First passage time distribution of multiple impatient particles with reversible binding

Optimizing Binding among Bimolecular Tethered Complexes

Tethered motion is ubiquitous in nature, offering controlled movement and spatial constraints to otherwise chaotic systems. The enhanced functionality and practical utility of tethers has been exploited in biotechnology, catalyzing the design of novel biosensors and molecular assembly techniques. While notable technological advances incorporating tethered motifs have been made, a theoretical gap persists within the paradigm, hindering a comprehensive understanding of tethered-based technologies. In this work, we focus on the characterization of the binding kinetics of two tethered molecules functionalized to a hard surface. Using a mean-field approximation, the binding time of such bimolecular system is determined analytically. Furthermore, estimates of the grafting site separation and polymer lengths which expedite binding are provided. These estimates, along with the analytical theories and frameworks established here, have the potential to improve efficacy in self-assembly methods in DNA nanotechnology and can be extended to more biologically specific endeavors including targeted drug-delivery and molecular sensing.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

First-passage times under frequent stochastic resetting

We determine the full distribution and moments of the first passage time for a wide class of stochastic search processes in the limit of frequent stochastic resetting. Our results apply to any system whose short-time behavior of the search process without resetting can be estimated. In addition to the typical case of exponentially distributed resetting times, we prove our results for a variety of resetting time distributions. We illustrate our results in several examples and show that the errors of our approximations vanish exponentially fast in typical scenarios of diffusive search.

Slowest first passage times, redundancy, and menopause timing

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.

Microscopic theory of adsorption kinetics

Adsorption is the accumulation of a solute at an interface that is formed between a solution and an additional gas, liquid, or solid phase. The macroscopic theory of adsorption dates back more than a century and is now well-established. Yet, despite recent advancements, a detailed and self-contained theory of single-particle adsorption is still lacking. Here, we bridge this gap by developing a microscopic theory of adsorption kinetics, from which the macroscopic properties follow directly. One of our central achievements is the derivation of the microscopic version of the seminal Ward-Tordai relation, which connects the surface and subsurface adsorbate concentrations via a universal equation that holds for arbitrary adsorption dynamics. Furthermore, we present a microscopic interpretation of the Ward-Tordai relation that, in turn, allows us to generalize it to arbitrary dimension, geometry, and initial conditions. The power of our approach is showcased on a set of hitherto unsolved adsorption problems to which we present exact analytical solutions. The framework developed herein sheds fresh light on the fundamentals of adsorption kinetics, which opens new research avenues in surface science with applications to artificial and biological sensing and to the design of nano-scale devices.

Optimality of intercellular signaling: Direct transport versus diffusion

Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: Diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that direct transport is favored over diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiencies of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.

Binding kinetics of harmonically confined random walkers

Diffusion-mediated binding of molecules under the influence of discrete spatially confining potentials is a commonly encountered scenario in systems subjected to explicit fields or implicit fields arising from tethering restraints. Here, we derive analytical expressions for the mean binding time of two random walkers geometrically confined by means of two harmonic potentials in one- and two-dimensional systems, which show excellent agreement with Brownian dynamics simulations. As a demonstration of its utility, we use this theory to maximize the communication speed in existing DNA walkers, obtaining quantitative agreement with previously reported experimental findings. The analytical expressions derived in this paper are broadly applicable to diverse systems, providing ways to characterize communication processes and optimize the rate of signal propagation for sensing and computing applications at the nanoscale.

Reversible target-binding kinetics of multiple impatient particles

Certain biochemical reactions can only be triggered after binding a sufficient number of particles to a specific target region such as an enzyme or a protein sensor. We investigate the distribution of the reaction time, i.e., the first instance when all independently diffusing particles are bound to the target. When each particle binds irreversibly, this is equivalent to the first-passage time of the slowest (last) particle. In turn, reversible binding to the target renders the problem much more challenging and drastically changes the distribution of the reaction time. We derive the exact solution of this problem and investigate the short-time and long-time asymptotic behaviors of the reaction time probability density. We also analyze how the mean reaction time depends on the unbinding rate and the number of particles. Our exact and asymptotic solutions are compared to Monte Carlo simulations.

First detection of threshold crossing events under intermittent sensing

The time taken by a random variable to cross a threshold for the first time, known as the first passage time, is of interest in many areas of sciences and engineering. Conventionally, there is an implicit assumption that the notional "sensor" monitoring the threshold crossing event is always active. In many realistic scenarios, the sensor monitoring the stochastic process works intermittently. Then, the relevant quantity of interest is the first detection time, which denotes the time when the sensor detects the random variable to be above the threshold for the first time. In this Letter, a birth-death process monitored by a random intermittent sensor is studied for which the first detection time distribution is obtained. In general, it is shown that the first detection time is related to and is obtainable from the first passage time distribution. Our analytical results display an excellent agreement with simulations. Furthermore, this framework is demonstrated in several applications-the susceptible infected susceptible compartmental and logistic models and birth-death processes with resetting. Finally, we solve the practically relevant problem of inferring the first passage time distribution from the first detection time.

Thermodynamics of the Isomerization of Monoterpene Epoxides

In this contribution, the thermodynamic analysis of α- and β-pinene epoxide isomerization over Fe and Cu supported on MCM-41 is presented using computational chemistry and group contribution methods (GCMs). Some physical-chemical data (T c, P c, v c, Z c, ω, T b, T fus) and thermodynamic (S°298.15, C p,298.15 °, C v,298.15 °, ΔH f,298.15 °, ΔG f,298.15 °, ΔH vb °, ΔH fus, C pL) properties obtained by different GCMs are reported for several monoterpenes and monoterpenoids, which significantly contribute to the knowledge of the properties of these compounds. Density functional theory (DFT), PBE-D3/6-311G(d,p), was employed for determining the Gibbs free energy and the heat of reaction associated with the transformation of monoterpene epoxides into aldehydes, ketones, and related oxygenated compounds in the presence of different solvents and at several temperatures. The calculations were compared with available data reported and the experimental results of the catalytic reactions. The transformation of α- and β-pinene epoxides into aldehydes appears to be more spontaneous and favorable than their transformations into alcohols in a wide range of temperatures. These results are in agreement with the experiments over Fe/MCM-41 and Cu/MCM-41, where α-pinene epoxide isomerization yields campholenic aldehyde (50-80% selectivity) as the main product. The 1.7Fe/MCM-41 material was more active in all solvents than 1.3Cu/MCM-41 for both α- and β-pinene epoxide isomerization. However, perillyl alcohol (20-70% selectivity) was the most favored for the isomerization reaction, except when ethyl acetate was the solvent. Enthalpy and Gibbs free energy of the studied reactions estimated by both GCMs and DFT calculations did not show large differences for most of the reactions at evaluated temperatures.

A first-passage approach to diffusion-influenced reversible binding and its insights into nanoscale signaling at the presynapse

Synaptic transmission between neurons is governed by a cascade of stochastic calcium ion reaction–diffusion events within nerve terminals leading to vesicular release of neurotransmitter. Since experimental measurements of such systems are challenging due to their nanometer and sub-millisecond scale, numerical simulations remain the principal tool for studying calcium-dependent neurotransmitter release driven by electrical impulses, despite the limitations of time-consuming calculations. In this paper, we develop an analytical solution to rapidly explore dynamical stochastic reaction–diffusion problems based on first-passage times. This is the first analytical model that accounts simultaneously for relevant statistical features of calcium ion diffusion, buffering, and its binding/unbinding reaction with a calcium sensor for synaptic vesicle fusion. In particular, unbinding kinetics are shown to have a major impact on submillisecond sensor occupancy probability and therefore cannot be neglected. Using Monte Carlo simulations we validated our analytical solution for instantaneous calcium influx and that through voltage-gated calcium channels. We present a fast and rigorous analytical tool that permits a systematic exploration of the influence of various biophysical parameters on molecular interactions within cells, and which can serve as a building block for more general cell signaling simulators.

Extreme first-passage times for random walks on networks

Many biological, social, and communication systems can be modeled by "searchers" moving through a complex network. For example, intracellular cargo is transported on tubular networks, news and rumors spread through online social networks, and the rapid global spread of infectious diseases occurs through passengers traveling on the airport network. To understand the timescale of search (or "transport" or "spread"), one commonly studies the first-passage time (FPT) of a single searcher (or "transporter" or "spreader") to a target. However, in many scenarios the relevant timescale is not the FPT of a single searcher to a target, but rather the FPT of the fastest searcher to a target out of many searchers. For example, many processes in cell biology are triggered by the first molecule to find a target out of many, and the time it takes an infectious disease to reach a particular city depends on the first infected traveler to arrive out of potentially many infected travelers. Such fastest FPTs are called extreme FPTs. In this paper, we study extreme FPTs for a general class of continuous-time random walks on networks (which includes continuous-time Markov chains). In the limit of many searchers, we find explicit formulas for the probability distribution and all the moments of the kth fastest FPT for any fixed k≥1. These rigorous formulas depend only on network parameters along a certain geodesic path(s) from the starting location to the target since the fastest searchers take a direct route to the target. Hence, the extreme FPTs are independent of the details of the network outside this geodesic(s) and can be drastically faster and less variable than conventional FPTs of single searchers. Furthermore, our results allow one to estimate if a particular system is in a regime characterized by fast extreme FPTs. We also prove similar results for mortal searchers on a network that are conditioned to find the target before a fast inactivation time. We illustrate our results with numerical simulations and uncover potential pitfalls of modeling diffusive or subdiffusive processes involving extreme statistics. In particular, we find that the many searcher limit does not commute with the diffusion limit for random walks, and thus care must be taken when choosing spatially continuous versus spatially discrete diffusion models.

Universal formula for extreme first passage statistics of diffusion

The timescales of many physical, chemical, and biological processes are determined by first passage times (FPTs) of diffusion. The overwhelming majority of FPT research studies the time it takes a single diffusive searcher to find a target. However, the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers. This fastest FPT depends on extremely rare events and has a drastically faster timescale than the FPT of a given single searcher. In this work, we prove a simple explicit formula for every moment of the fastest FPT. The formula is remarkably universal, as it holds for d-dimensional diffusion processes (i) with general space-dependent diffusivities and force fields, (ii) on Riemannian manifolds, (iii) in the presence of reflecting obstacles, and (iv) with partially absorbing targets. Our results rigorously confirm, generalize, correct, and unify various conjectures and heuristics about the fastest FPT.

Boundary homogenization for trapping patchy particles

Many systems in chemical and biological physics involve diffusing particles being trapped by absorbing patches on otherwise reflecting surfaces. Such systems are commonly studied by boundary homogenization, in which the heterogenous boundary condition on the patchy surface is replaced by a uniform boundary condition involving a single parameter which encapsulates the effective trapping properties of the surface. In prior works on boundary homogenization, the surface is patchy and the diffusing particles are homogeneous. In this paper, we consider the opposite scenario in which a homogenous surface traps patchy particles, which could model proteins with localized binding sites, cells with membrane receptors, or patchy colloids or nanoparticles. We derive an explicit formula for the effective trapping rate which reveals a fundamental interplay between the translational and rotational diffusivities of the patchy particle, a phenomenon not typically seen in boundary homogenization. Motivated by receptors on the cell membrane, our analysis also allows for the possibility that the patches diffuse on the surface of the particle. We formulate the system in terms of a high-dimensional, time-dependent, anisotropic diffusion equation and employ matched asymptotic analysis to derive the effective trapping rate. We confirm our results by detailed numerical simulations.

Target search on DNA by interacting molecules: First-passage approach

Gene regulation is one of the most important fundamental biological processes in living cells. It involves multiple protein molecules that locate specific sites on DNA and assemble gene initiation or gene repression multimolecular complexes. While the protein search dynamics for DNA targets has been intensively investigated, the role of intermolecular interactions during the genetic activation or repression remains not well quantified. Here, we present a simple one-dimensional model of target search for two interacting molecules that can reversibly form a dimer molecular complex, which also participates in the search process. In addition, the proteins have finite residence times on specific target sites, and the gene is activated or repressed when both proteins are simultaneously present at the target. The model is analyzed using first-passage analytical calculations and Monte Carlo computer simulations. It is shown that the search dynamics exhibit a complex behavior depending on the strength of intermolecular interactions and on the target residence times. We also found that the search time shows a nonmonotonic behavior as a function of the dissociation rate for the molecular complex. Physical-chemical arguments to explain these observations are presented. Our theoretical approach highlights the importance of molecular interactions in the complex process of gene activation/repression by multiple transcription factor proteins.