Uniform asymptotic approximation of diffusion to a small target: Generalized reaction models

Searching for a partially absorbing target by a run-and-tumble particle in a confined space

A random search of a partially absorbing target by a run-and-tumble particle in a confined one-dimensional space is investigated. We analytically obtain the mean searching time, which shows a nonmonotonic behavior as a function of the self-propulsion speed of the active particle, indicating the existence of an optimal speed, when the absorption strength of the target is finite. In the limit of large and small absorption strengths, respectively, asymptotes of the mean searching time and the optimal speed are found. We also demonstrate that the first-passage problem of a diffusive run-and-tumble particle in high dimensions can be mapped into a one-dimensional problem with a partially absorbing target. Finally, as a practical application exploiting the existence of the optimal speed, we propose a filtering device to extract active particles with a desired speed and evaluate how the resolution of the filtering device depends on the absorption strength.

Imperfect narrow escape problem

We consider the kinetics of the imperfect narrow escape problem, i.e., the time it takes for a particle diffusing in a confined medium of generic shape to reach and to be adsorbed by a small, imperfectly reactive patch embedded in the boundary of the domain, in two or three dimensions. Imperfect reactivity is modeled by an intrinsic surface reactivity κ of the patch, giving rise to Robin boundary conditions. We present a formalism to calculate the exact asymptotics of the mean reaction time in the limit of large volume of the confining domain. We obtain exact explicit results in the two limits of large and small reactivities of the reactive patch, and a semianalytical expression in the general case. Our approach reveals an anomalous scaling of the mean reaction time as the inverse square root of the reactivity in the large-reactivity limit, valid for an initial position near the extremity of the reactive patch. We compare our exact results with those obtained within the "constant flux approximation"; we show that this approximation turns out to give exactly the next-to-leading-order term of the small-reactivity limit, and provides a good approximation of the reaction time far from the reactive patch for all reactivities, but not in the vicinity of the boundary of the reactive patch due to the above-mentioned anomalous scaling. These results thus provide a general framework to quantify the mean reaction times for the imperfect narrow escape problem.

Universal kinetics of imperfect reactions in confinement

Chemical reactions generically require that particles come into contact. In practice, reaction is often imperfect and can necessitate multiple random encounters between reactants. In confined geometries, despite notable recent advances, there is to date no general analytical treatment of such imperfect transport-limited reaction kinetics. Here, we determine the kinetics of imperfect reactions in confining domains for any diffusive or anomalously diffusive Markovian transport process, and for different models of imperfect reactivity. We show that the full distribution of reaction times is obtained in the large confining volume limit from the knowledge of the mean reaction time only, which we determine explicitly. This distribution for imperfect reactions is found to be identical to that of perfect reactions upon an appropriate rescaling of parameters, which highlights the robustness of our results. Strikingly, this holds true even in the regime of low reactivity where the mean reaction time is independent of the transport process, and can lead to large fluctuations of the reaction time - even in simple reaction schemes. We illustrate our results for normal diffusion in domains of generic shape, and for anomalous diffusion in complex environments, where our predictions are confirmed by numerical simulations.

Strong intracellular signal inactivation produces sharper and more robust signaling from cell membrane to nucleus

For a chemical signal to propagate across a cell, it must navigate a tortuous environment involving a variety of organelle barriers. In this work we study mathematical models for a basic chemical signal, the arrival times at the nuclear membrane of proteins that are activated at the cell membrane and diffuse throughout the cytosol. Organelle surfaces within human B cells are reconstructed from soft X-ray tomographic images, and modeled as reflecting barriers to the molecules’ diffusion. We show that signal inactivation sharpens signals, reducing variability in the arrival time at the nuclear membrane. Inactivation can also compensate for an observed slowdown in signal propagation induced by the presence of organelle barriers, leading to arrival times at the nuclear membrane that are comparable to models in which the cytosol is treated as an open, empty region. In the limit of strong signal inactivation this is achieved by filtering out molecules that traverse non-geodesic paths.

The inside of cells is a complex spatial environment, filled with organelles, filaments and proteins. It is an open question how cell signaling pathways function robustly in the presence of such spatial heterogeneity. In this work we study how organelle barriers influence the most basic of chemical signals; the diffusive propagation of an activated protein from the cell membrane to nucleus. Three-dimensional B cell organelle and membrane geometries reconstructed from soft X-ray tomographic images are used in building mathematical models of the signal propagation process. Our models demonstrate that organelle barriers significantly increase the time required for a diffusing protein to traverse from the cell membrane to nucleus when compared to a cell with an empty cytosolic space. We also show that signal inactivation, a fundamental component of all signaling pathways, can provide robustness in the signal arrival time in two ways. Increasing rates of signal inactivation reduce variability in the arrival time, while also dramatically reducing the degree to which organelle barriers increase the arrival time (in comparison to a cell with an empty cytosol).

Reaction time for trimolecular reactions in compartment-based reaction-diffusion models

Trimolecular reaction models are investigated in the compartment-based (lattice-based) framework for stochastic reaction-diffusion modeling. The formulae for the first collision time and the mean reaction time are derived for the case where three molecules are present in the solution under periodic boundary conditions. For the case of reflecting boundary conditions, similar formulae are obtained using a computer-assisted approach. The accuracy of these formulae is further verified through comparison with numerical results. The presented derivation is based on the first passage time analysis of Montroll [J. Math. Phys. 10, 753 (1969)]. Montroll's results for two-dimensional lattice-based random walks are adapted and applied to compartment-based models of trimolecular reactions, which are studied in one-dimensional or pseudo one-dimensional domains.

Efficient Low-Order Approximation of First-Passage Time Distributions

We consider the problem of computing first-passage time distributions for reaction processes modeled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerization process and show good agreement with stochastic simulations.