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. Similarly, the notion of a forgiving drug in pharmacology is a medication that maintains therapeutic effect despite lapses in patient adherence to the prescribed regimen. What makes a network robust to stochastic noise? This question is challenging due to the many network parameters (size, topology, rate constants) and many types of noisy inputs. In this paper, we propose a summary statistic to describe the robustness of a network of linear differential equations (i.e. a first-order mass-action system). This statistic is the variance of a certain random walk passage time on the network. This statistic can be quickly computed on a modern computer, even for complex networks with thousands of nodes. Furthermore, we use this statistic to prove theorems about how certain network motifs increase robustness. Importantly, our analysis provides intuition for why a network is or is not robust to noise. We illustrate our results on thousands of randomly generated networks with a variety of stochastic inputs.

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. Our results show that such cover times depend only on properties of the network along the shortest paths to the most distant parts of the target. This mere local dependence contrasts with the well-known result that cover times for single searchers depend on global properties of the network. We illustrate our rigorous results by stochastic simulations.

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. In this paper, we construct and analyze a stochastic pharmacokinetic/pharmacodynamic (PK/PD) model to quantify and understand drug forgiveness. The model parameterizes a medication merely by an effective rate of onset of effect when the medication is taken (on-rate) and an effective rate of loss of effect when a dose is missed (off-rate). Patient dosing is modeled by a stochastic process that allows for correlations in missed doses. We analyze this "on/off" model and derive explicit formulas that show how treatment efficacy depends on drug parameters and patient adherence. As a case study, we compare the effects of nonadherence on the efficacy of various antihypertensive medications. Our analysis shows how different drugs can have identical efficacies under perfect adherence, but vastly different efficacies for adherence patterns typical of actual patients. We further demonstrate that complex PK/PD models can indeed be parameterized in terms of effective on-rates and off-rates. Finally, we have created an online app to allow pharmacometricians to explore the implications of our model and analysis.

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. STUDY DESIGN: Based on the existing biological data on follicle loss rates, a stochastic model of primordial follicle wastage was developed to determine the years of delay in menopause (denoted by D) by ovarian tissue cryopreservation and transplantation near menopause. Our model accounted for (1) age at ovarian tissue harvest (21-40 years), (2) the amount of ovarian cortex harvested, (3) transplantation of harvested tissues in single vs multiple procedures (fractionation), and (4) posttransplant follicle survival (40% [conservative] vs 80% [improved] vs 100% [ideal or hypothetical]). RESULTS: Our model predicted that, for most women aged <40 years, ovarian tissue cryopreservation and transplantation would result in a significant delay in menopause. The advantage is greater if the follicle loss after transplant can be minimized. As an example, the delay in menopause (D) for a woman with a median ovarian reserve who cryopreserves 25% of her ovarian cortex at the age of 25 years and for whom 40% of follicles survive after transplantation would be approximately 11.8 years, but this extends to 15.5 years if the survival is 80%. As another novel finding, spreading the same amount of tissue to repetitive transplants significantly extends the benefit. For example, for the same 25-year-old woman with a median ovarian reserve, 25% cortex removal, and 40% follicle survival, fractionating the transplants to 3 or 6 procedures would result in the corresponding delay in menopause (D) of 23 or 31 years. The same conditions (3 or 6 procedures) would delay menopause as much as 47 years if posttransplant follicle survival is improved to 80% with modern approaches. An interactive Web tool was created to test all variables and the feasibility of ovarian tissue freezing and transplantation to delay ovarian aging (here). CONCLUSION: Our model predicts that with harvesting at earlier adult ages and better transplant techniques, a significant menopause postponement and, potentially, fertile life span extension can be achieved by ovarian tissue cryopreservation and transplantation in healthy women.

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). Here, we have probed how stochastic control of primordial follicle loss might relate to the distribution of the preceding menopausal transition (MT), when women begin to experience menstrual cycle irregularity. We show that identical random walk model conditions produce both the reported MT distribution and the ANM distribution when thresholds are set for growing follicle availability. The MT and ANM are shown to correspond to gaps when primordial follicles fail to grow for 7 and 12 days, respectively. Modeling growing follicle supply is shown to precisely recapitulate epidemiological data and provides quantitative criteria for the MT and ANM in humans.

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. In contrast, agents who decide last make decisions as if they were initially unbiased, and hence make better choices. We obtain asymptotic expressions in the large population limit that quantify how agents' initial inclinations shape early decisions. Our analysis shows how bias, information quality, and decision order interact in non-trivial ways to determine the reliability of decisions in a group.

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.

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. 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 that 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.

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. In this paper, we propose that the oversupply of PFs at birth enables a simple stochastic PFGA mechanism to yield a steady supply of growing follicles that lasts for several decades. Assuming stochastic PFGA, we apply extreme value theory to histological PF count data to show that the supply of growing follicles is remarkably robust to a variety of perturbations and that the timing of ovarian function cessation (age of natural menopause) is surprisingly tightly controlled. Though stochasticity is often viewed as an obstacle in physiology and PF oversupply has been called "wasteful," this analysis suggests that stochastic PFGA and PF oversupply function together to ensure robust and reliable female reproductive aging.

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. In this article, we extend a popular reaction-diffusion model of FRAP to the case of subdiffusion modeled by a fractional diffusion equation. By analyzing this reaction-subdiffusion model, we show that FRAP data are consistent with both diffusive and subdiffusive motion in many scenarios. We illustrate this general result by fitting our model to FRAP data from glucocorticoid receptors in a cell nucleus. We further show that the assumed model of molecular motion (normal diffusion or subdiffusion) strongly impacts the biological parameter values inferred from a given experimentally observed FRAP curve. We additionally analyze our model in three simplified parameter regimes and discuss parameter identifiability for varying subdiffusion exponents.

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. Because cell cycle entry of (pre)granulosa cells is required for PF growth activation (PFGA), we propose that rare ISR checkpoint resolution allows individual PFs to begin to grow. Fluctuating ISR activity within individual PFs can be described by a random process. In this article, we model ISR activity of individual PFs by one-dimensional random walks (RWs) and monitor the rate at which simulated checkpoint resolution and thus PFGA threshold crossing occurs. We show that the simultaneous recapitulation of (i) the loss of PFs over time within simulated subjects, and (ii) the timing of PF depletion in populations of simulated subjects equivalent to the distribution of the human age of natural menopause can be produced using this approach. In the RW model, the probability that individual PFs grow is influenced by regionally fluctuating conditions, that over time manifests in the known pattern of PFGA. Considered at the level of the ovary, randomness appears to be a key, purposeful feature of human ovarian aging.

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. We suppose that the patient forgets to take a dose and then realizes some time later and must decide what remedial steps to take. Using mathematical analysis, we derive several metrics which quantify the effects of taking the dose late. The metrics involve the difference between the drug concentration time courses for the case that the dose is taken late and the case that the dose is taken on time. In particular, the metrics are the integral of the absolute difference over all time, the maximum of the difference, and the maximum of the integral of the difference over any single dosing interval. We apply these general mathematical formulas to levothyroxine, atorvastatin, and immediate release and extended release formulations of lamotrigine. We further show how population variability can be immediately incorporated into these results. Finally, we use this analysis to propose general principles and strategies for dealing with dosing irregularities.

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. In previous work we have shown that during this flutter phase, no matter how small the percentage of time that the spiracles are open, the insect can absorb almost as much oxygen as if the spiracle were always open, if the insect flutters fast enough. This left open the question of water loss during the flutter phase, which is the question addressed in this paper. We formulate a stochastic diffusion-convection model for the concentration of water vapor in the tracheae. Mathematical analysis of the model yields an explicit formula for water loss as a function of six non-dimensional parameters and we use experimental data from various insects to show that, for parameters in the physiological ranges, water loss during the flutter phase is approximately proportional to the percentage of time open. This means that the insect can solve the oxygen uptake versus water loss problem by choosing to have their spiracles open a small percentage of time during the flutter phase and fluttering rapidly.

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. We assume that the drug effect is far from maximal and consider direct effect, effect compartment (biophase), and indirect response PD models. We prove that the average drug effect relative to the clinically desired effect is simply the fraction of prescribed doses actually taken by the patient. Hence, under these assumptions, drug forgiveness cannot be defined in terms of the average effect. We argue that forgiveness should instead be understood in terms of effect fluctuations. We prove that the rates of PK absorption, PK elimination, and PD elimination are exactly equivalent for determining effect fluctuations. We prove all the aforementioned results for any pattern of nonadherence, including late doses, missed doses, drug holidays, extra doses, etc. To obtain quantitative estimates of effect fluctuations, we consider a simple statistical pattern of nonadherence and analytically calculate the coefficient of variation of effect. We further show how effect fluctuations can be reduced by taking an extra "make up" dose following a missed dose if any one of the aforementioned PK/PD rates is sufficiently slow. We illustrate some of our results for a nonlinear indirect response model of metformin.

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. Our model takes the form of the standard single compartment pharmacokinetic model with first-order absorption and elimination, except that the patient takes medication only at a given proportion of the prescribed dosing times. Doses are missed randomly, and we use stochastic analysis to study the resulting random drug level in the body. We then use our mathematical results to propose principles for designing drug regimens that are robust to nonadherence. In particular, we quantify the resilience of extended release drugs to nonadherence, which is quite significant in some circumstances, and we show the benefit of taking a double dose following a missed dose if the drug absorption or elimination rate is slow compared to the dosing interval. We further use our results to compare some antiepileptic and antipsychotic drug regimens.

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. We find exact formulas for drug concentration statistics, including the mean, the coefficient of variation, and the deviation from perfect adherence. We determine how adherence rates translate into drug concentrations, and how this depends on the drug half-life, the dosing interval, and how missed doses are handled. While clinical recommendations require extensive validation and should depend on drug and patient specifics, as a general principle our theory suggests that nonadherence is best mitigated by taking double doses following missed doses if the drug has a long half-life. This conclusion contradicts some existing recommendations that cite long drug half-lives as the reason to avoid a double dose after a missed dose. Furthermore, we show that a patient who takes double doses after missed doses can have at most only slightly more drug in their body than a perfectly adherent patient if the drug half-life is long. We also investigate other ways of handling missed doses, including taking an extra fractional dose following a missed dose. We discuss our results in the context of hypothyroid patients taking levothyroxine.

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. 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 because, 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.

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. In this paper, we study the subdiffusion-limited model, which is defined by mesoscopic equations with fractional derivatives applied to both the movement and the reaction terms. Assuming that the reaction terms are affine functions, we show that the solution to the fractional system is the expectation of a random time change of the solution to the corresponding integer order system. This result yields a simple and explicit algebraic relationship between the fractional and integer order solutions in Laplace space. We then find the microscopic Langevin description of individual molecules that corresponds to such mesoscopic equations and give a computer simulation method to generate their stochastic trajectories. This analysis identifies some precise microscopic conditions that dictate when this type of mesoscopic model is or is not appropriate. We apply our results to several scenarios in cell biology which, despite the ubiquity of subdiffusion in cellular environments, have been modeled almost exclusively by normal diffusion. Specifically, we consider subdiffusive models of morphogen gradient formation, fluctuating mobility, and fluorescence recovery after photobleaching (FRAP) experiments. We also apply our results to fractional ordinary differential equations.

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. In this paper, we first give a short and elementary proof of these previous results. We then show how this argument gives the evolution equations for more general cases, including subdiffusion following any fractional Fokker-Planck equation in an arbitrary d-dimensional spatial domain with time-dependent reactions between infinitely many discrete states. In contrast to previous works which employed a variety of technical mathematical methods, our analysis reveals that the evolution equations follow from (1) the probabilistic independence of the stochastic spatial and discrete processes describing a single particle and (2) the linearity of the integro-differential operators describing spatial movement. We also apply our results to systems combining reactions with superdiffusion.