Moments of action provide insight into critical times for advection-diffusion-reaction processes

Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether these data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach; and, second, we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods, we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.

Diffusion in heterogeneous discs and spheres: New closed-form expressions for exit times and homogenization formulas

Mathematical models of diffusive transport underpin our understanding of chemical, biochemical, and biological transport phenomena. Analysis of such models often focuses on relatively simple geometries and deals with diffusion through highly idealized homogeneous media. In contrast, practical applications of diffusive transport theory inevitably involve dealing with more complicated geometries as well as dealing with heterogeneous media. One of the most fundamental properties of diffusive transport is the concept of mean particle lifetime or mean exit time, which are particular applications of the concept of first passage time and provide the mean time required for a diffusing particle to reach an absorbing boundary. Most formal analysis of mean particle lifetime applies to relatively simple geometries, often with homogeneous (spatially invariant) material properties. In this work, we present a general framework that provides exact mathematical insight into the mean particle lifetime, and higher moments of particle lifetime, for point particles diffusing in heterogeneous discs and spheres with radial symmetry. Our analysis applies to geometries with an arbitrary number and arrangement of distinct layers, where transport in each layer is characterized by a distinct diffusivity. We obtain exact closed-form expressions for the mean particle lifetime for a diffusing particle released at an arbitrary location, and we generalize these results to give exact, closed-form expressions for any higher-order moment of particle lifetime for a range of different boundary conditions. Finally, using these results, we construct new homogenization formulas that provide an accurate simplified description of diffusion through heterogeneous discs and spheres.

Finite transition times for multispecies diffusion in heterogeneous media coupled via first-order reaction networks

Calculating how long a coupled multispecies reactive-diffusive transport process in a heterogeneous medium takes to effectively reach steady state is important in many applications. In this paper, we show how the time required for such processes to transition to within a small specified tolerance of steady state can be calculated accurately without having to solve the governing time-dependent model equations. Our approach is valid for general first-order reaction networks and an arbitrary number of species. Three numerical examples are presented to confirm the analysis and investigate the efficacy of the approach. A key finding is that for sequential reactions our approach works better provided the two smallest reaction rates are well separated.

New homogenization approaches for stochastic transport through heterogeneous media

The diffusion of molecules in complex intracellular environments can be strongly influenced by spatial heterogeneity and stochasticity. A key challenge when modelling such processes using stochastic random walk frameworks is that negative jump coefficients can arise when transport operators are discretized on heterogeneous domains. Often this is dealt with through homogenization approximations by replacing the heterogeneous medium with an effective homogeneous medium. In this work, we present a new class of homogenization approximations by considering a stochastic diffusive transport model on a one-dimensional domain containing an arbitrary number of layers with different jump rates. We derive closed form solutions for the kth moment of particle lifetime, carefully explaining how to deal with the internal interfaces between layers. These general tools allow us to derive simple formulae for the effective transport coefficients, leading to significant generalisations of previous homogenization approaches. Here, we find that different jump rates in the layers give rise to a net bias, leading to a non-zero advection, for the entire homogenized system. Example calculations show that our generalized approach can lead to very different outcomes than traditional approaches, thereby having the potential to significantly affect simulation studies that use homogenization approximations.

Characteristic time scales for diffusion processes through layers and across interfaces

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

Calculating Groundwater Response Times for Flow in Heterogeneous Porous Media

Predicting the amount of time required for a transient groundwater response to take place is a practical question that is of interest in many situations. This time scale is often called the response time. In the groundwater hydrology literature, there are two main methods used to calculate the response time: (1) both the transient and steady-state groundwater flow equations are solved, and the response time is taken to be amount of time required for the transient solution to approach the steady solution within some tolerance; and (2) simple scaling arguments are adopted. Certain limitations restrict both of these approaches. In this study, we outline a third method, based on the theory of mean action time. We derive the governing boundary value problem for both the mean and variance of action time for confined flow in two-dimensional heterogeneous porous media. Importantly, we show that these boundary value problems can be solved using widely available software. Applying these methods to a test case reveals the advantages of the theory of mean action time relative to standard methods.

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a finite transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady-state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time, (ii) mean plus one standard deviation of action time, and (iii) an approach we derive by approximating the large time asymptotic behavior of the cumulative distribution function. Our approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance δ and the (k-1)th and kth moments (k≥1) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. First, although the first two approaches are useful at characterizing the time scale of the transition, they do not provide accurate estimates for diffusion processes. Second, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits using only the moments with the accuracy increasing as the index k is increased.

Kinetics of receptor occupancy during morphogen gradient formation

During embryogenesis, sheets of cells are patterned by concentration profiles of morphogens, molecules that act as dose-dependent regulators of gene expression and cell differentiation. Concentration profiles of morphogens can be formed by a source-sink mechanism, whereby an extracellular protein is secreted from a localized source, diffuses through the tissue and binds to cell surface receptors. A morphogen molecule bound to its receptor can either dissociate or be internalized by the cell. The effects of morphogens on cells depend on the occupancy of surface receptors, which in turn depends on morphogen concentration. In the simplest case, the local concentrations of the morphogen and morphogen-receptor complexes monotonically increase with time from zero to their steady-state values. Here, we derive analytical expressions for the time scales which characterize the formation of the steady-state concentrations of both the diffusible morphogen molecules and morphogen-receptor complexes at a given point in the patterned tissue.

Simplified approach for calculating moments of action for linear reaction-diffusion equations

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time underapproximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.