Proxitaxis: Proximity-Driven Search Dynamics

Updated 6 July 2026

Proxitaxis is a distance-only adaptive search strategy that combines local exploration with stochastic resets and inspection moves based solely on proximity.
The method optimizes resetting rates and adaptivity exponents to maximize capture probability, exhibiting distinct phase transitions as target distance varies.
Proxitaxis underlies various proximity-mediated guidance mechanisms in biology and engineered systems, linking search theory with active matter migration.

Searching arXiv for recent and foundational papers relevant to proxitaxis and closely related guidance mechanisms. arXiv search query: "proxitaxis OR topotaxis active particles obstacle sliding guidance chemotaxis resetting proximity search"

Proxitaxis denotes guidance by proximity rather than by a fully resolved directional cue. In the strict formulation introduced as an adaptive search strategy, the searcher has access only to the instantaneous distance to a target, not its bearing, and combines distance-dependent local motion, stochastic resetting to a reference position, and inspection moves that update that reference position when a closer location is discovered (Vecchio et al., 8 Jul 2025). Several earlier studies are mechanistically close to this idea because they generate directed migration from nearby structures, local contact rules, or localized sources rather than from direct directional sensing; this broader usage overlaps with obstacle-mediated topotaxis, trail following, adhesion-gradient crawling, and source-localized phoretic guidance (Sadjadi et al., 2023).

1. Definition and conceptual scope

In its explicit search-theoretic sense, proxitaxis is a distance-only search protocol. The operative signal is the scalar proximity $R(t)=\|\vec R(t)\|$ , not a gradient vector, so the searcher cannot perform gradient descent in the usual chemotactic sense. The strategy therefore replaces directional steering by a combination of three control elements: local adaptive exploration, stochastic returns to a memorized location, and intermittent updating of that memory when a better location is found (Vecchio et al., 8 Jul 2025).

This sharply distinguishes proxitaxis from chemotaxis, where a searcher or cell uses directional information from a chemical field, and from standard stochastic resetting, where resets occur to a fixed initial location rather than to a dynamically updated best-so-far position. It also differs from infotaxis, which is usually cast as an information-maximization problem, and from run-and-tumble search, where control is exerted primarily through persistence and reorientation rather than through memory-based reset geometry (Vecchio et al., 8 Jul 2025).

A broader mechanistic reading is suggested by adjacent literature. In that reading, proxitaxis describes migration biased by proximity-dependent interactions with nearby obstacles, surfaces, trails, ligands, or source regions. In eukaryotic cells, local extracellular signals can bias protrusion competition through Ras and PI3K–PIP $_3$ modules; in bacterial trail following, local gradients of deposited polymer bias both translation and orientation; in ratchet assays, asymmetric protrusion success rectifies otherwise stochastic protrusive activity (Khamviwath et al., 2013). This suggests that “proxitaxis” can function both as a formal search strategy and as a descriptive label for proximity-mediated guidance more generally.

2. Formal search-theoretic formulation

The formal model is posed in $d$ spatial dimensions with a spherical target of radius $\epsilon$ centered at the origin. Search proceeds in intervals $\{\tau_1,\tau_2,\dots\}$ , independently drawn from the exponential law

$p(\tau)=b\,e^{-b\tau},$

where $b$ is the inspection rate (Vecchio et al., 8 Jul 2025).

Within a given interval, the searcher starts from a reference position $\vec R_0$ with $R_0=\|\vec R_0\|>\epsilon$ . During a small time increment, it either resets to $\vec R_0$ with probability $_3$ 0 or performs local diffusion with distance-dependent diffusivity

$_3$ 1

so that motion becomes more active at smaller target distance (Vecchio et al., 8 Jul 2025). The corresponding radial Itô process is

$_3$ 2

with $_3$ 3 (Vecchio et al., 8 Jul 2025).

At the end of an interval, an inspection move updates the reset point. If the current position is closer to the target than the present reference point, the new reference becomes the current position; otherwise the previous reference is retained. Operationally, proxitaxis therefore stores a best-so-far location in terms of scalar proximity and uses resetting to erase unfavorable excursions (Vecchio et al., 8 Jul 2025).

The single-interval capture probability is defined from the survival probability $_3$ 4 as

$_3$ 5

and the renewal relation for resetting gives

$_3$ 6

For the solvable family $_3$ 7, with

$_3$ 8

the exact capture probability becomes

$_3$ 9

where $d$ 0 is the modified Bessel function of the second kind (Vecchio et al., 8 Jul 2025).

3. Optimal control and phase structure

The strategy is optimized over the resetting rate $d$ 1 and the adaptivity exponent $d$ 2. A central result is that the capture probability has a unique global maximum for each parameter set, and that maximizing $d$ 3 is equivalent to minimizing the mean first-passage time with resetting: $d$ 4 Thus the optimal proxitactic strategy is simultaneously the optimal interval-wise capture strategy and the optimal resetting-based first-passage strategy (Vecchio et al., 8 Jul 2025).

The optimized controls $d$ 5 do not vary smoothly everywhere. Instead, the formalism exhibits multiple phase transitions as a function of the initial distance $d$ 6 and the inspection rate $d$ 7. In one dimension, a threshold

$d$ 8

separates two distinct phase topologies (Vecchio et al., 8 Jul 2025).

Inspection regime	Distance regime	Optimal control
$d$ 9	$\epsilon$ 0	$\epsilon$ 1
$\epsilon$ 2	$\epsilon$ 3	$\epsilon$ 4
$\epsilon$ 5	$\epsilon$ 6	$\epsilon$ 7
$\epsilon$ 8	$\epsilon$ 9	$\{\tau_1,\tau_2,\dots\}$ 0
$\{\tau_1,\tau_2,\dots\}$ 1	$\{\tau_1,\tau_2,\dots\}$ 2	$\{\tau_1,\tau_2,\dots\}$ 3
$\{\tau_1,\tau_2,\dots\}$ 4	$\{\tau_1,\tau_2,\dots\}$ 5	$\{\tau_1,\tau_2,\dots\}$ 6

For $\{\tau_1,\tau_2,\dots\}$ 7, the intermediate optimum is ordinary diffusion with resetting. For $\{\tau_1,\tau_2,\dots\}$ 8, the intermediate optimum instead uses adaptive diffusion without resetting. This is the sense in which the model exhibits multiple phase transitions: as $\{\tau_1,\tau_2,\dots\}$ 9 decreases, the optimal search protocol changes nonanalytically twice, and the identity of the intermediate phase itself changes at $p(\tau)=b\,e^{-b\tau},$ 0 (Vecchio et al., 8 Jul 2025).

The same qualitative structure is reported in two and three dimensions. Representative critical values include $p(\tau)=b\,e^{-b\tau},$ 1 and $p(\tau)=b\,e^{-b\tau},$ 2 in $p(\tau)=b\,e^{-b\tau},$ 3 for $p(\tau)=b\,e^{-b\tau},$ 4, $p(\tau)=b\,e^{-b\tau},$ 5, and $p(\tau)=b\,e^{-b\tau},$ 6 and $p(\tau)=b\,e^{-b\tau},$ 7 in $p(\tau)=b\,e^{-b\tau},$ 8 for the same $p(\tau)=b\,e^{-b\tau},$ 9 and $b$ 0. This suggests that the multi-transition structure is not a one-dimensional artifact (Vecchio et al., 8 Jul 2025).

Near the target, the optimum becomes singular in a universal way: $b$ 1 as $b$ 2. The leading-order behavior is reported to hold in all dimensions and to be independent of $b$ 3. For $b$ 4, the optimum freezes to $b$ 5 and $b$ 6, and the optimized capture probability tends to unity (Vecchio et al., 8 Jul 2025).

4. Broader mechanistic usage: proximity-mediated guidance

Several research programs are described as highly relevant to the idea of proxitaxis even though they do not use the term in the strict resetting-based sense. What they share is a conversion of local, proximity-dependent interactions into directional bias.

Mechanism	Operative cue	Representative source
Obstacle-contact topotaxis	Sliding-angle heterogeneity at nearby pillars	(Sadjadi et al., 2023)
Eukaryotic directional sensing	Local membrane input biasing protrusion competition	(Khamviwath et al., 2013)
Trail following	Local Psl density and gradient on a surface	(Gelimson et al., 2016)
Protrusion ratchet guidance	Asymmetric success of protrusions across adhesive gaps	(Caballero et al., 2013)
Multivalent crawling	Ligand-density gradient on a substrate	(Sleath et al., 2023)
Source-localized self-phoresis	Localized fuel patch near a wall	(Mancuso et al., 2024)

In a pillar array with uniform obstacle density, active particles can accumulate in regions with larger sliding angle $b$ 7 solely because the obstacle-contact rule varies across space. The mechanism is not a density gradient but an interface-scattering asymmetry induced by different near-obstacle sliding dynamics; in the authors’ effective description, the key quantities are asymmetric inter-region crossing probabilities $b$ 8 and $b$ 9 (Sadjadi et al., 2023). This is a paradigmatic example of proximity-mediated redirection by nearby structures.

In eukaryotic cells, the Ras adaptive module and the PI3K–PIP $\vec R_0$ 0 amplifier convert weak local extracellular asymmetries into strongly localized protrusive domains. The model emphasizes that movement is not directly prescribed as a velocity law; rather, a local signal biases which protrusions survive and dominate. This suggests a mechanistic link between proximity to a cue and emergent migration direction without a fixed steering rule (Khamviwath et al., 2013).

In Pseudomonas aeruginosa, deposited Psl trails constitute a persistent substrate-bound memory. The stochastic many-body model includes both translational drift up local trail gradients and an orientational response

$\vec R_0$ 1

and the orientational term is reported as crucial for sustained trail following. This is proxitaxis-like in the sense that guidance is local, persistent, and mediated by contact-relevant environmental modification rather than long-range field sensing (Gelimson et al., 2016).

Ratchet assays on asymmetric fibronectin micropatterns provide a cell-scale version of the same logic. There, directional motion is predicted from the asymmetry of “efficient protrusions,” summarized by the direction index

$\vec R_0$ 2

with $\vec R_0$ 3. Long-term motion is captured by a biased persistent random walk whose transition probabilities are determined from protrusion statistics (Caballero et al., 2013). This suggests that proxitactic bias can emerge from asymmetric protrusion success even when the immediate cue is geometrical rather than scalar distance.

A recurring theme in engineered and biomimetic systems is that proximity-biased guidance often requires an externalized memory or a source-relative physical interaction. One explicit example is self-assisted amoeboid navigation in mazes, where a cell follows an effective concentration

$\vec R_0$ 4

combining an external chemoattractant with a self-secreted repellent. In the reported maze geometries, success rates increase from $\vec R_0$ 5 to as high as $\vec R_0$ 6, and from $\vec R_0$ 7 to $\vec R_0$ 8 and $\vec R_0$ 9 in two other configurations, when the self-secreted marker is included. The paper interprets this marker as a memory of visited regions (Hecht et al., 2011). This is not the same algorithm as proxitaxis with resetting, but it is closely related in function: local information is converted into a recursive avoidance of unproductive states.

Collective repulsive autochemotaxis provides a many-body counterpart. Searchers release a chemical clue and move away from high concentrations, thereby avoiding already explored or crowded regions. The reported effect is an improvement in mean first-passage time by orders of magnitude depending on coupling strength, driven by increased persistence length and by a more homogeneous distribution of searchers (Meyer et al., 2024). A plausible implication is that proxitaxis-like search can be generalized from target approach to distributed exploration using environmental memory.

Mechanically coupled active particles show that proximity bias can arise without explicit sensing. For an active-passive dimer in an activity gradient, the crossover from anti-chemotaxis to chemotaxis occurs at

$R_0=\|\vec R_0\|>\epsilon$ 0

with $R_0=\|\vec R_0\|>\epsilon$ 1 in three dimensions, and the short-time uphill drift is maximal at

$R_0=\|\vec R_0\|>\epsilon$ 2

which gives $R_0=\|\vec R_0\|>\epsilon$ 3 in three dimensions (Vuijk et al., 2020). The mechanism is the asymmetric exploration of the activity gradient by the active unit while it is bound to a load.

Localized-source guidance is made especially explicit for a self-phoretic Janus particle near a patch source of fuel on a wall. Depending on the Damköhler number $R_0=\|\vec R_0\|>\epsilon$ 4, the phoretic mobilities $R_0=\|\vec R_0\|>\epsilon$ 5 and $R_0=\|\vec R_0\|>\epsilon$ 6, and the patch size $R_0=\|\vec R_0\|>\epsilon$ 7, the particle may be attracted to the source, repelled from it, or settle into a stable hovering state above the patch center. In one cap-up spherical case, the crossover height is fitted by

$R_0=\|\vec R_0\|>\epsilon$ 8

For a prolate spheroid with $R_0=\|\vec R_0\|>\epsilon$ 9, $\vec R_0$ 0, $\vec R_0$ 1, and $\vec R_0$ 2, numerical trajectories show migration toward the patch, reorientation to the wall normal, and convergence to hovering (Mancuso et al., 2024). This is a concrete source-localization instance of proxitaxis in the broader sense.

Multivalent vesicles crawling along substrate ligand gradients provide an adhesion-mediated realization. Motion is biased by the multivalent free-energy gradient $\vec R_0$ 3, but the paper emphasizes that dynamics are reaction-limited, with drift scaling as

$\vec R_0$ 4

rather than being determined solely by a static adhesion preference. Reported drift velocities are $\vec R_0$ 5 for $\vec R_0$ 6 nt and $\vec R_0$ 7 for $\vec R_0$ 8 nt, while $\vec R_0$ 9 nt yields negligible drift (Sleath et al., 2023). This indicates that proximity-based guidance can be thermodynamically biased yet kinetically gated.

6. Detection, classification, and open problems

Because many distinct mechanisms can generate target-directed or source-relative motion, detecting proxitaxis-like behavior from trajectories is not equivalent to identifying its mechanism. A general statistical test for long-range attraction computes, for each near-target movement step, a p-value

$_3$ 00

interpreted as the probability that a move at least as target-directed as observed could arise from that cell’s target-independent migration statistics. The observed distribution of $_3$ 01 is then compared against a randomized-target reference (Metzner, 2019). This provides an operational test for attraction-like behavior, but it does not distinguish chemotaxis from trail following, mechanical guidance, or other proxitaxis-like processes.

Mechanistic ambiguity is further underscored by state-switching transport models. In a ligand or substrate gradient, an ensemble can move up-gradient when the bound-state diffusivity is lower than the free-state diffusivity and down-gradient when it is higher. The effective transport law is generated by spatially varying bound-state occupancy rather than by an explicit steering force (Mandal et al., 2021). Likewise, mechanically mediated migration can produce chemotactic drift mainly by modulating persistence times rather than instantaneous speed; in the oscillating-multipole model of soft active objects, the relevant asymmetry lies in the barrier-controlled run times $_3$ 02, not primarily in the velocity magnitudes (Leoni et al., 2015). This suggests that proxitaxis is better treated as an operational description of proximity-biased migration than as a single mechanistic class.

A further classification issue concerns the scope of the term itself. The 2025 search theory defines proxitaxis narrowly as a distance-only adaptive strategy with resetting and inspection (Vecchio et al., 8 Jul 2025). Earlier and adjacent works motivate a broader usage in which nearby structures, localized sources, or substrate-bound traces bias motion without furnishing a full directional cue. This suggests a two-level taxonomy: a strict algorithmic sense and a broader mechanistic sense.

The explicit search formalism points to immediate extensions to multiple targets and multiple searchers (Vecchio et al., 8 Jul 2025). The surrounding literature suggests additional open directions: noisy distance sensing, moving targets, heterogeneous media, quenched disorder, and collective environments in which the relevant “proximity” variable is itself dynamically written by the searchers. Taken together, these works place proxitaxis at the intersection of first-passage theory, adaptive control, active matter, and migration through structured environments.