Stochastic model for Soj relocation dynamics in Bacillus subtilis

Influence of global gene regulatory networks on single cell heterogeneity of green fluorescent protein production in Bacillus subtilis

BACKGROUND

Gram-positive bacterium Bacillus subtilis has been extensively studied as a microbial cell factory for high-level producing a wide range of interesting products. Green fluorescent protein (GFP) is commonly used as a marker for determining the strength of a given promoter or for the subcellular localization of a fusion protein. However, the inherent heterogeneity of GFP expression among individual cells that can arise from global regulation differences in the expression host, has not yet been systematically assessed. B. subtilis strains with single mutation(s) in the two major transcriptional regulators CcpA and/or CodY were earlier found to improve overall heterologous protein production levels. Here, we investigate the dynamic production performance of GFP in the reporter strains with chromosomally integrated Physpank-sfGFP(Sp).

RESULTS

The mutation R214C in the DNA-binding domain of CodY effectively enhances GFP production at the population level relative to two other strains, i.e. wildtype (WT) and CcpAT19S. During the late stationary phase, the high- and low-level GFP-producing cells coexist in the WT population, while the CodYR214C population at the single-cell level shows higher phenotypic homogeneity of fluorescence signals.

CONCLUSION

Expression of GFP is prominently heterogeneous in the WT B. subtilis cells, and this phenotypic heterogeneity can be significantly reduced by CodYR214C mutation. The rates of production heterogeneity show a high correlation to the overall GFP yields. Moreover, the toolkit of flow cytometry and fluorescence microscopy that can achieve real-time profiles of GFP production performance in various strains may facilitate the further use of B. subtilis as a cell factory.

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.

Physical model of protein cluster positioning in growing bacteria

Chemotaxic receptors in bacteria form clusters at cell poles and also laterally, and this clustering plays an important role in signal transduction. These clusters were found to be periodically arranged on the surface of the bacterium Escherichia coli, independent of any known positioning mechanism. In this work we extend a model based on diffusion and aggregation to more realistic geometries and present a means based on "bursty" protein production to distinguish spontaneous positioning from an independently existing positioning mechanism. We also consider the case of isotropic cellular growth and characterize the degree of order arising spontaneously. Our model could also be relevant for other examples of periodically positioned protein clusters in bacteria.

Effects of molecular noise on bistable protein distributions in rod-shaped bacteria

The distributions of many proteins in rod-shaped bacteria are far from homogeneous. Often they accumulate at the cell poles or in the cell centre. At the same time, the copy number of proteins in a single cell is relatively small making the patterns noisy. To explore limits to protein patterns due to molecular noise, we studied a generic mechanism for spontaneous polar protein assemblies in rod-shaped bacteria, which are based on cooperative binding of proteins to the cytoplasmic membrane. For mono-polar assemblies, we find that the switching time between the two poles increases exponentially with the cell length and with the protein number. This feature could be beneficial to organelle maintenance in ageing bacteria.

Defining the role of ATP hydrolysis in mitotic segregation of bacterial plasmids

Hydrolysis of ATP by partition ATPases, although considered a key step in the segregation mechanism that assures stable inheritance of plasmids, is intrinsically very weak. The cognate centromere-binding protein (CBP), together with DNA, stimulates the ATPase to hydrolyse ATP and to undertake the relocation that incites plasmid movement, apparently confirming the need for hydrolysis in partition. However, ATP-binding alone changes ATPase conformation and properties, making it difficult to rigorously distinguish the substrate and cofactor roles of ATP in vivo. We had shown that mutation of arginines R36 and R42 in the F plasmid CBP, SopB, reduces stimulation of SopA-catalyzed ATP hydrolysis without changing SopA-SopB affinity, suggesting the role of hydrolysis could be analyzed using SopA with normal conformational responses to ATP. Here, we report that strongly reducing SopB-mediated stimulation of ATP hydrolysis results in only slight destabilization of mini-F, although the instability, as well as an increase in mini-F clustering, is proportional to the ATPase deficit. Unexpectedly, the reduced stimulation also increased the frequency of SopA relocation over the nucleoid. The increase was due to drastic shortening of the period spent by SopA at nucleoid ends; average speed of migration per se was unchanged. Reduced ATP hydrolysis was also associated with pronounced deviations in positioning of mini-F, though time-averaged positions changed only modestly. Thus, by specifically targeting SopB-stimulated ATP hydrolysis our study reveals that even at levels of ATPase which reduce the efficiency of splitting clusters and the constancy of plasmid positioning, SopB still activates SopA mobility and plasmid positioning, and sustains near wild type levels of plasmid stability.

Genes enabling bacteria to survive and thrive in challenging environments are very often found on small, non-essential DNA molecules called plasmids. Many plasmids are naturally present in the cell in very few copies and so risk being lost from one of the daughter cells upon division. These plasmids elaborate a partition system, functionally similar to mitosis, which assures their faithful inheritance. Chromosomes also generally possess such systems. We know that partition systems involve two proteins, that one (B) stimulates the other (A) to hydrolyse ATP, and that upon binding to A protein ATP confers properties needed for partition. ATP's double action, as hydrolysis substrate and cofactor, complicates definition of its role in the mechanism. The novelty of our approach lies in use of B protein mutants that do not stimulate hydrolysis. Our results reveal that the major function of ATP hydrolysis is not to displace plasmid molecules to their positions in each cell half, as generally thought, but to split initial sibling plasmid pairs and prevent their reforming. This study is the first to dissect ATPase activity in vivo using normal A-protein ATPase, and so opens a new avenue to exploration of the mechanisms that ensure plasmid and chromosome inheritance.

Perspective: Stochastic algorithms for chemical kinetics

We outline our perspective on stochastic chemical kinetics, paying particular attention to numerical simulation algorithms. We first focus on dilute, well-mixed systems, whose description using ordinary differential equations has served as the basis for traditional chemical kinetics for the past 150 years. For such systems, we review the physical and mathematical rationale for a discrete-stochastic approach, and for the approximations that need to be made in order to regain the traditional continuous-deterministic description. We next take note of some of the more promising strategies for dealing stochastically with stiff systems, rare events, and sensitivity analysis. Finally, we review some recent efforts to adapt and extend the discrete-stochastic approach to systems that are not well-mixed. In that currently developing area, we focus mainly on the strategy of subdividing the system into well-mixed subvolumes, and then simulating diffusional transfers of reactant molecules between adjacent subvolumes together with chemical reactions inside the subvolumes.

How mathematical modelling elucidates signalling in Bacillus subtilis

Appropriate stimulus perception, signal processing and transduction ensure optimal adaptation of bacteria to environmental challenges. In the Gram-positive model bacterium Bacillus subtilis signalling networks and molecular interactions therein are well-studied, making this species a suitable candidate for the application of mathematical modelling. Here, we review systems biology approaches, focusing on chemotaxis, sporulation, σ(B) -dependent general stress response and competence. Processes like chemotaxis and Z-ring assembly depend critically on the subcellular localization of proteins. Environmental response strategies, including sporulation and competence, are characterized by phenotypic heterogeneity in isogenic cultures. The examples of mathematical modelling also include investigations that have demonstrated how operon structure and signalling dynamics are intricately interwoven to establish optimal responses. Our review illustrates that these interdisciplinary approaches offer new insights into the response of B. subtilis to environmental challenges. These case studies reveal modelling as a tool to increase the understanding of complex systems, to help formulating hypotheses and to guide the design of more directed experiments that test predictions.

Pushing and pulling in prokaryotic DNA segregation

In prokaryotes, DNA can be segregated by three different types of cytoskeletal filaments. The best-understood type of partitioning (par) locus encodes an actin homolog called ParM, which forms dynamically unstable filaments that push plasmids apart in a process reminiscent of mitosis. However, the most common type of par locus, which is present on many plasmids and most bacterial chromosomes, encodes a P loop ATPase (ParA) that distributes plasmids equidistant from one another on the bacterial nucleoid. A third type of par locus encodes a tubulin homolog (TubZ) that forms cytoskeletal filaments that move rapidly with treadmill dynamics.

Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length

Self-organization of proteins in space and time is of crucial importance for the functioning of cellular processes. Often, this organization takes place in the presence of strong random fluctuations due to the small number of molecules involved. We report on stochastic switching of the Min-protein distributions between the two cell halves in short Escherichia coli cells. A computational model provides strong evidence that the macroscopic switching is rooted in microscopic noise on the molecular scale. In longer bacteria, the switching turns into regular oscillations that are required for positioning of the division plane. As the pattern becomes more regular, cell-to-cell variability also lessens, indicating cell length-dependent regulation of Min-protein activity.

The diffusive finite state projection algorithm for efficient simulation of the stochastic reaction-diffusion master equation

We have developed a computational framework for accurate and efficient simulation of stochastic spatially inhomogeneous biochemical systems. The new computational method employs a fractional step hybrid strategy. A novel formulation of the finite state projection (FSP) method, called the diffusive FSP method, is introduced for the efficient and accurate simulation of diffusive transport. Reactions are handled by the stochastic simulation algorithm.

An experimentalist's guide to computational modelling of the Min system

Spatio-temporal oscillations of the Min proteins are essential for selecting the cell division site in Escherichia coli. These oscillations are a key example of a biological phenomenon that can only be understood on a systems level rather than on the level of its individual components. Here, we review the key concepts that mathematical modelling has added to our understanding of the Min system. While several different mechanisms have been proposed, in all cases the oscillations emerge from a dynamic instability of a uniform protein distribution. To generate this instability, however, the various mechanisms rely on different features of Min protein interactions and transport. We critically evaluate these mechanisms in light of recent experimental evidence. We also review the effects of fluctuations caused by low cellular concentration of Min proteins, and describe how stochastic effects may potentially influence Min protein dynamics.

Deletion of the parA (soj) homologue in Pseudomonas aeruginosa causes ParB instability and affects growth rate, chromosome segregation, and motility

The parA and parB genes of Pseudomonas aeruginosa are located approximately 8 kb anticlockwise from oriC. ParA is a cytosolic protein present at a level of around 600 molecules per cell in exponential phase, but the level drops about fivefold in stationary phase. Overproduction of full-length ParA or the N-terminal 85 amino acids severely inhibits growth of P. aeruginosa and P. putida. Both inactivation of parA and overexpression of parA in trans in P. aeruginosa also lead to accumulation of anucleate cells and changes in motility. Inactivation of parA also increases the turnover rate (degradation) of ParB. This may provide a mechanism for controlling the level of ParB in response to the growth rate and expression of the parAB operon.

Computational methods for diffusion-influenced biochemical reactions

MOTIVATION

We compare stochastic computational methods accounting for space and discrete nature of reactants in biochemical systems. Implementations based on Brownian dynamics (BD) and the reaction-diffusion master equation are applied to a simplified gene expression model and to a signal transduction pathway in Escherichia coli.

RESULTS

In the regime where the number of molecules is small and reactions are diffusion-limited predicted fluctuations in the product number vary between the methods, while the average is the same. Computational approaches at the level of the reaction-diffusion master equation compute the same fluctuations as the reference result obtained from the particle-based method if the size of the sub-volumes is comparable to the diameter of reactants. Using numerical simulations of reversible binding of a pair of molecules we argue that the disagreement in predicted fluctuations is due to different modeling of inter-arrival times between reaction events. Simulations for a more complex biological study show that the different approaches lead to different results due to modeling issues. Finally, we present the physical assumptions behind the mesoscopic models for the reaction-diffusion systems.

AVAILABILITY

Input files for the simulations and the source code of GMP can be found under the following address: http://www.cwi.nl/projects/sic/bioinformatics2007/

Unusual features of the cell cycle in mycobacteria: Polar-restricted growth and the snapping-model of cell division

Cell division patterns in mycobacteria have been examined in order to further our understanding of how these important organisms grow in the apparent absence of key systems required for the growth of rod-shaped bacteria. Analysis of the distribution of cell lengths in the population during different phases of growth showed that the modal cell length decreases during later phases of growth, declining from 3.5 to 2.5 microm for Mycobacterium bovis BCG cells sampled in log phase and stationary phase, respectively. The population also became more homogeneous, as indicated by the proportion of cells in the most common class increasing from 15% to 28%. Similar patterns were observed for Mycobacterium smegmatis and Mycobacterium tuberculosis. Consistent with other actinomycetes, and in contrast to most rod-shaped bacteria, the deposition of newly synthesised peptidoglycan in mycobacteria is restricted to the poles of the cell, as evidenced by staining with fluorescently labelled vancomycin. A "V-form" of bacteria was observed in cultures at all stages of growth, but the proportion decreased in older cultures. The V-shape appears to be a result of the uneven splitting of the exterior cell envelope at the new septum; this exposes the new peptidoglycan which is illustrated by spots of fluorescent vancomycin staining associated with the exterior side of the "V", and supports the 'snapping division model'. The restriction of growth to the poles of the cell differs from the pattern observed in other rod-shaped bacteria, in which the cell poles are inert and lateral growth occurs by deposition of peptidoglycan along the body of the cylinder. The mechanisms that maintain the shape of mycobacteria and that identify the mid-point for cell division remain to be determined.

Dynamic filaments of the bacterial cytoskeleton

Bacterial cells contain a variety of structural filamentous proteins necessary for the spatial regulation of cell shape, cell division, and chromosome segregation, analogous to the eukaryotic cytoskeletal proteins. The molecular mechanisms by which these proteins function are beginning to be revealed, and these proteins show numerous three-dimensional structural features and biochemical properties similar to those of eukaryotic actin and tubulin, revealing their evolutionary relationship. Recent technological advances have illuminated links between cell division and chromosome segregation, suggesting a higher complexity and organization of the bacterial cell than was previously thought.

Systems biology and its impact on anti-infective drug development

Systems biology offers the potential for more effective selection of novel targets for anti-infective drugs. In contrast to conventional reductionist biology, a systems approach allows targets to be viewed in a wider context of the entire physiology of the cell, with the potential to identify key susceptible nodes and to predict synergistic effects of blocking multiple pathways. In addition to the holistic perspective provided by systems biology, the emphasis on quantitative analysis is likely to add further rigour to the process of target selection. Systems biology also offers the potential to incorporate different levels of information into the selection process. Consideration of data from microbial population biology may be important in the context of predicting future drug-resistance profiles associated with targeting a particular pathway, for example. This chapter provides an overview of major themes in the developing field of systems biology, summarising the core technologies and the strategies used to translate datasets into useful quantitative models capable of predicting complex biological behaviour.

Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli

MOTIVATION

Many biochemical networks involve reactions localized on the cell membrane. This can give rise to spatial gradients of the concentration of cytosolic species. Moreover, the number of membrane molecules can be small and stochastic effects can become relevant. Pathways usually consist of a complex interaction network and are characterized by a large set of parameters. The inclusion of spatial and stochastic effects is a major challenge in developing quantitative and dynamic models of pathways.

RESULTS

We have developed a particle-based spatial stochastic method (GMP) to simulate biochemical networks in space, including fluctuations from the diffusion of particles and reactions. Gradients emerging from membrane reactions can be resolved. As case studies for the GMP method we used a simple gene expression system and the phosphoenolpyruvate:glucose phosphotransferase system pathway.

AVAILABILITY

The source code for the GMP method is available at http://www.science.uva.nl/research/scs/CellMath/GMP.

Noise-induced Min phenotypes in E. coli

The spatiotemporal oscillations of the Escherichia coli proteins MinD and MinE direct cell division to the region between the chromosomes. Several quantitative models of the Min system have been suggested before, but no one of them accounts for the behavior of all documented mutant phenotypes. We analyzed the stochastic reaction-diffusion kinetics of the Min proteins for several E. coli mutants and compared the results to the corresponding deterministic mean-field description. We found that wild-type (wt) and filamentous (ftsZ⁻) cells are well characterized by the mean-field model, but that a stochastic model is necessary to account for several of the characteristics of the spherical (rodA⁻) and phospathedylethanolamide-deficient (PE⁻) phenotypes. For spherical cells, the mean-field model is bistable, and the system can get trapped in a non-oscillatory state. However, when the intrinsic noise is considered, only the experimentally observed oscillatory behavior remains. The stochastic model also reproduces the change in oscillation directions observed in the spherical phenotype and the occasional gliding of the MinD region along the inner membrane. For the PE⁻ mutant, the stochastic model explains the appearance of randomly localized and dense MinD clusters as a nucleation phenomenon, in which the stochastic kinetics at low copy number causes local discharges of the high MinD^ATP to MinD^ADP potential. We find that a simple five-reaction model of the Min system can explain all documented Min phenotypes, if stochastic kinetics and three-dimensional diffusion are accounted for. Our results emphasize that local copy number fluctuation may result in phenotypic differences although the total number of molecules of the relevant species is high.

Many molecules inside a living cell do not have time to diffuse through the whole cell in-between reactions. Furthermore, the chemical reactions are random and discrete events. In this study, the authors study an example in which these aspects of intracellular chemistry need to be considered when we try to understand how a biological system works.

The authors have investigated the spatial oscillation patterns that are displayed by the Min system of Escherichia coli. In wild-type E. coli, the Min proteins oscillate back and forth between the cell poles to help the bacterium find its middle before cell division. The authors used computer simulations to explain why the oscillation patterns change the way they do in different mutants of E. coli. They find that two of the mutant phenotypes can only be explained if one considers the randomness and discreteness of chemical reactions in addition to the spatial characteristics of the process. Particularly interesting is the phospathedylethanolamide-deficient phenotype, in which large dense clusters of MinD protein appear for some time at random locations on the membrane. The authors believe that this phenotype is due to a nucleation phenomenon, in which the stochastic kinetics at low copy number is amplified to macroscopic proportions.

Subcellular Positioning of F Plasmid Mediated by Dynamic Localization of SopA and SopB

SopA, SopB proteins and the cis-acting sopC DNA region of F plasmid are essential for partitioning of the plasmid, ensuring proper subcellular positioning of the plasmid DNA molecules. We have analyzed by immunofluorescence microscopy the subcellular localization of SopA and SopB. The majority of SopB molecules formed foci, which localized frequently with F plasmid DNA molecules. The foci increased in number in proportion to the cell length. Interestingly, beside the foci formation, SopB formed a spiral structure that was dependent on SopA, which also formed a spiral structure, independent of the presence of SopB, and these two structures partially overlapped. On the basis of these results and previous biochemical studies together with our simulations, we propose a theoretical model named "the reaction-diffusion partitioning model", using reaction-diffusion equations that explain the dynamic subcellular localization of SopA and SopB proteins and the subcellular positioning of F plasmid. We hypothesized that sister copies of plasmid DNA compete with each other for sites at which SopB multimer is at the optimum concentration. The plasmid incompatibility mediated by the Sop system might be explained clearly by this hypothesis.

A stochastic model of Min oscillations inEscherichia coliand Min protein segregation during cell division

The Min system in Escherichia coli directs division to the centre of the cell through pole-to-pole oscillations of the MinCDE proteins. We present a one-dimensional stochastic model of these oscillations which incorporates membrane polymerization of MinD into linear chains. This model reproduces much of the observed phenomenology of the Min system, including pole-to-pole oscillations of the Min proteins. We then apply this model to investigate the Min system during cell division. Oscillations continue initially unaffected by the closing septum, before cutting off rapidly. The fractions of Min proteins in the daughter cells vary widely, from 50%-50% up to 85%-15% of the total from the parent cell, suggesting that there may be another mechanism for regulating these levels in vivo.