, these constructions provide two complementary ways of viewing conditioning. Retrospectively, conditioning selects from the original ensemble those trajectories that satisfy a prescribed outcome. Dynamically, the same conditioning can be represented by new effective transition rates that generate the conditioned ensemble directly.

VII. DISCUSSION

The eagle casts for prey; the poet and scientist cast for word and idea. Neurons wander through tissue, holding out for just the right synaptic partners to make us, us. Many of the feats that most animate life are ultimately accomplished by exploratory trial and error. Organisms, cells, and molecules—in stark contrast to the granular deterministic control algorithms that dominate machines and computers—are often not controlled by microscopic instructions that steer their instantaneous trajectories. Instead, a different mechanism commonly organizes such systems: individuals sample variation by exploring randomly and widely, until fortunate trajectories are stabilized by selection and environmental forces. This picture of exploratory dynamics, visible recursively on all scales of biological function, invites the understanding that the magic of compounded variation and selection is available not only to Darwinian evolution on generational timescales, but also to physiological processes on much shorter timescales. In such cases, biological function can be verified once reached rather than micromanaged at every step of a trajectory¹⁰⁴, a principle also alive in algorithmic complexity theory¹³⁵. This view casts the variability in molecular and organismal activity pervading biology as a potential computational resource, and the energy it consumes not as a cost, but rather as a mechanism.

The biological diversity, architectural complexity, and new ingredients of these exploratory dynamics demand fresh quantitative thinking. In this work, we explored how common mathematics can unify a wide variety of biological settings steered by such selective, not instructive, rules. Using the typical time until success as a central measure, a common mathematical protocol describes how biochemical turnovers or resets rescue (or retard) the success of biological searching. Counterintuitively, these results explain (with fresh transparency) how continually erasing progress in a search, even when that progress is nearing triumph, can sometimes be of crucial help on average. This analysis is motivated by myriad examples where biological components are dynamically created and destroyed or subjected to aborted trajectories, from dynamic microtubule instability to kinetic proofreading to misfolded protein degradation. We also explored how the presence of local cues or drift velocities in the seeker’s motion accelerates search and introduces new physics to the (im)prudence of resets. Next, we showed how checkpoint mechanisms (allowing retention of partial progress) can reduce the average time to success from exponential in the distance to a target to subexponential or even linear in this distance (see Table II). These sundry results are usefully organized by the Péclet number, which partitions at least three surprising regimes in searcher behavior, as explored by Fig. 22. For instance, counterintuitively, our modeling reveals the surprise that faster resetting is more prudent than the optimal resetting rate without such cue. These results inspire provocative questions and beg much future work. For instance, do biological systems guided by cues bear sensitivity to such rich physical regimes and their transitions?

We tackled how mathematics for exploration smoothly generalizes to the case of multiple exploratory searchers or multiple required targets. We showed how intraflagellar transport gives a surprising manifestation of search under reset to shape biochemicals in space, focusing in that case on the concentration field created by multiple walkers. Such searches that are only permitted to explore within a typical reset or turnover time before restarting reflect a specific type of conditioning on ventured trajectories. In fact, broader types of macroscopic conditioning can also impose microscopic order on trajectories. To explore this more general idea, we first investigated trajectories conditioned to start at one position and end at another at a specified time, yielding a Brownian bridge. We explored how this concept enlarges to describe the ensemble of trajectories that end at a specified position specifically for the first time at a particular end time. Armed with these continuous random walk results, we showed how these concepts can be generalized to random walks on graphs conditioned on starting and ending states at particular times, thus generating what we might call a Schrödinger bridge on a graph. These graph bridges enjoy informational interpretations that quantify how much conditioning on a final state narrows an ensemble of realizable trajectories. Collectively, these results articulate how simple macroscopic conditioning can convert simple search dynamics bearing rich signatures of nonlocal forces compelled by compatibility with future conditions).

A vast and lovely primary literature exists on many of the themes we explore in this work. We enthusiastically refer the reader to sources such as^{3,26–28,93,115} on stochastic reset,⁴⁰ on conditional probability, and¹²⁵ for an excellent discussion of Schrödinger bridges and their ilk. Throughout this paper, our intention was to be concrete, careful, expository, and intuitive complements to these formative works. To our knowledge, we also obtain several new results, including surprises about optimal reset policies under a variety of cue characteristics and explicit constrained random walk propagators on small concrete graphs.

Some useful mathematical procedures recur throughout such analyses. A common mathematical protocol begins with the Markovian property of a search trajectory, which leads to a recurrence relation for its typical time to first reach a target state (Fig. 9). A continuum approximation turns this recurrence into a differential equation equipped with the boundary conditions appropriate to the problem. Its analytical, asymptotic, or numerical solutions then reveal how biophysical control knobs such as reset, catastrophic rescue, partial progress saving, and cues shape the typical time to biological success.

To accommodate biological complications such as multiple searchers or targets, or checkpoints, this protocol can be complemented by quantities such as the survival probability that an individual searcher trajectory has not yet succeeded by some time. For independent searchers, this immediately gives the probability that at least one has succeeded and hence the first success time of the group.

Precise descriptions and predictions about the typical timing of successful exploratory trajectories, like central results in this work, should give empirical leverage to interpret experiments. For instance, consider the magnificent case study of synaptogenesis. How can only a few tens of thousands (few × 10⁴) of human genes give rise to almost one hundred billion (≈ 9 × 10¹⁰) neurons, which form up to 10¹⁵ synaptic connections? The specificity of such connections is a staggering puzzle: in neurodevelopment, growing neurons (microns in width) can crawl past extraordinary reaches of tissue—up to tens of centimeters—to find a particular sensorimotor synaptic partner cell⁵⁶. One key to this extraordinary wiring or healing of nervous connections is a search and stabilize process, paved by specific chemical cues and stabilizing feedbacks. This occurs along the ideas of a so-called chemoaffinity hypothesis, articulated by Roger Sperry in the 1940’s and elaborated by many modern investigators. In a series of literally incisive experiments, Sperry and colleagues cut nerves attaching to the eye of fish and reconnected them in scrambled orientations¹²¹. They found that neural cells grew to reconnect to their original partner tissue cells and regions, ignoring wide swaths of possible alternative synaptic partners along the way^76,121. The fact that not only particular synaptic destinies, but the paths by which optic fibers reconnected to their partner regions, was so stereotyped implicates a strong, putatively chemically specified encoding of many of these connections. Subsequent mechanistic work, including molecular-resolution imaging of neuronal connectivity over time, has sharpened this picture^63,69,70,140. Yet, the central exploratory character of cellular attachment trajectories, stabilized by suitable and impressively specific tissue cues, is ratified by many sources of evidence. Throughout this intellectual history, mathematics has served important guiding roles in sharpening empirical questions about the details of the establishment of such neural maps³⁶. However, importantly, most such models adjudicate possible mechanisms purely on the basis of the feasible structures of the ultimate input-output maps between potential neuronal partners, and their compatibility with wild-type and perturbation experiments. We propose—in light of the type of statistical mechanics of trajectories this present paper gives simple examples of—that candidate models might also be discriminated on additional empirical bases. Both the typical trajectory dynamics and timescales in setting up such developmental maps should vary dramatically among proposed mechanisms, and thus could be high-leverage tools to help distinguish between otherwise competitively plausible biological universes. We speculate that careful framing of theoretical knobs powering exploratory dynamics, and temporal experimental statistics in developmental wiring, can be a rewarding further direction.

Our analyses of simplified biological searchers often probed the optimal resetting, checkpoint, multiplicity, or cue strategy to speed up the typical time to biological success. The presence of optimal routines under some conditions raises the tantalizing question about whether various biological systems might have tuned their presiding parameters to pursue the vicinity of such optima, as in a longer philosophical tradition of biological thinking^65,83,95. However, both naturalism and mathematics encourage caution and humility with prejudice in pursuing this question. The objective function that organisms follow in their searches is not straightforwardly only the expected first time to success, but very plausibly includes other dynamical features of trajectories. For instance, many (though not all!) of the disparate biological mechanisms that accomplish resets or effective turnovers are dissipative in character—a famous and very literal example being the hydrolysis of GTP driving the dynamic collapse of microtubules⁵⁶. Clearly, the prudence of a search strategy might attend to both time and energy. Even armed with better understanding, living organisms probably do not care so monotonically or unidimensionally about speed or energy cost as computer science paradigms might formulate most emphatically. The multivariate objective functions truly attending different organisms, including perhaps preventing that ultimate adsorbing boundary condition of death, merit respect and future inquiry.

This said, the sweep of parametric variation revealed by theoretical predictions is valuable not only to inquire about point optima per se. Even if biological systems are more guided by a mechanics of tinkering⁴⁸, and not necessarily optimal in a global sense, mechanisms like reset and ingenious conditionality can powerfully facilitate solutions that are “good enough,” or enable new control knobs for their regulation.

Studies of exploratory dynamics invite many urgent conceptual, mathematical, biological, and experimental extensions. Our analyses of random walk under drift or with exponential cues factorized away a crucial layer of physics. Specifically, these descriptions baldly coarse-grained the means by which chemical feedbacks are literally transduced into differential motion towards a target. That transduction encompasses the magnificent area of biophysics we call chemotaxis, clearly with hugely rich behaviors. An integrated synthesis of both organismal sensitivities to cues, and the downstream search processes we explore, is surely required to understand the whole space of functionality relevant to organisms.

A related tradition asks how organisms might handle only sporadic information about targets, as opposed to continuous gradients about where targets are to be found relative to a seeker’s current position. One beautiful entry in this tradition is the suggestion that simple “infotaxis” algorithms can confront the inherently sporadic information delivered by e.g. turbulent transport of biochemical bearings. This situates an interesting conceptual spectrum in connection with our work. At one end, an informative continuous cue is more or less always available to a biological agent—this is the domain of classical bacterial chemotaxis studies. At the other end, no cue is available to a biological searcher at all locally until they arrive at the terminal condition. This setting is perhaps well typified by the capture of kinetochores by microtubules, or conceivably by the establishment of Ran protein patterns⁸⁹; relevant motions in these systems execute totally blind search until a terminal condition, where simple gradient descent or infotaxis would fail. In the middle lie situations where cues are available, but weakly or sporadically, and strategies such as infotactic search should gain purchase and perhaps explanatory relevance. We suspect dissecting these distinctions and their serious consequences will be a rewarding program in further work.

Interestingly, different biological contexts may encode the parameters and algorithmic structure of search processes at different scales. For example, a bacterium’s pursuit of sugar molecules might be largely steered by its internal molecular architecture. Yet the way a bee pursues a fragrant flower is not only sculpted by its own decisions, but actively architected by the flower as well: both searcher and target are under strong, and deeply reciprocal, selection. A mathematics which better accommodates the duality of both searcher and target adaptiveness is a rich further direction.

We showed how multiple independent, statistically-identical, searchers can accelerate various search processes. Biology probably exploits parallel searchers in yet stronger and deeper ways. For example, experiments tracking the migration of T cells in zebrafish, exploratory search dynamics that support responses to antigens and infections, report that T cells show a wide spectrum of speeds, directional persistence, and turning rates⁵². One hypothesis is that this heterogeneity is crucial for pursuing targets at many characteristic distances in disordered positions. Another conceptual tradition points out that when step sizes of individual random walks vary widely say according to a power law, in a so called Lévy flight), search can be much more efficient, as studied extensively in ecology, including birds hunting prey and sporadic food¹²⁹. How resets, checkpoints, and other architectural decorations to search might complement these effects will be a rich future direction.

Many of our calculations idealize the triumphant cessation of a biological trajectory by the first arrival to a special target state of biological interest. This is a reasonable abstraction for biological events in which downstream activity is much faster than the search itself, but is not a universally appropriate description. For instance, in the earlier example of synaptogenesis, many neurons initially attach to some tentative partner cells, then execute competition and feedbacked effects to accomplish the ultimate pruning to singular partnership^33,56. Thus, at least some of the functional magic of exploratory dynamics lies not only in variation and selection, but also the third force of competition, or feedback, or precise executions of imperfect selection. Describing the mathematics of such graded activations in some unified way is a worthy future enterprise.

Our last examples show how macroscopic conditioning is a much more general organizing concept than just restricting random walk excursions to be at most a certain typical length (as in resetting). Brownian bridges and Schrödinger bridges give exact results for how distributions over trajectories are constrained by modest macroscopic criteria. They show sometimes strange and counterintuitive behaviors, including nonmonotonic changes in trajectory variances over time and nonlocal forcing from compatibility with their future boundary conditions, as in fictitious forces in noninertial reference frames in classical mechanics. One way to view their behaviors is as precise epistemic statements, namely how just knowing one thing about a macroscopic constraint gives you further gifts of microscopic knowledge about the character of trajectories. This is a view close in spirit to Jaynes’s union of statistical mechanics and information theory^49,50, and allows the collapse of trajectory diversity under conditioning to be quantified directly in informational terms. Whether or how biological systems execute conditionality of the sorts communicated by these abstractions more literally and mechanistically—for example, constraining trajectories of cell states in between pinned time endpoints in a cell cycle is an exciting frontier for future work. In a separate domain, very recent and stimulating work shows that the power of conditionality is particularly vivid in examples of competing self-replicating populations¹⁰⁴ and error correcting polymer duplication mechanisms¹⁰⁵. These settings invoke rich new physics, such as any functional benefits in stereotyped trajectories; consequences when the number of searchers change in time; and how time delay penalties might correlate with errors.

Finally, we note that another dynamic infusing exploratory biology awaits elegant and unifying mathematical exposition: time-dependent protocols or aging effects. Take the charismatic example of a phalanx of hundreds of thousands of ants, spreading out in massively parallel search of food. At first, no pheromones significantly organize or complicate the search, and the exploratory statistics are close to a random walk with resets to a home base. When enough trails have occurred, however, individual ants respond to a chemical patina of history. After a food source has been found and a feedback accelerates a highway of entrainment towards that source, the search dynamic is close to random seeking with cues. Eventually, even these cues dissipate. These typify a sort of epochal transition in the character of the search problem, its environmental inputs, and its transition rates. Smoothly and tractably accommodating the many feedbacks and hysteresis truly attending such dynamics will be a science question minting discoveries for long to come.

DATA AVAILABILITY

The data that support the findings of this study will be openly available at https://github.com/RPGroup-PBoC/2026_math_exploratory_dynamics.