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Paul C Bressloff

Publications and source records attributed to Paul C Bressloff.

At least 37 records · Page 2Linked to original sources
arXiv2023

Truncated stochastically switching processes

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or to exceed a given number of resets, respectively.

cond-mat.stat-mech↗
arXiv2023

Semipermeable interfaces and the target problem

In this chapter, we review our recent work on first passage time (FPT) problems for absorption by a target whose interface is semipermeable. For pedagogical reasons, we focus on a single Brownian particle searching for a single target in a bounded domain. We begin by writing down the forward diffusion equation for the target problem, and define various quantities of interest such as the survival probability, absorption flux, and the FPT density. We also present a general method of solution based on Green's functions and the spectral decomposition of so-called Dirichlet-to-Neumann (D-to-N) operators. We then use an encounter-based approach to extend the theory to the case of non-Markovian absorption within the target interior. Encounter-based models consider the joint probability density or generalized propagator for particle position and the amount of particle-target contact time prior to absorption. In the case of a partially absorbing target interior, the contact time is given by a Brownian functional known as the occupation time. Finally, we develop a more general probabilistic model of single-particle diffusion through semi-permeable interfaces, by combining the encounter-based approach with so-called snapping out Brownian motion (BM). Snapping out BM sews together successive rounds of partially absorbing BMs that are restricted to either the interior or the exterior of the semipermeable interface. The rule for terminating each round is implemented using encounter-based model of partially absorbing BM. We show that this results in a time-dependent permeability that can be heavy-tailed.

cond-mat.stat-mech↗
arXiv2023

Transition path theory for diffusive search with stochastic resetting

Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain $A$ and then delivering it to a target domain $B$. The rate of resource accumulation at the target, $k_{AB}$, is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating $k_{AB}$ involves determining the total probability flux of direct transport paths across a dividing surface $S$ between $A$ and $B$. This requires taking into account discontinuous jumps across $S$ due to resetting. We derive a general expression for $k_{AB}$ and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which $k_{AB}$ is maximized. We explore how this feature depends on model parameters.

cond-mat.stat-mech↗
arXiv2023

Asymptotic analysis of conversion-limited phase separation

Liquid-liquid phase separation plays a major role in the formation and maintenance of various membrane-less subcellular structures in the cytoplasm and nucleus of cells. Biological condensates contain enhanced concentrations of proteins and RNA, many of which can be continually exchanged with the surrounding medium. Coarsening is an important step in the kinetics of phase separation, whereby an emulsion of polydisperse condensates transitions to a single condensate in thermodynamic equilibrium with a surrounding dilute phase. A key feature of biological phase separation is the co-existence of multiple condensates over significant time scales, which is consistent with experimental observations showing a slowing of coarsening rates. It has recently been proposed that one rate limiting step could be the slow interfacial conversion of a molecular constituent between the dilute and dense phases. In this paper we analyze conversion-limited phase separation within the framework of diffusion in singularly perturbed domains, which exploits the fact that biological condensates tend to be much smaller than the size of a cell. Using matched asymptotic analysis, we solve the quasi-static diffusion equation for the concentration in the dilute phase, and then derive kinetic equations for the slow growth/shrinkage of the condensates. This provides a systematic way of obtaining corrections to mean-field theory that take into account the geometry of the cell and the locations of all the condensates.

cond-mat.soft↗
arXiv2023

Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time $\ell(t)$. If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density $ψ(\ell)$. The marginal probability density for particle position $\X(t)$ prior to absorption depends on $ψ$ and the joint probability density for the pair $(\X(t),\ell(t))$, also known as the local time propagator. In the case of normal diffusion one can use a Feynman-Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval $[0,L]$ with a reflecting boundary at $x=L$. In particular, we determine how the choice of function $ψ$ affects the large-$t$ power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.

cond-mat.stat-mech↗
arXiv2023

Encounter-based reaction-subdiffusion model II: partially absorbing traps and the occupation time propagator

In this paper we develop an encounter-based model of reaction-subdiffusion in a domain $Ω$ with a partially absorbing interior trap $\calU\subset Ω$. We assume that the particle can freely enter and exit $\calU$, but is only absorbed within $\calU$. We take the probability of absorption to depend on the amount of time a particle spends within the trap, which is specified by a Brownian functional known as the occupation time $A(t)$. The first passage time (FPT) for absorption is identified with the point at which the occupation time crosses a random threshold $\widehat{A}$ with probability density $ψ(a)$. Non-Markovian models of absorption can then be incorporated by taking $ψ(a)$ to be non-exponential. The marginal probability density for particle position $\X(t)$ prior to absorption depends on $ψ$ and the joint probability density for the pair $(\X(t),A(t))$, also known as the occupation time propagator. In the case of normal diffusion one can use a Feynman-Kac formula to derive an evolution equation for the propagator. However, care must be taken when combining fractional diffusion with chemical reactions in the same medium. Therefore, we derive the occupation time propagator equation from first principles by taking the continuum limit of a heavy-tailed CTRW. We then use the solution of the propagator equation to investigate conditions under which the mean FPT (MFPT) for absorption within a trap is finite. We show that this depends on the choice of threshold density $ψ(a)$ and the subdiffusivity. Hence, as previously found for evanescent reaction-subdiffusion models, the processes of subdiffusion and absorption are intermingled.

cond-mat.stat-mech↗
arXiv2023

A numerical method for solving snapping out Brownian motion in 2D bounded domains

Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, including gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, thermal conduction in composite media, synaptic receptor trafficking, and intercellular gap junctions. At the single particle level, diffusion across a semi-permeable interface can be formulated in terms of so-called snapping out Brownian motion (SNOBM). The latter sews together successive rounds of reflected BM, each of which is restricted to one side of the interface. Each round of reflected BM is killed when the local time at the interface exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of the interface.) The particle then immediately resumes reflected BM on the same side or the other side of the interface according to a stochastic switch, and the process is iterated. In this paper, we develop a Monte Carlo algorithm for simulating a two-dimensional version of SNOBM, which is used to solve a first passage time (FPT) problem for diffusion in a domain with semi-permeable partially absorbing traps. Our method combines a walk-on-spheres (WOS) method with an efficient algorithm for computing the boundary local time that uses a Skorokhod integral representation of the latter. We validate our algorithm by comparing the Monte Carlo estimates of the MFPT to the exact solution for a single circular trap, and show that our simulations are consistent with asymptotic results obtained for the 2D narrow capture problem involving multiple small circular targets. We also discuss extensions to higher dimensions.

cond-mat.stat-mech↗
arXiv2023

Close encounters of the sticky kind: Brownian motion at absorbing boundaries

Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. Prior to absorption, the surface is typically assumed to act as a totally reflecting boundary, which means that the contact time is determined by a Brownian functional known as the boundary local time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, active matter, finance, and human crowd dynamics. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of non-sticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. Different models of absorption correspond to different choices for the random threshold probability distribution. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities of interest depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP). The diffusion limit is well-defined provided that the velocity of the RTP is biased and spatially varying within a boundary layer around the origin.

cond-mat.stat-mech↗
arXiv2023

Trapping of an active Brownian particle at a partially absorbing wall

Active matter concerns the self-organization of energy consuming elements such as motile bacteria or self-propelled colloids. A canonical example is an active Brownian particle (ABP) that moves at constant speed while its direction of motion undergoes rotational diffusion. When ABPs are confined within a channel, they tend to accumulate at the channel walls, even when inter-particle interactions are ignored. Each particle pushes on the boundary until a tumble event reverses its direction. The wall thus acts as a sticky boundary. In this paper we consider a natural extension of sticky boundaries that allows for a particle to be permanently killed (absorbed) whilst attached to a wall. In particular, we investigate the first passage time (FPT) problem for an ABP in a two-dimensional channel where one of the walls is partially absorbing. Calculating the exact FPT statistics requires solving a non-trivial two-way diffusion boundary value problem (BVP). We follow a different approach by separating out the dynamics away from the absorbing wall from the dynamics of absorption and escape whilst attached to the wall. Using probabilistic methods, we derive an explicit expression for the MFPT of absorption, assuming that the arrival statistics of particles at the wall are known. Our method also allows us to incorporate a more general encounter-based model of absorption.

cond-mat.stat-mech↗
arXiv2022

2D interfacial diffusion model of inhibitory synaptic receptor dynamics

The lateral diffusion and trapping of protein receptors within the postsynaptic membrane of a neuron plays a key role in determining the strength of synaptic connections and their regulation during learning and memory. In this paper we construct and analyze a 2D interfacial diffusion model of inhibitory synaptic receptor dynamics. The model involves three major components. First, the boundary of each synapse is treated as a semi-permeable interface due to the effects of cytoskeletal structures. Second, the effective diffusivity within a synapse is taken to be smaller than the extrasynaptic diffusivity due to the temporary binding to scaffold protein buffers within the synapse. Third, receptors from intracellular pools are inserted into the membrane extrasynaptically and internalized extrasynaptically and synaptically. We first solve the model equations for a single synapse in an unbounded domain and explore how the non-equilibrium steady-state number of synaptic receptors depends on model parameters. We then use matched asymptotic analysis to solve the corresponding problem of multiple synapses in a large, bounded domain. Finally, treating a synapse as a phase separated condensate of scaffold proteins, we describe how diffusion of individual scaffold proteins can also be modeled in terms of interfacial diffusion. We thus establish interfacial diffusion as a general paradigm for exploring synaptic dynamics and plasticity.

cond-mat.stat-mech↗
arXiv2022

The 3D narrow capture problem for traps with semipermeable interfaces

In this paper we analyze the narrow capture problem for a single Brownian particle diffusing in a three-dimensional (3D) bounded domain containing a set of small, spherical traps. The boundary surface of each trap is taken to be a semipermeable membrane. That is, the continuous flux across the interface is proportional to an associated jump discontinuity in the probability density. The constant of proportionality is identified with the permeability $κ$. In addition, we allow for discontinuities in the diffusivity and chemical potential across each interface; the latter introduces a directional bias. We also assume that the particle can be absorbed (captured) within the interior of each trap at some Poisson rate $γ$. In the small-trap limit, we use matched asymptotics and Green's function methods to calculate the splitting probabilities and unconditional MFPT to be absorbed by one of the traps. However, the details of the analysis depend on how various parameters scale with the characteristic trap radius $ε$. Under the scalings $γ=O(1/ε^2)$ and $κ=O(1/ε)$, we show that the semipermeable membrane reduces the effective capacitance $\calC$ of each spherical trap compared to the standard example of totally absorbing traps. Finally, we consider the unidirectional limit in which each interface only allows particles to flow into a trap. The traps then act as partially absorbing surfaces with a constant reaction rate $κ$. Combining asymptotic analysis with the encounter-based formulation of partially reactive surfaces, we show how a generalized surface absorption mechanism (non-Markovian) can be analyzed in terms of the capacitances $\calC$. We thus establish that a wide range of narrow capture problems can be characterized in terms of the effective capacitances of the traps.

cond-mat.stat-mech↗
arXiv2022

Diffusion with stochastic resetting screened by a semipermeable interface

In this paper we consider the diffusive search for a bounded target $Ω\in \R^d$ with its boundary $\partial Ω$ totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface $\partial \calM$ with $Ω\subset \calM\subset \R^d$. That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position $\x_0$ according to a Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be outside the interface, $\x_0\in \calM^c$, which means that resetting does not occur when the particle is within the interior of $\partial \calM$. Hence, the semipermeable interface also screens out the effects of resetting. We first solve the boundary value problem (BVP) for diffusion on the half-line $x\in [0,\infty)$ with an absorbing boundary at $x=0$, a semipermeable barrier at $x=L$, and stochastic resetting to $x_0>L$ for all $x>L$. We calculate the mean first passage time (MFPT) to be absorbed by the target and explore its behavior as a function of the permeability $κ_0$ of the interface and its spatial position $L$. We then perform the analogous calculations for a three-dimensional (3D) spherically symmetric interface and target, and show that the MFPT exhibits the same qualitative behavior as the 1D case. Finally, we introduce a stochastic single-particle realization of the search process based on a generalization of so-called snapping out BM. The latter sews together successive rounds of reflecting Brownian motion on either side of the interface. The main challenge is establishing that the probability density generated by the snapping out BM satisfies the permeable boundary conditions at the interface. We show how this can be achieved using renewal theory.

cond-mat.stat-mech↗
arXiv2022

Renewal equations for single-particle diffusion through a semipermeable interface

Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional diffusion through a barrier with constant permeability $κ_0$ can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or right of the barrier. Each round is killed (absorbed) at the barrier when its Brownian local time exceeds an exponential random variable parameterized by $κ_0$. A new round is then immediately started in either direction with equal probability. It has recently been shown that the probability density for snapping out BM satisfies a renewal equation that relates the full density to the probability densities of partially reflected BM on either side of the barrier. Moreover, generalized versions of the renewal equation can be constructed that incorporate non-Markovian, encounter-based models of absorption. In this paper we extend the renewal theory of snapping out BM to single-particle diffusion in bounded domains and higher spatial dimensions. We also consider an example of an asymmetric interface in which the directional switching after each absorption event is biased. Finally, we show how to incorporate an encounter-based model of absorption for single-particle diffusion through a spherically symmetric interface. We find that, even when the same non-Markovian model of absorption applies on either side of the interface, the resulting permeability is an asymmetric time-dependent function with memory. Moreover, the permeability functions tend to be heavy-tailed.

cond-mat.stat-mech↗
arXiv2022

Encounter-based model of a run-and-tumble particle

In this paper we extend the encounter-based model of diffusion-mediated surface absorption to the case of an unbiased run-and-tumble particle (RTP) confined to a finite interval $[0,L]$ and switching between two constant velocity states $\pm v$ at a rate $α$. The encounter-based formalism is motivated by the observation that various surface-based reactions are better modeled in terms of a reactivity that is a function of the amount of time that a particle spends in a neighborhood of an absorbing surface, which is specified by a functional known as the boundary local time. The effects of surface reactions are taken into account by identifying the first passage time (FPT) for absorption with the event that the local time crosses some random threshold $\widehat{\ell}$. In the case of a Brownian particle, the local time $\ell(t)$ is a continuous non-decreasing function of the time $t$. Taking $Ψ(\ell)\equiv ¶[\widehat{\ell}>\ell]$ to be an exponential distribution, $Ψ[\ell]=\e^{-κ_0\ell}$, is equivalent to imposing a Robin boundary condition with a constant rate of absorption $κ_0$. One major difference in the encounter-based model of an RTP is that the boundary local time $\ell(t)$ is a now a discrete random variable that counts the number of collisions of the RTP with the boundary. Given this modification, we show that in the case of a geometric distribution $Ψ(\ell)=z^{\ell}$, $z=1/(1+κ_0/v)$, we recover the RTP analog of the Robin boundary condition. This allows us to solve the boundary value problem (BVP) for the joint probability density for particle position and the local time, and thus incorporate more general models of absorption based on non-geometric distributions $Ψ(\ell)$.

cond-mat.stat-mech↗
arXiv2022

A probabilistic model of diffusion through a semi-permeable barrier

Diffusion through semipermeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogs such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability $κ_0$ is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by $κ_0$. A new round is then immediately started in either direction with equal probability. In this paper we use a combination of renewal theory, Laplace transforms and Green's function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modeling diffusion through a semipermeable barrier. This includes modifications of the diffusion process away from the barrier (eg. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to time-dependent permeabilities.

cond-mat.stat-mech↗
arXiv2022

Morphogen gradient formation in partially absorbing media

Morphogen gradients play an essential role in the spatial regulation of cell patterning during early development. The classical mechanism of morphogen gradient formation involves the diffusion of morphogens away from a localized source combined with some form of bulk absorption. Morphogen gradient formation plays a crucial role during early development, whereby a spatially varying concentration of morphogen protein drives a corresponding spatial variation in gene expression during embryogenesis. In most models, the absorption rate is taken to be a constant multiple of the local concentration. In this paper, we explore a more general class of diffusion-based model in which absorption is formulated probabilistically in terms of a stopping time condition. Absorption of each particle occurs when its time spent within the bulk domain (occupation time) exceeds a randomly distributed threshold $a$; the classical model with a constant rate of absorption is recovered by taking the threshold distribution $Ψ(a)=\e^{-κ_0 a}$. We explore how the choice of $Ψ(a)$ affects the steady-state concentration gradient, and the relaxation to steady-state as determined by the accumulation time. In particular, we show that the more general model can generate similar concentration profiles to the classical case, while significantly reducing the accumulation time.

cond-mat.stat-mech↗
arXiv2022

Diffusion in a partially absorbing medium with position and occupation time resetting

In this paper we consider diffusion in a domain $Ω$ containing a partially absorbing target $\calM$ with position and occupation time resetting. The occupation time $A_t$ is a Brownian functional that determines the amount of time that the particle spends in $ \calM$ over the time interval $[0,t]$. We assume that there exists some internal state $\calU_t$ of the particle at time $t$ which is modified whenever the particle is diffusing within $\calM $. The state $\calU_t$ is taken to be a monotonically increasing function of $A_t$, and absorption occurs as soon as $\calU_t$ crosses some fixed threshold. We first show how to analyze threshold absorption in terms of the joint probability density or generalized propagator $P(\x,a,t|\x_0)$ for the pair $(\X_t,A_t)$ in the case of a non-absorbing substrate $\calM$, where $\X_t$ is the particle position at time $t$ and $\x_0$ is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position $\X_t$ and the internal state $\calU_t$ are reset to their initial values, $\X_t\rightarrow \x_0$ and $\calU_t\rightarrow 0$, at a Poisson rate $r$. The latter is mathematically equivalent to resetting the occupation time, $A_t\rightarrow 0$. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the double Laplace transform of $P(\x,a,t|\x_0)$ with respect to $t$ and $a$. In order to develop the basic theory, we focus on one-dimensional (1D) diffusion with $\calM$ given by a finite or semi-infinite interval, and explore how the MFPT with resetting depends on various model parameters. We also compare the threshold mechanism with the classical case of a constant absorption rate.

cond-mat.stat-mech↗
arXiv2022

Accumulation times for diffusion-mediated surface reactions

In this paper we consider a multiparticle version of a recent probabilistic framework for studying diffusion-mediated surface reactions. The basic idea of the probabilistic approach is to consider the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a totally reflecting boundary; the effects of surface reactions are then incorporated via an appropriate stopping condition for the local time. The propagator is determined by solving a Robin boundary value problem, in which the constant rate of reactivity is identified as the Laplace variable $z$ conjugate to the local time, and then inverting the solution with respect to $z$. Here we reinterpret the propagator as a particle concentration in which surface absorption is counterbalanced by particle source terms. We investigate conditions under which there exists a non-trivial steady state solution, and analyze the relaxation to steady state by calculating the corresponding accumulation time. In particular, we show that the first two moments of the stopping local time density have to be finite.

cond-mat.stat-mech↗