Highly canalized MinD transfer and MinE sequestration explain the origin of robust MinCDE-protein dynamics

Parallelization of Particle-Based Reaction-Diffusion Simulations Using MPI

Particle-based reaction-diffusion offers a high-resolution alternative to the continuum reaction-diffusion approach, capturing the volume-excluding nature of molecules undergoing stochastic dynamics. This is essential for simulating self-assembly into higher-order structures like filaments, lattices, or macromolecular complexes. Applications of self-assembly are ubiquitous in chemistry, biology, and materials science, but these higher-resolution methods increase computational cost. Here, we present a parallel implementation of the particle-based NERDSS software using the message passing interface (MPI), achieving close to linear scaling for up to 96 processors. By using a spatial decomposition of the system across processors, our approach extends to very large simulation volumes. The scalability of parallel NERDSS is evaluated for reversible reactions and several examples of higher-order self-assembly in 3D and 2D, with all test cases producing accurate solutions. Parallel efficiency depends on the system size, timescales, and reaction network, showing optimal scaling for smaller assemblies with slower timescales. We provide parallel NERDSS code open-source, supporting development and extension to other particle-based reaction-diffusion software.

Parallelization of particle-based reaction-diffusion simulations using MPI

Particle-based reaction-diffusion models offer a high-resolution alternative to the continuum reaction-diffusion approach, capturing the discrete and volume-excluding nature of molecules undergoing stochastic dynamics. These methods are thus uniquely capable of simulating explicit self-assembly of particles into higher-order structures like filaments, spherical cages, or heterogeneous macromolecular complexes, which are ubiquitous across living systems and in materials design. The disadvantage of these high-resolution methods is their increased computational cost. Here we present a parallel implementation of the particle-based NERDSS software using the Message Passing Interface (MPI) and spatial domain decomposition, achieving close to linear scaling for up to 96 processors in the largest simulation systems. The scalability of parallel NERDSS is evaluated for bimolecular reactions in 3D and 2D, for self-assembly of trimeric and hexameric complexes, and for protein lattice assembly from 3D to 2D, with all parallel test cases producing accurate solutions. We demonstrate how parallel efficiency depends on the system size, the reaction network, and the limiting timescales of the system, showing optimal scaling only for smaller assemblies with slower timescales. The formation of very large assemblies represents a challenge in evaluating reaction updates across processors, and here we restrict assembly sizes to below the spatial decomposition size. We provide the parallel NERDSS code open source, with detailed documentation for developers and extension to other particle-based reaction-diffusion software.

Plasmid partitioning driven by collective migration of ParA between nucleoid lobes

The ParABS system is crucial for the faithful segregation and inheritance of many bacterial chromosomes and low-copy-number plasmids. However, despite extensive research, the spatiotemporal dynamics of the ATPase ParA and its connection to the dynamics and positioning of the ParB-coated cargo have remained unclear. In this study, we utilize high-throughput imaging, quantitative data analysis, and computational modeling to explore the in vivo dynamics of ParA and its interaction with ParB-coated plasmids and the nucleoid. As previously observed, we find that F-plasmid ParA undergoes collective migrations ("flips") between cell halves multiple times per cell cycle. We reveal that a constricting nucleoid is required for these migrations and that they are triggered by a plasmid crossing into the cell half with greater ParA. Using simulations, we show that these dynamics can be explained by the combination of nucleoid constriction and cooperative ParA binding to the DNA, in line with the behavior of other ParA proteins. We further show that these ParA flips act to equally partition plasmids between the two lobes of the constricted nucleoid and are therefore important for plasmid stability, especially in fast growth conditions for which the nucleoid constricts early in the cell cycle. Overall, our work identifies a second mode of action of the ParABS system and deepens our understanding of how this important segregation system functions.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.

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.

Redundancy and the role of protein copy numbers in the cell polarization machinery of budding yeast

How can a self-organized cellular function evolve, adapt to perturbations, and acquire new sub-functions? To make progress in answering these basic questions of evolutionary cell biology, we analyze, as a concrete example, the cell polarity machinery of Saccharomyces cerevisiae. This cellular module exhibits an intriguing resilience: it remains operational under genetic perturbations and recovers quickly and reproducibly from the deletion of one of its key components. Using a combination of modeling, conceptual theory, and experiments, we propose that multiple, redundant self-organization mechanisms coexist within the protein network underlying cell polarization and are responsible for the module's resilience and adaptability. Based on our mechanistic understanding of polarity establishment, we hypothesize that scaffold proteins, by introducing new connections in the existing network, can increase the redundancy of mechanisms and thus increase the evolvability of other network components. Moreover, our work gives a perspective on how a complex, redundant cellular module might have evolved from a more rudimental ancestral form.

Kinetic principles of ParA2-ATP cycling guide dynamic subcellular localizations in Vibrio cholerae

Dynamic protein gradients are exploited for the spatial organization and segregation of replicated chromosomes. However, mechanisms of protein gradient formation and how that spatially organizes chromosomes remain poorly understood. Here, we have determined the kinetic principles of subcellular localizations of ParA2 ATPase, an essential spatial regulator of chromosome 2 segregation in the multichromosome bacterium, Vibrio cholerae. We found that ParA2 gradients self-organize in V. cholerae cells into dynamic pole-to-pole oscillations. We examined the ParA2 ATPase cycle and ParA2 interactions with ParB2 and DNA. In vitro, ParA2-ATP dimers undergo a rate-limiting conformational switch, catalysed by DNA to achieve DNA-binding competence. This active ParA2 state loads onto DNA cooperatively as higher order oligomers. Our results indicate that the midcell localization of ParB2-parS2 complexes stimulate ATP hydrolysis and ParA2 release from the nucleoid, generating an asymmetric ParA2 gradient with maximal concentration toward the poles. This rapid dissociation coupled with slow nucleotide exchange and conformational switch provides for a temporal lag that allows the redistribution of ParA2 to the opposite pole for nucleoid reattachment. Based on our data, we propose a 'Tug-of-war' model that uses dynamic oscillations of ParA2 to spatially regulate symmetric segregation and positioning of bacterial chromosomes.

Directing Min protein patterns with advective bulk flow

The Min proteins constitute the best-studied model system for pattern formation in cell biology. We theoretically predict and experimentally show that the propagation direction of in vitro Min protein patterns can be controlled by a hydrodynamic flow of the bulk solution. We find downstream propagation of Min wave patterns for low MinE:MinD concentration ratios, upstream propagation for large ratios, but multistability of both propagation directions in between. Whereas downstream propagation can be described by a minimal model that disregards MinE conformational switching, upstream propagation can be reproduced by a reduced switch model, where increased MinD bulk concentrations on the upstream side promote protein attachment. Our study demonstrates that a differential flow, where bulk flow advects protein concentrations in the bulk, but not on the surface, can control surface-pattern propagation. This suggests that flow can be used to probe molecular features and to constrain mathematical models for pattern-forming systems.

Cell-Size Space Regulates the Behavior of Confined Polymers: From Nano- and Micromaterials Science to Biology

Polymer micromaterials in a liquid or gel phase covered with a surfactant membrane are widely used materials in pharmaceuticals, cosmetics, and foods. In particular, cell-sized micromaterials of biopolymer solutions covered with a lipid membrane have been studied as artificial cells to understand cells from a physicochemical perspective. The characteristics and phase transitions of polymers confined to a microscopic space often differ from those in bulk systems. The effect that causes this difference is referred to as the cell-size space effect (CSE), but the specific physicochemical factors remain unclear. This study introduces the analysis of CSE on molecular diffusion, nanostructure transition, and phase separation and presents their main factors, i.e., short- and long-range interactions with the membrane surface and small volume (finite element nature). This serves as a guide for determining the dominant factors of CSE. Furthermore, we also introduce other factors of CSE such as spatial closure and the relationships among space size, the characteristic length of periodicity, the structure size, and many others produced by biomolecular assemblies through the analysis of protein reaction-diffusion systems and biochemical reactions.

Bridging scales in a multiscale pattern-forming system

Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.

Mode selection mechanism in traveling and standing waves revealed by Min wave reconstituted in artificial cells

Reaction-diffusion coupling (RDc) generates spatiotemporal patterns, including two dynamic wave modes: traveling and standing waves. Although mode selection plays a substantial role in the spatiotemporal organization of living cell molecules, the mechanism for selecting each wave mode remains elusive. Here, we investigated a wave mode selection mechanism using Min waves reconstituted in artificial cells, emerged by the RDc of MinD and MinE. Our experiments and theoretical analysis revealed that the balance of membrane binding and dissociation from the membrane of MinD determines the mode selection of the Min wave. We successfully demonstrated that the transition of the wave modes can be regulated by controlling this balance and found hysteresis characteristics in the wave mode transition. These findings highlight a previously unidentified role of the balance between activators and inhibitors as a determinant of the mode selection of waves by RDc and depict an unexplored mechanism in intracellular spatiotemporal pattern formations.

Unraveling the physiochemical nature of colloidal motion waves among silver colloids

Traveling waves are common in biological and synthetic systems, including the recent discovery that silver (Ag) colloids form traveling motion waves in H₂O₂ and under light. Here, we show that this colloidal motion wave is a heterogeneous excitable system. The Ag colloids generate traveling chemical waves via reaction-diffusion, and either self-propel through self-diffusiophoresis ("ballistic waves") or are advected by diffusio-osmotic flows from gradients of neutral molecules ("swarming waves"). Key results include the experimental observation of traveling waves of OH^- with pH-sensitive fluorescent dyes and a Rogers-McCulloch model that qualitatively and quantitatively reproduces the key features of colloidal waves. These results are a step forward in elucidating the Ag-H₂O₂-light oscillatory system at individual and collective levels. In addition, they pave the way for using colloidal waves either as a platform for studying nonlinear phenomena, or as a tool for colloidal transport and for information transmission in microrobot ensembles.

Pattern forming systems coupling linear bulk diffusion to dynamically active membranes or cells

Some analytical and numerical results are presented for pattern formation properties associated with novel types of reaction-diffusion (RD) systems that involve the coupling of bulk diffusion in the interior of a multi-dimensional spatial domain to nonlinear processes that occur either on the domain boundary or within localized compartments that are confined within the domain. The class of bulk-membrane system considered herein is derived from an asymptotic analysis in the limit of small thickness of a thin domain that surrounds the bulk medium. When the bulk domain is a two-dimensional disk, a weakly nonlinear analysis is used to characterize Turing and Hopf bifurcations that can arise from the linearization around a radially symmetric, but spatially non-uniform, steady-state of the bulk-membrane system. In a singularly perturbed limit, the existence and linear stability of localized membrane-bound spike patterns is analysed for a Gierer-Meinhardt activator-inhibitor model that includes bulk coupling. Finally, the emergence of collective intracellular oscillations is studied for a class of PDE-ODE bulk-cell model in a bounded two-dimensional domain that contains spatially localized, but dynamically active, circular cells that are coupled through a linear bulk diffusion field. Applications of such coupled bulk-membrane or bulk-cell systems to some biological systems are outlined, and some open problems in this area are discussed. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Bulk-surface coupling identifies the mechanistic connection between Min-protein patterns in vivo and in vitro

Self-organisation of Min proteins is responsible for the spatial control of cell division in Escherichia coli, and has been studied both in vivo and in vitro. Intriguingly, the protein patterns observed in these settings differ qualitatively and quantitatively. This puzzling dichotomy has not been resolved to date. Using reconstituted proteins in laterally wide microchambers with a well-controlled height, we experimentally show that the Min protein dynamics on the membrane crucially depend on the micro chamber height due to bulk concentration gradients orthogonal to the membrane. A theoretical analysis shows that in vitro patterns at low microchamber height are driven by the same lateral oscillation mode as pole-to-pole oscillations in vivo. At larger microchamber height, additional vertical oscillation modes set in, marking the transition to a qualitatively different in vitro regime. Our work reveals the qualitatively different mechanisms of mass transport that govern Min protein-patterns for different bulk heights and thus shows that Min patterns in cells are governed by a different mechanism than those in vitro.

Speed limits of protein assembly with reversible membrane localization

Protein assembly is often studied in a three-dimensional solution, but a significant fraction of binding events involve proteins that can reversibly bind and diffuse along a two-dimensional surface. In a recent study, we quantified how proteins can exploit the reduced dimensionality of the membrane to trigger complex formation. Here, we derive a single expression for the characteristic timescale of this multi-step assembly process, where the change in dimensionality renders rates and concentrations effectively time-dependent. We find that proteins can accelerate dimer formation due to an increase in relative concentration, driving more frequent collisions, which often win out over slow-downs due to diffusion. Our model contains two protein populations that dimerize with one another and use a distinct site to bind membrane lipids, creating a complex reaction network. However, by identifying two major rate-limiting pathways to reach an equilibrium steady-state, we derive an excellent approximation for the mean first passage time when lipids are in abundant supply. Our theory highlights how the "sticking rate" or effective adsorption coefficient of the membrane is central in controlling timescales. We also derive a corrected localization rate to quantify how the geometry of the system and diffusion can reduce rates of membrane localization. We validate and test our results using kinetic and particle-based reaction-diffusion simulations. Our results establish how the speed of key assembly steps can shift by orders-of-magnitude when membrane localization is possible, which is critical to understanding mechanisms used in cells.

Exploring dissipative sources of non-Markovian biochemical reaction systems

Many biological processes including important intracellular processes are governed by biochemical reaction networks. Usually, these reaction systems operate far from thermodynamic equilibrium, implying free-energy dissipation. On the other hand, single reaction events happen often in a memory manner, leading to non-Markovian kinetics. A question then arises: how do we calculate free-energy dissipation (defined as the entropy production rate) in this physically real case? We derive an analytical formula for calculating the energy consumption of a general reaction system with molecular memory characterized by nonexponential waiting-time distributions. It shows that this dissipation is composed of two parts: one from broken detailed balance of an equivalent Markovian system with the same topology and substrates, and the other from the direction-time dependence of waiting-time distributions. But, if the system is in a detailed balance and the waiting-time distribution is direction-time independent, there is no energy dissipation even in the non-Markovian case. These general results provide insights into the physical mechanisms underlying nonequilibrium processes. A continuous-time random-walk model and a generalized model of stochastic gene expression are chosen to clearly show dissipative sources and the relationship between energy dissipation and molecular memory.

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.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

Molecular interactions and their predictive roles in cell pole determination in bacteria

Bacterial cell cycle is divided into well-coordinated phases; chromosome duplication and segregation, cell elongation, septum formation, and cytokinesis. The temporal separation of these phases depends upon the growth rates and doubling time in different bacteria. The entire process of cell division starts with the assembly of divisome complex at mid-cell position followed by constriction of the cell wall and septum formation. In the mapping of mid-cell position for septum formation, the gradient of oscillating Min proteins across the poles plays a pivotal role in several bacteria genus. The cues in the cell that defines the poles and plane of cell division are not fully characterized in cocci. Recent studies have shed some lights on molecular interactions at the poles and the underlying mechanisms involved in pole determination in non-cocci. In this review, we have brought forth recent findings on these aspects together, which would suggest a model to explain the mechanisms of pole determination in rod shaped bacteria and could be extrapolated as a working model in cocci.

Probing Biomolecular Interactions by a Pattern-Forming Peptide-Conjugate Sensor

As a key mechanism underpinning many biological processes, protein self-organization has been extensively studied. However, the potential to apply the distinctive, nonlinear biochemical properties of such self-organizing systems to biotechnological problems such as the facile detection and characterization of biomolecular interactions has not yet been explored. Here, we describe an in vitro assay in a 96-well plate format that harnesses the emergent behavior of the Escherichia coli Min system to provide a readout of biomolecular interactions. Crucial for the development of our approach is a minimal MinE-derived peptide that stimulates MinD ATPase activity only when dimerized. We found that this behavior could be induced via any pair of foreign, mutually binding molecular entities fused to the minimal MinE peptide. The resulting MinD ATPase activity and the spatiotemporal nature of the produced protein patterns quantitatively correlate with the affinity of the fused binding partners, thereby enabling a highly sensitive assay for biomolecular interactions. Our assay thus provides a unique means of quantitatively visualizing biomolecular interactions and may prove useful for the assessment of domain interactions within protein libraries and for the facile investigation of potential inhibitors of protein–protein interactions.

Turing Patterning in Stratified Domains

Reaction-diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal-mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction-diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction-diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

Exponential volume dependence of entropy-current fluctuations at first-order phase transitions in chemical reaction networks

In chemical reaction networks, bistability can only occur far from equilibrium. It is associated with a first-order phase transition where the control parameter is the thermodynamic force. At the bistable point, the entropy production is known to be discontinuous with respect to the thermodynamic force. We show that the fluctuations of the entropy production have an exponential volume-dependence when the system is bistable. At the phase transition, the exponential prefactor is the height of the effective potential barrier between the two fixed-points. Our results obtained for Schlögl's model can be extended to any chemical network.

Flow Induced Symmetry Breaking in a Conceptual Polarity Model

Important cellular processes, such as cell motility and cell division, are coordinated by cell polarity, which is determined by the non-uniform distribution of certain proteins. Such protein patterns form via an interplay of protein reactions and protein transport. Since Turing’s seminal work, the formation of protein patterns resulting from the interplay between reactions and diffusive transport has been widely studied. Over the last few years, increasing evidence shows that also advective transport, resulting from cytosolic and cortical flows, is present in many cells. However, it remains unclear how and whether these flows contribute to protein-pattern formation. To address this question, we use a minimal model that conserves the total protein mass to characterize the effects of cytosolic flow on pattern formation. Combining a linear stability analysis with numerical simulations, we find that membrane-bound protein patterns propagate against the direction of cytoplasmic flow with a speed that is maximal for intermediate flow speed. We show that the mechanism underlying this pattern propagation relies on a higher protein influx on the upstream side of the pattern compared to the downstream side. Furthermore, we find that cytosolic flow can change the membrane pattern qualitatively from a peak pattern to a mesa pattern. Finally, our study shows that a non-uniform flow profile can induce pattern formation by triggering a regional lateral instability.

Stochastic thermodynamics of chemical reactions coupled to finite reservoirs: A case study for the Brusselator

Biomolecular processes are typically modeled using chemical reaction networks coupled to infinitely large chemical reservoirs. A difference in chemical potential between these reservoirs can drive the system into a non-equilibrium steady-state (NESS). In reality, these processes take place in finite systems containing a finite number of molecules. In such systems, a NESS can be reached with the help of an externally driven pump for which we introduce a simple model. The crucial parameters are the pumping rate and the finite size of the chemical reservoir. We apply this model to a simple biochemical oscillator, the Brusselator, and quantify the performance using the number of coherent oscillations. As a surprising result, we find that higher precision can be achieved with finite-size reservoirs even though the corresponding current fluctuations are larger than in the ideal infinite case.

Conformational equilibrium of MinE regulates the allowable concentration ranges of a protein wave for cell division

The Min system for determining the cell division position at the center in bacteria has a unique character that uses a protein wave (Min wave) that emerges from its components (MinD and MinE). The Min wave emerges under the coupling of chemical reactions and molecular diffusions of MinDE and appears when the concentrations of MinD and MinE are similar. However, the nanoscale mechanism to determine their concentration ranges has remained elusive. In this study, by using artificial cells as a mimic of cells, we showed that the dominant MinE conformations determined the allowable concentration ranges for the emergence of the Min wave. Furthermore, the deletion of the membrane-binding region of MinE indicated that the region was essential for limiting the concentration ranges to be narrower. These findings illustrate a parameter tuning mechanism underlying complex molecular systems at the nanoscale for spatiotemporal regulation in living cells and show a possibility that the regulation of the equilibrium among molecular conformations can work as a switch for cell division.

Self-organization Assay for Min Proteins of Escherichia coli in Micro-droplets Covered with Lipids

The Min system determines the cell division plane of bacteria. As a cue of spatiotemporal regulation, the Min system uses wave propagation of MinD protein (Min wave). Therefore, the reconstitution of the Min wave in cell-sized closed space will lead to the creation of artificial cells capable of cell division. The Min waves emerge via coupling between the reactions among MinD, MinE, and ATP and the differences in diffusion rate on the cell membrane and in the cytoplasm. Because Min waves appear only under the balanced condition of the reaction-diffusion coupling, special attentions are needed towards several technical points for the reconstitution of Min waves in artificial cells. This protocol describes a technical method for stably generating Min waves in artificial cells.

Local Self-Enhancement of MinD Membrane Binding in Min Protein Pattern Formation

The proteins MinD, MinE and MinC are constitutive for the spatiotemporal organization of cell division in Escherichia coli, in particular, for positioning the division machinery at mid-cell. To achieve this function, the ATPase MinD and the ATPase-activating protein MinE undergo coordinated pole-to-pole oscillations and have thus become a paradigm for protein pattern formation in biology. The exact molecular mechanisms enabling MinDE self-organization, and particularly the role of cooperativity in the membrane binding of MinD, thought to be a key requirement, have remained poorly understood. However, for bottom-up synthetic biology aiming at a de novo design of key cellular features, elucidating these mechanisms is of great relevance. By combining in vitro reconstitution with rationally guided mutagenesis of MinD, we found that when bound to membranes, MinD displays new interfaces for multimerization, which are distinct from the canonical MinD dimerization site. We propose that these additional transient interactions contribute to the local self-enhancement of MinD at the membrane, while their relative lability maintains the structural plasticity required for MinDE wave propagation. This could represent a powerful structural regulation feature not reported so far for self-organizing proteins.

Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov-Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in mass-conserved cell polarity models.

Pattern localization to a domain edge

The formation of protein patterns inside cells is generically described by reaction-diffusion models. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g., membrane-bound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud site in budding yeast and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can "sense" the edge of the template. A mechanism for how this edge sensing may work remains elusive. Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a two-component mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems and for the design of synthetic pattern forming systems.

Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans

In the Caenorhabditis elegans zygote, PAR protein patterns, driven by mutual anatagonism, determine the anterior-posterior axis and facilitate the redistribution of proteins for the first cell division. Yet, the factors that determine the selection of the polarity axis remain unclear. We present a reaction-diffusion model in realistic cell geometry, based on biomolecular reactions and accounting for the coupling between membrane and cytosolic dynamics. We find that the kinetics of the phosphorylation-dephosphorylation cycle of PARs and the diffusive protein fluxes from the cytosol towards the membrane are crucial for the robust selection of the anterior-posterior axis for polarisation. The local ratio of membrane surface to cytosolic volume is the main geometric cue that initiates pattern formation, while the choice of the long-axis for polarisation is largely determined by the length of the aPAR-pPAR interface, and mediated by processes that minimise the diffusive fluxes of PAR proteins between cytosol and membrane.

Design of biochemical pattern forming systems from minimal motifs

Although molecular self-organization and pattern formation are key features of life, only very few pattern-forming biochemical systems have been identified that can be reconstituted and studied in vitro under defined conditions. A systematic understanding of the underlying mechanisms is often hampered by multiple interactions, conformational flexibility and other complex features of the pattern forming proteins. Because of its compositional simplicity of only two proteins and a membrane, the MinDE system from Escherichia coli has in the past years been invaluable for deciphering the mechanisms of spatiotemporal self-organization in cells. Here, we explored the potential of reducing the complexity of this system even further, by identifying key functional motifs in the effector MinE that could be used to design pattern formation from scratch. In a combined approach of experiment and quantitative modeling, we show that starting from a minimal MinE-MinD interaction motif, pattern formation can be obtained by adding either dimerization or membrane-binding motifs. Moreover, we show that the pathways underlying pattern formation are recruitment-driven cytosolic cycling of MinE and recombination of membrane-bound MinE, and that these differ in their in vivo phenomenology.

The E. coli MinCDE system in the regulation of protein patterns and gradients

Molecular self-organziation, also regarded as pattern formation, is crucial for the correct distribution of cellular content. The processes leading to spatiotemporal patterns often involve a multitude of molecules interacting in complex networks, so that only very few cellular pattern-forming systems can be regarded as well understood. Due to its compositional simplicity, the Escherichia coli MinCDE system has, thus, become a paradigm for protein pattern formation. This biological reaction diffusion system spatiotemporally positions the division machinery in E. coli and is closely related to ParA-type ATPases involved in most aspects of spatiotemporal organization in bacteria. The ATPase MinD and the ATPase-activating protein MinE self-organize on the membrane as a reaction matrix. In vivo, these two proteins typically oscillate from pole-to-pole, while in vitro they can form a variety of distinct patterns. MinC is a passenger protein supposedly operating as a downstream cue of the system, coupling it to the division machinery. The MinCDE system has helped to extract not only the principles underlying intracellular patterns, but also how they are shaped by cellular boundaries. Moreover, it serves as a model to investigate how patterns can confer information through specific and non-specific interactions with other molecules. Here, we review how the three Min proteins self-organize to form patterns, their response to geometric boundaries, and how these patterns can in turn induce patterns of other molecules, focusing primarily on experimental approaches and developments.

Protein Recruitment through Indirect Mechanochemical Interactions

Some of the key proteins essential for important cellular processes are capable of recruiting other proteins from the cytosol to phospholipid membranes. The physical basis for this cooperativity of binding is, surprisingly, still unclear. Here, we suggest a general feedback mechanism that explains cooperativity through mechanochemical coupling mediated by the mechanical properties of phospholipid membranes. Our theory predicts that protein recruitment, and therefore also protein pattern formation, involves membrane deformation and is strongly affected by membrane composition.

Interlinked GTPase cascades provide a motif for both robust switches and oscillators

GTPases regulate a wide range of cellular processes, such as intracellular vesicular transport, signal transduction and protein translation. These hydrolase enzymes operate as biochemical switches by toggling between an active guanosine triphosphate (GTP)-bound state and an inactive guanosine diphosphate (GDP)-bound state. We compare two network motifs, a single-species switch and an interlinked cascade that consists of two species coupled through positive and negative feedback loops. We find that interlinked cascades are closer to the ideal all-or-none switch and are more robust against fluctuating signals. While the single-species switch can only achieve bistability, interlinked cascades can be converted into oscillators by tuning the cofactor concentrations, which catalyse the activity of the cascade. These regimes can only be achieved with sufficient chemical driving provided by GTP hydrolysis. In this study, we present a thermodynamically consistent model that can achieve bistability and oscillations with the same feedback motif.

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.

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’.

Reverse and forward engineering of protein pattern formation

Living systems employ protein pattern formation to regulate important life processes in space and time. Although pattern-forming protein networks have been identified in various prokaryotes and eukaryotes, their systematic experimental characterization is challenging owing to the complex environment of living cells. In turn, cell-free systems are ideally suited for this goal, as they offer defined molecular environments that can be precisely controlled and manipulated. Towards revealing the molecular basis of protein pattern formation, we outline two complementary approaches: the biochemical reverse engineering of reconstituted networks and the de novo design, or forward engineering, of artificial self-organizing systems. We first illustrate the reverse engineering approach by the example of the Escherichia coli Min system, a model system for protein self-organization based on the reversible and energy-dependent interaction of the ATPase MinD and its activating protein MinE with a lipid membrane. By reconstituting MinE mutants impaired in ATPase stimulation, we demonstrate how large-scale Min protein patterns are modulated by MinE activity and concentration. We then provide a perspective on the de novo design of self-organizing protein networks. Tightly integrated reverse and forward engineering approaches will be key to understanding and engineering the intriguing phenomenon of protein pattern formation.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'.

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.

Stationary Patterns in a Two-Protein Reaction-Diffusion System

Patterns formed by reaction-diffusion mechanisms are crucial for the development or sustenance of most organisms in nature. Patterns include dynamic waves, but are more often found as static distributions, such as animal skin patterns. Yet, a simplistic biological model system to reproduce and quantitatively investigate static reaction-diffusion patterns has been missing so far. Here, we demonstrate that the Escherichia coli Min system, known for its oscillatory behavior between the cell poles, is under certain conditions capable of transitioning to quasi-stationary protein distributions on membranes closely resembling Turing patterns. We systematically titrated both proteins, MinD and MinE, and found that removing all purification tags and linkers from the N-terminus of MinE was critical for static patterns to occur. At small bulk heights, dynamic patterns dominate, such as in rod-shaped microcompartments. We see implications of this work for studying pattern formation in general, but also for creating artificial gradients as downstream cues in synthetic biology applications.

Plasmonic Nanosensors Reveal a Height Dependence of MinDE Protein Oscillations on Membrane Features

Single-particle plasmon spectroscopy has become a standard technique to detect and quantify the presence of unlabeled macromolecules. Here, we extend this method to determine their exact distance from the plasmon sensors with sub-nanometer resolution by systematically varying the sensing range into the surrounding by adjusting the size of the plasmonic nanoparticles. We improved current single-particle plasmon spectroscopy to record continuously for hours the scattering spectra of thousands of nanoparticles of different sizes simultaneously with 1.8 s time resolution. We apply this technique to study the interaction dynamics of bacterial Min proteins with supported lipid membranes of different composition. Our experiments reveal a surprisingly flexible operating mode of the Min proteins: In the presence of cardiolipin and membrane curvature induced by nanoparticles, the protein oscillation occurs on top of a stationary MinD patch. Our results reveal the need to consider membrane composition and local curvature as important parameters to quantitatively understand the Min protein system and could be extrapolated to other macromolecular systems. Our label-free method is generally easily implementable and well suited to measure distances of interacting biological macromolecules.

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.

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.

In Vitro Reconstitution of Self-Organizing Protein Patterns on Supported Lipid Bilayers

Many aspects of the fundamental spatiotemporal organization of cells are governed by reaction-diffusion type systems. In vitro reconstitution of such systems allows for detailed studies of their underlying mechanisms which would not be feasible in vivo. Here, we provide a protocol for the in vitro reconstitution of the MinCDE system of Escherichia coli, which positions the cell division septum in the cell middle. The assay is designed to supply only the components necessary for self-organization, namely a membrane, the two proteins MinD and MinE and energy in the form of ATP. We therefore fabricate an open reaction chamber on a coverslip, on which a supported lipid bilayer is formed. The open design of the chamber allows for optimal preparation of the lipid bilayer and controlled manipulation of the bulk content. The two proteins, MinD and MinE, as well as ATP, are then added into the bulk volume above the membrane. Imaging is possible by many optical microscopies, as the design supports confocal, wide-field and TIRF microscopy alike. In a variation of the protocol, the lipid bilayer is formed on a patterned support, on cell-shaped PDMS microstructures, instead of glass. Lowering the bulk solution to the rim of these compartments encloses the reaction in a smaller compartment and provides boundaries that allow mimicking of in vivo oscillatory behavior. Taken together, we describe protocols to reconstitute the MinCDE system both with and without spatial confinement, allowing researchers to precisely control all aspects influencing pattern formation, such as concentration ranges and addition of other factors or proteins, and to systematically increase system complexity in a relatively simple experimental setup.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

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.