arXiv subjects

Radek Erban

Publications and source records attributed to Radek Erban.

At least 37 records · Page 2Linked to original sources

arXiv2019

Limiting stochastic processes of shift-periodic dynamical systems

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences $x_{n+1}=F(x_n)$ generated by such maps display rich dynamical behaviour. The integer parts $\lfloor x_n \rfloor$ give a discrete-time random walk for a suitable initial distribution of $x_0$ and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on $[0,1]$, convergence of trajectories to a continuous-time random walk is shown in a limit.

math.DS↗

arXiv2019

Multi-resolution dimer models in heat baths with short-range and long-range interactions

This work investigates multi-resolution methodologies for simulating dimer models. The solvent particles which make up the heat bath interact with the monomers of the dimer either through direct collisions (short-range) or through harmonic springs (long-range). Two types of multi-resolution methodologies are considered in detail: (a) describing parts of the solvent far away from the dimer by a coarser approach; (b) describing each monomer of the dimer by using a model with different level of resolution. These methodologies are then utilised to investigate the effect of a shared heat bath versus two uncoupled heat baths, one for each monomer. Furthermore the validity of the multi-resolution methods is discussed by comparison to dynamics of macroscopic Langevin equations.

physics.comp-ph↗

arXiv2018

Noise-induced Mixing and Multimodality in Reaction Networks

We analyze a class of chemical reaction networks under mass-action kinetics and involving multiple time-scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks, and consist of a slow-subnetwork, describing conversions among the different gene states, and fast-subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a novel phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast-subnetworks which are `mixed' by the slow-subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

q-bio.MN↗

arXiv2017

Multi-resolution polymer Brownian dynamics with hydrodynamic interactions

A polymer model given in terms of beads, interacting through Hookean springs and hydrodynamic forces, is studied. Brownian dynamics description of this bead-spring polymer model is extended to multiple resolutions. Using this multiscale approach, a modeller can efficiently look at different regions of the polymer in different spatial and temporal resolutions with scalings given for the number of beads, statistical segment length and bead radius in order to maintain macro-scale properties of the polymer filament. The Boltzmann distribution of a Gaussian chain for differing statistical segment lengths gives a Langevin equation for the multi-resolution model with a mobility tensor for different bead sizes. Using the pre-averaging approximation, the translational diffusion coefficient is obtained as a function of the inverse of a matrix and then in closed form in the long-chain limit. This is then confirmed with numerical experiments.

physics.chem-ph↗

arXiv2017

Noise Control for DNA Computing

Synthetic biology is a growing interdisciplinary field, with far-reaching applications, which aims to design biochemical systems that behave in a desired manner. With the advancement of strand-displacement DNA computing, a large class of abstract biochemical networks may be physically realized using DNA molecules. Methods for systematic design of the abstract systems with prescribed behaviors have been predominantly developed at the (less-detailed) deterministic level. However, stochastic effects, neglected at the deterministic level, are increasingly found to play an important role in biochemistry. In such circumstances, methods for controlling the intrinsic noise in the system are necessary for a successful network design at the (more-detailed) stochastic level. To bridge the gap, the noise-control algorithm for designing biochemical networks is developed in this paper. The algorithm structurally modifies any given reaction network under mass-action kinetics, in such a way that (i) controllable state-dependent noise is introduced into the stochastic dynamics, while (ii) the deterministic dynamics are preserved. The capabilities of the algorithm are demonstrated on a production-decay reaction system, and on an exotic system displaying bistability. For the production-decay system, it is shown that the algorithm may be used to redesign the network to achieve noise-induced multistability. For the exotic system, the algorithm is used to redesign the network to control the stochastic switching, and achieve noise-induced oscillations.

q-bio.MN↗

arXiv2017

Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability

Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations and multistability. The results are then used for construction of two reaction systems, which are at the deterministic level described by two-dimensional third-degree kinetic ODEs. The first system displays a homoclinic bifurcation, and a coexistence of a stable critical point and a stable limit cycle in the phase plane. The second system displays a multiple limit cycle bifurcation, and a coexistence of two stable limit cycles. The deterministic solutions (obtained by solving the kinetic ODEs) and stochastic solutions (noisy time-series generating by the Gillespie algorithm, and the underlying probability distributions obtained by solving the chemical master equation (CME)) of the constructed systems are compared, and the observed differences highlighted. The constructed systems are proposed as test problems for statistical methods, which are designed to detect and classify properties of given noisy time-series arising from biological applications.

q-bio.MN↗

arXiv2016

Varying the resolution of the Rouse model on temporal and spatial scales: application to multiscale modelling of DNA dynamics

A multi-resolution bead-spring model for polymer dynamics is developed as a generalization of the Rouse model. A polymer chain is described using beads of variable sizes connected by springs with variable spring constants. A numerical scheme which can use different timesteps to advance the positions of different beads is presented and analyzed. The position of a particular bead is only updated at integer multiples of the timesteps associated with its connecting springs. This approach extends the Rouse model to a multiscale model on both spatial and temporal scales, allowing simulations of localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modelling approach to describe the rest of the polymer chain. A method for changing the model resolution on-the-fly is developed using the Metropolis-Hastings algorithm. It is shown that this approach maintains key statistics of the end-to-end distance and diffusion of the polymer filament and makes computational savings when applied to a model for the binding of a protein to the DNA filament.

physics.bio-ph↗

arXiv2016

Steric effects induce geometric remodeling of actin bundles in filopodia

Filopodia are ubiquitous fingerlike protrusions, spawned by many eukaryotic cells, to probe and interact with their environments. Polymerization dynamics of actin filaments, comprising the structural core of filopodia, largely determine their instantaneous lengths and overall lifetimes. The polymerization reactions at the filopodial tip require transport of G-actin, which enter the filopodial tube from the filopodial base and diffuse toward the filament barbed ends near the tip. Actin filaments are mechanically coupled into a tight bundle by cross-linker proteins. Interestingly, many of these proteins are relatively short, restricting the free diffusion of cytosolic G-actin throughout the bundle and, in particular, its penetration into the bundle core. To investigate the effect of steric restrictions on G-actin diffusion by the porous structure of filopodial actin filament bundle, we used a particle-based stochastic simulation approach. We discovered that excluded volume interactions result in partial and then full collapse of central filaments in the bundle, leading to a hollowed-out structure. The latter may further collapse radially due to the activity of cross-linking proteins, hence producing conical-shaped filament bundles. Interestingly, electron microscopy experiments on mature filopodia indeed frequently reveal actin bundles that are narrow at the tip and wider at the base. Overall, our work demonstrates that excluded volume effects in the context of reaction-diffusion processes in porous networks may lead to unexpected geometric growth patterns and complicated, history-dependent dynamics of intermediate metastable configurations.

q-bio.SC↗

arXiv2016

Particle-based Multiscale Modeling of Calcium Puff Dynamics

Intracellular calcium is regulated in part by the release of Ca$^{2+}$ ions from the endoplasmic reticulum via inositol-4,5-triphosphate receptor (IP$_3$R) channels (among other possibilities such as RyR and L-type calcium channels). The resulting dynamics are highly diverse, lead to local calcium "puffs" as well as global waves propagating through cells, as observed in {\it Xenopus} oocytes, neurons, and other cell types. Local fluctuations in the number of calcium ions play a crucial role in the onset of these features. Previous modeling studies of calcium puff dynamics stemming from IP$_3$R channels have predominantly focused on stochastic channel models coupled to deterministic diffusion of ions, thereby neglecting local fluctuations of the ion number. Tracking of individual ions is computationally difficult due to the scale separation in the Ca$^{2+}$ concentration when channels are in the open or closed states. In this paper, a spatial multiscale model for investigating of the dynamics of puffs is presented. It couples Brownian motion (diffusion) of ions with a stochastic channel gating model. The model is used to analyze calcium puff statistics. Concentration time traces as well as channel state information are studied. We identify the regime in which puffs can be found and develop a mean-field theory to extract the boundary of this regime. Puffs are only possible when the time scale of channel inhibition is sufficiently large. Implications for the understanding of puff generation and termination are discussed.

q-bio.SC↗

arXiv2015

Hybrid framework for the simulation of stochastic chemical kinetics

Stochasticity plays a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be modelled as Markov processes, typically simulated using the Gillespie Stochastic Simulation Algorithm (SSA). While easy to implement and exact, the computational cost of using the Gillespie SSA to simulate such systems can become prohibitive as the frequency of reaction events increases. This has motivated numerous coarse-grained schemes, where the "fast" reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation when all reactants are abundant, the approximation breaks down when one or more species exist only in small concentrations and the fluctuations arising from the discrete nature of the reactions becomes significant. This is particularly problematic when using such methods to compute statistics of extinction times for chemical species, as well as simulating non-equilibrium systems such as cell-cycle models in which a single species can cycle between abundance and scarcity. In this paper, a hybrid jump-diffusion model for simulating well- mixed stochastic kinetics is derived. It acts as a bridge between the Gillespie SSA and the chemical Langevin equation. For low reactant reactions the underlying behaviour is purely discrete, while purely diffusive when the concentrations of all species is large, with the two different behaviours coexisting in the intermediate region. A bound on the weak error in the classical large volume scaling limit is obtained, and three different numerical discretizations of the jump-diffusion model are described. The benefits of such a formalism are illustrated using computational examples.

q-bio.MN↗

arXiv2015

Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics

A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial absorption condition, which uses a Robin boundary condition in the separation coordinate between two possible reactants. This boundary condition can be interpreted as an idealization of a reactive interaction potential model, in which a potential barrier must be surmounted before reactions can occur. In this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation, where molecules react instantly upon reaching the peak of the barrier. The limiting boundary condition is derived by the method of matched asymptotic expansions, and shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased. Limiting boundary conditions for the same interaction potential in both the overdamped Fokker-Planck equation (Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are investigated. It is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit where the Kramers equation solution converges to the solution of the Fokker-Planck equation.

physics.bio-ph↗

arXiv2015

Chemical Reaction Systems with a Homoclinic Bifurcation: an Inverse Problem

An inverse problem framework for constructing reaction systems with prescribed properties is presented. Kinetic transformations are defined and analysed as a part of the framework, allowing an arbitrary polynomial ordinary differential equation to be mapped to the one that can be represented as a reaction network. The framework is used for construction of specific two- and three-dimensional bistable reaction systems undergoing a supercritical homoclinic bifurcation, and the topology of their phase spaces is discussed.

math.DS↗

arXiv2015

Coupling all-atom molecular dynamics simulations of ions in water with Brownian dynamics

Molecular dynamics (MD) simulations of ions (K$^+$, Na$^+$, Ca$^{2+}$ and Cl$^-$) in aqueous solutions are investigated. Water is described using the SPC/E model. A stochastic coarse-grained description for ion behaviour is presented and parameterized using MD simulations. It is given as a system of coupled stochastic and ordinary differential equations, describing the ion position, velocity and acceleration. The stochastic coarse-grained model provides an intermediate description between all-atom MD simulations and Brownian dynamics (BD) models. It is used to develop a multiscale method which uses all-atom MD simulations in parts of the computational domain and (less detailed) BD simulations in the remainder of the domain.

physics.comp-ph↗

arXiv2015

On Cucker-Smale model with noise and delay

A generalization of the Cucker-Smale model for collective animal behaviour is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modelling: (i) stochasticity (imperfections) of individual behaviour; and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a byproduct, a new result regarding the asymptotic behaviour of delayed geometric Brownian motion is obtained. In the second part of the paper results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behaviour of the system - namely, that an introduction of intermediate time delay may facilitate flocking.

math.DS↗

arXiv2015

ADM-CLE approach for detecting slow variables in continuous time Markov chains and dynamic data

A method for detecting intrinsic slow variables in high-dimensional stochastic chemical reaction networks is developed and analyzed. It combines anisotropic diffusion maps (ADM) with approximations based on the chemical Langevin equation (CLE). The resulting approach, called ADM-CLE, has the potential of being more efficient than the ADM method for a large class of chemical reaction systems, because it replaces the computationally most expensive step of ADM (running local short bursts of simulations) by using an approximation based on the CLE. The ADM-CLE approach can be used to estimate the stationary distribution of the detected slow variable, without any a-priori knowledge of it. If the conditional distribution of the fast variables can be obtained analytically, then the resulting ADM-CLE approach does not make any use of Monte Carlo simulations to estimate the distributions of both slow and fast variables.

cs.CE↗

arXiv2015

Parameter estimation and bifurcation analysis of stochastic models of gene regulatory networks: tensor-structured methods

Stochastic modelling provides an indispensable tool for understanding how random events at the molecular level influence cellular functions. In practice, the common challenge is to calibrate a large number of model parameters against the experimental data. A related problem is to efficiently study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation). In this paper, tensor-structured parametric analysis (TPA) is presented. It is based on recently proposed low-parametric tensor-structured representations of classical matrices and vectors. This approach enables simultaneous computation of the model properties for all parameter values within a parameter space. This methodology is exemplified to study the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks. The TPA has been implemented in Matlab and the codes are available at http://www.stobifan.org .

q-bio.MN↗

arXiv2014

Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics

Several different methods exist for efficient approximation of paths in multiscale stochastic chemical systems. Another approach is to use bursts of stochastic simulation to estimate the parameters of a stochastic differential equation approximation of the paths. In this paper, multiscale methods for approximating paths are used to formulate different strategies for estimating the dynamics by diffusion processes. We then analyse how efficient and accurate these methods are in a range of different scenarios, and compare their respective advantages and disadvantages to other methods proposed to analyse multiscale chemical networks.

math.NA↗

arXiv2014

Reduction of chemical systems by delayed quasi-steady state assumptions

Mathematical analysis of mass action models of large complex chemical systems is typically only possible if the models are reduced. The most common reduction technique is based on quasi-steady state assumptions. To increase the accuracy of this technique we propose delayed quasi-steady state assumptions (D-QSSA) which yield systems of delay differential equations. We define the approximation based on D-QSSA, prove the corresponding error estimate, and show how it approximates the invariant manifold. Then we define a class of well mixed chemical systems and formulate assumptions enabling the application of D-QSSA. We also apply the D-QSSA to a model of Hes1 expression and to a cell-cycle model to illustrate the improved accuracy of the D-QSSA with respect to the standard quasi-steady state assumptions.

math.DS↗