Some Results on Brownian Motion Perturbed by Alternating Jumps in Biological Modeling

We consider the model of random evolution on the real line consisting in a Brownian motion perturbed by alternating jumps. We give the probability density of the process and pinpoint a connection with the limit density of a telegraph process subject to alternating jumps. We study the first-crossing-time probability in two special cases, in the presence of a constant upper boundary

A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Shaping bursting by electrical coupling and noise

Gap-junctional coupling is an important way of communication between neurons and other excitable cells. Strong electrical coupling synchronizes activity across cell ensembles. Surprisingly, in the presence of noise synchronous oscillations generated by an electrically coupled network may differ qualitatively from the oscillations produced by uncoupled individual cells forming the network. A prominent example of such behavior is the synchronized bursting in islets of Langerhans formed by pancreatic \beta-cells, which in isolation are known to exhibit irregular spiking. At the heart of this intriguing phenomenon lies denoising, a remarkable ability of electrical coupling to diminish the effects of noise acting on individual cells. In this paper, we derive quantitative estimates characterizing denoising in electrically coupled networks of conductance-based models of square wave bursting cells. Our analysis reveals the interplay of the intrinsic properties of the individual cells and network topology and their respective contributions to this important effect. In particular, we show that networks on graphs with large algebraic connectivity or small total effective resistance are better equipped for implementing denoising. As a by-product of the analysis of denoising, we analytically estimate the rate with which trajectories converge to the synchronization subspace and the stability of the latter to random perturbations. These estimates reveal the role of the network topology in synchronization. The analysis is complemented by numerical simulations of electrically coupled conductance-based networks. Taken together, these results explain the mechanisms underlying synchronization and denoising in an important class of biological models

Single particle trajectories reveal active endoplasmic reticulum luminal flow

The endoplasmic reticulum (ER), a network of membranous sheets and pipes, supports functions encompassing biogenesis of secretory proteins and delivery of functional solutes throughout the cell[1, 2]. Molecular mobility through the ER network enables these functionalities, but diffusion alone is not sufficient to explain luminal transport across supramicrometre distances. Understanding the ER structure–function relationship is critical in light of mutations in ER morphology-regulating proteins that give rise to neurodegenerative disorders[3, 4]. Here, super-resolution microscopy and analysis of single particle trajectories of ER luminal proteins revealed that the topological organization of the ER correlates with distinct trafficking modes of its luminal content: with a dominant diffusive component in tubular junctions and a fast flow component in tubules. Particle trajectory orientations resolved over time revealed an alternating current of the ER contents, while fast ER super-resolution identified energy-dependent tubule contraction events at specific points as a plausible mechanism for generating active ER luminal flow. The discovery of active flow in the ER has implications for timely ER content distribution throughout the cell, particularly important for cells with extensive ER-containing projections such as neurons.Wellcome Trust - 3-3249/Z/16/Z and 089703/Z/09/Z [Kaminski] UK Demential Research Institute [Avezov] Wellcome Trust - 200848/Z/16/Z, WT: UNS18966 [Ron] FRM Team Research Grant [Holcman] Engineering and Physical Sciences Research Council (EPSRC) - EP/L015889/1 and EP/H018301/1 [Kaminski] Medical Research Council (MRC) - MR/K015850/1 and MR/K02292X/1 [Kaminski

A Stochastic Compartmental Model for Fast Axonal Transport

In this paper we develop a probabilistic micro-scale compartmental model and use it to study macro-scale properties of axonal transport, the process by which intracellular cargo is moved in the axons of neurons. By directly modeling the smallest scale interactions, we can use recent microscopic experimental observations to infer all the parameters of the model. Then, using techniques from probability theory, we compute asymptotic limits of the stochastic behavior of individual motor-cargo complexes, while also characterizing both equilibrium and non-equilibrium ensemble behavior. We use these results in order to investigate three important biological questions: (1) How homogeneous are axons at stochastic equilibrium? (2) How quickly can axons return to stochastic equilibrium after large local perturbations? (3) How is our understanding of delivery time to a depleted target region changed by taking the whole cell point-of-view

On the Nature and Shape of Tubulin Trails: Implications on Microtubule Self-Organization

Microtubules, major elements of the cell skeleton are, most of the time, well organized in vivo, but they can also show self-organizing behaviors in time and/or space in purified solutions in vitro. Theoretical studies and models based on the concepts of collective dynamics in complex systems, reaction-diffusion processes and emergent phenomena were proposed to explain some of these behaviors. In the particular case of microtubule spatial self-organization, it has been advanced that microtubules could behave like ants, self-organizing by 'talking to each other' by way of hypothetic (because never observed) concentrated chemical trails of tubulin that are expected to be released by their disassembling ends. Deterministic models based on this idea yielded indeed like-looking spatio-temporal self-organizing behaviors. Nevertheless the question remains of whether microscopic tubulin trails produced by individual or bundles of several microtubules are intense enough to allow microtubule self-organization at a macroscopic level. In the present work, by simulating the diffusion of tubulin in microtubule solutions at the microscopic scale, we measure the shape and intensity of tubulin trails and discuss about the assumption of microtubule self-organization due to the production of chemical trails by disassembling microtubules. We show that the tubulin trails produced by individual microtubules or small microtubule arrays are very weak and not elongated even at very high reactive rates. Although the variations of concentration due to such trails are not significant compared to natural fluctuations of the concentration of tubuline in the chemical environment, the study shows that heterogeneities of biochemical composition can form due to microtubule disassembly. They could become significant when produced by numerous microtubule ends located in the same place. Their possible formation could play a role in certain conditions of reaction. In particular, it gives a mesoscopic basis to explain the collective dynamics observed in excitable microtubule solutions showing the propagation of concentration waves of microtubules at the millimeter scale, although we doubt that individual microtubules or bundles can behave like molecular ants

Large Scale Stochastic Dynamics

In focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps. More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, dynamical phase transitions, metastability, large deviations, to mention only a few key items