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Spheres with absorbing rings, shown in brown. The ring size increases from left to right. The most right sphere is perfectly absorbing.
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This paper focuses on trapping of diffusing particles by a sphere with an absorbing cap of arbitrary size on the otherwise reflecting surface. We approach the problem using boundary homogenization which is an approximate replacement of non-uniform boundary conditions on the surface of the sphere by an effective uniform boundary condition with appro...
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Topologically non-trivial Hamiltonians with periodic boundary conditions are characterized by strictly quantized invariants. Open questions and fundamental challenges concern their existence, and the possibility of measuring them in systems with open boundary conditions and limited spatial extension. Here, we consider transport in Hofstadter strips...
Citations
... More generally, trapping rates have been derived in a great variety of scenarios, including particles diffusing above patchy planes [3,6,7,9,31,34], patchy particles diffusing above patchy planes [38], bimolecular reactions between two diffusive particles where either one is patchy [15,29,30,48] or both are patchy [37], and a patchy particle diffusing above an entirely reactive plane [27]. ...
... Prior work on boundary homogenization has primarily considered perfectly reactive patches [3,6,7,9,15,27,29,30,31,34,37,38,48]. Recently, some other groups have considered diffusive interactions with partially reactive patches and targets [11,13,42]. ...
A wide variety of physical, chemical, and biological processes involve diffusive particles interacting with surfaces containing reactive patches. The theory of boundary homogenization seeks to encapsulate the effective reactivity of such a patchy surface by a single trapping rate parameter. In this paper, we derive the trapping rate for partially reactive patches occupying a small fraction of a surface. We use matched asymptotic analysis, double perturbation expansions, and homogenization theory to derive formulas for the trapping rate in terms of the far-field behavior of solutions to certain partial differential equations (PDEs). We then develop kinetic Monte Carlo (KMC) algorithms to rapidly compute these far-field behaviors. These KMC algorithms depend on probabilistic representations of PDE solutions, including using the theory of Brownian local time. We confirm our results by comparing to KMC simulations of the full stochastic system. We further compare our results to prior heuristic approximations.
... In these systems, cells infer the spatial location of an external source through the noisy arrivals of diffusing molecules (searchers) to membrane receptors (targets). Dating back to the seminal work of Berg and Purcell (1977), there is a long history of mathematical modeling of such systems (Zwanzig 1990;Zwanzig and Szabo 1991;Bernoff et al. 2018;Lindsay et al. 2017;Berezhkovskii et al. 2004Berezhkovskii et al. , 2006Dagdug et al. 2016;Eun 2017;Muratov and Shvartsman 2008;Lawley and Miles 2019;Eun 2020;Handy and Lawley 2021). A recent work investigated the theoretical limits of what a cell could infer about the source location from the number of arrivals at different membrane receptors . ...
Biological events are often initiated when a random “searcher” finds a “target,” which is called a first passage time (FPT). In some biological systems involving multiple searchers, an important timescale is the time it takes the slowest searcher(s) to find a target. For example, of the hundreds of thousands of primordial follicles in a woman’s ovarian reserve, it is the slowest to leave that trigger the onset of menopause. Such slowest FPTs may also contribute to the reliability of cell signaling pathways and influence the ability of a cell to locate an external stimulus. In this paper, we use extreme value theory and asymptotic analysis to obtain rigorous approximations to the full probability distribution and moments of slowest FPTs. Though the results are proven in the limit of many searchers, numerical simulations reveal that the approximations are accurate for any number of searchers in typical scenarios of interest. We apply these general mathematical results to models of ovarian aging and menopause timing, which reveals the role of slowest FPTs for understanding redundancy in biological systems. We also apply the theory to several popular models of stochastic search, including search by diffusive, subdiffusive, and mortal searchers.
... N 1 /(4πR 2 ) is the density of patches). Since the seminal derivation of the leading order result in (3), a large literature has been devoted to obtaining higher order corrections which account for details such as patch shape, patch arrangement and clustering, surface curvature, and patch diffusivity [11][12][13][14][15][16][17][18][19][20][21] . ...
... This work is related to a long line of previous studies. Boundary homogenization has been used to study the reaction kinetics for patchy particles [18][19][20][21]25 and patchy surfaces 11,12,15,16,20,32,33 , where the patchy object (particle or surface) interacts with a uniformly reactive particle or surface. Related work on interactions of two patchy spherical particles includes Refs. ...
Trapping diffusive particles at surfaces is a key step in many systems in chemical and biological physics. Trapping often occurs via reactive patches on the surface and/or the particle. The theory of boundary homogenization has been used in many prior works to estimate the effective trapping rate for such a system in the case that either (i) the surface is patchy and the particle is uniformly reactive or (ii) the particle is patchy and the surface is uniformly reactive. In this paper, we estimate the trapping rate for the case that the surface and the particle are both patchy. In particular, the particle diffuses translationally and rotationally and reacts with the surface when a patch on the particle contacts a patch on the surface. We first formulate a stochastic model and derive a five-dimensional partial differential equation describing the reaction time. We then use matched asymptotic analysis to derive the effective trapping rate assuming the patches are roughly evenly distributed and occupy a small fraction of the surface and the particle. This trapping rate involves the electrostatic capacitance of a four-dimensional duocylinder, which we compute using a kinetic Monte Carlo algorithm. We further use Brownian local time theory to derive a simple heuristic estimate of the trapping rate and show that it is remarkably close to the asymptotic estimate. Finally, we develop a kinetic Monte Carlo algorithm to simulate the full stochastic system and then use these simulations to confirm the accuracy of our trapping rate estimates and homogenization theory.
... In these systems, cells infer the spatial location of an external source through the noisy arrivals of diffusing molecules (searchers) to membrane receptors (targets). Dating back to the seminal work of Berg and Purcell [49], there is a long history of mathematical modeling of such systems [50][51][52][53][54][55][56][57][58][59][60][61]. ...
Biological events are often initiated when a random ``searcher'' finds a ``target,'' which is called a first passage time (FPT). In some biological systems involving multiple searchers, an important timescale is the time it takes the slowest searcher(s) to find a target. For example, of the hundreds of thousands of primordial follicles in a woman's ovarian reserve, it is the slowest to leave that trigger the onset of menopause. Such slowest FPTs may also contribute to the reliability of cell signaling pathways and influence the ability of a cell to locate an external stimulus. In this paper, we use extreme value theory and asymptotic analysis to obtain rigorous approximations to the full probability distribution and moments of slowest FPTs. Though the results are proven in the limit of many searchers, numerical simulations reveal that the approximations are accurate for any number of searchers in typical scenarios of interest. We apply these general mathematical results to a recent model of ovarian aging and menopause timing, which reveals the role of slowest FPTs for understanding redundancy in biological systems. We also apply the theory to several popular models of stochastic search, including search by diffusive, subdiffusive, and mortal searchers.
... By a different method based on pseudopotentials, Isaacson and Newby developed a uniform asymptotic approximation of diffusion to a small target [43]. When the target is located on the boundary, homogenization techniques can be applied [44][45][46][47][48][49][50][51][52] (see also discussion in [53]). In some geometric settings, one can go further and develop self-consistent approximations for the mean reaction time and its whole distribution [54][55][56][57][58]. ...
... The trapping length L from Eqs.(49,59) of prolate (lines) and oblate (symbols) spheroids for several dimensions d. Note that L/b = 1/(d − 2) at a/b = 1. ...
... Symbols present the numerical computation by a finite-elements method (see Appendix D), whereas thick lines show the approximate relation(5). In three dimensions, thick blue line presents Eq. (5) with the "corrected" capacity C ′ from Eq. (16), whereas thin blue line corresponds to the capacity C. The trapping length L given by Eqs.(49,59) is 0.1300, 0.0596, 0.0379, 0.0276 for prolate spheroids, and 0.1669, 0.0864, 0.0588, 0.0447 for oblate spheroids, with d = 3, 4, 5, 6, respectively. ...
We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient, and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time, the long-time decay of the survival probability, and the overall reaction rate on that target. We identify the relevant lengthscale of the target, which determines its trapping capacity, and we investigate its relation to the target shape. In particular, we study the effect of target anisotropy on the principal eigenvalue by computing the harmonic capacity of prolate and oblate spheroids in various space dimensions. Some implications of these results in chemical physics and biophysics are briefly discussed.
... In this section, we assume a point TX and derive i) the expected molecule hitting rate, ii) the expected fraction of absorbed molecules, and iii) the expected asymptotic fraction of absorbed molecules as t → ∞ at the AP-based RX by applying boundary homogenization [19]. To this end, we first derive the expected CIR of an RX with uniform surface reaction rate. ...
... By solving (21), we obtain (20). By substituting w e into (4), (5), and (10), we obtain (19). ...
... with ̟(w e ) = γ(w e ) √ D σ . Proof: When molecules are released from an arbitrary point ν on the membrane of a spherical TX, the expected molecule hitting rate at the RX is obtained by replacing r 0 with r ν in h p (t) in (19), where r ν is the distance between point ν and the center of the RX. Following [14, Appendix B], we derive h s (t) by computing the surface integral of h p (t, r ν ) over the TX membrane, which is given by ...
This paper investigates the channel impulse response (CIR), i.e., the molecule hitting rate, of a molecular communication (MC) system employing an absorbing receiver (RX) covered by multiple non overlapping receptors. In this system, receptors are heterogeneous, i.e., they may have different sizes and arbitrary locations. Furthermore, we consider two types of transmitter (TX), namely a point TX and a membrane fusion (MF)-based spherical TX. We assume the point TX or the center of the MF-based TX has a fixed distance to the center of the RX. Given this fixed distance, the TX can be at different locations and the CIR of the RX depends on the exact location of the TX. By averaging over all possible TX locations, we analyze the expected molecule hitting rate at the RX as a function of the sizes and locations of the receptors, where we assume molecule degradation may occur during the propagation of the signaling molecules. Notably, our analysis is valid for different numbers, a wide range of sizes, and arbitrary locations of the receptors, and its accuracy is confirmed via particle-based simulations. Exploiting our numerical results, we show that the expected number of absorbed molecules at the RX increases with the number of receptors, when the total area on the RX surface covered by receptors is fixed. Based on the derived analytical expressions, we compare different geometric receptor distributions by examining the expected number of absorbed molecules at the RX. We show that evenly distributed receptors result in a larger number of absorbed molecules than other distributions. We further compare three models that combine different types of TXs and RXs.
... By a different method based on pseudopotentials, Isaacson and Newby developed a uniform asymptotic approximation of diffusion to a small target [43]. When the target is located on the boundary, homogenization techniques can be applied [44][45][46][47][48][49][50][51][52] (see also discussion in [53]). In some geometric settings, one can go further and develop self-consistent approximations for the mean reaction time and its whole distribution [54][55][56][57][58]. ...
... Substituting Eqs. (49,53) into Eq. (6), we get the trapping length: ...
... Using the Pfaff transformation, one can rewrite this expression as Eq. (49). To our knowledge, such a compact expression for the capacity of the oblate spheroid in R d was not earlier reported. ...
We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time, the long-time decay of the survival probability, and the overall reaction rate on that target. We identify the relevant length scale of the target, which determines its trapping capacity, and investigate its relation to the target shape. In particular, we study the effect of target anisotropy on the principal eigenvalue by computing the harmonic capacity of prolate and oblate spheroids in various space dimensions. Some implications of these results in chemical physics and biophysics are briefly discussed.
... In this section, we derive i) the expected molecule hitting rate, ii) the expected fraction of absorbed molecules, and iii) the expected asymptotic fraction of absorbed molecules as t → ∞ at a RX with multiple APs by applying boundary homogenization [13]. To this end, we first derive the expected CIR of a RX with a uniform surface reaction rate. ...
... with l ′ i = l i /r R . In (13), O(·) represents the infinitesimal of higher order, which is omitted during calculation. ...
... When all APs have the same size, (13) can be further simplied as presented in the following corollary. ...
This paper analyzes the channel impulse response of an absorbing receiver (RX) covered by multiple non-overlapping receptors with different sizes and arbitrary locations in a molecular communication system. In this system, a point transmitter (TX) is assumed to be uniformly located on a virtual sphere at a fixed distance from the RX. Considering molecule degradation during the propagation from the TX to the RX, the expected molecule hitting rate at the RX over varying locations of the TX is analyzed as a function of the size and location of each receptor. Notably, this analytical result is applicable for different numbers, sizes, and locations of receptors, and its accuracy is demonstrated via particle-based simulations. Numerical results show that (i) the expected number of absorbed molecules at the RX increases with an increasing number of receptors, when the total area of receptors on the RX surface is fixed, and (ii) evenly distributed receptors lead to the largest expected number of absorbed molecules.
... (2.7) and (5.2). These results were obtained in the theory of diffusion-limited reactions by the method of boundary homogenization, which allows one to get relatively simple formulas for the effective trapping rate by inhomogeneous surfaces in highly nontrivial geometries [13,[18][19][20][21][22][23][24]. Here we have demonstrated how some of these formulas can be used beyond the scope of diffusion-limited kinetics. ...
In this paper we analyze diffusive transport of noninteracting electrically uncharged solute molecules through a cylindrical membrane channel with a constriction located in the middle of the channel. The constriction is modeled by an infinitely thin partition with a circular hole in its center. The focus is on how the presence of the partition slows down the transport governed by the difference in the solute concentrations in the two reservoirs separated by the membrane. It is assumed that the solutions in both reservoirs are well stirred. To quantify the effect of the constriction we use the notion of diffusion resistance defined as the ratio of the concentration difference to the steady-state flux. We show that when the channel length exceeds its radius, the diffusion resistance is the sum of the diffusion resistance of the cylindrical channel without a partition and an additional diffusion resistance due to the presence of the partition. We derive an expression for the additional diffusion resistance as a function of the tube radius and that of the hole in the partition. The derivation involves the replacement of the nonpermeable partition with the hole by an effective uniform semipermeable partition with a properly chosen permeability. Such a replacement makes it possible to reduce the initial three-dimensional diffusion problem to a one-dimensional one that can be easily solved. To determine the permeability of the effective partition, we take advantage of the results found earlier for trapping of diffusing particles by inhomogeneous surfaces, which were obtained with the method of boundary homogenization. Brownian dynamics simulations are used to corroborate our approximate analytical results and to establish the range of their applicability.
... For example, Zwanzig used an effective medium formalism to account for the effects of interference between receptors [15]. Other works have modified Eq. 1 to include other effects, including receptor arrangement, cell membrane curvature, and receptor motion [16][17][18][19][20][21][22][23][24][25][26]. In one particularly important study, Wagner et al. [27] extended Eq. 1 to non-spherical geometries and used this analysis to argue that the cylindrical morphology of cell envelope extensions serves to increase nutrient uptake. ...
... In particular, the concentration is constant in the angular variables outside a boundary layer, where the width of this layer depends on the length scale of the surface heterogeneity. Many sophisticated methods have been developed to choose the trapping rate κ in Eq. 7 in order to incorporate the number, size, and arrangement of receptors [17][18][19][20][21][22][23][24][25][26]. If the receptors occupy a small fraction of the cell surface, then the trapping rate is linear in the number of receptors N and is given by [16,19,28] ...
... Therefore, the remarkable result of Berg and Purcell that a cell requires only a small receptor surface fraction f in order to have uptake near the maximum J max still holds in the case of finite receptor kinetics. As mentioned in the Introduction, many previous works have sought to modify and refine the Berg-Purcell formula to incorporate various details in the problem [16][17][18][19][20][21][22][23][24][25][26]. It is therefore worth pointing out that the discrepancy between J * and J bp is much greater than some previous modifications of J bp . ...
From nutrient uptake, to chemoreception, to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules since, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many particle stochastic system.
