Protein concentration gradients and switching diffusions

Truncated stochastically switching processes

There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.

Stochastically switching diffusion with partially reactive surfaces

In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a two-state model in which the probability of surface absorption depends on the current particle state and the amount of time the particle has spent in a neighborhood of the surface in each state. The latter is determined by a pair of local times ℓ_{n,t}, n=0,1, which are Brownian functionals that keep track of particle-surface encounters over the time interval [0,t]. We proceed by constructing a differential Chapman-Kolmogorov equation for a pair of generalized propagators P_{n}(x,ℓ_{0},ℓ_{1},t), where P_{n} is the joint probability density for the set (X_{t},ℓ_{0,t},ℓ_{1,t}) when N_{t}=n, where X_{t} denotes the particle position and N_{t} is the corresponding conformational state. Performing a double Laplace transform with respect to ℓ_{0},ℓ_{1} yields an effective system of equations describing diffusion in a bounded domain Ω, in which there is switching between two Robin boundary conditions on ∂Ω. The corresponding constant reactivities are κ_{j}=Dz_{j} and j=0,1, where z_{j} is the Laplace variable corresponding to ℓ_{j} and D is the diffusivity. Given the solution for the propagators in Laplace space, we construct a corresponding probabilistic model for partial absorption, which requires finding the inverse Laplace transform with respect to z_{0},z_{1}. We illustrate the theory by considering diffusion of a particle on the half-line with the boundary at x=0 effectively switching between a totally reflecting and a partially absorbing state. We calculate the flux due to absorption and use this to compute the resulting MFPT in the presence of a renewal-based stochastic resetting protocol. The latter resets the position and conformational state of the particle as well as the corresponding local times. Finally, we indicate how to extend the analysis to higher spatial dimensions using the spectral theory of Dirichlet-to-Neumann operators.

Local accumulation times in a diffusion-trapping model of receptor dynamics at proximal axodendritic synapses

The lateral diffusion and trapping of neurotransmitter receptors within the postsynaptic membrane of a neuron play a key role in determining synaptic strength and plasticity. Trapping is mediated by the reversible binding of receptors to scaffolding proteins (slots) within a synapse. In this paper we introduce a method for analyzing the transient dynamics of proximal axodendritic synapses in a diffusion-trapping model of receptor trafficking. Given a population of spatially distributed synapses, each of which has a fixed number of slots, we calculate the rate of relaxation to the steady-state distribution of bound slots (synaptic weights) in terms of a set of local accumulation times. Assuming that the rates of exocytosis and endocytosis are sufficiently slow, we show that the steady-state synaptic weights are independent of each other (purely local). On the other hand, the local accumulation time of a given synapse depends on the number of slots and the spatial location of all the synapses, indicating a form of transient heterosynaptic plasticity. This suggests that local accumulation time measurements could provide useful information regarding the distribution of synaptic weights within a dendrite.

Modeling protein dynamics in Caenorhabditis elegans embryos reveals that the PLK-1 gradient relies on weakly coupled reaction-diffusion mechanisms

Intracellular gradients have essential roles in cell and developmental biology, but their formation is not fully understood. We have developed a computational approach facilitating interpretation of protein dynamics and gradient formation. We have combined this computational approach with experiments to understand how Polo-Like Kinase 1 (PLK-1) forms a cytoplasmic gradient in Caenorhabditis elegans embryos. Although the PLK-1 gradient depends on the Muscle EXcess-5/6 (MEX-5/6) proteins, we reveal differences in PLK-1 and MEX-5 gradient formation that can be explained by a model with two components, PLK-1 bound to MEX-5 and unbound PLK-1. Our combined approach suggests that a weak coupling between PLK-1 and MEX-5 reaction–diffusion mechanisms dictates the dynamic exchange of PLK-1 with the cytoplasm, explaining PLK-1 high diffusivity and smooth gradient.

Protein gradients have fundamental roles in cell and developmental biology. In the one-cell Caenorhabditis elegans embryo, the mitotic Polo-Like Kinase 1 (PLK-1) forms an anterior-rich cytoplasmic gradient, which is crucial for asymmetric cell division and embryonic development. The PLK-1 gradient depends on the RNA-binding Muscle-EXcess-5 protein (MEX-5), whose slow-diffusing complexes accumulate in the anterior via a reaction–diffusion mechanism. Here, we combine experiments and a computational approach to investigate the dynamics of PLK-1 gradient formation. We find that the gradient of PLK-1 initiates later, is less steep, and forms with slower dynamics than does the MEX-5 gradient. The data show that PLK-1 diffuses faster than MEX-5 in both anterior and posterior cytoplasmic regions. Our simulations suggest that binding to slow-diffusing MEX-5 is required for PLK-1 gradient formation, but that a significant fraction of unbound PLK-1 is necessary to justify the different gradient dynamics. We provide a computational tool able to predict gradient establishment prior to cell division and show that a two-component, bound and unbound, model of PLK-1 dynamics recapitulates the experimental observations.

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

A general problem of current interest is the analysis of diffusion problems in singularly perturbed domains, within which small subdomains are removed from the domain interior and boundary conditions imposed on the resulting holes. One major application is to intracellular diffusion, where the holes could represent organelles or biochemical substrates. In this paper, we use a combination of matched asymptotic analysis and Green’s function methods to calculate the so-called accumulation time for relaxation to steady state. The standard measure of the relaxation rate is in terms of the principal non-zero eigenvalue of the negative Laplacian. However, this global measure does not account for possible differences in the relaxation rate at different spatial locations, is independent of the initial conditions, and relies on the assumption that the eigenvalues have sufficiently large spectral gaps. As previously established for diffusion-based morphogen gradient formation, the accumulation time provides a better measure of the relaxation process.

Local accumulation time for diffusion in cells with gap junction coupling

In this paper we analyze the relaxation to steady state of intracellular diffusion in a pair of cells with gap-junction coupling. Gap junctions are prevalent in most animal organs and tissues, providing a direct diffusion pathway for both electrical and chemical communication between cells. Most analytical models of gap junctions focus on the steady-state diffusive flux and the associated effective diffusivity. Here we investigate the relaxation to steady state in terms of the so-called local accumulation time. The latter is commonly used to estimate the time to form a protein concentration gradient during morphogenesis. The basic idea is to treat the fractional deviation from the steady-state concentration as a cumulative distribution for the local accumulation time. One of the useful features of the local accumulation time is that it takes into account the fact that different spatial regions can relax at different rates. We consider both static and dynamic gap junction models. The former treats the gap junction as a resistive channel with effective permeability μ, whereas the latter represents the gap junction as a stochastic gate that randomly switches between an open and closed state. The local accumulation time is calculated by solving the diffusion equation in Laplace space and then taking the small-s limit. We show that the accumulation time is a monotonically increasing function of spatial position, with a jump discontinuity at the gap junction. This discontinuity vanishes in the limit μ→∞ for a static junction and β→0 for a stochastically gated junction, where β is the rate at which the gate closes. Finally, our results are generalized to the case of a linear array of cells with nearest-neighbor gap junction coupling.

Biophysical Models of PAR Cluster Transport by Cortical Flow in C. elegans Early Embryogenesis

The clustering of membrane-bound proteins facilitates their transport by cortical actin flow in early Caenorhabditis elegans embryo cell polarity. PAR-3 clustering is critical for this process, yet the biophysical processes that couple protein clusters to cortical flow remain unknown. We develop a discrete, stochastic agent-based model of protein clustering and test four hypothetical models for how clusters may interact with the flow. Results show that the canonical way to assess transport characteristics from single-particle tracking data used thus far in this area, the Péclet number, is insufficient to distinguish these hypotheses and that all models can account for transport characteristics quantified by this measure. However, using this model, we demonstrate that these different cluster-cortex interactions may be distinguished using a different metric, namely the scalar projection of cluster displacement on to the flow displacement vector. Our results thus provide a testable way to use existing single-particle tracking data to test how endogenous protein clusters may interact with the cortical flow to localize during polarity establishment. To facilitate this investigation, we also develop both improved simulation and semi-analytic methodologies to quantify motion summary statistics (e.g., Péclet number and scalar projection) for these stochastic models as a function of biophysical parameters.

Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics

A novel approach to simulate simple protein-ligand systems at large time and length scales is to couple Markov state models (MSMs) of molecular kinetics with particle-based reaction-diffusion (RD) simulations, MSM/RD. Currently, MSM/RD lacks a mathematical framework to derive coupling schemes, is limited to isotropic ligands in a single conformational state, and lacks multiparticle extensions. In this work, we address these needs by developing a general MSM/RD framework by coarse-graining molecular dynamics into hybrid switching diffusion processes. Given enough data to parameterize the model, it is capable of modeling protein-protein interactions over large time and length scales, and it can be extended to handle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules. To enable reproducibility, we have published our code in the MSM/RD software package.

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.