Narrow escape with imperfect reactions.

The imperfect narrow escape problem considers the mean first passage time (MFPT) of a Brownian particle through one of several small, partially reactive traps on an otherwise reflecting boundary within a confining domain. Mathematically, this problem is equivalent to Poisson's equation with mixed Neumann-Robin boundary conditions. Here, we obtain this MFPT in general three-dimensional domains by using strong localized perturbation theory in the small trap limit.

Fast decisions reflect biases; slow decisions do not.

Decisions are often made by heterogeneous groups of individuals, each with distinct initial biases and access to information of different quality. We show that in groups of independent agents who accumulate evidence the first to decide are those with the strongest initial biases. Their decisions align with their initial bias, regardless of the underlying truth.

Understanding and Quantifying Network Robustness to Stochastic Inputs.

A variety of biomedical systems are modeled by networks of deterministic differential equations with stochastic inputs. In some cases, the network output is remarkably constant despite a randomly fluctuating input. In the context of biochemistry and cell biology, chemical reaction networks and multistage processes with this property are called robust.

Cover times of many random walkers on a discrete network.

The speed of an exhaustive search can be measured by a cover time, which is defined as the time it takes a random searcher to visit every state in some target set. Cover times have been studied in both the physics and probability literatures, with most prior works focusing on a single searcher. In this paper, we prove an explicit formula for all the moments of the cover time for many searchers on an arbitrary discrete network.

Fast decisions reflect biases, slow decisions do not.

Decisions are often made by heterogeneous groups of individuals, each with distinct initial biases and access to information of different quality. We show that in large groups of independent agents who accumulate evidence the first to decide are those with the strongest initial biases. Their decisions align with their initial bias, regardless of the underlying truth.

Mathematical recapitulation of the end stages of human ovarian aging.

Ovarian aging in women can be described as highly unpredictable within individuals but predictable across large populations. We showed previously that modeling an individual woman's ovarian reserve of primordial follicles using mathematical random walks replicates the natural pattern of growing follicles exiting the reserve. Compiling many simulations yields the observed population distribution of the age at natural menopause (ANM).

How drug onset rate and duration of action affect drug forgiveness.

Medication nonadherence is one of the largest problems in healthcare today, particularly for patients undergoing long-term pharmacotherapy. To combat nonadherence, it is often recommended to prescribe so-called "forgiving" drugs, which maintain their effect despite lapses in patient adherence. Nevertheless, drug forgiveness is difficult to quantify and compare between different drugs.

Modeling delay of age at natural menopause with planned tissue cryopreservation and autologous transplantation.

Background: Ovarian tissue cryopreservation has been proven to preserve fertility against gonadotoxic treatments. It has not been clear how this procedure would perform if planned for slowing ovarian aging.

Objective: This study aimed to determine the feasibility of cryopreserving ovarian tissue to extend reproductive life span and delay menopause by autotransplantation near menopause.

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.

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.

Boundary homogenization for patchy surfaces trapping patchy particles.

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.

Why is there an "oversupply" of human ovarian follicles?†.

Women are born with hundreds of thousands to over a million primordial ovarian follicles (PFs) in their ovarian reserve. However, only a few hundred PFs will ever ovulate and produce a mature egg. Why are hundreds of thousands of PFs endowed around the time of birth when far fewer follicles are required for ongoing ovarian endocrine function and only a few hundred will survive to ovulate? Recent experimental, bioinformatics, and mathematical analyses support the hypothesis that PF growth activation (PFGA) is inherently stochastic.

Inferences from FRAP data are model dependent: A subdiffusive analysis.

Fluorescence recovery after photobleaching (FRAP) is a widely used biological experiment to study the kinetics of molecules that react and move randomly. Since the development of FRAP in the 1970s, many reaction-diffusion models have been used to interpret FRAP data. However, intracellular molecules are widely observed to move by anomalous subdiffusion instead of normal diffusion.

Recapitulating human ovarian aging using random walks.

Mechanism(s) that control whether individual human primordial ovarian follicles (PFs) remain dormant, or begin to grow, are all but unknown. One of our groups has recently shown that activation of the Integrated Stress Response (ISR) pathway can slow follicular granulosa cell proliferation by activating cell cycle checkpoints. Those data suggest that the ISR is active and fluctuates according to local conditions in dormant PFs.

Should patients skip late doses of medication? A pharmacokinetic perspective.

Missed doses, late doses, and other dosing irregularities are major barriers to effective pharmacotherapy, especially for the treatment of chronic conditions. What should a patient do if they did not take their last dose at the prescribed time? Should they take it late or skip it? In this paper, we investigate the pharmacokinetic effects of taking a late dose. We consider a single compartment model with linear absorption and elimination for a patient instructed to take doses at regular time intervals.

Spiracular fluttering decouples oxygen uptake and water loss: a stochastic PDE model of respiratory water loss in insects.

In insect respiration, oxygen from the air diffuses through a branching system of air-filled tubes to the cells of the body and carbon dioxide produced in cellular respiration diffuses out. The tracheal system has a very large surface area, so water loss is a potential threat and the question of how insects regulate oxygen uptake and water loss has been an important issue in insect physiology for the past century. The tracheal system starts at spiracles on the surface of the body that insects can open and close, and three phases are observed experimentally, open or closed for relatively long periods of time and opening and closing rapidly, which is called fluttering.

A pharmacokinetic and pharmacodynamic analysis of drug forgiveness.

Nonadherence to medication is a major public health problem. To combat nonadherence, some clinicians have suggested using "forgiving" drugs, which maintain efficacy in spite of delayed or missed doses. What pharmacokinetic (PK) and pharmacodynamic (PD) factors make a drug forgiving? In this paper, we address this question by analyzing a linear PK/PD model for a patient with imperfect adherence.

Designing Drug Regimens that Mitigate Nonadherence.

Medication adherence is a well-known problem for pharmaceutical treatment of chronic diseases. Understanding how nonadherence affects treatment efficacy is made difficult by the ethics of clinical trials that force patients to skip doses of the medication being tested, the unpredictable timing of missed doses by actual patients, and the many competing variables that can either mitigate or magnify the deleterious effects of nonadherence, such as pharmacokinetic absorption and elimination rates, dosing intervals, dose sizes, and adherence rates. In this paper, we formulate and analyze a mathematical model of the drug concentration in an imperfectly adherent patient.

What should patients do if they miss a dose of medication? A theoretical approach.

Medication adherence is a major problem for patients with chronic diseases that require long term pharmacotherapy. Many unanswered questions surround adherence, including how adherence rates translate into treatment efficacy and how missed doses of medication should be handled. To address these questions, we formulate and analyze a mathematical model of the drug concentration in a patient with imperfect adherence.

Revising Berg-Purcell for finite receptor kinetics.

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.

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.

Subdiffusion-limited fractional reaction-subdiffusion equations with affine reactions: Solution, stochastic paths, and applications.

In contrast to normal diffusion, there is no canonical model for reactions between chemical species which move by anomalous subdiffusion. Indeed, the type of mesoscopic equation describing reaction-subdiffusion systems depends on subtle assumptions about the microscopic behavior of individual molecules. Furthermore, the correspondence between mesoscopic and microscopic models is not well understood.

Anomalous reaction-diffusion equations for linear reactions.

Deriving evolution equations accounting for both anomalous diffusion and reactions is notoriously difficult, even in the simplest cases. In contrast to normal diffusion, reaction kinetics cannot be incorporated into evolution equations modeling subdiffusion by merely adding reaction terms to the equations describing spatial movement. A series of previous works derived fractional reaction-diffusion equations for the spatiotemporal evolution of particles undergoing subdiffusion in one space dimension with linear reactions between a finite number of discrete states.

Receptor Organization Determines the Limits of Single-Cell Source Location Detection.

Many types of cells require the ability to pinpoint the location of an external stimulus from the arrival of diffusing signaling molecules at cell-surface receptors. How does the organization (number and spatial configuration) of these receptors shape the limit of a cell's ability to infer the source location? In the idealized scenario of a spherical cell, we apply asymptotic analysis to compute splitting probabilities between individual receptors and formulate an information-theoretic framework to quantify the role of receptor organization. Clustered configurations of receptors provide an advantage in detecting sources aligned with the clusters, suggesting a possible multiscale mechanism for single-cell source inference.