Searcharxiv⌕ Search

arXiv subjects

David Holcman

Publications and source records attributed to David Holcman.

At least 37 records · Page 2Linked to original sources
arXiv2020

Exit versus escape in a stochastic dynamical system of neuronal networks explains heterogenous bursting intervals

Neuronal networks can generate burst events. It remains unclear how to analyse interburst periods and their statistics. We study here the phase-space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics and we identify that interburst corresponds to the escape from a basin of attraction. Using stochastic simulations, we report here that the distribution of the these durations do not match with the time to reach the boundary. We further analyse this phenomenon by studying a generic class of two-dimensional dynamical systems perturbed by small noise that exhibits two peculiar behaviors: 1- the maximum associated to the probability density function is not located at the point attractor, which came as a surprise. The distance between the maximum and the attractor increases with the noise amplitude $σ$, as we show using WKB approximation and numerical simulations. 2- For such systems, exiting from the basin of attraction is not sufficient to characterize the entire escape time, due to trajectories that can return several times inside the basin of attraction after crossing the boundary, before eventually escaping far away. To conclude, long-interburst durations are inherent properties of the dynamics and sould be expected in empirical time series.

cond-mat.stat-mech↗
arXiv2020

Modeling bursting in neuronal networks using facilitation-depression and afterhyperpolarization

In the absence of inhibition, excitatory neuronal networks can alternate between bursts and interburst intervals (IBI), with heterogeneous length distributions. As this dynamic remains unclear, especially the durations of each epoch, we develop here a bursting model based on synaptic depression and facilitation that also accounts for afterhyperpolarization (AHP), which is a key component of IBI. The framework is a novel stochastic three dimensional dynamical system perturbed by noise: numerical simulations can reproduce a succession of bursts and interbursts. Each phase corresponds to an exploration of a fraction of the phase-space, which contains three critical points (one attractor and two saddles) separated by a two-dimensional stable manifold $Σ$. We show here that bursting is defined by long deterministic excursions away from the attractor, while IBI corresponds to escape induced by random fluctuations. We show that the variability in the burst durations, depends on the distribution of exit points located on $Σ$ that we compute using WKB and the method of characteristics. Finally, to better characterize the role of several parameters such as the network connectivity or the AHP time scale, we compute analytically the mean burst and AHP durations in a linear approximation. To conclude the distribution of bursting and IBI could result from synaptic dynamics modulated by AHP.

q-bio.NC↗
arXiv2020

Escape from an attractor generated by recurrent exit

Kramer's theory of activation over a potential barrier consists in computing the mean exit time from the boundary of a basin of attraction of a randomly perturbed dynamical system. Here we report that for some systems, crossing the boundary is not enough, because stochastic trajectories return inside the basin with a high probability a certain number of times before escaping far away. This situation is due to a shallow potential. We compute the mean and distribution of escape times and show how this result explains the large distribution of interburst durations in neuronal networks.

cond-mat.stat-mech↗
arXiv2020

Computational methods and fast hybrid stochastic simulations for triangulation sensing and identifying principles of cell navigation in the brain

Brownian simulations can be used to generate statistics relevant for studying molecular interactions or trafficking. However, the concurrent simulation of many Brownian trajectories at can become computationally intractable. Replacing detailed Brownian simulations by a rate model was the basis of Gillespie's algorithm, but requires one to disregard spatial information. However, this information is crucial in molecular and cellular biology. Alternatively one can use a hybrid approach, generating Brownian paths only in a small region where the spatial organization is relevant and avoiding it in the remainder of the domain. Here we review the recent progress of hybrid methods and simulations in the context of cell sensing and guidance via external chemical gradients. Specifically, we highlight the reconstruction of the location of a point source in 2D and 3D from diffusion fluxes arriving at narrow windows located on the cell. We discuss cases in which these windows are located on the boundary of the 2D or 3D half-space, on a disk in free space, inside a 2D corridor, or a 3D ball. The hybrid method in question performs Brownian simulations only inside a region of interest. It uses the Neumann-Green's function for the mentioned geometries to generate exact mappings exit and entry points when the trajectory leaves the region, thus avoiding the explicit computation of Brownian paths in an infinite domain. Matched asymptotics is used to compute the probability fluxes to small windows and we review how such an approach can be used to reconstruct the location of a point source and estimating the uncertainty in the source reconstruction due to an additive perturbation present in the fluxes. We also review the influence of various window configurations on the source position recovery. Finally, we discuss potential applications in developmental cell biology and possible computational principles.

cond-mat.stat-mech↗
arXiv2020

Voltage distribution in a non-locally but globally electroneutral confined electrolyte medium: applications for nanophysiology

The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface.

cond-mat.soft↗
arXiv2019

Triangulation sensing: how cells recover a source from diffusing particles in three dimensions

How can cells embedded into a gradient concentration triangulate the position of the source and migrate toward their final destination? The source triangulation requires to recover the three dimensional coordinates of the source from the fluxes of diffusing cues at narrow windows (receptors) located on the surface of a cell. We develop here a method to address this question and we show in the limit of fast binding rate to the receptors, that at least three receptors are necessary. We solve the steady-state diffusion equation using an asymptotic approach, which agrees with hybrid stochastic-analytical simulations. Interestingly, with an accuracy of few percent, the source cannot be located if it is located at a distance tens of time the size of the cell. Finally, the precision of the source recovery increases with the number of receptors.

q-bio.CB↗
arXiv2018

Active unidirectional network flow generates a packet molecular transport in cells

Internet, social media, neuronal or blood vessel are organized in complex networks. These networks are characterized by several quantities such as the underlying graph connectivity (topology), how they grow in time, scaling laws or by the mean time a random search can cover a network and also by shortest path between two nodes. We present here a novel type of network property based on a unidirectional transport mechanism occurring in the Endoplasmic Reticulum network found in the cell cytoplasm. This mechanism is an active-waiting transportation, where molecules have to wait a random time before being transported from one node to the next one. We find that the consequence of this unusual network transportation is that molecules travel together by recurrent packets, which quite a large deviation behavior compared to classical propagation in graphs. To conclude, this form of transportation is associated with an efficient and robust molecular redistribution inside cells.

q-bio.SC↗
arXiv2018

Extreme Narrow Escape: shortest paths for the first particles to escape through a small window

What is the path associated with the fastest Brownian particle that reaches a narrow window located on the boundary of a domain? Although the distribution of the fastest arrival times has been well studied in dimension 1, much less is known in higher dimensions. Based on the Wiener path-integral, we derive a variational principle for the path associated with the fastest arrival particle. Specifically, we show that in a large ensemble of independent Brownian trajectories, the first moment of the shortest arrival time is associated with the minimization of the energy-action and the optimal trajectories are geodesics. Escape trajectories concentrate along these geodesics, as confirmed by stochastic simulations when an obstacle is positioned in front of the narrow window. To conclude paths in stochastic dynamics and their time scale can differ significantly from the mean properties, usually at the basis of the Smoluchowski's theory of chemical reactions.

cond-mat.stat-mech↗
arXiv2017

Do cells sense time by number of divisions?

Do biological cells sense time by the number of their divisions, a process that ends at senescence? We consider the question "can the cell's perception of time be expressed through the length of the shortest telomere?" The answer is that the absolute time before senescence cannot be expressed by the telomere's length and that a cell can survive many more divisions than intuitively expected. This apparent paradox is due to shortening and elongation of the telomere, which suggests a random walk model of the telomere's length. The model indicates two phases, first, a determinist drift of the length toward a quasi-equilibrium state, and second, persistence of the length near an attracting state for the majority of divisions prior to senescence. The measure of stability of the latter phase is the expected number of divisions at the attractor ("lifetime") prior to crossing a threshold to senescence. The telomerase regulates stability by creating an effective potential barrier that separates statistically the shortest lifetime from the next shortest. The random walk has to overcome the barrier in order to extend the range of the first regime. The model explains how random telomere dynamics underlies the extension of cell survival time.

q-bio.SC↗
arXiv2017

Asymptotics of extreme statistics of escape time in 1,2 and 3-dimensional diffusions

The first of $N$ identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target, sets the time scale of activation, which in general is much faster than the arrival to the target of only a single trajectory. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number of i.i.d. Brownian trajectories to a small target in 1,2, and 3 dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular transduction.

q-bio.SC↗
arXiv2017

Mixed analytical-stochastic simulation method for the recovery of a Brownian gradient source from probability fluxes to small windows

Is it possible to recover the position of a source from the steady-state fluxes of Brownian particles to small absorbing windows located on the boundary of a domain? To address this question, we develop a numerical procedure to avoid tracking Brownian trajectories in the entire infinite space. Instead, we generate particles near the absorbing windows, computed from the analytical expression of the exit probability. When the Brownian particles are generated by a steady-state gradient at a single point, we compute asymptotically the fluxes to small absorbing holes distributed on the boundary of half-space and on a disk in two dimensions, which agree with stochastic simulations. We also derive an expression for the splitting probability between small windows using the matched asymptotic method. Finally, when there are more than two small absorbing windows, we show how to reconstruct the position of the source from the diffusion fluxes. The present approach provides a computational first principle for the mechanism of sensing a gradient of diffusing particles, a ubiquitous problem in cell biology.

q-bio.CB↗
arXiv2017

Encounter time of two loci governed by polymer de-condensation and local chromatin interaction

The time for a DNA sequence to find its homologous depends on a long random search process inside the cell nucleus. Using polymer models, we model and compute here the mean first encounter time (MFET) between two sites located on two different polymer chains and confined by potential wells. We find that reducing the potential (tethering) forces results in a local polymer decondensation near the loci and numerical simulations of the polymer model show that these changes are associated with a reduction of the MFET by several orders of magnitude. We derive here new asymptotic formula for the MFET, confirmed by Brownian simulations. We conclude that the acceleration of the search process after local chromatin decondensation can be used to analyze the local search step during homology search.

physics.bio-ph↗
arXiv2017

Randomly cross-linked polymer models

Polymer models are used to describe chromatin, which can be folded at different spatial scales by binding molecules. By folding, chromatin generates loops of various sizes. We present here a randomly cross-linked (RCL) polymer model, where monomer pairs are connected randomly. We obtain asymptotic formulas for the steady-state variance, encounter probability, the radius of gyration, instantaneous displacement and the mean first encounter time between any two monomers. The analytical results are confirmed by Brownian simulations. Finally, the present results can be used to extract the minimum number of cross-links in a chromatin region from {conformation capture} data.

cond-mat.soft↗
arXiv2017

Reconstructing the gradient source position from steady-state fluxes to small receptors

Recovering the position of a source from the fluxes of diffusing particles through small receptors allows a biological cell to determine its relative position, spatial localization and guide it to a final target. However, how a source can be recovered from point fluxes remains unclear. Using the Narrow Escape Time approach for an open domain, we compute the diffusion fluxes of Brownian particles generated by a steady-state gradient from a single source through small holes distributed on a surface in two dimensions. {We find that the location of a source can be recovered when there are at least 3 receptors and the source is positioned no further than 10 cell radii away}, but this condition is not necessary in a narrow strip. The present approach provides a computational basis for the first step of direction sensing of a gradient at a single cell level.

q-bio.CB↗
arXiv2017

Geometrical effects on nonlinear electrodiffusion in cell physiology

We report here new electrical laws, derived from nonlinear electro-diffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck (PNP) equations for charge concentration and electric potential as a model of electro-diffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson's equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nano-pipettes and help to understand the local voltage changes inside dendrites and axons with heterogenous local geometry.

q-bio.SC↗
arXiv2017

Electrostatics of non-neutral biological microdomains

Voltage and charge distributions in cellular microdomains regulate communications, excitability, and signal transduction. We report here new electrical laws in a cell, which follow from a nonlinear electro-diffusion model. These newly discovered relations derive from the geometrical cell-membrane properties, such as membrane curvature, volume, and surface area. These electro-diffusion laws can now be used to predict and interpret voltage distribution in cellular microdomains.

physics.bio-ph↗
arXiv2016

100 years after Smoluchowski: stochastic processes in cell biology

100 years after Smoluchowski introduces his approach to stochastic processes, they are now at the basis of mathematical and physical modeling in cellular biology: they are used for example to analyse and to extract features from large number (tens of thousands) of single molecular trajectories or to study the diffusive motion of molecules, proteins or receptors. Stochastic modeling is a new step in large data analysis that serves extracting cell biology concepts. We review here the Smoluchowski's approach to stochastic processes and provide several applications for coarse-graining diffusion, studying polymer models for understanding nuclear organization and finally, we discuss the stochastic jump dynamics of telomeres across cell division and stochastic gene regulation.

physics.data-an↗