1
|
Winters A, Öttinger HC, Vermant J. Comparative analysis of fluctuations in viscoelastic stress: A comparison of the temporary network and dumbbell models. J Chem Phys 2024; 161:014901. [PMID: 38949587 DOI: 10.1063/5.0213660] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2024] [Accepted: 06/11/2024] [Indexed: 07/02/2024] Open
Abstract
Traditionally, stress fluctuations in flowing and deformed materials are overlooked, with an obvious focus on average stresses in a continuum mechanical approximation. However, these fluctuations, often dismissed as "noise," hold the potential to provide direct insights into the material structure and its structure-stress coupling, uncovering detailed aspects of fluid transport and relaxation behaviors. Despite advancements in experimental techniques allowing for the visualization of these fluctuations, their significance remains largely untapped as modeling efforts continue to target Newtonian fluids within the confines of Gaussian noise assumptions. In the present work, a comparative analysis of stress fluctuations in two distinct microstructural models is carried out: the temporary network model and the hydrodynamic dumbbell model. Despite both models conforming to the upper convected Maxwell model at a macroscopic level, the temporary network model predicts non-Gaussian fluctuations. We find that stress fluctuations within the temporary network model exhibit more pronounced abruptness at the local scale, with only an enlargement of the control volume leading to a gradual Gaussian-like noise, diminishing the differences between the two models. These findings underscore the heightened sensitivity of fluctuating rheology to microstructural details and the microstructure-flow coupling, beyond what is captured by macroscopically averaged stresses.
Collapse
Affiliation(s)
- Arturo Winters
- Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 5, Zurich 8093, Switzerland
| | | | - Jan Vermant
- Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 5, Zurich 8093, Switzerland
| |
Collapse
|
2
|
Montefusco A, Helfmann L, Okunola T, Winkelmann S, Schütte C. Partial mean-field model for neurotransmission dynamics. Math Biosci 2024; 369:109143. [PMID: 38220067 DOI: 10.1016/j.mbs.2024.109143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2023] [Revised: 12/07/2023] [Accepted: 01/09/2024] [Indexed: 01/16/2024]
Abstract
This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the spatial-stochastic particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios.
Collapse
Affiliation(s)
- Alberto Montefusco
- Mathematics of Complex Systems, Zuse-Institut Berlin, Takustraße 7, Berlin, 14195, Germany
| | - Luzie Helfmann
- Mathematics of Complex Systems, Zuse-Institut Berlin, Takustraße 7, Berlin, 14195, Germany
| | - Toluwani Okunola
- Mathematics of Complex Systems, Zuse-Institut Berlin, Takustraße 7, Berlin, 14195, Germany; Institute Of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany
| | - Stefanie Winkelmann
- Mathematics of Complex Systems, Zuse-Institut Berlin, Takustraße 7, Berlin, 14195, Germany.
| | - Christof Schütte
- Mathematics of Complex Systems, Zuse-Institut Berlin, Takustraße 7, Berlin, 14195, Germany; Institute of Mathematics, Freie Universität Berlin, Arnimallee 6, Berlin, 14195, Germany
| |
Collapse
|
3
|
Deinum EE, Jacobs B. Rho of Plants patterning: linking mathematical models and molecular diversity. JOURNAL OF EXPERIMENTAL BOTANY 2024; 75:1274-1288. [PMID: 37962515 PMCID: PMC10901209 DOI: 10.1093/jxb/erad447] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 11/08/2023] [Indexed: 11/15/2023]
Abstract
ROPs (Rho of Plants) are plant specific small GTPases involved in many membrane patterning processes and play important roles in the establishment and communication of cell polarity. These small GTPases can produce a wide variety of patterns, ranging from a single cluster in tip-growing root hairs and pollen tubes to an oriented stripe pattern controlling protoxylem cell wall deposition. For an understanding of what controls these various patterns, models are indispensable. Consequently, many modelling studies on small GTPase patterning exist, often focusing on yeast or animal cells. Multiple patterns occurring in plants, however, require the stable co-existence of multiple active ROP clusters, which does not occur with the most common yeast/animal models. The possibility of such patterns critically depends on the precise model formulation. Additionally, different small GTPases are usually treated interchangeably in models, even though plants possess two types of ROPs with distinct molecular properties, one of which is unique to plants. Furthermore, the shape and even the type of ROP patterns may be affected by the cortical cytoskeleton, and cortex composition and anisotropy differ dramatically between plants and animals. Here, we review insights into ROP patterning from modelling efforts across kingdoms, as well as some outstanding questions arising from these models and recent experimental findings.
Collapse
Affiliation(s)
- Eva E Deinum
- Mathematical and Statistical Methods (Biometris), Plant Science Group, Wageningen University, 6708 PB Wageningen, The Netherlands
| | - Bas Jacobs
- Mathematical and Statistical Methods (Biometris), Plant Science Group, Wageningen University, 6708 PB Wageningen, The Netherlands
| |
Collapse
|
4
|
Cornalba F, Fischer J. The Dean-Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 2023; 247:76. [PMID: 37547904 PMCID: PMC10403491 DOI: 10.1007/s00205-023-01903-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/13/2021] [Accepted: 07/13/2023] [Indexed: 08/08/2023]
Abstract
The Dean-Kawasaki equation-a strongly singular SPDE-is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N ≫ 1 . The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean-Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean-Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in N - 1 (in suitable weak metrics). In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.
Collapse
Affiliation(s)
- Federico Cornalba
- Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria
| | - Julian Fischer
- Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria
| |
Collapse
|
5
|
Hamster C, van Heijster P. Waves in a Stochastic Cell Motility Model. Bull Math Biol 2023; 85:70. [PMID: 37329390 PMCID: PMC10276800 DOI: 10.1007/s11538-023-01164-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2023] [Accepted: 05/04/2023] [Indexed: 06/19/2023]
Abstract
In Bhattacharya et al. (Sci Adv 6(32):7682, 2020), a set of chemical reactions involved in the dynamics of actin waves in cells was studied at two levels. The microscopic level, where the individual chemical reactions are directly modelled using Gillespie-type algorithms, and on a macroscopic level where a deterministic reaction-diffusion equation arises as the large-scale limit of the underlying chemical reactions. In this work, we derive, and subsequently study, the related mesoscopic stochastic reaction-diffusion system, or chemical Langevin equation, that arises from the same set of chemical reactions. We explain how the stochastic patterns that arise from this equation can be used to understand the experimentally observed dynamics from Bhattacharya et al. In particular, we argue that the mesoscopic stochastic model better captures the microscopic behaviour than the deterministic reaction-diffusion equation, while being more amenable for mathematical analysis and numerical simulations than the microscopic model.
Collapse
Affiliation(s)
- Christian Hamster
- Biometris, Wageningen University and Research, Droevendaalse steeg 1, 6708 PB Wageningen, The Netherlands
| | - Peter van Heijster
- Biometris, Wageningen University and Research, Droevendaalse steeg 1, 6708 PB Wageningen, The Netherlands
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD Australia
| |
Collapse
|
6
|
Moškon M, Komac R, Zimic N, Mraz M. Distributed biological computation: from oscillators, logic gates and switches to a multicellular processor and neural computing applications. Neural Comput Appl 2021. [DOI: 10.1007/s00521-021-05711-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
|
7
|
Carrero JI, Loaiza JS, Serna A. Stochastic reaction, stochastic diffusion. CHEMTEXTS 2020. [DOI: 10.1007/s40828-020-0108-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
|
8
|
Delle Site L, Praprotnik M, Bell JB, Klein R. Particle–Continuum Coupling and its Scaling Regimes: Theory and Applications. ADVANCED THEORY AND SIMULATIONS 2020. [DOI: 10.1002/adts.201900232] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Affiliation(s)
- Luigi Delle Site
- Freie Universität Berlin Institute of Mathematics Arnimallee 6, 14195 Berlin Germany
| | - Matej Praprotnik
- Laboratory for Molecular Modeling National Institute of Chemistry SI‐1001 Ljubljana, Slovenia & Department of Physics Faculty of Mathematics and Physics University of Ljubljana SI‐1000 Ljubljana Slovenia
| | - John B. Bell
- Lawrence Berkeley National Lab 1 Cyclotron Rd. Berkeley CA 94720 USA
| | - Rupert Klein
- Freie Universität Berlin Institute of Mathematics Arnimallee 6, 14195 Berlin Germany
| |
Collapse
|
9
|
Kang HW, Erban R. Multiscale Stochastic Reaction-Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations. Bull Math Biol 2019; 81:3185-3213. [PMID: 31165406 PMCID: PMC6677718 DOI: 10.1007/s11538-019-00613-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2018] [Accepted: 05/09/2019] [Indexed: 12/21/2022]
Abstract
Two multiscale algorithms for stochastic simulations of reaction-diffusion processes are analysed. They are applicable to systems which include regions with significantly different concentrations of molecules. In both methods, a domain of interest is divided into two subsets where continuous-time Markov chain models and stochastic partial differential equations (SPDEs) are used, respectively. In the first algorithm, Markov chain (compartment-based) models are coupled with reaction-diffusion SPDEs by considering a pseudo-compartment (also called an overlap or handshaking region) in the SPDE part of the computational domain right next to the interface. In the second algorithm, no overlap region is used. Further extensions of both schemes are presented, including the case of an adaptively chosen boundary between different modelling approaches.
Collapse
Affiliation(s)
- Hye-Won Kang
- Department of Mathematics and Statistics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 USA
| | - Radek Erban
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK
| |
Collapse
|
10
|
Kim C, Nonaka A, Bell JB, Garcia AL, Donev A. Fluctuating hydrodynamics of reactive liquid mixtures. J Chem Phys 2018; 149:084113. [PMID: 30193508 DOI: 10.1063/1.5043428] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Fluctuating hydrodynamics (FHD) provides a framework for modeling microscopic fluctuations in a manner consistent with statistical mechanics and nonequilibrium thermodynamics. This paper presents an FHD formulation for isothermal reactive incompressible liquid mixtures with stochastic chemistry. Fluctuating multispecies mass diffusion is formulated using a Maxwell-Stefan description without assuming a dilute solution, and momentum dynamics is described by a stochastic Navier-Stokes equation for the fluid velocity. We consider a thermodynamically consistent generalization for the law of mass action for non-dilute mixtures and use it in the chemical master equation (CME) to model reactions as a Poisson process. The FHD approach provides remarkable computational efficiency over traditional reaction-diffusion master equation methods when the number of reactive molecules is large, while also retaining accuracy even when there are as few as ten reactive molecules per hydrodynamic cell. We present a numerical algorithm to solve the coupled FHD and CME equations and validate it on both equilibrium and nonequilibrium problems. We simulate a diffusively driven gravitational instability in the presence of an acid-base neutralization reaction, starting from a perfectly flat interface. We demonstrate that the coupling between velocity and concentration fluctuations dominates the initial growth of the instability.
Collapse
Affiliation(s)
- Changho Kim
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Andy Nonaka
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - John B Bell
- Computational Research Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Alejandro L Garcia
- Department of Physics and Astronomy, San Jose State University, 1 Washington Square, San Jose, California 95192, USA
| | - Aleksandar Donev
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
| |
Collapse
|
11
|
Del Razo MJ, Qian H, Noé F. Grand canonical diffusion-influenced reactions: A stochastic theory with applications to multiscale reaction-diffusion simulations. J Chem Phys 2018; 149:044102. [PMID: 30068197 DOI: 10.1063/1.5037060] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
Smoluchowski-type models for diffusion-influenced reactions (A + B → C) can be formulated within two frameworks: the probabilistic-based approach for a pair A, B of reacting particles and the concentration-based approach for systems in contact with a bath that generates a concentration gradient of B particles that interact with A. Although these two approaches are mathematically similar, it is not straightforward to establish a precise mathematical relationship between them. Determining this relationship is essential to derive particle-based numerical methods that are quantitatively consistent with bulk concentration dynamics. In this work, we determine the relationship between the two approaches by introducing the grand canonical Smoluchowski master equation (GC-SME), which consists of a continuous-time Markov chain that models an arbitrary number of B particles, each one of them following Smoluchowski's probabilistic dynamics. We show that the GC-SME recovers the concentration-based approach by taking either the hydrodynamic or the large copy number limit. In addition, we show that the GC-SME provides a clear statistical mechanical interpretation of the concentration-based approach and yields an emergent chemical potential for nonequilibrium spatially inhomogeneous reaction processes. We further exploit the GC-SME robust framework to accurately derive multiscale/hybrid numerical methods that couple particle-based reaction-diffusion simulations with bulk concentration descriptions, as described in detail through two computational implementations.
Collapse
Affiliation(s)
- Mauricio J Del Razo
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA
| | - Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| |
Collapse
|
12
|
Abstract
We develop a Split Reactive Brownian Dynamics (SRBD) algorithm for particle simulations of reaction-diffusion systems based on the Doi or volume reactivity model, in which pairs of particles react with a specified Poisson rate if they are closer than a chosen reactive distance. In our Doi model, we ensure that the microscopic reaction rules for various association and dissociation reactions are consistent with detailed balance (time reversibility) at thermodynamic equilibrium. The SRBD algorithm uses Strang splitting in time to separate reaction and diffusion and solves both the diffusion-only and reaction-only subproblems exactly, even at high packing densities. To efficiently process reactions without uncontrolled approximations, SRBD employs an event-driven algorithm that processes reactions in a time-ordered sequence over the duration of the time step. A grid of cells with size larger than all of the reactive distances is used to schedule and process the reactions, but unlike traditional grid-based methods such as reaction-diffusion master equation algorithms, the results of SRBD are statistically independent of the size of the grid used to accelerate the processing of reactions. We use the SRBD algorithm to compute the effective macroscopic reaction rate for both reaction-limited and diffusion-limited irreversible association in three dimensions and compare to existing theoretical predictions at low and moderate densities. We also study long-time tails in the time correlation functions for reversible association at thermodynamic equilibrium and compare to recent theoretical predictions. Finally, we compare different particle and continuum methods on a model exhibiting a Turing-like instability and pattern formation. Our studies reinforce the common finding that microscopic mechanisms and correlations matter for diffusion-limited systems, making continuum and even mesoscopic modeling of such systems difficult or impossible. We also find that for models in which particles diffuse off lattice, such as the Doi model, reactions lead to a spurious enhancement of the effective diffusion coefficients.
Collapse
Affiliation(s)
- Aleksandar Donev
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Chiao-Yu Yang
- Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA
| | - Changho Kim
- Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| |
Collapse
|
13
|
Widmer LA, Stelling J. Bridging intracellular scales by mechanistic computational models. Curr Opin Biotechnol 2018; 52:17-24. [PMID: 29486391 DOI: 10.1016/j.copbio.2018.02.005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2017] [Accepted: 02/11/2018] [Indexed: 12/31/2022]
Abstract
The impact of intracellular spatial organization beyond classical compartments on processes such as cell signaling is increasingly recognized. A quantitative, mechanistic understanding of cellular systems therefore needs to account for different scales in at least three coordinates: time, molecular abundances, and space. Mechanistic mathematical models may span all these scales, but corresponding multi-scale models need to resolve mechanistic details on small scales while maintaining computational tractability for larger ones. This typically results in models that combine different levels of description: from a microscopic representation of chemical reactions up to continuum dynamics in space and time. We highlight recent progress in bridging these model classes and outline current challenges in multi-scale models such as active transport and dynamic geometries.
Collapse
Affiliation(s)
- Lukas Andreas Widmer
- Department of Biosystems Science and Engineering and Swiss Institute of Bioinformatics, ETH Zürich, Basel, Switzerland; Systems Biology PhD Program, Life Science Zurich Graduate School, Zurich, Switzerland
| | - Jörg Stelling
- Department of Biosystems Science and Engineering and Swiss Institute of Bioinformatics, ETH Zürich, Basel, Switzerland.
| |
Collapse
|
14
|
Abstract
We analyze the effects of an externally applied electric field on thermal fluctuations for a binary electrolyte fluid. We show that the fluctuating Poisson-Nernst-Planck (PNP) equations for charged multispecies diffusion coupled with the fluctuating fluid momentum equation result in enhanced charge transport via a mechanism distinct from the well-known enhancement of mass transport that accompanies giant fluctuations. Although the mass and charge transport occurs by advection by thermal velocity fluctuations, it can macroscopically be represented as electrodiffusion with renormalized electric conductivity and a nonzero cation-anion diffusion coefficient. Specifically, we predict a nonzero cation-anion Maxwell-Stefan coefficient proportional to the square root of the salt concentration, a prediction that agrees quantitatively with experimental measurements. The renormalized or effective macroscopic equations are different from the starting PNP equations, which contain no cross-diffusion terms, even for rather dilute binary electrolytes. At the same time, for infinitely dilute solutions the renormalized electric conductivity and renormalized diffusion coefficients are consistent and the classical PNP equations with renormalized coefficients are recovered, demonstrating the self-consistency of the fluctuating hydrodynamics equations. Our calculations show that the fluctuating hydrodynamics approach recovers the electrophoretic and relaxation corrections obtained by Debye-Huckel-Onsager theory, while elucidating the physical origins of these corrections and generalizing straightforwardly to more complex multispecies electrolytes. Finally, we show that strong applied electric fields result in anisotropically enhanced "giant" velocity fluctuations and reduced fluctuations of salt concentration.
Collapse
|