Noise-induced Min phenotypes in E. coli

A dynamic partitioning mechanism polarizes membrane protein distribution

The plasma membrane is widely regarded as the hub of the numerous signal transduction activities. Yet, the fundamental biophysical mechanisms that spatiotemporally compartmentalize different classes of membrane proteins remain unclear. Using multimodal live-cell imaging, here we first show that several lipid-anchored membrane proteins are consistently depleted from the membrane regions where the Ras/PI3K/Akt/F-actin network is activated. The dynamic polarization of these proteins does not depend upon the F-actin-based cytoskeletal structures, recurring shuttling between membrane and cytosol, or directed vesicular trafficking. Photoconversion microscopy and single-molecule measurements demonstrate that these lipid-anchored molecules have substantially dissimilar diffusion profiles in different regions of the membrane which enable their selective segregation. When these diffusion coefficients are incorporated into an excitable network-based stochastic reaction-diffusion model, simulations reveal that the altered affinity mediated selective partitioning is sufficient to drive familiar propagating wave patterns. Furthermore, normally uniform integral and lipid-anchored membrane proteins partition successfully when membrane domain-specific peptides are optogenetically recruited to them. We propose "dynamic partitioning" as a new mechanism that can account for large-scale compartmentalization of a wide array of lipid-anchored and integral membrane proteins during various physiological processes where membrane polarizes.

The mechanism of MinD stability modulation by MinE in Min protein dynamics

The patterns formed both in vivo and in vitro by the Min protein system have attracted much interest because of the complexity of their dynamic interactions given the apparent simplicity of the component parts. Despite both the experimental and theoretical attention paid to this system, the details of the biochemical interactions of MinD and MinE, the proteins responsible for the patterning, are still unclear. For example, no model consistent with the known biochemistry has yet accounted for the observed dual role of MinE in the membrane stability of MinD. Until now, a statistical comparison of models to the time course of Min protein concentrations on the membrane has not been carried out. Such an approach is a powerful way to test existing and novel models that are difficult to test using a purely experimental approach. Here, we extract time series from previously published fluorescence microscopy time lapse images of in vitro experiments and fit two previously described and one novel mathematical model to the data. We find that the novel model, which we call the Asymmetric Activation with Bridged Stability Model, fits the time-course data best. It is also consistent with known biochemistry and explains the dual MinE role via MinE-dependent membrane stability that transitions under the influence of rising MinE to membrane instability with positive feedback. Our results reveal a more complex network of interactions between MinD and MinE underlying Min-system dynamics than previously considered.

Quantitative characterization of membrane-protein reversible association using FCS

Functionally meaningful reversible protein-membrane interactions mediate many biological events. Fluorescence correlation spectroscopy (FCS) is increasingly used to quantitatively study the non-reversible binding of proteins to membranes using lipid vesicles in solution. However, the lack of a complete description of the phase and statistical equilibria in the case of reversible protein-membrane partitioning has hampered the application of FCS to quantify the partition coefficient (K_x). In this work, we further extend the theory that describes membrane-protein partitioning to account for spontaneous protein-membrane dissociation and reassociation to the same or a different lipid vesicle. We derive the probability distribution of proteins on lipid vesicles for reversible binding and demonstrate that FCS is a suitable technique for accurate K_x quantification of membrane-protein reversible association. We also establish the limits to K_x determination by FCS studying the Cramer-Rao bound on the variance of the retrieved parameters. We validate the mathematical formulation against reaction-diffusion simulations to study phase and statistical equilibria and compare the K_x obtained from a computational FCS titration experiment with the experimental ground truth. Finally, we demonstrate the application of our methodology studying the association of anti-HIV broadly neutralizing antibody (10E8-3R) to the membrane.

A dynamic partitioning mechanism polarizes membrane protein distribution

The plasma membrane is widely regarded as the hub of the signal transduction network activities that drives numerous physiological responses, including cell polarity and migration. Yet, the symmetry breaking process in the membrane, that leads to dynamic compartmentalization of different proteins, remains poorly understood. Using multimodal live-cell imaging, here we first show that multiple endogenous and synthetic lipid-anchored proteins, despite maintaining stable tight association with the inner leaflet of the plasma membrane, were unexpectedly depleted from the membrane domains where the signaling network was spontaneously activated such as in the new protrusions as well as within the propagating ventral waves. Although their asymmetric patterns resembled those of standard peripheral "back" proteins such as PTEN, unlike the latter, these lipidated proteins did not dissociate from the membrane upon global receptor activation. Our experiments not only discounted the possibility of recurrent reversible translocation from membrane to cytosol as it occurs for weakly bound peripheral membrane proteins, but also ruled out the necessity of directed vesicular trafficking and cytoskeletal supramolecular structure-based restrictions in driving these dynamic symmetry breaking events. Selective photoconversion-based protein tracking assays suggested that these asymmetric patterns instead originate from the inherent ability of these membrane proteins to "dynamically partition" into distinct domains within the plane of the membrane. Consistently, single-molecule measurements showed that these lipid-anchored molecules have substantially dissimilar diffusion profiles in different regions of the membrane. When these profiles were incorporated into an excitable network-based stochastic reaction-diffusion model of the system, simulations revealed that our proposed "dynamic partitioning" mechanism is sufficient to give rise to familiar asymmetric propagating wave patterns. Moreover, we demonstrated that normally uniform integral and lipid-anchored membrane proteins in Dictyostelium and mammalian neutrophil cells can be induced to partition spatiotemporally to form polarized patterns, by optogenetically recruiting membrane domain-specific peptides to these proteins. Together, our results indicate "dynamic partitioning" as a new mechanism of plasma membrane organization, that can account for large-scale compartmentalization of a wide array of lipid-anchored and integral membrane proteins in different physiological processes.

A Gaussian jump process formulation of the reaction–diffusion master equation enables faster exact stochastic simulations

We propose a Gaussian jump process model on a regular Cartesian lattice for the diffusion part of the Reaction–Diffusion Master Equation (RDME). We derive the resulting Gaussian RDME (GRDME) formulation from analogy with a kernel-based discretization scheme for continuous diffusion processes and quantify the limits of its validity relative to the classic RDME. We then present an exact stochastic simulation algorithm for the GRDME, showing that the accuracies of GRDME and RDME are comparable, but exact simulations of the GRDME require only a fraction of the computational cost of exact RDME simulations. We analyze the origin of this speedup and its scaling with problem dimension. The benchmarks suggest that the GRDME is a particularly beneficial model for diffusion-dominated systems in three dimensional spaces, often occurring in systems biology and cell biology.

CellDynaMo–stochastic reaction-diffusion-dynamics model: Application to search-and-capture process of mitotic spindle assembly

We introduce a Stochastic Reaction-Diffusion-Dynamics Model (SRDDM) for simulations of cellular mechanochemical processes with high spatial and temporal resolution. The SRDDM is mapped into the CellDynaMo package, which couples the spatially inhomogeneous reaction-diffusion master equation to account for biochemical reactions and molecular transport within the Langevin Dynamics (LD) framework to describe dynamic mechanical processes. This computational infrastructure allows the simulation of hours of molecular machine dynamics in reasonable wall-clock time. We apply SRDDM to test performance of the Search-and-Capture of mitotic spindle assembly by simulating, in three spatial dimensions, dynamic instability of elastic microtubules anchored in two centrosomes, movement and deformations of geometrically realistic centromeres with flexible kinetochores and chromosome arms. Furthermore, the SRDDM describes the mechanics and kinetics of Ndc80 linkers mediating transient attachments of microtubules to the chromosomal kinetochores. The rates of these attachments and detachments depend upon phosphorylation states of the Ndc80 linkers, which are regulated in the model by explicitly accounting for the reactions of Aurora A and B kinase enzymes undergoing restricted diffusion. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of the chromosome connections, that adding chromosome arms to kinetochores improve the accuracy by slowing down chromosome movements, that Aurora A and kinetochore deformations have a small positive effect on the attachment accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy of spindle assembly.

The CellDynaMo package models, in 3D, any cellular subsystem where sufficient detail of the macromolecular players and the kinetics of relevant reactions are available. The package is based on the Stochastic Reaction-Diffusion-Dynamics model that combines the stochastic description of chemical kinetics, Brownian diffusion-based description of molecular transport, and Langevin dynamics-based representation of mechanical processes most pertinent to the system. We apply the model to test the Search-and-Capture mechanism of mitotic spindle assembly. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of chromosome connections, that chromosome arms improve the attachment accuracy by slowing down chromosome movements, that Aurora A kinase and kinetochore deformations have small positive effects on the accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy.

A multiscale compartment-based model of stochastic gene regulatory networks using hitting-time analysis

Spatial stochastic models of single cell kinetics are capable of capturing both fluctuations in molecular numbers and the spatial dependencies of the key steps of intracellular regulatory networks. The spatial stochastic model can be simulated both on a detailed microscopic level using particle tracking and on a mesoscopic level using the reaction-diffusion master equation. However, despite substantial progress on simulation efficiency for spatial models in the last years, the computational cost quickly becomes prohibitively expensive for tasks that require repeated simulation of thousands or millions of realizations of the model. This limits the use of spatial models in applications such as multicellular simulations, likelihood-free parameter inference, and robustness analysis. Further approximation of the spatial dynamics is needed to accelerate such computational engineering tasks. We here propose a multiscale model where a compartment-based model approximates a detailed spatial stochastic model. The compartment model is constructed via a first-exit time analysis on the spatial model, thus capturing critical spatial aspects of the fine-grained simulations, at a cost close to the simple well-mixed model. We apply the multiscale model to a canonical model of negative-feedback gene regulation, assess its accuracy over a range of parameters, and demonstrate that the approximation can yield substantial speedups for likelihood-free parameter inference.

Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

Though the term noise usually carries negative connotations, it can also contribute positively to the characteristic dynamics of a system. In biological systems, where noise arises from the stochastic interactions between molecules, its study is usually confined to genetic regulatory systems in which copy numbers are small and fluctuations large. However, noise can have important roles when the number of signaling molecules is large. The extension of pseudopods and the subsequent motion of amoeboid cells arises from the noise-induced trigger of an excitable system. Chemoattractant signals bias this triggering thereby directing cell motion. To date, this paradigm has not been tested by mathematical models that account accurately for the noise that arises in the corresponding reactions. In this study, we employ a reaction-diffusion master equation approach to investigate the effects of noise. Using a modular approach and a three-dimensional cell model with specific subdomains attributed to the cell membrane and cortex, we explore the spatiotemporal dynamics of the system. Our simulations recreate many experimentally-observed cell behaviors thereby supporting the biased-excitable network hypothesis.

Dynamics of the Bacillus subtilis Min System

The molecular mechanisms that help to place the division septum in bacteria is of fundamental importance to ensure cell proliferation and maintenance of cell shape and size. The Min protein system, found in many rod-shaped bacteria, is thought to play a major role in division site selection.

Division site selection is a vital process to ensure generation of viable offspring. In many rod-shaped bacteria, a dynamic protein system, termed the Min system, acts as a central regulator of division site placement. The Min system is best studied in Escherichia coli, where it shows a remarkable oscillation from pole to pole with a time-averaged density minimum at midcell. Several components of the Min system are conserved in the Gram-positive model organism Bacillus subtilis. However, in B. subtilis, it is commonly believed that the system forms a stationary bipolar gradient from the cell poles to midcell. Here, we show that the Min system of B. subtilis localizes dynamically to active sites of division, often organized in clusters. We provide physical modeling using measured diffusion constants that describe the observed enrichment of the Min system at the septum. Mathematical modeling suggests that the observed localization pattern of Min proteins corresponds to a dynamic equilibrium state. Our data provide evidence for the importance of ongoing septation for the Min dynamics, consistent with a major role of the Min system in controlling active division sites but not cell pole areas.

Smart computational exploration of stochastic gene regulatory network models using human-in-the-loop semi-supervised learning

MOTIVATION

Discrete stochastic models of gene regulatory network models are indispensable tools for biological inquiry since they allow the modeler to predict how molecular interactions give rise to nonlinear system output. Model exploration with the objective of generating qualitative hypotheses about the workings of a pathway is usually the first step in the modeling process. It involves simulating the gene network model under a very large range of conditions, due to the large uncertainty in interactions and kinetic parameters. This makes model exploration highly computational demanding. Furthermore, with no prior information about the model behavior, labor-intensive manual inspection of very large amounts of simulation results becomes necessary. This limits systematic computational exploration to simplistic models.

RESULTS

We have developed an interactive, smart workflow for model exploration based on semi-supervised learning and human-in-the-loop labeling of data. The workflow lets a modeler rapidly discover ranges of interesting behaviors predicted by the model. Utilizing that similar simulation output is in proximity of each other in a feature space, the modeler can focus on informing the system about what behaviors are more interesting than others by labeling, rather than analyzing simulation results with custom scripts and workflows. This results in a large reduction in time-consuming manual work by the modeler early in a modeling project, which can substantially reduce the time needed to go from an initial model to testable predictions and downstream analysis.

AVAILABILITY AND IMPLEMENTATION

A python-package is available at https://github.com/Wrede/mio.git.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

C-terminal eYFP fusion impairs Escherichia coli MinE function

The Escherichia coli Min system plays an important role in the proper placement of the septum ring at mid-cell during cell division. MinE forms a pole-to-pole spatial oscillator with the membrane-bound ATPase MinD, resulting in MinD concentration being the lowest at mid-cell. MinC, the direct inhibitor of the septum initiator protein FtsZ, forms a complex with MinD at the membrane, mirroring its polar gradients. Therefore, MinC-mediated FtsZ inhibition occurs away from mid-cell. Min oscillations are often studied in living cells by time-lapse microscopy using fluorescently labelled Min proteins. Here, we show that, despite permitting oscillations to occur in a range of protein concentrations, the enhanced yellow fluorescent protein (eYFP) C-terminally fused to MinE impairs its function. Combining in vivo, in vitro and in silico approaches, we demonstrate that eYFP compromises the ability of MinE to displace MinC from MinD, to stimulate MinD ATPase activity and to directly bind to the membrane. Moreover, we reveal that MinE-eYFP is prone to aggregation. In silico analyses predict that other fluorescent proteins are also likely to compromise several functionalities of MinE, suggesting that the results presented here are not specific to eYFP.

Hierarchical algorithm for the reaction-diffusion master equation

We have developed an algorithm coupling mesoscopic simulations on different levels in a hierarchy of Cartesian meshes. Based on the multiscale nature of the chemical reactions, some molecules in the system will live on a fine-grained mesh, while others live on a coarse-grained mesh. By allowing molecules to transfer from the fine levels to the coarse levels when appropriate, we show that we can save up to three orders of magnitude of computational time compared to microscopic simulations or highly resolved mesoscopic simulations, without losing significant accuracy. We demonstrate this in several numerical examples with systems that cannot be accurately simulated with a coarse-grained mesoscopic model.

The Linear Noise Approximation for Spatially Dependent Biochemical Networks

An algorithm for computing the linear noise approximation (LNA) of the reaction-diffusion master equation (RDME) is developed and tested. The RDME is often used as a model for biochemical reaction networks. The LNA is derived for a general discretization of the spatial domain of the problem. If M is the number of chemical species in the network and N is the number of nodes in the discretization in space, then the computational work to determine approximations of the mean and the covariances of the probability distributions is proportional to [Formula: see text] in a straightforward implementation. In our LNA algorithm, the work is proportional to [Formula: see text]. Since N usually is larger than M, this is a significant reduction. The accuracy of the approximation in the algorithm is estimated analytically and evaluated in numerical experiments.

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.

Spatial Stochastic Intracellular Kinetics: A Review of Modelling Approaches

Models of chemical kinetics that incorporate both stochasticity and diffusion are an increasingly common tool for studying biology. The variety of competing models is vast, but two stand out by virtue of their popularity: the reaction-diffusion master equation and Brownian dynamics. In this review, we critically address a number of open questions surrounding these models: How can they be justified physically? How do they relate to each other? How do they fit into the wider landscape of chemical models, ranging from the rate equations to molecular dynamics? This review assumes no prior knowledge of modelling chemical kinetics and should be accessible to a wide range of readers.

Cell-sized confinement controls generation and stability of a protein wave for spatiotemporal regulation in cells

The Min system, a system that determines the bacterial cell division plane, uses changes in the localization of proteins (a Min wave) that emerges by reaction-diffusion coupling. Although previous studies have shown that space sizes and boundaries modulate the shape and speed of Min waves, their effects on wave emergence were still elusive. Here, by using a microsized fully confined space to mimic live cells, we revealed that confinement changes the conditions for the emergence of Min waves. In the microsized space, an increased surface-to-volume ratio changed the localization efficiency of proteins on membranes, and therefore, suppression of the localization change was necessary for the stable generation of Min waves. Furthermore, we showed that the cell-sized space strictly limits parameters for wave emergence because confinement inhibits both the instability and excitability of the system. These results show that confinement of reaction-diffusion systems has the potential to control spatiotemporal patterns in live cells.

Surface reaction-diffusion kinetics on lattice at the microscopic scale

Microscopic models of reaction-diffusion processes on the cell membrane can link local spatiotemporal effects to macroscopic self-organized patterns often observed on the membrane. Simulation schemes based on the microscopic lattice method (MLM) can model these processes at the microscopic scale by tracking individual molecules, represented as hard spheres, on fine lattice voxels. Although MLM is simple to implement and is generally less computationally demanding than off-lattice approaches, its accuracy and consistency in modeling surface reactions have not been fully verified. Using the Spatiocyte scheme, we study the accuracy of MLM in diffusion-influenced surface reactions. We derive the lattice-based bimolecular association rates for two-dimensional (2D) surface-surface reaction and one-dimensional (1D) volume-surface adsorption according to the Smoluchowski-Collins-Kimball model and random walk theory. We match the time-dependent rates on lattice with off-lattice counterparts to obtain the correct expressions for MLM parameters in terms of physical constants. The expressions indicate that the voxel size needs to be at least 0.6% larger than the molecule to accurately simulate surface reactions on triangular lattice. On square lattice, the minimum voxel size should be even larger, at 5%. We also demonstrate the ability of MLM-based schemes such as Spatiocyte to simulate a reaction-diffusion model that involves all dimensions: three-dimensional (3D) diffusion in the cytoplasm, 2D diffusion on the cell membrane, and 1D cytoplasm-membrane adsorption. With the model, we examine the contribution of the 2D reaction pathway to the overall reaction rate at different reactant diffusivity, reactivity, and concentrations.

A 3D Multiscale Model to Explore the Role of EGFR Overexpression in Tumourigenesis

The epidermal growth factor receptor (EGFR) signalling cascade is one of the main pathways that regulate the survival and division of mammalian cells. It is also one of the most altered transduction pathways in cancer. Acquired mutations in the EGFR/ERK pathway can cause the overexpression of EGFR on the surface of the cell, while others downregulate the inactivation of switched on intracellular proteins such as Ras and Raf. This upregulates the activity of ERK and promotes cell division. We develop a 3D multiscale model to explore the role of EGFR overexpression on tumour initiation. In this model, cells are described as individual objects that move, interact, divide, proliferate, and die by apoptosis. We use Brownian Dynamics to describe the extracellular and intracellular regulations of cells as well as the spatial and stochastic effects influencing them. The fate of each cell depends on the number of active transcription factors in the nucleus. We use numerical simulations to investigate the individual and combined effects of mutations on the intracellular regulation of individual cells. Next, we show that the distance between active receptors increase the level of EGFR/ERK signalling. We demonstrate the usefulness of the model by quantifying the impact of mutational alterations in the EGFR/ERK pathway on the growth rate of in silico tumours.

A hybrid method for micro-mesoscopic stochastic simulation of reaction-diffusion systems

The present paper introduces a new micro-meso hybrid algorithm based on the Ghost Cell Method concept in which the microscopic subdomain is governed by the Reactive Multi-Particle Collision (RMPC) dynamics. The mesoscopic subdomain is modeled using the Reaction-Diffusion Master Equation (RDME). The RDME is solved by means of the Inhomogeneous Stochastic Simulation Algorithm. No hybrid algorithm has hitherto used the RMPC dynamics for modeling reactions and the trajectories of each individual particle. The RMPC is faster than other molecular based methods and has the advantage of conserving mass, energy and momentum in the collision and free streaming steps. The new algorithm is tested on three reaction-diffusion systems. In all the systems studied, very good agreement with the deterministic solutions of the corresponding differential equations is obtained. In addition, it has been shown that proper discretization of the computational domain results in significant speed-ups in comparison with the full RMPC algorithm.

Self-organization principles of intracellular pattern formation

Dynamic patterning of specific proteins is essential for the spatio-temporal regulation of many important intracellular processes in prokaryotes, eukaryotes and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article, we review quantitative models for intracellular Min protein patterns in Escherichia coli, Cdc42 polarization in Saccharomyces cerevisiae and the bipolar PAR protein patterns found in Caenorhabditis elegans. By analysing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as ‘activators’, ‘inhibitors’ or ‘substrate depletion’. Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction–diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics.

This article is part of the theme issue ‘Self-organization in cell biology’.

The Min-protein oscillations in Escherichia coli: an example of self-organized cellular protein waves

In the rod-shaped bacterium Escherichia coli, selection of the cell centre as the division site involves pole-to-pole oscillations of the proteins MinC, MinD and MinE. This spatio-temporal pattern emerges from interactions among the Min proteins and with the cytoplasmic membrane. Combining experimental studies in vivo and in vitro together with theoretical analysis has led to a fairly good understanding of Min-protein self-organization. In different geometries, the system can, in addition to standing waves, also produce travelling planar and spiral waves as well as coexisting stable stationary distributions. Today it stands as one of the best-studied examples of cellular self-organization of proteins.This article is part of the theme issue 'Self-organization in cell biology'.

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.

Minimal in vitro systems shed light on cell polarity

Cell polarity - the morphological and functional differentiation of cellular compartments in a directional manner - is required for processes such as orientation of cell division, directed cellular growth and motility. How the interplay of components within the complexity of a cell leads to cell polarity is still heavily debated. In this Review, we focus on one specific aspect of cell polarity: the non-uniform accumulation of proteins on the cell membrane. In cells, this is achieved through reaction-diffusion and/or cytoskeleton-based mechanisms. In reaction-diffusion systems, components are transformed into each other by chemical reactions and are moving through space by diffusion. In cytoskeleton-based processes, cellular components (i.e. proteins) are actively transported by microtubules (MTs) and actin filaments to specific locations in the cell. We examine how minimal systems - in vitro reconstitutions of a particular cellular function with a minimal number of components - are designed, how they contribute to our understanding of cell polarity (i.e. protein accumulation), and how they complement in vivo investigations. We start by discussing the Min protein system from Escherichia coli, which represents a reaction-diffusion system with a well-established minimal system. This is followed by a discussion of MT-based directed transport for cell polarity markers as an example of a cytoskeleton-based mechanism. To conclude, we discuss, as an example, the interplay of reaction-diffusion and cytoskeleton-based mechanisms during polarity establishment in budding yeast.

Center Finding in E. coli and the Role of Mathematical Modeling: Past, Present and Future

We review the key role played by mathematical modeling in elucidating two center-finding patterning systems in Escherichia coli: midcell division positioning by the MinCDE system and DNA partitioning by the ParABS system. We focus particularly on how, despite much experimental effort, these systems were simply too complex to unravel by experiments alone, and instead required key injections of quantitative, mathematical thinking. We conclude the review by analyzing the frequency of modeling approaches in microbiology over time. We find that while such methods are increasing in popularity, they are still probably heavily under-utilized for optimal progress on complex biological questions.

Diffusion-dynamics laws in stochastic reaction networks

Many biological activities are induced by cellular chemical reactions of diffusing reactants. The dynamics of such systems can be captured by stochastic reaction networks. A recent numerical study has shown that diffusion can significantly enhance the fluctuations in gene regulatory networks. However, the universal relation between diffusion and stochastic system dynamics remains veiled. Within the approximation of reaction-diffusion master equation (RDME), we find general relation that the steady-state distribution in complex balanced networks is diffusion-independent. Here, complex balance is the nonequilibrium generalization of detailed balance. We also find that for a diffusion-included network with a Poisson-like steady-state distribution, the diffusion can be ignored at steady state. We then derive a necessary and sufficient condition for networks holding such steady-state distributions. Moreover, we show that for linear reaction networks the RDME reduces to the chemical master equation, which implies that the stochastic dynamics of networks is unaffected by diffusion at any arbitrary time. Our findings shed light on the fundamental question of when diffusion can be neglected, or (if nonnegligible) its effects on the stochastic dynamics of the reaction network.

Protein gradients on the nucleoid position the carbon-fixing organelles of cyanobacteria

Carboxysomes are protein-based bacterial organelles encapsulating key enzymes of the Calvin-Benson-Bassham cycle. Previous work has implicated a ParA-like protein (hereafter McdA) as important for spatially organizing carboxysomes along the longitudinal axis of the model cyanobacterium Synechococcus elongatus PCC 7942. Yet, how self-organization of McdA emerges and contributes to carboxysome positioning is unknown. Here, we identify a small protein, termed McdB that localizes to carboxysomes and drives emergent oscillatory patterning of McdA on the nucleoid. Our results demonstrate that McdB directly stimulates McdA ATPase activity and its release from DNA, driving carboxysome-dependent depletion of McdA locally on the nucleoid and promoting directed motion of carboxysomes towards increased concentrations of McdA. We propose that McdA and McdB are a previously unknown class of self-organizing proteins that utilize a Brownian-ratchet mechanism to position carboxysomes in cyanobacteria, rather than a cytoskeletal system. These results have broader implications for understanding spatial organization of protein mega-complexes and organelles in bacteria.

Cyanobacteria are tiny organisms that can harness the energy of the sun to power their cells. Many of the tools required for this complex photosynthetic process are packaged into small compartments inside the cell, the carboxysomes. In Synechococcus elongatus, a cyanobacterium that is shaped like a rod, the carboxysomes are positioned at regular intervals along the length of the cell. This ensures that, when the bacterium splits itself in half to reproduce, both daughter cells have the same number of carboxysomes.

Researchers know that, in S. elongatus, a protein called McdA can oscillate from one end of the cell to the other. This protein is responsible for the carboxysomes being in the right place, and some scientists believe that it helps to create an internal skeleton that anchors and drags the compartments into position.

Here, MacCready et al. propose another mechanism and, by combining various approaches, identify a new partner for McdA. This protein, called McdB, is present on the carboxysomes. McdB also binds to McdA, which itself attaches to the nucleoid – the region in the cell that contains the DNA. McdB forces McdA to release itself from DNA, causing the protein to reposition itself along the nucleoid. Because McdB attaches to McdA, the carboxysomes then follow suit, constantly seeking the highest concentrations of McdA bound to nearby DNA. Instead of relying on a cellular skeleton, these two proteins can organize themselves on their own using the nucleoid as a scaffold; in turn, they distribute carboxysomes evenly along the length of a cell.

Plants also obtain their energy from the sun via photosynthesis, but they do not carry carboxysomes. Scientists have tried to introduce these compartments inside plant cells, hoping that it could generate crops with higher yields. Knowing how carboxysomes are organized so they can be passed down from one generation to the next could be important for these experiments.

The Min Oscillator Defines Sites of Asymmetric Cell Division in Cyanobacteria during Stress Recovery

When resources are abundant, many rod-shaped bacteria reproduce through precise, symmetric divisions. However, realistic environments entail fluctuations between restrictive and permissive growth conditions. Here, we use time-lapse microscopy to study the division of the cyanobacterium Synechococcus elongatus as illumination intensity varies. We find that dim conditions produce elongated cells whose divisions follow a simple rule: cells shorter than ∼8 μm divide symmetrically, but above this length divisions become asymmetric, typically producing a short ∼3-μm daughter. We show that this division strategy is implemented by the Min system, which generates multi-node patterns and traveling waves in longer cells that favor the production of a short daughter. Mathematical modeling reveals that the feedback loops that create oscillatory Min patterns are needed to implement these generalized cell division rules. Thus, the Min system, which enforces symmetric divisions in short cells, acts to strongly suppress mid-cell divisions when S. elongatus cells are long.

Effects of geometry and topography on Min-protein dynamics

In the rod-shaped bacterium Escherichia coli, the center is selected by the Min-proteins as the site of cell division. To this end, the proteins periodically translocate between the two cell poles, where they suppress assembly of the cell division machinery. Ample evidence notably obtained from in vitro reconstitution experiments suggests that the oscillatory pattern results from self-organization of the proteins MinD and MinE in presence of a membrane. A mechanism built on cooperative membrane attachment of MinD and persistent MinD removal from the membrane induced by MinE has been shown to be able to reproduce the observed Min-protein patterns in rod-shaped E. coli and on flat supported lipid bilayers. Here, we report our results of a numerical investigation of patterns generated by this mechanism in various geoemtries. Notably, we consider the dynamics on membrane patches of different forms, on topographically structured lipid bilayers, and in closed geometries of various shapes. We find that all previously described patterns can be reproduced by the mechanism. However, it requires different parameter sets for reproducing the patterns in closed and in open geometries.

Regulation of Pom cluster dynamics in Myxococcus xanthus

Precise positioning of the cell division site is essential for the correct segregation of the genetic material into the two daughter cells. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a cluster on the chromosome that performs a biased random walk to midcell and positively regulates cell division there. PomZ, an ATPase, is necessary for tethering of the cluster to the nucleoid and regulates its movement towards midcell. It has remained unclear how the cluster dynamics change when the biochemical parameters, such as the attachment rates of PomZ dimers to the nucleoid and the cluster, the ATP hydrolysis rate of PomZ or the mobility of PomZ interacting with the nucleoid and cluster, are varied. To answer these questions, we investigate a one-dimensional model that includes the nucleoid, the Pom cluster and PomZ proteins. We find that a mechanism based on the diffusive PomZ fluxes on the nucleoid into the cluster can explain the latter's midnucleoid localization for a broad parameter range. Furthermore, there is an ATP hydrolysis rate that minimizes the time the cluster needs to reach midnucleoid. If the dynamics of PomZ on the nucleoid is slow relative to the cluster's velocity, we observe oscillatory cluster movements around midnucleoid. To understand midnucleoid localization, we developed a semi-analytical approach that dissects the net movement of the cluster into its components: the difference in PomZ fluxes into the cluster from either side, the force exerted by a single PomZ dimer on the cluster and the effective friction coefficient of the cluster. Importantly, we predict that the Pom cluster oscillates around midnucleoid if the diffusivity of PomZ on the nucleoid is reduced. A similar approach to that applied here may also prove useful for cargo localization in ParABS systems.

A Markovian Approach towards Bacterial Size Control and Homeostasis in Anomalous Growth Processes

Regardless of the progress achieved during recent years, the mechanisms coupling growth and division to attain cell size homeostasis in bacterial populations are still not well understood. In particular, there is a gap of knowledge about the mechanisms controlling anomalous growth events that are ubiquitous even in wild-type phenotypes. Thus, when cells exceed the doubling size the divisome dynamics sets a characteristic length scale that suggests a sizer property. Yet, it has been recently shown that the size at birth and the size increment still satisfy an adder-like correlation. Herein we propose a Markov chain model, that we complement with computational and experimental approaches, to clarify this issue. In this context, we show that classifying cells as a function of the characteristic size set by the divisome dynamics provides a compelling framework to understand size convergence, growth, and division at the large length scale, including the adaptation to, and rescue from, filamentation processes. Our results reveal the independence of size homeostasis on the division pattern of long cells and help to reconcile sizer concepts at the single cell level with an adder-like behavior at a population level.

Stochastic Simulation of Pattern Formation in Growing Tissue: A Multilevel Approach

We take up the challenge of designing realistic computational models of large interacting cell populations. The goal is essentially to bring Gillespie's celebrated stochastic methodology to the level of an interacting population of cells. Specifically, we are interested in how the gold standard of single-cell computational modeling, here taken to be spatial stochastic reaction-diffusion models, may be efficiently coupled with a similar approach at the cell population level. Concretely, we target a recently proposed set of pathways for pattern formation involving Notch-Delta signaling mechanisms. These involve cell-to-cell communication as mediated both via direct membrane contact sites and via cellular protrusions. We explain how to simulate the process in growing tissue using a multilevel approach and we discuss implications for future development of the associated computational methods.

The effect of cell geometry on polarization in budding yeast

The localization (or polarization) of proteins on the membrane during the mating of budding yeast (Saccharomyces cerevisiae) is an important model system for understanding simple pattern formation within cells. While there are many existing mathematical models of polarization, for both budding and mating, there are still many aspects of this process that are not well understood. In this paper we set out to elucidate the effect that the geometry of the cell can have on the dynamics of certain models of polarization. Specifically, we look at several spatial stochastic models of Cdc42 polarization that have been adapted from published models, on a variety of tip-shaped geometries, to replicate the shape change that occurs during the growth of the mating projection. We show here that there is a complex interplay between the dynamics of polarization and the shape of the cell. Our results show that while models of polarization can generate a stable polarization cap, its localization at the tip of mating projections is unstable, with the polarization cap drifting away from the tip of the projection in a geometry dependent manner. We also compare predictions from our computational results to experiments that observe cells with projections of varying lengths, and track the stability of the polarization cap. Lastly, we examine one model of actin polarization and show that it is unlikely, at least for the models studied here, that actin dynamics and vesicle traffic are able to overcome this effect of geometry.

Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach

We develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuatinghydrodynamics (FHD). For hydrodynamicsystems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poissonfluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamicfluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. By comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.

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.

Fundamental principles in bacterial physiology-history, recent progress, and the future with focus on cell size control: a review

Bacterial physiology is a branch of biology that aims to understand overarching principles of cellular reproduction. Many important issues in bacterial physiology are inherently quantitative, and major contributors to the field have often brought together tools and ways of thinking from multiple disciplines. This article presents a comprehensive overview of major ideas and approaches developed since the early 20th century for anyone who is interested in the fundamental problems in bacterial physiology. This article is divided into two parts. In the first part (sections 1-3), we review the first 'golden era' of bacterial physiology from the 1940s to early 1970s and provide a complete list of major references from that period. In the second part (sections 4-7), we explain how the pioneering work from the first golden era has influenced various rediscoveries of general quantitative principles and significant further development in modern bacterial physiology. Specifically, section 4 presents the history and current progress of the 'adder' principle of cell size homeostasis. Section 5 discusses the implications of coarse-graining the cellular protein composition, and how the coarse-grained proteome 'sectors' re-balance under different growth conditions. Section 6 focuses on physiological invariants, and explains how they are the key to understanding the coordination between growth and the cell cycle underlying cell size control in steady-state growth. Section 7 overviews how the temporal organization of all the internal processes enables balanced growth. In the final section 8, we conclude by discussing the remaining challenges for the future in the field.

MinE conformational switching confers robustness on self-organized Min protein patterns

Many fundamental cellular processes are spatially regulated by self-organized protein patterns, which are often based on nucleotide-binding proteins that switch their nucleotide state upon interaction with a second, activating protein. For reliable function, these protein patterns must be robust against parameter changes, although the basis for such robustness is generally elusive. Here we take a combined theoretical and experimental approach to the Escherichia coli Min system, a paradigmatic system for protein self-organization. By mathematical modeling and in vitro reconstitution of mutant proteins, we demonstrate that the robustness of pattern formation is dramatically enhanced by an interlinked functional switching of both proteins, rather than one. Such interlinked functional switching could be a generic means of obtaining robustness in biological pattern-forming systems.

Protein patterning is vital for many fundamental cellular processes. This raises two intriguing questions: Can such intrinsically complex processes be reduced to certain core principles and, if so, what roles do the molecular details play in individual systems? A prototypical example for protein patterning is the bacterial Min system, in which self-organized pole-to-pole oscillations of MinCDE proteins guide the cell division machinery to midcell. These oscillations are based on cycling of the ATPase MinD and its activating protein MinE between the membrane and the cytoplasm. Recent biochemical evidence suggests that MinE undergoes a reversible, MinD-dependent conformational switch from a latent to a reactive state. However, the functional relevance of this switch for the Min network and pattern formation remains unclear. By combining mathematical modeling and in vitro reconstitution of mutant proteins, we dissect the two aspects of MinE’s switch, persistent membrane binding and a change in MinE’s affinity for MinD. Our study shows that the MinD-dependent change in MinE’s binding affinity for MinD is essential for patterns to emerge over a broad and physiological range of protein concentrations. Mechanistically, our results suggest that conformational switching of an ATPase-activating protein can lead to the spatial separation of its distinct functional states and thereby confer robustness on an intracellular protein network with vital roles in bacterial cell division.

Size Laws and Division Ring Dynamics in Filamentous Escherichia coli cells

Our understanding of bacterial cell size control is based mainly on stress-free growth conditions in the laboratory [1-10]. In the real world, however, bacteria are routinely faced with stresses that produce long filamentous cell morphologies [11-28]. Escherichia coli is observed to filament in response to DNA damage [22-25], antibiotic treatment [11-14, 28], host immune systems [15, 16], temperature [17], starvation [20], and more [18, 19, 21], conditions which are relevant to clinical settings and food preservation [26]. This shape plasticity is considered a survival strategy [27]. Size control in this regime remains largely unexplored. Here we report that E. coli cells use a dynamic size ruler to determine division locations combined with an adder-like mechanism to trigger divisions. As filamentous cells increase in size due to growth, or decrease in size due to divisions, its multiple Fts division rings abruptly reorganize to remain one characteristic cell length away from the cell pole and two such length units away from each other. These rules can be explained by spatiotemporal oscillations of Min proteins. Upon removal of filamentation stress, the cells undergo a sequence of division events, randomly at one of the possible division sites, on average after the time required to grow one characteristic cell size. These results indicate that E. coli cells continuously keep track of absolute length to control size, suggest a wider relevance for the adder principle beyond the control of normally sized cells, and provide a new perspective on the function of the Fts and Min systems.

Mechanistic insights of the Min oscillator via cell-free reconstitution and imaging

The MinD and MinE proteins of Escherichia coli self-organize into a standing-wave oscillator on the membrane to help align division at mid-cell. When unleashed from cellular confines, MinD and MinE form a spectrum of patterns on artificial bilayers-static amoebas, traveling waves, traveling mushrooms, and bursts with standing-wave dynamics. We recently focused our cell-free studies on bursts because their dynamics recapitulate many features of Min oscillation observed in vivo. The data unveiled a patterning mechanism largely governed by MinE regulation of MinD interaction with membrane. We proposed that the MinD to MinE ratio on the membrane acts as a toggle switch between MinE-stimulated recruitment and release of MinD from the membrane. In this review, we summarize cell-free data on the Min system and expand upon a molecular mechanism that provides a biochemical explanation as to how these two 'simple' proteins can form the remarkable spectrum of patterns.

Computationally Efficient Modelling of Stochastic Spatio-Temporal Dynamics in Biomolecular Networks

Measurement techniques in biology are now able to provide data on the trajectories of multiple individual molecules simultaneously, motivating the development of techniques for the stochastic spatio-temporal modelling of biomolecular networks. However, standard approaches based on solving stochastic reaction-diffusion equations are computationally intractable for large-scale networks. We present a novel method for modeling stochastic and spatial dynamics in biomolecular networks using a simple form of the Langevin equation with noisy kinetic constants. Spatial heterogeneity in molecular interactions is decoupled into a set of compartments, where the distribution of molecules in each compartment is idealised as being uniform. The reactions in the network are then modelled by Langevin equations with correcting terms, that account for differences between spatially uniform and spatially non-uniform distributions, and that can be readily estimated from available experimental data. The accuracy and extreme computational efficiency of the approach is demonstrated on a model of the epidermal growth factor receptor network in the human mammary epithelial cell.

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.

An application of the Krylov-FSP-SSA method to parameter fitting with maximum likelihood

Monte Carlo methods such as the stochastic simulation algorithm (SSA) have traditionally been employed in gene regulation problems. However, there has been increasing interest to directly obtain the probability distribution of the molecules involved by solving the chemical master equation (CME). This requires addressing the curse of dimensionality that is inherent in most gene regulation problems. The finite state projection (FSP) seeks to address the challenge and there have been variants that further reduce the size of the projection or that accelerate the resulting matrix exponential. The Krylov-FSP-SSA variant has proved numerically efficient by combining, on one hand, the SSA to adaptively drive the FSP, and on the other hand, adaptive Krylov techniques to evaluate the matrix exponential. Here we apply this Krylov-FSP-SSA to a mutual inhibitory gene network synthetically engineered in Saccharomyces cerevisiae, in which bimodality arises. We show numerically that the approach can efficiently approximate the transient probability distribution, and this has important implications for parameter fitting, where the CME has to be solved for many different parameter sets. The fitting scheme amounts to an optimization problem of finding the parameter set so that the transient probability distributions fit the observations with maximum likelihood. We compare five optimization schemes for this difficult problem, thereby providing further insights into this approach of parameter estimation that is often applied to models in systems biology where there is a need to calibrate free parameters.

The PomXYZ Proteins Self-Organize on the Bacterial Nucleoid to Stimulate Cell Division

Cell division site positioning is precisely regulated to generate correctly sized and shaped daughters. We uncover the strategy used by the social bacterium Myxococcus xanthus to position the FtsZ cytokinetic ring at midcell. PomX, PomY, and the nucleoid-binding ParA/MinD ATPase PomZ self-assemble forming a large nucleoid-associated complex that localizes at the division site before FtsZ to directly guide and stimulate division. PomXYZ localization is generated through self-organized biased random motion on the nucleoid toward midcell and constrained motion at midcell. Experiments and theory show that PomXYZ motion is produced by diffusive PomZ fluxes on the nucleoid into the complex. Flux differences scale with the intracellular asymmetry of the complex and are converted into a local PomZ concentration gradient across the complex with translocation toward the higher PomZ concentration. At midcell, fluxes equalize resulting in constrained motion. Flux-based mechanisms may represent a general paradigm for positioning of macromolecular structures in bacteria.

Accurate reaction-diffusion operator splitting on tetrahedral meshes for parallel stochastic molecular simulations

Spatial stochastic molecular simulations in biology are limited by the intense computation required to track molecules in space either in a discrete time or discrete space framework, which has led to the development of parallel methods that can take advantage of the power of modern supercomputers in recent years. We systematically test suggested components of stochastic reaction-diffusion operator splitting in the literature and discuss their effects on accuracy. We introduce an operator splitting implementation for irregular meshes that enhances accuracy with minimal performance cost. We test a range of models in small-scale MPI simulations from simple diffusion models to realistic biological models and find that multi-dimensional geometry partitioning is an important consideration for optimum performance. We demonstrate performance gains of 1-3 orders of magnitude in the parallel implementation, with peak performance strongly dependent on model specification.

E. coli Cell Cycle Machinery

Cytokinesis in E. coli is organized by a cytoskeletal element designated the Z ring. The Z ring is formed at midcell by the coalescence of FtsZ filaments tethered to the membrane by interaction of FtsZ's conserved C-terminal peptide (CCTP) with two membrane-associated proteins, FtsA and ZipA. Although interaction between an FtsZ monomer and either of these proteins is of low affinity, high affinity is achieved through avidity - polymerization linked CCTPs interacting with the membrane tethers. The placement of the Z ring at midcell is ensured by antagonists of FtsZ polymerization that are positioned within the cell and target FtsZ filaments through the CCTP. The placement of the ring is reinforced by a protein network that extends from the terminus (Ter) region of the chromosome to the Z ring. Once the Z ring is established, additional proteins are recruited through interaction with FtsA, to form the divisome. The assembled divisome is then activated by FtsN to carry out septal peptidoglycan synthesis, with a dynamic Z ring serving as a guide for septum formation. As the septum forms, the cell wall is split by spatially regulated hydrolases and the outer membrane invaginates in step with the aid of a transenvelope complex to yield progeny cells.

Stochastic Simulation Service: Bridging the Gap between the Computational Expert and the Biologist

We present StochSS: Stochastic Simulation as a Service, an integrated development environment for modeling and simulation of both deterministic and discrete stochastic biochemical systems in up to three dimensions. An easy to use graphical user interface enables researchers to quickly develop and simulate a biological model on a desktop or laptop, which can then be expanded to incorporate increasing levels of complexity. StochSS features state-of-the-art simulation engines. As the demand for computational power increases, StochSS can seamlessly scale computing resources in the cloud. In addition, StochSS can be deployed as a multi-user software environment where collaborators share computational resources and exchange models via a public model repository. We demonstrate the capabilities and ease of use of StochSS with an example of model development and simulation at increasing levels of complexity.

Stochastic Simulation Service: Bridging the Gap between the Computational Expert and the Biologist

We present StochSS: Stochastic Simulation as a Service, an integrated development environment for modeling and simulation of both deterministic and discrete stochastic biochemical systems in up to three dimensions. An easy to use graphical user interface enables researchers to quickly develop and simulate a biological model on a desktop or laptop, which can then be expanded to incorporate increasing levels of complexity. StochSS features state-of-the-art simulation engines. As the demand for computational power increases, StochSS can seamlessly scale computing resources in the cloud. In addition, StochSS can be deployed as a multi-user software environment where collaborators share computational resources and exchange models via a public model repository. We demonstrate the capabilities and ease of use of StochSS with an example of model development and simulation at increasing levels of complexity.

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.