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Finite-size effects in molecular simulations: A physico-mathematical view
No full textMolecular simulation of condensed matter systems has always been characterized by the aim for an optimal balance between a precise physical description of the simulated substance, and the efficient use of computational resources. A major challenge for the accurate representation of a physical system in a simulation, therefore, consists in determining the appropriate size of the simulated sample. The latter must be sufficiently large in order to represent the bulk of the substance, and thus to reproduce its characteristic thermodynamic features. This problem is known under the name of “finite-size effects”, and several criteria have been adopted in order to determine these effects, thereby inferring about the validity of a simulation study. In this article, we discuss the application of a rigorous mathematical theorem, the so-called “two-sided Bogoliubov inequality”, to estimate the finite-size effects. The theorem provides upper and lower bounds for the free energy cost of partitioning a system into equivalent, non-interacting subsystems, and it can be used to obtain a rigorous definition of the minimal size of a system with its full thermodynamic features. The corresponding criterion based on this theorem is complementary to those existing in the literature, and it can be applied to both classical and quantum systems. The need for accurate and physically consistent results of current simulations is enormously increased by the use of simulation data in machine learning procedures. Physically inconsistent data, produced by simulations of insufficient size, results in a substantial error in the modeling procedure that propagates further into the study of several other systems or larger scales beyond the molecular one. Furthermore, the statistical nature of machine learning implies questions about the number of parameters and the size of the training set. Such problems are the equivalent of the size effects discussed in the first part of the review. Here this feature is treated
employing the same statistical mechanics framework developed for the first problem
Multi-Grid Reaction-Diffusion Master Equation: Applications to Morphogen Gradient Modelling
No full textThe multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem
Independently engaging protein tethers of different length enhance synaptic vesicle trafficking to the plasma membrane
No full textSynaptic vesicle (SV) trafficking toward the plasma membrane (PM) and subsequent SV maturation are essential for neurotransmitter release. These processes, including SV docking and priming, are co-ordinated by various proteins, such as SNAREs, Munc13 and synaptotagmin (Syt), which connect (tether) the SV to the PM. Here, we investigated how tethers of varying lengths mediate SV docking using a simplified mathematical model. The heights of the three tether types, as estimated from the structures of the SNARE complex, Munc13 and Syt, defined the SV–PM distance ranges for tether formation. Geometric considerations linked SV–PM distances to the probability and rate of tether formation. We assumed that SV tethering constrains SV motility and that multiple tethers are associated by independent interactions. The model predicted that forming multiple tethers favours shorter SV–PM distances. Although tethers acted independently in the model, their geometrical properties often caused their sequential assembly, from longer ones (Munc13/Syt), which accelerated SV movement towards the PM, to shorter ones (SNAREs), which stabilized PM-proximal SVs. Modifying tether lengths or numbers affected SV trafficking. The independent implementation of tethering proteins enabled their selective removal to mimic gene knockout (KO) situations. This showed that simulated SV–PM distance distributions qualitatively aligned with published electron microscopy studies upon removal of SNARE and Syt tethers, whereas Munc13 KO data were best approximated when assuming additional disruption of SNARE tethers. Thus, although salient features of SV docking can be accounted for by independent tethering alone, our results suggest that functional tether interactions not yet featured in our model are crucial for biological function
Coherent set identification via direct low rank maximum likelihood estimation
No full textWe analyze connections between two low rank modeling approaches from the last decade for treating dynamical data. The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called direct Bayesian model reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of
Open quantum systems and the grand canonical ensemble
No full textThe celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the
description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at
given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an
environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle
number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently,
solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the
corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad
equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical
potential is externally imposed rather than derived from first principles, and hence the corresponding density
operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of
grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and
complementary approach to the current literature that consists in using a generalized system Hamiltonian which
includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the
von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to
a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the
quantities involved are obtained from the physics of the system without any external assumptions
SeArcH schemes for Approximate stRing mAtching
No full textFinding approximate occurrences of a query in a text using a full-text index is a central problem in stringology with many applications, especially in bioinformatics. The recent work has shown significant speed-ups by combining bidirectional indices and employing variations of search schemes. Search schemes partition a query and describe how to search the resulting parts with a given error bound. The performance of search schemes can be approximated by the node count, which represents an upper bound of the number of search steps. Finding optimum search schemes is a difficult combinatorial optimization problem that becomes hard to solve for four and more errors. This paper improves on a few topics important to search scheme based searches: First, we show how search schemes can be used to model previously published approximate search strategies such as suffix filters, 01*0-seeds, or the pigeonhole principle. This unifies these strategies in the search scheme framework, makes them easily comparable and results in novel search schemes that allow for any number of errors. Second, we present a search scheme construction heuristic, which is on par with optimum search schemes and has a better node count than any known search scheme for equal or above four errors. Finally, using the different search schemes, we show that the node count measure is not an ideal performance metric and therefore propose an improved performance metric called the weighted node count, which approximates a search algorithm’s run time much more accurately
Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body
No full textThis paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier–Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields
A hybrid algorithm for systems of noninteracting particles with an external potential
No full textOur focus is on simulating the dynamics of noninteracting particles including the effects of an external potential, which, under certain assumptions, can be formally described by the Dean–Kawasaki equation. The Dean–Kawasaki equation can be solved numerically using standard finite volume methods. However, the numerical approximation implicitly requires a sufficiently large number of particles to ensure the positivity of the solution and accurate approximation of the stochastic flux. To address this challenge, we extend hybrid algorithms for particle systems to scenarios where the density is low. The aim is to create a hybrid algorithm that switches from a finite volume discretization to a particle-based method when the particle density falls below a certain threshold. We develop criteria for determining this threshold by comparing higher-order statistics obtained from the finite volume method with particle simulations. We then demonstrate the use of the resulting criteria for dynamic adaptation in both two- and three-dimensional spatial settings in the absence of an external potential. Finally we consider the dynamics when an external potential is included
Synchronisation for scalar conservation laws via Dirichlet boundary
No full textWe provide an elementary proof of geometric synchronisation for scalar conservation laws on a domain with Dirichlet boundary conditions. Unlike previous results, our proof does not rely on a strict maximum principle, and builds instead on a quantitative estimate of the dissipation at the boundary. We identify a coercivity condition under which the estimates are uniform over all initial conditions, via the construction of suitable super- and sub-solutions. In lack of such coercivity our results build on Lp energy estimates and a Lyapunov structure
A nonnegativity-preserving finite element method for a class of parabolic SPDEs with multiplicative noise
No full textWe consider a prototypical parabolic SPDE with finite-dimensional multiplicative noise, which, subject to a nonnegative initial datum, has a unique nonnegative solution. Inspired by well-established techniques in the deterministic case, we introduce a finite element discretization of this SPDE that is convergent and which, subject to a nonnegative initial datum and unconditionally with respect to the spatial discretization parameter, preserves nonnegativity of the numerical solution throughout the course of evolution. We perform a mathematical analysis of this method. In addition, in the associated linear setting, we develop a fully discrete scheme that also preserves nonnegativity, and we present numerical experiments that illustrate the advantages of the proposed method over alternative finite element and finite difference methods that were previously considered in the literature, which do not necessarily guarantee nonnegativity of the numerical solution