arXiv subjects

David Holcman

Publications and source records attributed to David Holcman.

At least 55 records · Page 3Linked to original sources

arXiv2016

Stochastic coagulation-fragmentation processes with a finite number of particles and applications

Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters. Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data is now available, but their interpretation remains challenging.} We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters. Using combinatorics properties and Markov chain representation, we compute steady-state distributions and moments for the number of particles per cluster in the case where the coagulation and fragmentation rates follow a detailed balance condition. We obtain explicit and asymptotic formulas for the cluster size and the number of clusters in terms of hypergeometric functions. To further characterize clustering, we introduce and discuss two mean times: one is time two particles spend together before they separate and other is the time they spend separated before they meet again for the first time. Finally we discuss applications of the present stochastic coagulation-fragmentation framework in cell biology.

math.PR↗

arXiv2016

Narrow escape to small windows on a small ball modeling the viral entry into the cell nucleus

A certain class of viruses replicates inside a cell if they can enter the nucleus through one of many small target pores, before being permanently trapped or degraded. We adopt for viral motion a switching stochastic process model and we estimate here the probability and the conditional mean first passage time for a viral particle to attain alive the nucleus. The cell nucleus is covered with thousands of small absorbing nuclear pores and the minimum distance between them defines the smallest spatial scale that limits the efficiency of stochastic simulations. Using the Neuman-Green's function method to solve the steady-state Fokker-Planck equation, we derive asymptotic formula for the probability and mean arrival time to a small window for various pores' distributions, that agree with stochastic simulations. These formulas reveal how key geometrical parameters defines the cytoplasmic stage of viral infection.

q-bio.SC↗

arXiv2015

Stochastic model of endosomal escape of Influenza virus

Influenza viruses enter a cell via endocytosis after binding to the surface. During the endosomal journey, acidification triggers a conformational change of the virus spike protein hemagglutinin (HA) that results in escape of the viral genome from the endosome to the cytoplasm. A quantitative understanding of the processes involved in HA mediated fusion with the endosome is still missing. We develop here a stochastic model to estimate the change of conformation of HAs inside the endosome nanodomain. Using a Markov-jump process to model the arrival of protons to HA binding sites, we compute the kinetics of their accumulation and the mean first time for HAs to be activated. This analysis reveals that HA proton binding sites possess a high chemical barrier, ensuring a stability of the spike protein at sub-acidic pH. Finally, we predict that activating more than 3 adjacent HAs is necessary to prevent a premature fusion.

q-bio.SC↗

arXiv2015

Recovering a stochastic process from noisy ensembles of many single particle trajectories

Recovering a stochastic process from noisy ensembles of single particle trajectories (SPTs) is resolved here using the Langevin equation as a model. The massive redundancy contained in SPTs data allows recovering local parameters of the underlying physical model. We use several parametric and non-parametric estimators to compute the first and second moment of the process and to recover the local drift, its derivative and the diffusion tensor. Using a local asymptotic expansion of the estimators and computing the empirical transition probability function, we develop here a method to deconvolve the instrumental from the physical noise. We use numerical simulations to explore the range of validity for the estimators. The present analysis allows characterizing what can exactly be recovered from the statistics of super-resolution microscopy trajectories used in molecular trafficking and underlying cellular function.

q-bio.SC↗

arXiv2014

Oscillatory survival probability and eigenvalues of the non-self adjoint Fokker-Planck operator

We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics $d\x_\eps=\mb{a}(\x_\eps)\, dt +\sqrt{2\eps}\,\mb{b}(\x_\eps)\,d\w$, which is activated by small noise over the boundary of the domain of attraction $D$ of a stable focus of the drift $\mb{a}(\x)$. The boundary $\p D$ of the domain is an unstable limit cycle of $\mb{a}(\x)$. The oscillations are explained by a singular perturbation expansion of the spectrum of the Dirichlet problem for the non-self adjoint Fokker-Planck operator in $D$ \[L_\eps u(\x)=\,\eps\sum_{i,j=1}^2 \frac{\p ^2\left[ σ^{i,j}\left(\x\right) u(\x) \right]}{\p x^i\p x^j}-\sum_{i=1}^2\frac {\p \left[ a^i\left(\x\right) u(\x)\right]} {\p x^i} =-λ_\eps u(\x),\] with $\mbσ(\x)=\mb{b}(\x)\mb{b}^T(\x)$. We calculate the leading-order asymptotic expansion of all eigenvalues $λ_\eps$ for small $\eps$. The principal eigenvalue is known to decay exponentially fast as $\eps\to0$. We find that for small $\eps$ the higher-order eigenvalues are given by $λ_{m,n}=nω_1+miω_2+O(\eps)$ for $n=1,2,\ldots,\,m=\pm1,\ldots$, where $ω_1$ and $ω_2$ are explicitly computed constants. We also find the asymptotic structure of the eigenfunctions of $L_\eps$ and of its adjoint $L^*_\eps$. We illustrate the oscillatory decay with a model of synaptic depression of neuronal network in neurobiology.

math.AP↗

arXiv2013

Computation of the Mean First-Encounter Time Between the Ends of a Polymer Chain

Using a novel theoretical approach, we study the mean first encounter time (MFET) between the two ends of a polymer. Previous approaches used various simplifications that reduced the complexity of the problem, leading, however to incompatible results. We construct here for the first time a general theory that allows us to compute the MFET. The method is based on estimating the mean time for a Brownian particle to reach a narrow domain in the polymer configuration space. In dimension two and three, we find that the MFET depends mainly on the first eigenvalue of the associated Fokker-Planck operator and provide precise estimates that are confirmed by Brownian simulations. Interestingly, although many time scales are involved in the encounter process, its distribution can be well approximated by a single exponential, which has several consequences for modeling chromosome dynamics in the nucleus. Another application of our result is computing the mean time for a DNA molecule to form a closed loop (when its two ends meet for the first time).

cond-mat.stat-mech↗

arXiv2012

Brownian Motion in Dire Straits

The passage of Brownian motion through a bottleneck in a bounded domain is a rare event and the mean time for such passage increases indefinitely as the bottleneck's radius shrinks to zero. Its calculation reveals the effect of geometry and smoothness on the flux through the bottleneck. We find new behavior of the narrow escape time through bottlenecks in planar and spatial domains and on a surface. Some applications are discussed.

math-ph↗

arXiv2012

The Complexity of Synaptic Transmission Revealed by a Multiscale Analysis Approach From The Molecular to The Cellular Level

Synaptic Transmission is a multiscale process, revealed by various approaches, going from the molecular to the cellular level. This correspondence points out to our recent contributions in this field. We now provide the physical and mathematical foundation, leading to the rational quantification of the analysis of the stochastic steps, underlying synaptic transmission.

q-bio.NC↗

arXiv2012

Coagulation-fragmentation for a finite number of particles and application to telomere clustering in the yeast nucleus

We develop a coagulation-fragmentation model to study a system composed of a small number of stochastic objects moving in a confined domain, that can aggregate upon binding to form local clusters of arbitrary sizes. A cluster can also dissociate into two subclusters with a uniform probability. To study the statistics of clusters, we combine a Markov chain analysis with a partition number approach. Interestingly, we obtain explicit formulas for the size and the number of clusters in terms of hypergeometric functions. Finally, we apply our analysis to study the statistical physics of telomeres (ends of chromosomes) clustering in the yeast nucleus and show that the diffusion-coagulation-fragmentation process can predict the organization of telomeres.

q-bio.SC↗

arXiv2011

Modeling the early steps of cytoplasmic trafficking in viral infection and gene delivery

Gene delivery of nucleic acid to the cell nucleus is a fundamental step in gene therapy. In this review of modeling drug and gene delivery, we focus on the particular stage of plasmid DNA or virus cytoplasmic trafficking. A challenging problem is to quantify the success of this limiting stage. We present some models and simulations of plasmid trafficking and of the limiting phase of DNA-polycation escape from an endosome and discuss virus cytoplasmic trafficking. The models can be used to assess the success of viral escape from endosomes, to quantify the early step of viral-cell infection, and to propose new simulation tools for designing new hybrid-viruses as synthetic vectors.

q-bio.SC↗

arXiv2011

The Mean First Rotation Time of a planar polymer

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on the square root of n and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations.Our analytical results are confirmed by Brownian simulations.

math.PR↗

arXiv2011

Transcription factor search for a DNA promoter in a three-states model

To ensure fast gene activation, Transcription Factors (TF) use a mechanism known as facilitated diffusion to find their DNA promoter site. Here we analyze such a process where a TF alternates between 3D and 1D diffusion. In the latter (TF bound to the DNA), the TF further switches between a fast translocation state dominated by interaction with the DNA backbone, and a slow examination state where interaction with DNA base pairs is predominant. We derive a new formula for the mean search time, and show that it is faster and less sensitive to the binding energy fluctuations compared to the case of a single sliding state. We find that for an optimal search, the time spent bound to the DNA is larger compared to the 3D time in the nucleus, in agreement with recent experimental data. Our results further suggest that modifying switching via phosphorylation or methylation of the TF or the DNA can efficiently regulate transcription.

q-bio.SC↗

arXiv2011

The Narrow Escape problem in a flat cylindrical microdomain with application to diffusion in the synaptic cleft

The mean first passage time (MFPT) for a Brownian particle to reach a small target in cellular microdomains is a key parameter for chemical activation. Although asymptotic estimations of the MFPT are available for various geometries, these formula cannot be applied to degenerated structures where one dimension of is much smaller compared to the others. Here we study the narrow escape time (NET) problem for a Brownian particle to reach a small target located on the surface of a flat cylinder, where the cylinder height is comparable to the target size, and much smaller than the cylinder radius. When the cylinder is sealed, we estimate the MFPT for a Brownian particle to hit a small disk located centrally on the lower surface. For a laterally open cylinder, we estimate the conditional probability and the conditional MFPT to reach the small disk before exiting through the lateral opening. We apply our results to diffusion in the narrow synaptic cleft, and compute the fraction and the mean time for neurotransmitters to find their specific receptors located on the postsynaptic terminal. Finally, we confirm our formulas with Brownian simulations.

q-bio.NC↗

arXiv2010

Stochastic modeling of gene activation and application to cell regulation

Transcription factors (TFs) are key regulators of gene expression. Based on the classical scenario in which the TF search process switches between one-dimensional motion along the DNA molecule and free Brownian motion in the nucleus, we study the arrival time of several TFs to multiple binding sites and derive, in the presence of competitive binding ligands, the probability that several target sites are bound. We then apply our results to the hunchback regulation by bicoid in the fly embryo and we propose a general mechanism that allows cells to read a morphogenetic gradient and specialize according to their position in the embryo.

q-bio.GN↗

arXiv2010

Estimating the synaptic current in a multi-conductance AMPA receptor model

A pre-synaptic neuron releases diffusing neurotransmitters such as glutamate that activate post-synaptic receptors. The amplitude of the post-synaptic current, mostly mediated by glutamatergic (AMPARs) receptors, is a fundamental signal that may generate an action potential. However, although various simulation results \cite{kullman,Barbour,Raghavachari} have addressed how synapses control the post-synaptic current, it is still unclear how this current depends analytically on factors such as the synaptic cleft geometry, the distribution, the number and the multi-conductance state of receptors, the geometry of post-synaptic density (PSD) and the neurotransmitter release location. To estimate the synaptic current maximal amplitude, we present a semi-analytical model of glutamate diffusing in the synaptic cleft. We modeled receptors as multi-conductance channels and we find that PSD morphological changes can significantly modulate the synaptic current, which is maximally reliable (the coefficient of variation is minimal) for an optimal size of the PSD, that depends on the vesicular release active zone. The existence of an optimal PSD size is related to nonlinear phenomena such as the multi-binding cooperativity of the neurotransmitter to the receptors. We conclude that changes in the PSD geometry can sustain a form of synaptic plasticity, independent of a change in the number of receptors.

q-bio.NC↗

arXiv2008

Semi-classical limits of the first eigenfunction and concentration on the recurrent sets of a dynamical system

Dear Reader, please find the third and last part of a series of papers on the singular perturbation of the first eigenfunction associated to a non self-adjoint second order elliptic operators. This series started in 1999 and we presented the early results in 2000 at Columbia University. We published two notes in CRAS in 2001 and 2005 summarizing our results. The present paper contains the proofs of the announced theorems and many open questions. We tried to publish these results in the the top tier of mathematical journals (Annals, Acta, Duke...) but our results were not deemed sufficiently interesting for them and probably not trendy enough. Some of you may like this work, so here it is. Best Regards, Ivan and David. We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold, when the diffusion constant $ε$ goes to zero. If the drift of the diffusion is given by a Morse-Smale vector field $b$, the limits of the eigenfunctions concentrate on the recurrent set of $b$. A blow-up analysis enables us to find the main properties of the limit measures on a recurrent set.

math-ph↗

arXiv2007

Effective Motion of a Virus Trafficking Inside a Biological Cell

Virus trafficking is fundamental for infection success and plasmid cytosolic trafficking is a key step of gene delivery. Based on the main physical properties of the cellular transport machinery such as microtubules, motor proteins, our goal here is to derive a mathematical model to study cytoplasmic trafficking. Because experimental results reveal that both active and passive movement are necessary for a virus to reach the cell nucleus, by taking into account the complex interactions of the virus with the microtubules, we derive here an estimate of the mean time a virus reaches the nucleus. In particular, we present a mathematical procedure in which the complex viral movement, oscillating between pure diffusion and a deterministic movement along microtubules, can be approximated by a steady state stochastic equation with a constant effective drift. An explicit expression for the drift amplitude is given as a function of the real drift, the density of microtubules and other physical parameters. The present approach can be used to model viral trafficking inside the cytoplasm, which is a fundamental step of viral infection, leading to viral replication and in some cases to cell damage.

q-bio.QM↗

arXiv2006

Dwell time of a Brownian interacting molecule in a cellular microdomain

The time spent by an interacting Brownian molecule inside a bounded microdomain has many applications in cellular biology, because the number of bounds is a quantitative signal, which can initiate a cascade of chemical reactions and thus has physiological consequences. In the present article, we propose to estimate the mean time spent by a Brownian molecule inside a microdomain $Ω$ which contains small holes on the boundary and agonist molecules located inside. We found that the mean time depends on several parameters such as the backward binding rate (with the agonist molecules), the mean escape time from the microdomain and the mean time a molecule reaches the binding sites (forward binding rate). In addition, we estimate the mean and the variance of the number of bounds made by a molecule before it exits $Ω$. These estimates rely on a boundary layer analysis of a conditional mean first passage time, solution of a singular partial differential equation. In particular, we apply the present results to obtain an estimate of the mean time spent (Dwell time) by a Brownian receptor inside a synaptic domain, when it moves freely by lateral diffusion on the surface of a neuron and interacts locally with scaffolding molecules.

physics.bio-ph↗