Diffusion in cells with stochastically gated gap junctions

We analyze a one-dimensional (1D) model of molecules diffusing along a line of N cells that are connected via stochastically gated gap junctions. Each gate switches between an open (n = 0) and a closed (n = 1) state according to a two-state Markov process, and the gates are treated as statistically independent. We proceed by spatially discretizing the stochastic diffusion equation using finite differences and constructing the Chapman–Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system. We thus generate a hierarchy of equations for the rth-order moments of the stochastic concentration, which in the continuum limit take the form of r-dimensional parabolic PDEs. We explicitly solve the first-order moment equations for N = 2 and calculate the effective permeability of the gap junction. For N > 2 (more than one gap junction), we show that the N-cell network has a completely different effective single-gate permeability when each particle (rather than each gate) independently switches between two conformational states n = 0, 1 and can only pass through a gate when n = 0. This difference is due to the fact that for switching gates, all particles diffuse in the same random environment, resulting in nontrivial statistical correlations. In both cases, the effective single-gate permeability has a nontrivial dependence on the number of cells N

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 gapjunction 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

Dynamically active compartments coupled by a stochastically gated gap junction

We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate

Residence times of a Brownian particle with temporal heterogeneity

We consider a diffusing particle that randomly switches conformational state. Motivated by various scenarios in cell biology, we suppose that (a) the diffusion coefficient depends on the conformational state and/or (b) the particle can only pass through a series of gates in the domain when it is in a particular conformational state. We develop probabilistic methods to analyze this case of diffusion with temporal heterogeneity, and use these methods to calculate the expected residence time in portions of the domain before absorption at a boundary. We find several new phenomena not seen in recent studies of diffusion with spatial heterogeneity, some of which are counterintuitive. In particular, the expected residence times can be non-monotonic functions of (i) the initial distance from the absorbing boundary and (ii) the diffusion coefficients. We focus on one-dimensional intervals, but show how the analysis can be extended to spherically symmetric d-dimensional domains

An Interneuron Circuit Reproducing Essential Spectral Features of Field Potentials

This document is the Accepted Manuscript version of the following article: Reinoud Maex, ‘An Interneuron Circuit Reproducing Essential Spectral Features of Field Potentials’, Neural Computation, March 2018. Under embargo until 22 June 2018. The final, definitive version of this paper is available online at doi: https://doi.org/10.1162/NECO_a_01068. © 2018 Massachusetts Institute of Technology. Content in the UH Research Archive is made available for personal research, educational, and non-commercial purposes only. Unless otherwise stated, all content is protected by copyright, and in the absence of an open license, permissions for further re-use should be sought from the publisher, the author, or other copyright holder.Recent advances in engineering and signal processing have renewed the interest in invasive and surface brain recordings, yet many features of cortical field potentials remain incompletely understood. In the present computational study, we show that a model circuit of interneurons, coupled via both GABA(A) receptor synapses and electrical synapses, reproduces many essential features of the power spectrum of local field potential (LFP) recordings, such as 1/f power scaling at low frequency (< 10 Hz) , power accumulation in the γ-frequency band (30–100 Hz), and a robust α rhythm in the absence of stimulation. The low-frequency 1/f power scaling depends on strong reciprocal inhibition, whereas the α rhythm is generated by electrical coupling of intrinsically active neurons. As in previous studies, the γ power arises through the amplifica- tion of single-neuron spectral properties, owing to the refractory period, by parameters that favour neuronal synchrony, such as delayed inhibition. The present study also confirms that both synaptic and voltage-gated membrane currents substantially contribute to the LFP, and that high-frequency signals such as action potentials quickly taper off with distance. Given the ubiquity of electrically coupled interneuron circuits in the mammalian brain, they may be major determinants of the recorded potentials.Peer reviewe

Effective permeability of a gap junction with age-structured switching

We analyze the diffusion equation in a bounded interval with a stochastically gated interior barrier at the center of the domain. This represents a stochastically gated gap junction linking a pair of identical cells. Previous work has modeled the switching of the gate as a two-state Markov process and used the theory of diffusion in randomly switching environments to derive an expression for the effective permeability of the gap junction. In this paper we extend the analysis of gap junction permeability to the case of a gate with age-structured switching. The latter could reflect the existence of a set of hidden internal states such that the statistics of the non-Markovian two-state model matches the statistics of a higher-dimensional Markov process. Using a combination of the method of characteristics and transform methods, we solve the partial differential equations for the expectations of the stochastic concentration, conditioned on the state of the gate and after integrating out the residence time of the age-structured process. This allows us to determine the jump discontinuity of the concentration at the gap junction and thus the effective permeability. We then use stochastic analysis to show that the solution to the stochastic PDE is a certain statistic of a single Brownian particle diffusing in a stochastically fluctuating environment. In addition to providing a simple probabilistic interpretation of the stochastic PDE, this representation enables an efficient numerical approximation of the solution of the PDE by Monte Carlo simulations of a single diffusing particle. The latter is used to establish that our analytical results match those obtained from Monte Carlo simulations for a variety of age-structured distributions

Stochastically gated diffusion model of selective nuclear transport

Nuclear pore complexes (NPCs) allow the selective exchange of molecules between the cytoplasm and cell nucleus. Although small molecules can diffuse freely through a NPC, the transport of proteins and nucleotides requires association with transport factors (kaps). The latter transiently bind to disordered flexible polymers within the NPC, known collectively as phenylalanine-glycine-nucleoporins (FG-Nups). It has recently been shown that transient binding combined with diffusion in the bound state is a sufficient mechanism for selective transport. However, selectivity is significantly reduced if the mobility of the bound state is too slow. In this paper we formulate the binding-diffusion mechanism of selective transport in terms of a “stochastically gated” diffusion process in which each bound particle undergoes confined diffusion within a subdomain of the NPC. This allows us to make explicit the fact that the diffusion of a particle when bound to a polymer tether is spatially confined rather than simply reduced. We calculate the selectivity of the NPC and explore its dependence on the size of the confinement domains. We then use probabilistic methods to determine the splitting probability and mean first passage time (MFPT) for an individual particle to pass through the pore. Our analysis establishes that spatial confinement can significantly reduce selectivity in a binding-diffusion model, suggesting that other biophysical mechanisms such as interchain transfer are required

Bioelectrical model of head-tail patterning based on cell ion channels and intercellular gap junctions

Robust control of anterior-posterior axial patterning during regeneration is mediated by bioelectric signaling. However, a number of systems-level properties of bioelectrochemical circuits, including stochastic outcomes such as seen in permanently de-stabilized "cryptic" flatworms, are not completely understood. We present a bioelectrical model for head-tail patterning that combines single-cell characteristics such as membrane ion channels with multicellular community effects via voltage-gated gap junctions. It complements the biochemically-focused models by describing the effects of intercellular electrochemical coupling, cutting plane, and gap junction blocking of the multicellular ensemble. We provide qualitative insights into recent experiments concerning planarian anterior/posterior polarity by showing that: (i) bioelectrical signals can help separated cell domains to know their relative position after injury and contribute to the transitions between the abnormal double-head state and the normal head-tail state; (ii) the bioelectrical phase-space of the system shows a bi-stability region that can be interpreted as the cryptic system state; and (iii) context-dependent responses are obtained depending on the cutting plane position, the initial bioelectrical state of the multicellular system, and the intercellular connectivity. The model reveals how simple bioelectric circuits can exhibit complex tissue-level patterning and suggests strategies for regenerative control in vivo and in synthetic biology contexts

Diffusion in an age-structured randomly switching environment

Age-structured processes are well-established in population biology, where birth and death rates often depend on the age of the underlying populations. Recently, however, different examples of age-structured processes have been considered in the context of cell motility or certain types of stochastically gated ion channels, where the state of the system is determined by a switching process with age-dependent transition rates. In this paper we consider the particular problem of diffusion on a finite interval, with randomly switching boundary conditions due to the presence of an age-structured stochastic gate at one end of the interval. When the gate is closed the particles are reflected, whereas when it is open the domain is in contact with a particle bath. We use a moments method to derive a partial differential equation for the expectations of the stochastic concentration, conditioned on the state of the gate. We then use transform methods to eliminate the residence time of the age-structured switching, resulting in non-Markovian equations for the expectations, and determine the effective steady-state concentration gradient. Our analytical results are shown to match those obtained using Monte Carlo simulations

Using Stochastic Differential Equations to Model Gap-Junction Gating Dynamics in Cardiac Myocytes

The cell-to-cell propagation of the cardiac action potential allows for the electro-mechanical coupling of cells, which promotes the coordinated contraction of cardiac tissue, often referred to as the heartbeat. The main structures that promote electrical coupling between adjacent cardiac cells are pore-like proteins called gap junctions that line the membranes of such cells, allowing a channel for electrically charged ions to travel between cells. It is known that the conformational, and hence conducting, properties of gap-junction channels change as a function of local gap-junctional voltage and local ionic concentrations and are stochastic in nature. Many previous models of gap junctions have made a constant-resistance approximation or used an ODE model relating gating state to a local voltage. In this thesis, we extend a previous ODE model of gap-junction gating state by Henriquez et al. and formulate it as a system of stochastic differential equations (SDEs) by deriving the expected change vector and covariance matrix of the model and integrating the covariance with respect to a stochastic process, the Wiener Process. In doing so, we construct the first SDE-based model of gap-junction gating dynamics. This SDE description of the electrical coupling between cardiac cells is integrated into a 1D cable model where intracellular current dynamics are described using the Luo-Rudy 1 formulation. Monte Carlo simulations are performed on the resulting model in order to gather data used to construct distributions of several model responses of interest, including conduction block, conduction velocity, gap-junction current and gap-junction conductance. We find a smoothing effect occurs as the number of gap junctions considered increases, but at small numbers of gap junctions, such as those observed in many diseased states, stochastic effects can be pronounced. In such decoupled regimes, stochastic effects are found to have a large effect on the occurrence of conduction block, the cessation of action potential propagation at some tissue location, and are found to increase the variance in conduction velocity from cell to cell. The waiting time between when two consecutive gap junctions reach their maximum current was found to conform to a gamma distribution, with shape and scale parameters a function of the number of gap junctions. As the number of gap junctions increases, the spread of the waiting time distributions decreases. Gap-junctional conductance was modeled as a time-dependent Gaussian distribution, with a temporal variance decreasing as a function of the elapsed time after depolarization. In the case of conduction block, we show that an emulator function can be constructed to estimate the probability of occurrence, thereby reducing the need for a large number of computationally intensive Monte Carlo simulations. Along with probabilistically describing the stochastic gap- junction model, these distributions can be leveraged in larger-scale tissue-level simulations to incorporate stochastic gap-junction gating at a reduced computational cost