Stochastic simulation of biological reactions, and its applications for studying actin polymerization

Molecular events in biological cells occur in local subregions, where the molecules tend to be small in number. The cytoskeleton, which is important for both the structural changes of cells and their functions, is also a countable entity because of its long fibrous shape. To simulate the local environment using a computer, stochastic simulations should be run. We herein report a new method of stochastic simulation based on random walk and reaction by the collision of all molecules. The microscopic reaction rate P(r) is calculated from the macroscopic rate constant k. The formula involves only local parameters embedded for each molecule. The results of the stochastic simulations of simple second-order, polymerization, Michaelis-Menten-type and other reactions agreed quite well with those of deterministic simulations when the number of molecules was sufficiently large. An analysis of the theory indicated a relationship between variance and the number of molecules in the system, and results of multiple stochastic simulation runs confirmed this relationship. We simulated Ca2(+) dynamics in a cell by inward flow from a point on the cell surface and the polymerization of G-actin forming F-actin. Our results showed that this theory and method can be used to simulate spatially inhomogeneous events.     

Steady state length distribution of F-actin under controlled fragmentation and mechanism of length redistribution following fragmentation

Under controlled levels of fragmentation by sonication in the presence of ATP, actin filaments reached different length distributions corresponding to different steady states of polymerization. When the sonication was interrupted, a spontaneous decrease in the number of filaments and corresponding increase in length was observed. The kinetics of the process of length redistribution subsequent to fragmentation was studied for the simpler case of F-actin filaments at equilibrium in the presence of ADP. Analysis of the data shows that a diffusion-like random walk mechanism of length redistribution quantitatively accounts for the observations much better than the previously proposed end-to-end reannealing of filaments. The involvement of this process in the kinetics of actin polymerization and in establishing the length distribution at steady state is discussed.     

The Effect of Tropomyosin Mutations on Actin-Tropomyosin Binding: In Search of Lost Time

The initial binding of tropomyosin onto actin filaments and then its polymerization into continuous cables on the filament surface must be precisely tuned to overall thin-filament structure, function, and performance. Low-affinity interaction of tropomyosin with actin has to be sufficiently strong to localize the tropomyosin on actin, yet not so tight that regulatory movement on filaments is curtailed. Likewise, head-to-tail association of tropomyosin molecules must be favorable enough to promote tropomyosin cable formation but not so tenacious that polymerization precedes filament binding. Arguably, little molecular detail on early tropomyosin binding steps has been revealed since Wegner’s seminal studies on filament assembly almost 40 years ago. Thus, interpretation of mutation-based actin-tropomyosin binding anomalies leading to cardiomyopathies cannot be described fully. In vitro, tropomyosin binding is masked by explosive tropomyosin polymerization once cable formation is initiated on actin filaments. In contrast, in silico analysis, characterizing molecular dynamics simulations of single wild-type and mutant tropomyosin molecules on F-actin, is not complicated by tropomyosin polymerization at all. In fact, molecular dynamics performed here demonstrates that a midpiece tropomyosin domain is essential for normal actin-tropomyosin interaction and that this interaction is strictly conserved in a number of tropomyosin mutant species. Elsewhere along these mutant molecules, twisting and bending corrupts the tropomyosin superhelices as they “lose their grip” on F-actin. We propose that residual interactions displayed by these mutant tropomyosin structures with actin mimic ones that occur in early stages of thin-filament generation, as if the mutants are recapitulating the assembly process but in reverse. We conclude therefore that an initial binding step in tropomyosin assembly onto actin involves interaction of the essential centrally located domain.     

Coarse-grained modeling of the actin filament derived from atomistic-scale simulations

A coarse-grained (CG) procedure that incorporates the information obtained from all-atom molecular dynamics (MD) simulations is presented and applied to actin filaments (F-actin). This procedure matches the averaged values and fluctuations of the effective internal coordinates that are used to define a CG model to the values extracted from atomistic MD simulations. The fluctuations of effective internal coordinates in a CG model are computed via normal-mode analysis (NMA), and the computed fluctuations are matched with the atomistic MD results in a self-consistent manner. Each actin monomer (G-actin) is coarse-grained into four sites, and each site corresponds to one of the subdomains of G-actin. The potential energy of a CG G-actin contains three bonds, two angles, and one dihedral angle; effective harmonic bonds are used to describe the intermonomer interactions in a CG F-actin. The persistence length of a CG F-actin was found to be sensitive to the cut-off distance of assigning intermonomer bonds. Effective harmonic bonds for a monomer with its third nearest neighboring monomers are found to be necessary to reproduce the values of persistence length obtained from all-atom MD simulations. Compared to the elastic network model, incorporating the information of internal coordinate fluctuations enhances the accuracy and robustness for a CG model to describe the shapes of low-frequency vibrational modes. Combining the fluctuation-matching CG procedure and NMA, the achievable time- and length scales of modeling actin filaments can be greatly enhanced. In particular, a method is described to compute the force-extension curve using the CG model developed in this work and NMA. It was found that F-actin is easily buckled under compressive deformation, and a writhing mode is developed as a result. In addition to the bending and twisting modes, this novel writhing mode of F-actin could also play important roles in the interactions of F-actin with actin-binding proteins and in the force-generation process via polymerization.     

Cooperative effects of cofilin (ADF) on actin structure suggest allosteric mechanism of cofilin function

Using site-specific fluorescence probes and cross-linking we demonstrated that cofilin (ADF), a key regulator of actin cellular dynamics, weakens longitudinal contacts in F-actin in a cooperative manner. Differential scanning calorimetry detected a dual nature of cofilin effects on F-actin conformation. At sub-stoichiometric cofilin to actin ratios, cofilin stabilized sterically and non-cooperatively protomers at the points of attachment, and destabilized allosterically and cooperatively protomers in the cofilin-free parts of F-actin. This destabilizing effect had a long range, with one cofilin molecule affecting more than 100 protomers, and concentration-dependent amplitude that reached maximum at about 1:2 molar ratio of cofilin to actin. In contrast to existing models, our results suggest an allosteric mechanism of actin depolymerization by cofilin. We propose that cofilin is less likely to sever actin filaments at the points of attachment as thought previously. Instead, due to its dual structural effect, spontaneous fragmentation occurs most likely in cofilin-free segments of filaments weakened allosterically by nearby cofilin molecules.     

Models for the length distributions of actin filaments: II. Polymerization and fragmentation by gelsolin acting together

In a previous paper, we studied elementary models for polymerization, depolymerization, and fragmentation of actin filaments (Edelstein-Keshet and Ermentrout, 1998, Bull. Math. Biol. 60, 449–475). When these processes act together, more complicated dynamics occur. We concentrate on a particular case study, using the actin-fragmenting protein gelsolin. A set of biological parameter values (drawn from the experimental literature) is used in computer simulations of the kinetics of gelsolin-mediated actin filament fragmentation.     

Real-time measurement of F-actin remodelling during exocytosis using Lifeact-EGFP transgenic animals

F-actin remodelling is essential for a wide variety of cell processes. It is important in exocytosis, where F-actin coats fusing exocytic granules. The purpose of these F-actin coats is unknown. They may be important in stabilizing the fused granules, they may play a contractile role and promote expulsion of granule content and finally may be important in endocytosis. To elucidate these functions of F-actin remodelling requires a reliable method to visualize F-actin dynamics in living cells. The recent development of Lifeact-EGFP transgenic animals offers such an opportunity. Here, we studied the characteristics of exocytosis in pancreatic acinar cells obtained from the Lifeact-EGFP transgenic mice. We show that the time-course of agonist-evoked exocytic events and the kinetics of each single exocytic event are the same for wild type and Lifeact-EGFP transgenic animals. We conclude that Lifeact-EGFP animals are a good model to study of exocytosis and reveal that F-actin coating is dependent on the de novo synthesis of F-actin and that development of actin polymerization occurs simultaneously in all regions of the granule. Our insights using the Lifeact-EGFP mice demonstrate that F-actin coating occurs after granule fusion and is a granule-wide event.     

Control of actin turnover by a salmonella invasion protein

Salmonella force their way into nonphagocytic host intestinal cells to initiate infection. Uptake is triggered by delivery into the target cell of bacterial effector proteins that stimulate cytoskeletal rearrangements and membrane ruffling. The Salmonella invasion protein A (SipA) effector is an actin binding protein that enhances uptake efficiency by promoting actin polymerization. SipA-bound actin filaments (F-actin) are also resistant to artificial disassembly in vitro. Using biochemical assays of actin dynamics and actin-based motility models, we demonstrate that SipA directly arrests cellular mechanisms of actin turnover. SipA inhibits ADF/cofilin-directed depolymerization both by preventing binding of ADF and cofilin and by displacing them from F-actin. SipA also protects F-actin from gelsolin-directed severing and reanneals gelsolin-severed F-actin fragments. These data suggest that SipA focuses host cytoskeletal reorganization by locally inhibiting both ADF/cofilin- and gelsolin-directed actin disassembly, while simultaneously stimulating pathogen-induced actin polymerization.     

Human disease mortality kinetics are explored through a chain model embodying principles of extreme value theory and competing risks.

The distributions for human disease-specific mortality exhibit two striking characteristics: survivorship curves that intersect near the longevity limit; and, the clustering of best-fitting Weibull shape parameter values into groups centered on integers. Correspondingly, we have hypothesized that the distribution intersections result from either competitive processes or population partitioning and the integral clustering in the shape parameter results from the occurrence of a small number of rare, rate-limiting events in disease progression. In this report we initiate a theoretical examination of these questions by exploring serial chain model dynamics and parameteric competing risks theory. The links in our chain models are composed of more than one bond, where the number of bonds in a link are denoted the link size and are the number of events necessary to break the link and, hence, the chain. We explored chains with all links of the same size or with segments of the chain composed of different size links (competition). Simulations showed that chain breakage dynamics depended on the weakest-link principle and followed kinetics of extreme-values which were very similar to human mortality kinetics. In particular, failure distributions for simple chains were Weibull-type extreme-value distributions with shape parameter values that were identifiable with the integral link size in the limit of infinite chain length. Furthermore, for chains composed of several segments of differing link size, the survival distributions for the various segments converged at a point in the S(t) tails indistinguishable from human data. This was also predicted by parameteric competing risks theory using Weibull underlying distributions. In both the competitive chain simulations and the parametric competing risks theory, however, the shape values for the intersecting distributions deviated from the integer values typical of human data. We conclude that rare events can be the source of integral shapes in human mortality, that convergence is a salient feature of multiple endpoints, but that pure competition may not be the best explanation for the exact type of convergence observable in human mortality. Finally, while the chain models were not motivated by any specific biological structures, interesting biological correlates to them may be useful in gerontological research.     

A mechanochemical model of actin filaments

In eukaryotic cells, actin filaments are involved in important processes such as motility, division, cell shape regulation, contractility, and mechanosensation. Actin filaments are polymerized chains of monomers, which themselves undergo a range of chemical events such as ATP hydrolysis, polymerization, and depolymerization. When forces are applied to F-actin, in addition to filament mechanical deformations, the applied force must also influence chemical events in the filament. We develop an intermediate-scale model of actin filaments that combines actin chemistry with filament-level deformations. The model is able to compute mechanical responses of F-actin during bending and stretching. The model also describes the interplay between ATP hydrolysis and filament deformations, including possible force-induced chemical state changes of actin monomers in the filament. The model can also be used to model the action of several actin-associated proteins, and for large-scale simulation of F-actin networks. All together, our model shows that mechanics and chemistry must be considered together to understand cytoskeletal dynamics in living cells.     

Nanoscale dynamics of actin filaments in the red blood cell membrane skeleton

Red blood cell (RBC) shape and deformability are supported by a planar network of short actin filament (F-actin) nodes (∼37 nm length, 15–18 subunits) interconnected by long spectrin strands at the inner surface of the plasma membrane. Spectrin-F-actin network structure underlies quantitative modeling of forces controlling RBC shape, membrane curvature, and deformation, yet the nanoscale organization and dynamics of the F-actin nodes in situ are not well understood. We examined F-actin distribution and dynamics in RBCs using fluorescent-phalloidin labeling of F-actin imaged by multiple microscopy modalities. Total internal reflection fluorescence and Zeiss Airyscan confocal microscopy demonstrate that F-actin is concentrated in multiple brightly stained F-actin foci ∼200–300 nm apart interspersed with dimmer F-actin staining regions. Single molecule stochastic optical reconstruction microscopy imaging of Alexa 647-phalloidin-labeled F-actin and computational analysis also indicates an irregular, nonrandom distribution of F-actin nodes. Treatment of RBCs with latrunculin A and cytochalasin D indicates that F-actin foci distribution depends on actin polymerization, while live cell imaging reveals dynamic local motions of F-actin foci, with lateral movements, appearance and disappearance. Regulation of F-actin node distribution and dynamics via actin assembly/disassembly pathways and/or via local extension and retraction of spectrin strands may provide a new mechanism to control spectrin-F-actin network connectivity, RBC shape, and membrane deformability.     

Brownian dynamics of interactions between glyceraldehyde-3-phosphate dehydrogenase (GAPDH) mutants and F-actin

Brownian dynamics simulations of computer models of GAPDH mutants interacting with F-actin emphasized the electrostatic nature of such interactions, and confirmed the importance of four previously identified lysine residues on the GAPDH structure in these interactions. Mutants were GAPDH models in which one or more of the previously identified lysines had been replaced with alanine. Simulations showed reduced binding of these mutants to F-actin compared to wild-type GAPDH. Binding was significantly reduced by mutating the four lysines; the specific electrostatic interaction energy of the quadruple mutant was -7.3 +/- 1.0 compared to -11.4 +/- 0.5 kcal/mol for the wild enzyme. The BD simulations also reaffirmed the importance of quaternary structure for GAPDH binding F-actin.     

Biomechanics of actin filaments: a computational multi-level study

The actin microfilament (F-actin) is a structural and functional component of the cell cytoskeleton. Notwithstanding the primary role it plays for the mechanics of the cell, the mechanical behaviour of F-actin is still not totally explored. In particular, the relationship between the mechanics of F-actin and its molecular architecture is not completely understood. In this study, the mechanical properties of F-actin were related to the molecular topology of its building monomers (G-actin) by employing a computational multi-level approach. F-actins with lengths up to 500 nm were modelled and characterized, using a combination of equilibrium molecular dynamics (MD) simulations and normal mode analysis (NMA). MD simulations were performed to analyze the molecular rearrangements of G-actin in physiological conditions; NMA was applied to compute the macroscopic properties of F-actin from its vibrational modes of motion. Results from this multi-level approach showed that bending stiffness, bending modulus and persistence length are independent from the length of F-actin. On the contrary, the orientations and motions of selected groups of residues of G-actin play a primary role in determining the filament flexibility. In conclusion, this study (i) demonstrated that a combined computational approach of MD and NMA allows to investigate the biomechanics of F-actin taking into account the molecular topology of the filament (i.e., the molecular conformations of G-actin) and (ii) that this can be done using only crystallographic G-actin, without the need of introducing experimental parameters nor of reducing the number of residues.     

Fragmentation is crucial for the steady-state dynamics of actin filaments

Despite the recognition that actin filaments are important for numerous cellular processes, and decades of investigation, the dynamics of in vitro actin filaments are still not completely understood. Here, we follow the time evolution of the length distribution of labeled actin reporter filaments in an unlabeled F-actin solution via fluorescence microscopy. Whereas treadmilling and diffusive length fluctuations cannot account for the observed dynamics, our results suggest that at low salt conditions, spontaneous fragmentation is crucial.     

Differences in internal dynamics of actin under different structural states detected by neutron scattering

F-actin, a helical polymer formed by polymerization of the monomers (G-actin), plays crucial roles in various aspects of cell motility. Flexibility of F-actin has been suggested to be important for such a variety of functions. Understanding the flexibility of F-actin requires characterization of a hierarchy of dynamical properties, from internal dynamics of the actin monomers through domain motions within the monomers and relative motions between the monomers within F-actin to large-scale motions of F-actin as a whole. As a first step toward this ultimate purpose, we carried out elastic incoherent neutron scattering experiments on powders of F-actin and G-actin hydrated with D₂O and characterized the internal dynamics of F-actin and G-actin. Well established techniques and analysis enabled the extraction of mean-square displacements and their temperature dependence in F-actin and in G-actin. An effective force constant analysis with a model consisting of three energy states showed that two dynamical transitions occur at ∼150 K and ∼245 K, the former of which corresponds to the onset of anharmonic motions and the latter of which couples with the transition of hydration water. It is shown that behavior of the mean-square displacements is different between G-actin and F-actin, such that G-actin is “softer” than F-actin. The differences in the internal dynamics are detected for the first time between the different structural states (the monomeric state and the polymerized state). The different behavior observed is ascribed to the differences in dynamical heterogeneity between F-actin and G-actin. Based on structural data, the assignment of the differences observed in the two samples to dynamics of specific loop regions involved in the polymerization of G-actin into F-actin is proposed.     

Structural modeling and molecular dynamics simulation of the actin filament

Actin is a major structural protein of the eukaryotic cytoskeleton and enables cell motility. Here, we present a model of the actin filament (F-actin) that not only incorporates the global structure of the recently published model by Oda et al. but also conserves internal stereochemistry. A comparison is made using molecular dynamics simulation of the model with other recent F-actin models. A number of structural determents such as the protomer propeller angle, the number of hydrogen bonds, and the structural variation among the protomers are analyzed. The MD comparison is found to reflect the evolution in quality of actin models over the last 6 years. In addition, simulations of the model are carried out in states with both ADP or ATP bound and local hydrogen-bonding differences characterized.     

A mathematical model of actin filament turnover for fitting FRAP data

A novel mathematical model of the actin dynamics in living cells under steady-state conditions has been developed for fluorescence recovery after photobleaching (FRAP) experiments. As opposed to other FRAP fitting models, which use the average lifetime of actins in filaments and the actin turnover rate as fitting parameters, our model operates with unbiased actin association/dissociation rate constants and accounts for the filament length. The mathematical formalism is based on a system of stochastic differential equations. The derived equations were validated on synthetic theoretical data generated by a stochastic simulation algorithm adapted for the simulation of FRAP experiments. Consistent with experimental findings, the results of this work showed that (1) fluorescence recovery is a function of the average filament length, (2) the F-actin turnover and the FRAP are accelerated in the presence of actin nucleating proteins, (3) the FRAP curves may exhibit both a linear and non-linear behaviour depending on the parameters of actin polymerisation, and (4) our model resulted in more accurate parameter estimations of actin dynamics as compared with other FRAP fitting models. Additionally, we provide a computational tool that integrates the model and that can be used for interpretation of FRAP data on actin cytoskeleton.     

Modeling Forest Mortality Caused by Drought Stress: Implications for Climate Change

Climate change is expected to affect forest landscape dynamics in many ways, but it is possible that the most important direct impact of climate change will be drought stress. We combined data from weather stations and forest inventory plots (FIA) across the upper Great Lakes region (USA) to study the relationship between measures of drought stress and mortality for four drought sensitivity species groups using a weight-of-evidence approach. For all groups, the model that predicted mortality as a function of mean drought length had the greatest plausibility. Model tests confirmed that the models for all groups except the most drought tolerant had predictive value. We assumed that no relationship exists between drought and mortality for the drought-tolerant group. We used these empirical models to develop a drought extension for the forest landscape disturbance and succession model LANDIS-II, and applied the model in Oconto county, Wisconsin (USA) to assess the influence of drought on forest dynamics relative to other factors such as stand-replacing disturbance and site characteristics. The simulations showed that drought stress does affect species composition and total biomass, but effects on age classes, spatial pattern, and productivity were insignificant. We conclude that (for the upper Midwest) (1) a drought-induced tree mortality signal can be detected using FIA data, (2) tree species respond primarily to the length of drought events rather than their severity, (3) the differences in drought tolerance of tree species can be quantified, (4) future increases in drought can potentially change forest composition, and (5) drought is a potentially important factor to include in forest dynamics simulations because it affects forest composition and carbon storage.     

The breakdown of genomic ancestry blocks in hybrid lineages given a finite number of recombination sites

When a lineage originates from hybridization genomic blocks of contiguous ancestry from different ancestors are fragmented through genetic recombination. The resulting blocks are delineated by so called junctions, which accumulate with every generation that passes. Modeling the accumulation of ancestry block junctions can elucidate processes and timeframes of genomic admixture. Previous models have not addressed ancestry block dynamics for chromosomes that consist of a finite number of recombination sites. However, genomic data typically consist of informative markers that are interspersed with fragments for which no ancestry information is available. Hence, repeated recombination events may occur between markers, effectively removing existing junctions. Here, we present an analytical treatment of the dynamics of the mean number of junctions over time, taking into account the number of recombination sites per chromosome, population size, genetic map length, and the frequency of the ancestral species in the founding hybrid swarm. We describe the expected number of junctions using equidistant molecular markers and estimate the number of junctions using random markers. This extended theory of junctions thus reflects properties of empirical data and can serve to study the genomic patterns following admixture.     

Probing the Flexibility of Tropomyosin and Its Binding to Filamentous Actin Using Molecular Dynamics Simulations

Tropomyosin (Tm) is a coiled-coil protein that binds to filamentous actin (F-actin) and regulates its interactions with actin-binding proteins like myosin by moving between three positions on F-actin (the blocked, closed, and open positions). To elucidate the molecular details of Tm flexibility in relation to its binding to F-actin, we conducted extensive molecular dynamics simulations for both Tm alone and Tm-F-actin complex in the presence of explicit solvent (total simulation time >400 ns). Based on the simulations, we systematically analyzed the local flexibility of the Tm coiled coil using multiple parameters. We found a good correlation between the regions with high local flexibility and a number of destabilizing regions in Tm, including six clusters of core alanines. Despite the stabilization by F-actin binding, the distribution of local flexibility in Tm is largely unchanged in the absence and presence of F-actin. Our simulations showed variable fluctuations of individual Tm periods from the closed position toward the open position. In addition, we performed Tm-F-actin binding calculations based on the simulation trajectories, which support the importance of Tm flexibility to Tm-F-actin binding. We identified key residues of Tm involved in its dynamic interactions with F-actin, many of which have been found in recent mutational studies to be functionally important, and the rest of which will make promising targets for future mutational experiments.