Space-time histories approach to fast stochastic simulation of bimolecular reactions.

Computational models of reaction-diffusion systems involving low copy numbers or strongly heterogeneous molecular spatial distributions, such as those frequently found in cellular signaling pathways, require approaches that account for the stochastic dynamics of individual particles, as opposed to approaches representing them through their average concentrations. Efforts to remedy the high computational cost associated with particle-based stochastic approaches by taking advantage of Green's functions are hampered by the need to draw random numbers from complicated, and therefore costly, non-standard probability distributions to update particle positions. Here, we introduce an approach that permits the reconstruction of entire molecular trajectories, including bimolecular encounters, in retrospect, after a simulated time step, while avoiding inefficient draws from non-standard distributions. This means that highly accurate stochastic simulations can be performed for system sizes that would be prohibitively costly to simulate with conventional Green's function based methods. The algorithm applies equally well to one, two, and three dimensional systems and can be readily extended to include deterministic forces specified by an interaction potential, such as the Coulomb potential.

Migrating Myeloid Cells Sense Temporal Dynamics of Chemoattractant Concentrations.

Chemoattractant-mediated recruitment of hematopoietic cells to sites of pathogen growth or tissue damage is critical to host defense and organ homeostasis. Chemotaxis is typically considered to rely on spatial sensing, with cells following concentration gradients as long as these are present. Utilizing a microfluidic approach, we found that stable gradients of intermediate chemokines (CCL19 and CXCL12) failed to promote persistent directional migration of dendritic cells or neutrophils. Instead, rising chemokine concentrations were needed, implying that temporal sensing mechanisms controlled prolonged responses to these ligands. This behavior was found to depend on G-coupled receptor kinase-mediated negative regulation of receptor signaling and contrasted with responses to an end agonist chemoattractant (C5a), for which a stable gradient led to persistent migration. These findings identify temporal sensing as a key requirement for long-range myeloid cell migration to intermediate chemokines and provide insights into the mechanisms controlling immune cell motility in complex tissue environments.

Unified path integral approach to theories of diffusion-influenced reactions.

Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we develop a general approach for computational modeling of diffusion-influenced reactions that is capable of capturing not only the classical Smoluchowski picture but also alternative theories, as is here exemplified by a volume reactivity model. In particular, we prove the path decomposition expansion of various Green's functions describing the irreversible and reversible reaction of an isolated pair of molecules. To this end, we exploit a connection between boundary value and interaction potential problems with δ- and δ^{'}-function perturbation. We employ a known path-integral-based summation of a perturbation series to derive a number of exact identities relating propagators and survival probabilities satisfying different boundary conditions in a unified and systematic manner. Furthermore, we show how the path decomposition expansion represents the propagator as a product of three factors in the Laplace domain that correspond to quantities figuring prominently in stochastic spatially resolved simulation algorithms. This analysis will thus be useful for the interpretation of current and the design of future algorithms. Finally, we discuss the relation between the general approach and the theory of Brownian functionals and calculate the mean residence time for the case of irreversible and reversible reactions.

Environment Tunes Propagation of Cell-to-Cell Variation in the Human Macrophage Gene Network.

Cell-to-cell variation in gene expression and the propagation of such variation (PoV or "noise propagation") from one gene to another in the gene network, as reflected by gene-gene correlation across single cells, are commonly observed in single-cell transcriptomic studies and can shape the phenotypic diversity of cell populations. While gene network "rewiring" is known to accompany cellular adaptation to different environments, how PoV changes between environments and its underlying regulatory mechanisms are less understood. Here, we systematically explored context-dependent PoV among genes in human macrophages, utilizing different cytokines as natural perturbations of multiple molecular parameters that may influence PoV. Our single-cell, epigenomic, computational, and stochastic simulation analyses reveal that environmental adaptation can tune PoV to potentially shape cellular heterogeneity by changing parameters such as the degree of phosphorylation and transcription factor-chromatin interactions. This quantitative tuning of PoV may be a widespread, yet underexplored, property of cellular adaptation to distinct environments.

Rate coefficients, binding probabilities, and related quantities for area reactivity models.

We further develop the general theory of the area reactivity model that describes the diffusion-influenced reaction of an isolated receptor-ligand pair in terms of a generalized Feynman-Kac equation and that provides an alternative to the classical contact reactivity model. Analyzing both the irreversible and reversible reaction, we derive the equation of motion of the survival probability as well as several relationships between single pair quantities and the reactive flux at the encounter distance. Building on these relationships, we derive the equation of motion of the many-particle survival probability for irreversible pseudo-first-order reactions. Moreover, we show that the usual definition of the rate coefficient as the reactive flux is deficient in the area reactivity model. Numerical tests for our findings are provided through Brownian Dynamics simulations. We calculate exact and approximate expressions for the irreversible rate coefficient and show that this quantity behaves differently from its classical counterpart. Furthermore, we derive approximate expressions for the binding probability as well as the average lifetime of the bound state and discuss on- and off-rates in this context. Throughout our approach, we point out similarities and differences between the area reactivity model and its classical counterpart, the contact reactivity model. The presented analysis and obtained results provide a theoretical framework that will facilitate the comparison of experiment and model predictions.

The area reactivity model of geminate recombination.

We investigate the reversible diffusion-influenced reaction of an isolated pair in the context of the area reactivity model that describes the reversible binding of a single molecule in the presence of a binding site in terms of a generalized version of the Feynman-Kac equation in two dimensions. We compute the corresponding exact Green's function in the Laplace domain for both the initially unbound and bound molecule. We discuss convolution relations that facilitate the calculation of the binding and survival probabilities. Furthermore, we calculate an exact analytical expression for the Green's function in the time domain by inverting the Laplace transform via the Bromwich contour integral and derive expressions for the binding and survival probability in the time domain as well. We numerically confirm the accuracy of the obtained expressions by propagating the generalized Feynman-Kac equation in the time domain. Our results should be useful for comparing the area reactivity model with the contact reactivity model.

Reversible diffusion-influenced reactions of an isolated pair on some two dimensional surfaces.

We investigate reversible diffusion-influenced reactions of an isolated pair in two dimensions. To this end, we employ convolution relations that permit deriving the survival probability of the reversible reaction directly in terms of the survival probability of the irreversible reaction. Furthermore, we make use of the mean reaction time approximation to write the irreversible survival probability in restricted spaces as a single exponential. In this way, we obtain exact and approximative expressions in the time domain for the reversible survival probability for three different two dimensional spatial domains: The infinite plane, the annular domain, and the surface of a sphere. Our obtained results should prove useful in the context of membrane-bound reversible diffusion-influenced reactions in cell biology.

Theory of reversible diffusion-influenced reactions with non-Markovian dissociation in two space dimensions.

We investigate the reversible diffusion-influenced reaction of an isolated pair in the presence of a non-Markovian generalization of the backreaction boundary condition in two space dimensions. Following earlier work by Agmon and Weiss, we consider residence time probability densities that decay slower than an exponential and that are characterized by a single parameter 0 < σ ≤ 1. We calculate an exact expression for a Green's function of the two-dimensional diffusion equation subject to a non-Markovian backreaction boundary condition that is valid for arbitrary σ and for all times. We use the obtained expression to derive the survival probability for the initially unbound pair and we calculate an exact expression for the probability S(t[line]*) that the initially bound particle is unbound. Finally, we obtain an approximate solution for long times. In particular, we show that the ultimate fate of the bound state is complete dissociation, as in the Markovian case. However, the limiting value is approached quite differently: Instead of a ~t(-1) decay, we obtain 1 - S(t[line]*) ~ t(-σ)ln t. The derived expressions should be relevant for a better understanding of reversible membrane-bound reactions in cell biology.

Exact Green's function of the reversible diffusion-influenced reaction for an isolated pair in two dimensions.

We derive an exact Green's function of the diffusion equation for a pair of disk-shaped interacting particles in two dimensions subject to a backreaction boundary condition. Furthermore, we use the obtained function to calculate exact expressions for the survival probability and the time-dependent rate coefficient for the initially unbound pair and the survival probability of the bound state. The derived expressions will be of particular utility for the description of reversible membrane-bound reactions in cell biology.

Computational modeling of cellular signaling processes embedded into dynamic spatial contexts.

Cellular signaling processes depend on spatiotemporal distributions of molecular components. Multicolor, high-resolution microscopy permits detailed assessment of such distributions, providing input for fine-grained computational models that explore mechanisms governing dynamic assembly of multimolecular complexes and their role in shaping cellular behavior. However, it is challenging to incorporate into such models both complex molecular reaction cascades and the spatial localization of signaling components in dynamic cellular morphologies. Here we introduce an approach to address these challenges by automatically generating computational representations of complex reaction networks based on simple bimolecular interaction rules embedded into detailed, adaptive models of cellular morphology. Using examples of receptor-mediated cellular adhesion and signal-induced localized mitogen-activated protein kinase (MAPK) activation in yeast, we illustrate the capacity of this simulation technique to provide insights into cell biological processes. The modeling algorithms, implemented in a new version of the Simmune toolset, are accessible through intuitive graphical interfaces and programming libraries.