Multiscale reaction-diffusion simulations with Smoldyn

From homogeneity to heterogeneity: Refining stochastic simulations of gene regulation

Cellular processes are intricately controlled through gene regulation, which is significantly influenced by intrinsic noise due to the small number of molecules involved. The Gillespie algorithm, a widely used stochastic simulation method, is pervasively employed to model these systems. However, this algorithm typically assumes that DNA is homogeneously distributed throughout the nucleus, which is not realistic. In this study, we evaluated whether stochastic simulations based on the assumption of spatial homogeneity can accurately capture the dynamics of gene regulation. Our findings indicate that when transcription factors diffuse slowly, these simulations fail to accurately capture gene expression, highlighting the necessity to account for spatial heterogeneity. However, incorporating spatial heterogeneity considerably increases computational time. To address this, we explored various stochastic quasi-steady-state approximations (QSSAs) that simplify the model and reduce simulation time. While both the stochastic total quasi-steady state approximation (stQSSA) and the stochastic low-state quasi-steady-state approximation (slQSSA) reduced simulation time, only the slQSSA provided an accurate model reduction. Our study underscores the importance of utilizing appropriate methods for efficient and accurate stochastic simulations of gene regulatory dynamics, especially when incorporating spatial heterogeneity.

Particle-based MR modeling with diffusion, microstructure, and enzymatic reactions

PURPOSE

To develop a novel particle-based in silico MR model and demonstrate applications of this model to signal mechanisms which are affected by the spatial organization of particles, including metabolic reaction kinetics, microstructural effects on diffusion, and radiofrequency (RF) refocusing effects in gradient-echo sequences.

METHODS

The model was developed by integrating a forward solution of the Bloch equations with a Brownian dynamics simulator. Simulation configurations were then designed to model MR signal dynamics of interest, with a primary focus on hyperpolarized 13C MRI methods. Phantom scans and spectrophotometric assays were conducted to validate model results in vitro.

RESULTS

The model accurately reproduced the reaction kinetics of enzyme-mediated conversion of pyruvate to lactate. When varying proportions of restrictive structure were added to the reaction volume, nonlinear changes in the reaction rate measured in vitro were replicated in silico. Modeling of RF refocusing effects characterized the degree of diffusion-weighted contribution from preserved residual magnetization in nonspoiled gradient-echo sequences.

CONCLUSIONS

These results show accurate reproduction of a range of MR signal mechanisms, establishing the model's capability to investigate the multifactorial signal dynamics such as those underlying hyperpolarized 13C MRI data.

Ultraliser: a framework for creating multiscale, high-fidelity and geometrically realistic 3D models for in silico neuroscience

Ultraliser is a neuroscience-specific software framework capable of creating accurate and biologically realistic 3D models of complex neuroscientific structures at intracellular (e.g. mitochondria and endoplasmic reticula), cellular (e.g. neurons and glia) and even multicellular scales of resolution (e.g. cerebral vasculature and minicolumns). Resulting models are exported as triangulated surface meshes and annotated volumes for multiple applications in in silico neuroscience, allowing scalable supercomputer simulations that can unravel intricate cellular structure-function relationships. Ultraliser implements a high-performance and unconditionally robust voxelization engine adapted to create optimized watertight surface meshes and annotated voxel grids from arbitrary non-watertight triangular soups, digitized morphological skeletons or binary volumetric masks. The framework represents a major leap forward in simulation-based neuroscience, making it possible to employ high-resolution 3D structural models for quantification of surface areas and volumes, which are of the utmost importance for cellular and system simulations. The power of Ultraliser is demonstrated with several use cases in which hundreds of models are created for potential application in diverse types of simulations. Ultraliser is publicly released under the GNU GPL3 license on GitHub (BlueBrain/Ultraliser).

SIGNIFICANCE

There is crystal clear evidence on the impact of cell shape on its signaling mechanisms. Structural models can therefore be insightful to realize the function; the more realistic the structure can be, the further we get insights into the function. Creating realistic structural models from existing ones is challenging, particularly when needed for detailed subcellular simulations. We present Ultraliser, a neuroscience-dedicated framework capable of building these structural models with realistic and detailed cellular geometries that can be used for simulations.

A novel stochastic simulation approach enables exploration of mechanisms for regulating polarity site movement

Cells polarize their movement or growth toward external directional cues in many different contexts. For example, budding yeast cells grow toward potential mating partners in response to pheromone gradients. Directed growth is controlled by polarity factors that assemble into clusters at the cell membrane. The clusters assemble, disassemble, and move between different regions of the membrane before eventually forming a stable polarity site directed toward the pheromone source. Pathways that regulate clustering have been identified but the molecular mechanisms that regulate cluster mobility are not well understood. To gain insight into the contribution of chemical noise to cluster behavior we simulated clustering using the reaction-diffusion master equation (RDME) framework to account for molecular-level fluctuations. RDME simulations are a computationally efficient approximation, but their results can diverge from the underlying microscopic dynamics. We implemented novel concentration-dependent rate constants that improved the accuracy of RDME-based simulations, allowing us to efficiently investigate how cluster dynamics might be regulated. Molecular noise was effective in relocating clusters when the clusters contained low numbers of limiting polarity factors, and when Cdc42, the central polarity regulator, exhibited short dwell times at the polarity site. Cluster stabilization occurred when abundances or binding rates were altered to either lengthen dwell times or increase the number of polarity molecules in the cluster. We validated key results using full 3D particle-based simulations. Understanding the mechanisms cells use to regulate the dynamics of polarity clusters should provide insights into how cells dynamically track external directional cues.

Cells localize polarity molecules in a small region of the plasma membrane forming a polarity cluster that directs functions such as migration, reproduction, and growth. Guided by external signals, these clusters move across the membrane allowing cells to reorient growth or motion. The polarity molecules continuously and randomly shuttle between the cluster and the cell cytosol and, as a result, the number and distribution of molecules at the cluster constantly changes. Here we present an improved stochastic simulation algorithm to investigate how such molecular-scale fluctuations induce cluster movement across the cell membrane. Unexpectedly, cluster mobility does not correlate with variations in total molecule abundance within the cluster, but rather with changes in the spatial distribution of molecules that form the cluster. Cluster motion is faster when polarity molecules are scarce and when they shuttle rapidly between the cluster and the cytosol. Our results suggest that cells control cluster mobility by regulating the abundance of polarity molecules and biochemical reactions that affect the time molecules spend at the cluster. We provide insights into how cells harness random molecular behavior to perform functions important for survival, such as detecting the direction of external signals.

Stochastic self-tuning hybrid algorithm for reaction-diffusion systems

Many biochemical phenomena involve reactants with vastly different concentrations, some of which are amenable to continuum-level descriptions, while the others are not. We present a hybrid self-tuning algorithm to model such systems. The method combines microscopic (Brownian) dynamics for diffusion with mesoscopic (Gillespie-type) methods for reactions and remains efficient in a wide range of regimes and scenarios with large variations of concentrations. Its accuracy, robustness, and versatility are balanced by redefining propensities and optimizing the mesh size and time step. We use a bimolecular reaction to demonstrate the potential of our method in a broad spectrum of scenarios: from almost completely reaction-dominated systems to cases where reactions rarely occur or take place very slowly. The simulation results show that the number of particles present in the system does not degrade the performance of our method. This makes it an accurate and computationally efficient tool to model complex multireaction systems.

Coarse-graining molecular dynamics: stochastic models with non-Gaussian force distributions

Incorporating atomistic and molecular information into models of cellular behaviour is challenging because of a vast separation of spatial and temporal scales between processes happening at the atomic and cellular levels. Multiscale or multi-resolution methodologies address this difficulty by using molecular dynamics (MD) and coarse-grained models in different parts of the cell. Their applicability depends on the accuracy and properties of the coarse-grained model which approximates the detailed MD description. A family of stochastic coarse-grained (SCG) models, written as relatively low-dimensional systems of nonlinear stochastic differential equations, is presented. The nonlinear SCG model incorporates the non-Gaussian force distribution which is observed in MD simulations and which cannot be described by linear models. It is shown that the nonlinearities can be chosen in such a way that they do not complicate parametrization of the SCG description by detailed MD simulations. The solution of the SCG model is found in terms of gamma functions.

Multiscale Stochastic Reaction-Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations

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.

Rule-Based Modeling Using Wildcards in the Smoldyn Simulator

Many biological molecules exist in multiple variants, such as proteins with different posttranslational modifications, DNAs with different sequences, and phospholipids with different chain lengths. Representing these variants as distinct species, as most biochemical simulators do, leads to the problem that the number of species, and chemical reactions that interconvert them, typically increase combinatorially with the number of ways that the molecules can vary. This can be alleviated by "rule-based modeling methods," in which software generates the chemical reaction network from relatively simple "rules." This chapter presents a new approach to rule-based modeling. It is based on wildcards that match to species names, much as wildcards can match to file names in computer operating systems. It is much simpler to use than the formal rule-based modeling approaches developed previously but can lead to unintended consequences if not used carefully. This chapter demonstrates rule-based modeling with wildcards through examples for signaling systems, protein complexation, polymerization, nucleic acid sequence copying and mutation, the "SMILES" chemical notation, and others. The method is implemented in Smoldyn, a spatial and stochastic biochemical simulator, for both generate-first and on-the-fly expansion, meaning whether the reaction network is generated before or during the simulation.

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 used 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.

eGFRD in all dimensions

Biochemical reactions often occur at low copy numbers but at once in crowded and diverse environments. Space and stochasticity therefore play an essential role in biochemical networks. Spatial-stochastic simulations have become a prominent tool for understanding how stochasticity at the microscopic level influences the macroscopic behavior of such systems. While particle-based models guarantee the level of detail necessary to accurately describe the microscopic dynamics at very low copy numbers, the algorithms used to simulate them typically imply trade-offs between computational efficiency and biochemical accuracy. eGFRD (enhanced Green's Function Reaction Dynamics) is an exact algorithm that evades such trade-offs by partitioning the N-particle system into M ≤ N analytically tractable one- and two-particle systems; the analytical solutions (Green's functions) then are used to implement an event-driven particle-based scheme that allows particles to make large jumps in time and space while retaining access to their state variables at arbitrary simulation times. Here we present "eGFRD2," a new eGFRD version that implements the principle of eGFRD in all dimensions, thus enabling efficient particle-based simulation of biochemical reaction-diffusion processes in the 3D cytoplasm, on 2D planes representing membranes, and on 1D elongated cylinders representative of, e.g., cytoskeletal tracks or DNA; in 1D, it also incorporates convective motion used to model active transport. We find that, for low particle densities, eGFRD2 is up to 6 orders of magnitude faster than conventional Brownian dynamics. We exemplify the capabilities of eGFRD2 by simulating an idealized model of Pom1 gradient formation, which involves 3D diffusion, active transport on microtubules, and autophosphorylation on the membrane, confirming recent experimental and theoretical results on this system to hold under genuinely stochastic conditions.

Algorithm for Mesoscopic Advection-Diffusion

In this paper, an algorithm is presented to calculate the transition rates between adjacent mesoscopic subvolumes in the presence of flow and diffusion. These rates can be integrated in stochastic simulations of reaction-diffusion systems that follow a mesoscopic approach, i.e., that partition the environment into homogeneous subvolumes and apply the spatial stochastic simulation algorithm (spatial SSA). The rates are derived by integrating Fick's second law over a single subvolume in 1D and are also shown to apply in 3D. The proposed algorithm corrects the derived rates to ensure that they are physically meaningful and it is implemented in the AcCoRD Simulator (Actor-based Communication via Reaction-Diffusion). Simulations using the proposed method are compared with a naive mesoscopic approach, microscopic simulations that track every molecule, and analytical results that are exact in 1D and an approximation in 3D. By choosing subvolumes that are sufficiently small, such that the Péclet number associated with a subvolume is sufficiently less than two, the accuracy of the proposed method is comparable with microscopic method, thus enabling the simulation of advection-reaction-diffusion systems with the spatial SSA.

Reaction rates for reaction-diffusion kinetics on unstructured meshes

The reaction-diffusion master equation is a stochastic model often utilized in the study of biochemical reaction networks in living cells. It is applied when the spatial distribution of molecules is important to the dynamics of the system. A viable approach to resolve the complex geometry of cells accurately is to discretize space with an unstructured mesh. Diffusion is modeled as discrete jumps between nodes on the mesh, and the diffusion jump rates can be obtained through a discretization of the diffusion equation on the mesh. Reactions can occur when molecules occupy the same voxel. In this paper, we develop a method for computing accurate reaction rates between molecules occupying the same voxel in an unstructured mesh. For large voxels, these rates are known to be well approximated by the reaction rates derived by Collins and Kimball, but as the mesh is refined, no analytical expression for the rates exists. We reduce the problem of computing accurate reaction rates to a pure preprocessing step, depending only on the mesh and not on the model parameters, and we devise an efficient numerical scheme to estimate them to high accuracy. We show in several numerical examples that as we refine the mesh, the results obtained with the reaction-diffusion master equation approach those of a more fine-grained Smoluchowski particle-tracking model.

Simulating biological processes: stochastic physics from whole cells to colonies

The last few decades have revealed the living cell to be a crowded spatially heterogeneous space teeming with biomolecules whose concentrations and activities are governed by intrinsically random forces. It is from this randomness, however, that a vast array of precisely timed and intricately coordinated biological functions emerge that give rise to the complex forms and behaviors we see in the biosphere around us. This seemingly paradoxical nature of life has drawn the interest of an increasing number of physicists, and recent years have seen stochastic modeling grow into a major subdiscipline within biological physics. Here we review some of the major advances that have shaped our understanding of stochasticity in biology. We begin with some historical context, outlining a string of important experimental results that motivated the development of stochastic modeling. We then embark upon a fairly rigorous treatment of the simulation methods that are currently available for the treatment of stochastic biological models, with an eye toward comparing and contrasting their realms of applicability, and the care that must be taken when parameterizing them. Following that, we describe how stochasticity impacts several key biological functions, including transcription, translation, ribosome biogenesis, chromosome replication, and metabolism, before considering how the functions may be coupled into a comprehensive model of a 'minimal cell'. Finally, we close with our expectation for the future of the field, focusing on how mesoscopic stochastic methods may be augmented with atomic-scale molecular modeling approaches in order to understand life across a range of length and time scales.

Bridging intracellular scales by mechanistic computational models

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.

Multi-Algorithm Particle Simulations with Spatiocyte

As quantitative biologists get more measurements of spatially regulated systems such as cell division and polarization, simulation of reaction and diffusion of proteins using the data is becoming increasingly relevant to uncover the mechanisms underlying the systems. Spatiocyte is a lattice-based stochastic particle simulator for biochemical reaction and diffusion processes. Simulations can be performed at single molecule and compartment spatial scales simultaneously. Molecules can diffuse and react in 1D (filament), 2D (membrane), and 3D (cytosol) compartments. The implications of crowded regions in the cell can be investigated because each diffusing molecule has spatial dimensions. Spatiocyte adopts multi-algorithm and multi-timescale frameworks to simulate models that simultaneously employ deterministic, stochastic, and particle reaction-diffusion algorithms. Comparison of light microscopy images to simulation snapshots is supported by Spatiocyte microscopy visualization and molecule tagging features. Spatiocyte is open-source software and is freely available at http://spatiocyte.org .

Mesoscopic-microscopic spatial stochastic simulation with automatic system partitioning

The reaction-diffusion master equation (RDME) is a model that allows for efficient on-lattice simulation of spatially resolved stochastic chemical kinetics. Compared to off-lattice hard-sphere simulations with Brownian dynamics or Green's function reaction dynamics, the RDME can be orders of magnitude faster if the lattice spacing can be chosen coarse enough. However, strongly diffusion-controlled reactions mandate a very fine mesh resolution for acceptable accuracy. It is common that reactions in the same model differ in their degree of diffusion control and therefore require different degrees of mesh resolution. This renders mesoscopic simulation inefficient for systems with multiscale properties. Mesoscopic-microscopic hybrid methods address this problem by resolving the most challenging reactions with a microscale, off-lattice simulation. However, all methods to date require manual partitioning of a system, effectively limiting their usefulness as "black-box" simulation codes. In this paper, we propose a hybrid simulation algorithm with automatic system partitioning based on indirect a priori error estimates. We demonstrate the accuracy and efficiency of the method on models of diffusion-controlled networks in 3D.

Smoldyn: particle-based simulation with rule-based modeling, improved molecular interaction and a library interface

Motivation

Smoldyn is a spatial and stochastic biochemical simulator. It treats each molecule of interest as an individual particle in continuous space, simulating molecular diffusion, molecule-membrane interactions and chemical reactions, all with good accuracy. This article presents several new features.

Results

Smoldyn now supports two types of rule-based modeling. These are a wildcard method, which is very convenient, and the BioNetGen package with extensions for spatial simulation, which is better for complicated models. Smoldyn also includes new algorithms for simulating the diffusion of surface-bound molecules and molecules with excluded volume. Both are exact in the limit of short time steps and reasonably good with longer steps. In addition, Smoldyn supports single-molecule tracking simulations. Finally, the Smoldyn source code can be accessed through a C/C ++ language library interface.

Availability and Implementation

Smoldyn software, documentation, code, and examples are at http://www.smoldyn.org .

Contact

steven.s.andrews@gmail.com.

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium 'sparks' as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.

Analysis of Ligand-Receptor Association and Intermediate Transfer Rates in Multienzyme Nanostructures with All-Atom Brownian Dynamics Simulations

We present the second-generation GeomBD Brownian dynamics software for determining interenzyme intermediate transfer rates and substrate association rates in biomolecular complexes. Substrate and intermediate association rates for a series of enzymes or biomolecules can be compared between the freely diffusing disorganized configuration and various colocalized or complexed arrangements for kinetic investigation of enhanced intermediate transfer. In addition, enzyme engineering techniques, such as synthetic protein conjugation, can be computationally modeled and analyzed to better understand changes in substrate association relative to native enzymes. Tools are provided to determine nonspecific ligand-receptor association residence times, and to visualize common sites of nonspecific association of substrates on receptor surfaces. To demonstrate features of the software, interenzyme intermediate substrate transfer rate constants are calculated and compared for all-atom models of DNA origami scaffold-bound bienzyme systems of glucose oxidase and horseradish peroxidase. Also, a DNA conjugated horseradish peroxidase enzyme was analyzed for its propensity to increase substrate association rates and substrate local residence times relative to the unmodified enzyme. We also demonstrate the rapid determination and visualization of common sites of nonspecific ligand-receptor association by using HIV-1 protease and an inhibitor, XK263. GeomBD2 accelerates simulations by precomputing van der Waals potential energy grids and electrostatic potential grid maps, and has a flexible and extensible support for all-atom and coarse-grained force fields. Simulation software is written in C++ and utilizes modern parallelization techniques for potential grid preparation and Brownian dynamics simulation processes. Analysis scripts, written in the Python scripting language, are provided for quantitative simulation analysis. GeomBD2 is applicable to the fields of biophysics, bioengineering, and enzymology in both predictive and explanatory roles.

Multi-class and multi-scale models of complex biological phenomena

Computational modeling has significantly impacted our ability to analyze vast (and exponentially increasing) quantities of experimental data for a variety of applications, such as drug discovery and disease forecasting. Single-scale, single-class models persist as the most common group of models, but biological complexity often demands more sophisticated approaches. This review surveys modeling approaches that are multi-class (incorporating multiple model types) and/or multi-scale (accounting for multiple spatial or temporal scales) and describes how these models, and combinations thereof, should be used within the context of the problem statement. We end by highlighting agent-based models as an intuitive, modular, and flexible framework within which multi-scale and multi-class models can be implemented.

SpringSaLaD: A Spatial, Particle-Based Biochemical Simulation Platform with Excluded Volume

We introduce Springs, Sites, and Langevin Dynamics (SpringSaLaD), a comprehensive software platform for spatial, stochastic, particle-based modeling of biochemical systems. SpringSaLaD models biomolecules in a coarse-grained manner as a group of linked spherical sites with excluded volume. This mesoscopic approach bridges the gap between highly detailed molecular dynamics simulations and the various methods used to study network kinetics and diffusion at the cellular level. SpringSaLaD is a standalone tool that supports model building, simulation, visualization, and data analysis, all through a user-friendly graphical user interface that should make it more accessible than tools built into more comprehensive molecular dynamics infrastructures. Importantly, for bimolecular reactions we derive an exact expression relating the macroscopic on-rate to the various microscopic parameters with the inclusion of excluded volume; this makes SpringSaLaD more accurate than other tools, which rely on approximate relationships between these parameters.

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

Molecular dynamics (MD) simulations of ions (K+, Na+, Ca2+ 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 parametrized 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.

Systems Biology Approaches for Understanding Genome Architecture

The linear and three-dimensional arrangement and composition of chromatin in eukaryotic genomes underlies the mechanisms directing gene regulation. Understanding this organization requires the integration of many data types and experimental results. Here we describe the approach of integrating genome-wide protein-DNA binding data to determine chromatin states. To investigate spatial aspects of genome organization, we present a detailed description of how to run stochastic simulations of protein movements within a simulated nucleus in 3D. This systems level approach enables the development of novel questions aimed at understanding the basic mechanisms that regulate genome dynamics.