Stochastic simulation of actin dynamics reveals the role of annealing and fragmentation

Recent observations of F-actin dynamics call for theoretical models to interpret and understand the quantitative data. A number of existing models rely on simplifications and do not take into account F-actin fragmentation and annealing. We use Gillespie's algorithm for stochastic simulations of the F-actin dynamics including fragmentation and annealing. The simulations vividly illustrate that fragmentation and annealing have little influence on the shape of the polymerization curve and on nucleotide profiles within filaments but drastically affect the F-actin length distribution, making it exponential. We find that recent surprising measurements of high length diffusivity at the critical concentration cannot be explained by fragmentation and annealing events unless both fragmentation rates and frequency of undetected fragmentation and annealing events are greater than previously thought. The simulations compare well with experimentally measured actin polymerization data and lend additional support to a number of existing theoretical models.     

Structural insights into the tropomodulin assembly at the pointed ends of actin filaments

Tropomodulins are a family of important regulators of actin dynamics at the pointed ends of actin filaments. Four isoforms of tropomodulin, Tmod1‐Tmod4, are expressed in vertebrates. Binding of tropomodulin to the pointed end is dependent on tropomyosin, an actin binding protein that itself is represented in mammals by up to 40 isoforms. The understanding of the regulatory role of the tropomodulin/tropomyosin molecular diversity has been limited due to the lack of a three‐dimensional structure of the tropomodulin/tropomyosin complex. In this study, we mapped tropomyosin residues interacting with two tropomyosin‐binding sites of tropomodulin and generated a three‐dimensional model of the tropomodulin/tropomyosin complex for each of these sites. The models were refined by molecular dynamics simulations and validated via building a self‐consistent three‐dimensional model of tropomodulin assembly at the pointed end. The model of the pointed‐end Tmod assembly offers new insights in how Tmod binding ensures tight control over the pointed end dynamics.     

Discrete mechanical model of lamellipodial actin network implements molecular clutch mechanism and generates arcs and microspikes

Mechanical forces, actin filament turnover, and adhesion to the extracellular environment regulate lamellipodial protrusions. Computational and mathematical models at the continuum level have been used to investigate the molecular clutch mechanism, calculating the stress profile through the lamellipodium and around focal adhesions. However, the forces and deformations of individual actin filaments have not been considered while interactions between actin networks and actin bundles is not easily accounted with such methods. We develop a filament-level model of a lamellipodial actin network undergoing retrograde flow using 3D Brownian dynamics. Retrograde flow is promoted in simulations by pushing forces from the leading edge (due to actin polymerization), pulling forces (due to molecular motors), and opposed by viscous drag in cytoplasm and focal adhesions. Simulated networks have densities similar to measurements in prior electron micrographs. Connectivity between individual actin segments is maintained by permanent and dynamic crosslinkers. Remodeling of the network occurs via the addition of single actin filaments near the leading edge and via filament bond severing. We investigated how several parameters affect the stress distribution, network deformation and retrograde flow speed. The model captures the decrease in retrograde flow upon increase of focal adhesion strength. The stress profile changes from compression to extension across the leading edge, with regions of filament bending around focal adhesions. The model reproduces the observed reduction in retrograde flow speed upon exposure to cytochalasin D, which halts actin polymerization. Changes in crosslinker concentration and dynamics, as well as in the orientation pattern of newly added filaments demonstrate the model’s ability to generate bundles of filaments perpendicular (actin arcs) or parallel (microspikes) to the protruding direction.     

A mathematical approach to cytoskeletal assembly

The cytoskeleton is a fundamental and important part of cell's structure, and is known to play a large role in controlling the shape, function, division, and motility of the cell. In recent years, the traditional biological and biophysical experimental work on the cytoskeleton has been enhanced by a variety of theoretical, physical and mathematical approaches. Many of these approaches have been developed in the traditional frameworks of physico-chemical and statistical mechanics or equilibrium thermodynamic principles. An alternative is to use kinetic modelling and couch the analysis in terms of differential equations which describe mean field properties of cytoskeletal networks or assemblies. This paper describes two such recent efforts. In the first part of the paper, a summary of work on the kinetics of polymerization, fragmentation, and dynamics of actin and polymers in the presence of gelsolin (which nulceates, fragments, and caps the filaments) is given. In the second part, some of the kinetic models aimed at elucidating the spatio-angular density distribution of actin filaments interacting via crosslinks is described. This model given insight into effects that govern the formation of clusters and bundles of actin filaments, and their spatial distribution. Received: 7 January 1998 / Revised version: 4 March 1998 / Accepted: 7 March 1998     

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.     

Orchestrating nonmuscle myosin II filament assembly at the onset of cytokinesis

Contractile forces in the actomyosin cortex are required for cellular morphogenesis. This includes the invagination of the cell membrane during division, where filaments of nonmuscle myosin II (NMII) are responsible for generating contractile forces in the cortex. However, how NMII heterohexamers form filaments in vivo is not well understood. To quantify NMII filament assembly dynamics, we imaged the cortex of Caenorhabditis elegans embryos at high spatial resolution around the time of the first division. We show that during the assembly of the cytokinetic ring, the number of NMII filaments in the cortex increases and more NMII motors are assembled into each filament. These dynamics are influenced by two proteins in the RhoA GTPase pathway, the RhoA-dependent kinase LET-502 and the myosin phosphatase MEL-11. We find that these two proteins differentially regulate NMII activity at the anterior and at the division site. We show that the coordinated action of these regulators generates a gradient of free NMII in the cytoplasm driving a net diffusive flux of NMII motors toward the cytokinetic ring. Our work highlights how NMII filament assembly and disassembly dynamics are orchestrated over space and time to facilitate the up-regulation of cortical contractility during cytokinesis.     

In Vitro Reconstitution of the Initial Stages of the Bacterial Cell Division Machinery

Fission of many prokaryotes as well as some eukaryotic organelles depends on the self-assembly of the FtsZ protein into a membrane-associated ring structure early in the division process. Different components of the machinery are then sequentially recruited. Although the assembly order has been established, the molecular interactions and the understanding of the force-generating mechanism of this dividing machinery have remained elusive. It is desirable to develop simple reconstituted systems that attempt to reproduce, at least partially, some of the stages of the process. High-resolution studies of Escherichia coli FtsZ filaments’ structure and dynamics on mica have allowed the identification of relevant interactions between filaments that suggest a mechanism by which the polymers could generate force on the membrane. Reconstituting the membrane-anchoring protein ZipA on E. coli lipid membrane on surfaces is now providing information on how the membrane attachment regulates FtsZ polymer dynamics and indicates the important role played by the lipid composition of the membrane.     

Mechanical Heterogeneity Favors Fragmentation of Strained Actin Filaments

We present a general model of actin filament deformation and fragmentation in response to compressive forces. The elastic free energy density along filaments is determined by their shape and mechanical properties, which were modeled in terms of bending, twisting, and twist-bend coupling elasticities. The elastic energy stored in filament deformation (i.e., strain) tilts the fragmentation-annealing reaction free-energy profile to favor fragmentation. The energy gradient introduces a local shear force that accelerates filament intersubunit bond rupture. The severing protein, cofilin, renders filaments more compliant in bending and twisting. As a result, filaments that are partially decorated with cofilin are mechanically heterogeneous (i.e., nonuniform) and display asymmetric shape deformations and energy profiles distinct from mechanically homogenous (i.e., uniform), bare actin, or saturated cofilactin filaments. The local buckling strain depends on the relative size of the compliant segment as well as the bending and twisting rigidities of flanking regions. Filaments with a single bare/cofilin-decorated boundary localize energy and force adjacent to the boundary, within the compliant cofilactin segment. Filaments with small cofilin clusters were predicted to fragment within the compliant cofilactin rather than at boundaries. Neglecting contributions from twist-bend coupling elasticity underestimates the energy density and gradients along filaments, and thus the net effects of filament strain to fragmentation. Spatial confinement causes compliant cofilactin segments and filaments to adopt higher deformation modes and store more elastic energy, thereby promoting fragmentation. The theory and simulations presented here establish a quantitative relationship between actin filament fragmentation thermodynamics and elasticity, and reveal how local discontinuities in filament mechanical properties introduced by regulatory proteins can modulate both the severing efficiency and location along filaments. The emergent behavior of mechanically heterogeneous filaments, particularly under confinement, emphasizes that severing in cells is likely to be influenced by multiple physical and chemical factors.     

Estimating the division rate for the growth-fragmentation equation

Growth-fragmentation equations arise in many different contexts, ranging from cell division, protein polymerization, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest to develop theoretical and numerical methods to recover reaction rates and parameters of the equation from indirect observation of the solution. Following the work done in Perthame and Zubelli (Inverse Probl 23:1037–1052, 2007) and Doumic et al. (2009) for the specific case of the cell division equation, we address here the general question of recovering the fragmentation rate of the equation from the observation of the time-asymptotic solution, when the fragmentation kernel and the growth rates are fully general. We give both theoretical results and numerical methods, and discuss the remaining issues.     

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.     

Agent‐based modeling of the effects of forest dynamics,selective logging,and fragment size on epiphyte communities

1. Forest canopies play a crucial role in structuring communities of vascular epiphytes by providing substrate for colonization, by locally varying microclimate, and by causing epiphyte mortality due to branch or tree fall. However, as field studies in the three‐dimensional habitat of epiphytes are generally challenging, our understanding of how forest structure and dynamics influence the structure and dynamics of epiphyte communities is scarce.
2. Mechanistic models can improve our understanding of epiphyte community dynamics. We present such a model that couples dispersal, growth, and mortality of individual epiphytes with substrate dynamics, obtained from a three‐dimensional functional–structural forest model, allowing the study of forest–epiphyte interactions. After validating the epiphyte model with independent field data, we performed several theoretical simulation experiments to assess how (a) differences in natural forest dynamics, (b) selective logging, and (c) forest fragmentation could influence the long‐term dynamics of epiphyte communities.
3. The proportion of arboreal substrate occupied by epiphytes (i.e., saturation level) was tightly linked with forest dynamics and increased with decreasing forest turnover rates. While species richness was, in general, negatively correlated with forest turnover rates, low species numbers in forests with very‐low‐turnover rates were due to competitive exclusion when epiphyte communities became saturated. Logging had a negative impact on epiphyte communities, potentially leading to a near‐complete extirpation of epiphytes when the simulated target diameters fell below a threshold. Fragment size had no effect on epiphyte abundance and saturation level but correlated positively with species numbers.
4. Synthesis: The presented model is a first step toward studying the dynamic forest–epiphyte interactions in an agent‐based modeling framework. Our study suggests forest dynamics as key factor in controlling epiphyte communities. Thus, both natural and human‐induced changes in forest dynamics, for example, increased mortality rates or the loss of large trees, pose challenges for epiphyte conservation.

     

Recruitment of ZipA to the division site by interaction with FtsZ

ZipA is an essential cell division protein in Escherichia coli that is recruited to the division site early in the division cycle. As it is anchored to the membrane and interacts with FtsZ, it is a candidate for tethering FtsZ filaments to the membrane during the formation of the Z ring. In this study, we have investigated the requirements for ZipA localization to the division site. ZipA requires FtsZ, but not FtsA or FtsI, to be localized, indicating that it is recruited by FtsZ. Consistent with this, apparently normal Z rings are formed in the absence of ZipA. The interaction between FtsZ and ZipA occurs through their carboxy-terminal domains. Although a MalE-ZipA fusion binds to FtsZ filaments, it does not affect the GTPase activity or dynamics of the filaments. These results are consistent with ZipA acting after Z ring formation, possibly to link the membrane to FtsZ filaments during invagination of the septum.     

Emergence and maintenance of variable-length actin filaments in a limiting pool of building blocks

Actin is one of the key structural components of the eukaryotic cytoskeleton that regulates cellular architecture and mechanical properties. Dynamic regulation of actin filament length and organization is essential for the control of many physiological processes including cell adhesion, motility and division. While previous studies have mostly focused on the mechanisms controlling the length of single actin filaments, it remains poorly understood how distinct actin filament populations in cells maintain different lengths using the same set of molecular building blocks. Here, we develop a theoretical model for the length regulation of multiple actin filaments by nucleation and growth-rate modulation by actin-binding proteins in a limiting pool of monomers. We first show that spontaneous nucleation of actin filaments naturally leads to heterogeneities in filament length distribution. We then investigate the effects of filament growth inhibition by capping proteins and growth promotion by formin proteins on filament length distribution. We find that filament length heterogeneity can be increased by growth inhibition, whereas growth promoters do not significantly affect length heterogeneity. Interestingly, a competition between filament growth inhibitors and growth promoters can give rise to bimodal filament length distribution as well as a highly heterogeneous length distribution with large statistical dispersion. We quantitatively predict how heterogeneity in actin filament length can be modulated by tuning filamentous actin nucleation and growth rates in order to create distinct filament subpopulations with different lengths.     

Understanding Nucleotide-Regulated FtsZ Filament Dynamics and the Monomer Assembly Switch with Large-Scale Atomistic Simulations

Bacterial cytoskeletal protein FtsZ assembles in a head-to-tail manner, forming dynamic filaments that are essential for cell division. Here, we study their dynamics using unbiased atomistic molecular simulations from representative filament crystal structures. In agreement with experimental data, we find different filament curvatures that are supported by a nucleotide-regulated hinge motion between consecutive FtsZ monomers. Whereas GTP-FtsZ filaments bend and twist in a preferred orientation, thereby burying the nucleotide, the differently curved GDP-FtsZ filaments exhibit a heterogeneous distribution of open and closed interfaces between monomers. We identify a coordinated Mg²⁺ ion as the key structural element in closing the nucleotide site and stabilizing GTP filaments, whereas the loss of the contacts with loop T7 from the next monomer in GDP filaments leads to open interfaces that are more prone to depolymerization. We monitored the FtsZ monomer assembly switch, which involves opening/closing of the cleft between the C-terminal domain and the H7 helix, and observed the relaxation of isolated and filament minus-end monomers into the closed-cleft inactive conformation. This result validates the proposed switch between the low-affinity monomeric closed-cleft conformation and the active open-cleft FtsZ conformation within filaments. Finally, we observed how the antibiotic PC190723 suppresses the disassembly switch and allosterically induces closure of the intermonomer interfaces, thus stabilizing the filament. Our studies provide detailed structural and dynamic insights into modulation of both the intrinsic curvature of the FtsZ filaments and the molecular switch coupled to the high-affinity end-wise association of FtsZ monomers.     

Analysis of turnover dynamics of the submembranous actin cortex

The cell cortex is a thin network of actin, myosin motors, and associated proteins that underlies the plasma membrane in most eukaryotic cells. It enables cells to resist extracellular stresses, perform mechanical work, and change shape. Cortical structural and mechanical properties depend strongly on the relative turnover rates of its constituents, but quantitative data on these rates remain elusive. Using photobleaching experiments, we analyzed the dynamics of three classes of proteins within the cortex of living cells: a scaffold protein (actin), a cross-linker (α-actinin), and a motor (myosin). We found that two filament subpopulations with very different turnover rates composed the actin cortex: one with fast turnover dynamics and polymerization resulting from addition of monomers to free barbed ends, and one with slow turnover dynamics with polymerization resulting from formin-mediated filament growth. Our data suggest that filaments in the second subpopulation are on average longer than those in the first and that cofilin-mediated severing of formin-capped filaments contributes to replenishing the filament subpopulation with free barbed ends. Furthermore, α-actinin and myosin minifilaments turned over significantly faster than F-actin. Surprisingly, only one-fourth of α-actinin dimers were bound to two actin filaments. Taken together, our results provide a quantitative characterization of essential mechanisms under­lying actin cortex homeostasis.     

Understanding Nucleotide-Regulated FtsZ Filament Dynamics and the Monomer Assembly Switch with Large-Scale Atomistic Simulations

Bacterial cytoskeletal protein FtsZ assembles in a head-to-tail manner, forming dynamic filaments that are essential for cell division. Here, we study their dynamics using unbiased atomistic molecular simulations from representative filament crystal structures. In agreement with experimental data, we find different filament curvatures that are supported by a nucleotide-regulated hinge motion between consecutive FtsZ monomers. Whereas GTP-FtsZ filaments bend and twist in a preferred orientation, thereby burying the nucleotide, the differently curved GDP-FtsZ filaments exhibit a heterogeneous distribution of open and closed interfaces between monomers. We identify a coordinated Mg²⁺ ion as the key structural element in closing the nucleotide site and stabilizing GTP filaments, whereas the loss of the contacts with loop T7 from the next monomer in GDP filaments leads to open interfaces that are more prone to depolymerization. We monitored the FtsZ monomer assembly switch, which involves opening/closing of the cleft between the C-terminal domain and the H7 helix, and observed the relaxation of isolated and filament minus-end monomers into the closed-cleft inactive conformation. This result validates the proposed switch between the low-affinity monomeric closed-cleft conformation and the active open-cleft FtsZ conformation within filaments. Finally, we observed how the antibiotic PC190723 suppresses the disassembly switch and allosterically induces closure of the intermonomer interfaces, thus stabilizing the filament. Our studies provide detailed structural and dynamic insights into modulation of both the intrinsic curvature of the FtsZ filaments and the molecular switch coupled to the high-affinity end-wise association of FtsZ monomers.     

IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: insights from modeling CFSE data

It is generally accepted that IL-2 influences the dynamics of populations of T cells in vitro and in vivo. However, which parameters for cell division and/or death are affected by IL-2 is not well understood. To get better insights into the potential ways of how IL-2 may influence the population dynamics of T cells, we analyze data on the dynamics of CFSE-labeled polyclonal CD4(+) T lymphocytes in vitro after anti-CD3 stimulation at different concentrations of exogenous IL-2. Inferring cell division and death rates from CFSE-delabeling experiments is not straightforward and requires the use of mathematical models. We find that to adequately describe the dynamics of T cells at low concentrations of exogenous IL-2, the death rate of divided cells has to increase with the number of divisions cells have undergone. IL-2 hardly affects the average interdivision time. At low IL-2 concentrations 1) fewer cells are recruited into the response and successfully complete their first division; 2) the stochasticity of cell division is increased; and 3) the rate, at which the death rate increases with the division number, increases. Summarizing, our mathematical reinterpretation suggests that the main effect of IL-2 on the in vitro dynamics of naive CD4(+) T cells occurs by affecting the rate of cell death and not by changing the rate of cell division.     

On the application of compartmental models to radioactive tracer kinetic studies of in vivo protein turnover in animals

A mathematical framework is presented for unifying and extending the various compartmental models and formulae used to calculate fractional protein synthesis and degradation rates in animals from data obtained by infusing labelled amino acids. It is shown how the various schemes can be derived as special cases of the product-precursor model or some three-pool variant. Three-compartment representations, which circumvent the need to measure the specific radioactivity of the precursor pool, are proposed. The mathematical solutions are generally presented in a form that is amenable to parameter estimation by non-linear least squares. The problems of measuring the true precursor pool for protein synthesis are addressed, and theoretical consideration is given to assaying aminoacyl-tRNA.     

In silico reconstitution of Listeria propulsion exhibits nano-saltation

To understand how the actin-polymerization-mediated movements in cells emerge from myriad individual protein–protein interactions, we developed a computational model of Listeria monocytogenes propulsion that explicitly simulates a large number of monomer-scale biochemical and mechanical interactions. The literature on actin networks and L. monocytogenes motility provides the foundation for a realistic mathematical/computer simulation, because most of the key rate constants governing actin network dynamics have been measured. We use a cluster of 80 Linux processors and our own suite of simulation and analysis software to characterize salient features of bacterial motion. Our “in silico reconstitution” produces qualitatively realistic bacterial motion with regard to speed and persistence of motion and actin tail morphology. The model also produces smaller scale emergent behavior; we demonstrate how the observed nano-saltatory motion of L. monocytogenes, in which runs punctuate pauses, can emerge from a cooperative binding and breaking of attachments between actin filaments and the bacterium. We describe our modeling methodology in detail, as it is likely to be useful for understanding any subcellular system in which the dynamics of many simple interactions lead to complex emergent behavior, e.g., lamellipodia and filopodia extension, cellular organization, and cytokinesis.     

Contribution of solvent water to the solution X-ray scattering profile of proteins

A theoretical framework is presented to analyze how solvent water contributes to the X-ray scattering profile of protein solution. Molecular dynamics simulations were carried out on pure water and an aqueous solution of myoglobin to determine the spatial distribution of water molecules in each of them. Their solution X-ray scattering (SXS) profiles were numerically evaluated with obtained atomic-coordinate data. It is shown that two kinds of contributions from solvent water must be considered to predict the SXS profile of a solution accurately. One is the excluded solvent scattering originating in exclusion of water molecules from the space occupied by solutes. The other is the hydration effect resulting from formation of a specific distribution of water around solutes. Explicit consideration of only two molecular layers of water is practically enough to incorporate the hydration effect. Care should be given to using an approximation in which an averaged electron density distribution is assumed for the structure factor because it may predict profiles considerably deviating from the correct profile at large K.