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.     

A Highlights from MBoC Selection: Mathematical Modeling of Endocytic Actin Patch Kinetics in Fission Yeast: Disassembly Requires Release of Actin Filament Fragments

We used the dendritic nucleation hypothesis to formulate a mathematical model of the assembly and disassembly of actin filaments at sites of clathrin-mediated endocytosis in fission yeast. We used the wave of active WASp recruitment at the site of the patch formation to drive assembly reactions after activation of Arp2/3 complex. Capping terminated actin filament elongation. Aging of the filaments by ATP hydrolysis and γ-phosphate dissociation allowed actin filament severing by cofilin. The model could simulate the assembly and disassembly of actin and other actin patch proteins using measured cytoplasmic concentrations of the proteins. However, to account quantitatively for the numbers of proteins measured over time in the accompanying article (Sirotkin et al., 2010 , MBoC 21: 2792–2802), two reactions must be faster in cells than in vitro. Conditions inside the cell allow capping protein to bind to the barbed ends of actin filaments and Arp2/3 complex to bind to the sides of filaments faster than the purified proteins in vitro. Simulations also show that depolymerization from pointed ends cannot account for rapid loss of actin filaments from patches in 10 s. An alternative mechanism consistent with the data is that severing produces short fragments that diffuse away from the patch.     

A stochastic analysis of actin polymerization in the presence of twinfilin and gelsolin

We develop an efficient stochastic simulation algorithm for analyzing actin filament growth and decay in the presence of various actin-binding proteins. The evolution of nucleotide profiles of filaments can be tracked and the resulting feedback to actin-binding proteins is incorporated. The computational efficiency of the new method enables us to focus on experimentally realistic problems, and as one example we use it to analyze the experimental data of Helfer et al. [(2006). Mammalian twinfilin sequesters ADP-G-actin and caps filament barbed ends: implications in motility. EMBO J. 25, 1184-1195] on the capping and G-actin sequestering activity of twinfilin. We show that the binding specificity of twinfilin for ADP-G-actin is crucial for the observed biphasic evolution of the filament length distribution in the presence of twinfilin, and we demonstrate that twinfilin can be an essential part of the molecular machinery for regulating filament lengths after a short burst of polymerization. Significantly, our simulations indicate that the pyrenyl-actin fluorescence experiments would fail to report the emergence of large filaments under certain experimental conditions.     

Distribution of attachment events relative to actin binding sites as evidenced in a bidirectional actomyosin interaction model

Optical trapping is one of the most evolving technologies that measures biophysical quantities and provides insights into some of the fundamental questions in the study of molecular motor proteins such as myosin. Several laboratories have successfully used this technique to observe and score nanometre-size displacements produced by myosin on interacting with actin. We have studied the distribution of attachment events for two myosin molecules with different orientations interacting with an actin filament within the framework of a Langevin-type bidirectional mathematical model. When myosin is detached from actin, our model predicts Brownian displacements centred at 0 ± 8 nm (mean ± SD, n = 251058). When attached, the time-averaged displacements of the actin filament system produced step sizes with peaks of 8 ± 6 nm (mean ± SD, n = 22174) (forward displacements) and −8 ± 6 nm (mean ± SD, n = 26769) (reverse displacements). We infer from our results that the population distribution of attachment events is strongly dependent on (i) the magnitude of the Brownian displacements, (ii) the location of the actin binding sites relative to the myosin molecules, (iii) the orientation of the myosin molcules, and (iv) the relative kinetics (rate constants) for the forward and reverse displacement events.     

Models of the actin monomer and filament from fluorescence resonance-energy transfer.

We have developed algorithms for combining fluorescence resonance-energy transfer (FRET) efficiency measurements into structural models which predict the relative positions of the chemical groups used in FRET. We used these algorithms to construct models of the actin monomer and filament derived solely from FRET measurements based on seven distinct loci. We found a mirror-image pair of monomer models which best fit the FRET data. One of these models agrees well with the atomic-resolution crystal structure recently published by Kabsch et al. in Heidelberg [Kabsch, W., Mannherz, H. G., Suck, D., Pai, E. F. & Holmes, K. C. (1990) Nature 347, 37-44]. The root-mean-square deviation between this FRET model and the crystal structure was about 0.9 nm. Other macromolecular models assembled from FRET measurements are likely to have a similar resolution. The largest discrepancy was for the Cys10 locus which deviated 1.44 nm from the crystal position. We discuss the limitations of the FRET method that may have contributed to this discrepancy, and conclude that the Cys10 FRET data have probably located Cys10 incorrectly in the FRET monomer model. Using the FRET monomer models, we found three orientations in the filament which best fit the intermonomer FRET data. These orientations differ substantially from the atomic-resolution filament model proposed by the Heidelberg group [Holmes, K., Popp, D., Gebhard, W. & Kabsch, W. (1990) Nature 347, 44-49], largely because of the discrepancies in the Cys10 data. These data should probably be excluded from the analysis; however, this would leave too few measurements to assemble a filament model. In the near future, we hope to obtain additional FRET measurements to other actin loci so that the filament modelling can be done without the Cys10 data.     

Actin filament dynamics in the actomyosin VI complex is regulated allosterically by calcium-calmodulin light chain

The contractile and enzymatic activities of myosin VI are regulated by calcium binding to associated calmodulin (CaM) light chains. We have used transient phosphorescence anisotropy to monitor the microsecond rotational dynamics of erythrosin-iodoacetamide-labeled actin with strongly bound myosin VI (MVI) and to evaluate the effect of MVI-bound CaM light chain on actin filament dynamics. MVI binding lowers the amplitude but accelerates actin filament microsecond dynamics in a Ca²⁺- and CaM-dependent manner, as indicated from an increase in the final anisotropy and a decrease in the correlation time of transient phosphorescence anisotropy decays. MVI with bound apo-CaM or Ca²⁺-CaM weakly affects actin filament microsecond dynamics, relative to other myosins (e.g., muscle myosin II and myosin Va). CaM dissociation from bound MVI damps filament rotational dynamics (i.e., increases the torsional rigidity), such that the perturbation is comparable to that induced by other characterized myosins. Analysis of individual actin filament shape fluctuations imaged by fluorescence microscopy reveals a correlated effect on filament bending mechanics. These data support a model in which Ca²⁺-dependent CaM binding to the IQ domain of MVI is linked to an allosteric reorganization of the actin binding site(s), which alters the structural dynamics and the mechanical rigidity of actin filaments. Such modulation of filament dynamics may contribute to the Ca²⁺- and CaM-dependent regulation of myosin VI motility and ATP utilization.     

Structure from NMR and molecular dynamics: Distance restraining inhibits motion in the essential subspace

Summary We address the question how well proteins can be modelled on the basis of NMR data, when these data are incorporated into the protein model using distance restraints in a molecular dynamics simulation. We found, using HPr as a model protein, that distance restraining freezes the essential motion of proteins, as defined by Amadei et al. [Amadei, A., Linssen, A.B.M. and Berendsen, H.J.C. (1993) Protein Struct. Funct. Genet., 17, 412–425]. We discuss how modelling protocols can be improved in order to solve this problem.     

Growing actin networks form lamellipodium and lamellum by self-assembly

Many different cell types are able to migrate by formation of a thin actin-based cytoskeletal extension. Recently, it became evident that this extension consists of two distinct substructures, designated lamellipodium and lamellum, which differ significantly in their kinetic and kinematic properties as well as their biochemical composition. We developed a stochastic two-dimensional computer simulation that includes chemical reaction kinetics, G-actin diffusion, and filament transport to investigate the formation of growing actin networks in migrating cells. Model parameters were chosen based on experimental data or theoretical considerations. In this work, we demonstrate the system's ability to form two distinct networks by self-organization. We found a characteristic transition in mean filament length as well as a distinct maximum in depolymerization flux, both within the first 1-2 μm. The separation into two distinct substructures was found to be extremely robust with respect to initial conditions and variation of model parameters. We quantitatively investigated the complex interplay between ADF/cofilin and tropomyosin and propose a plausible mechanism that leads to spatial separation of, respectively, ADF/cofilin- or tropomyosin-dominated compartments. Tropomyosin was found to play an important role in stabilizing the lamellar actin network. Furthermore, the influence of filament severing and annealing on the network properties is explored, and simulation data are compared to existing experimental data.     

Co-loss of profilin I, II and cofilin with actin from maturing phagosomes inDictyostelium discoideum

Summary Although it is known that actin polymerizes rapidly at the plasma membrane during the ingestion phase of phagocytosis, not yet fully understood are the mechanisms by which actin is recruited to form a phagoeytic cup and subsequently is dissociated from the phagosome. The aim of this study was to identify actin-binding proteins that mediated actin filament dynamics during phagosome formation and processing. We report that profilins I and II, which promote filament assembly, and cofilin, which stimulates filament disassembly, were constituents of phagosomes isolated fromDictyostelium discoideum fed latex beads, and associated with actin. Biochemical analyses detected one isoform only of cofilin, which bound actin in unstimulated cells as well as in cells engaged in phagocytosis, subjected to various stress treatments, and through development. At membranes of young phagosomes, profilins I and II colocalized with monomeric actin labeled with fluorescent DNase I, and cofilin colocalized with filamentous actin labeled with rhodamine phalloidin. Both immunocytochemical and quantitative immunoblotting data indicated that the kinetic loss of profilins I, II, and cofilin of maturing phagosomes closely followed the falling levels of actin associated with the vesicles. As evidence of vesicle processing,D. discoideum crystal protein (an esterase) was recruited rapidly to phagosomes and its levels increased while those of actin, profilins I, II, and cofilin jointly decreased. The localization data and concurrent losses of profilins and cofilin with actin from phagosomes are consistent with the roles of these actin-binding proteins in filament dynamics and indicated that they were involved in regulating the assembly and disassembly of the actin coat of phagosomes.Abbreviations DNase deoxyribonuclease - FITC fluorescein isothiocyanate - NEpHGE nonequilibrium pH gradient gel electrophoresis - SDS-PAGE sodium dodecyl sulfate-polyacrylamide gel electrophoresis     

Direct Measurements of Local Coupling between Myosin Molecules Are Consistent with a Model of Muscle Activation

Muscle contracts due to ATP-dependent interactions of myosin motors with thin filaments composed of the proteins actin, troponin, and tropomyosin. Contraction is initiated when calcium binds to troponin, which changes conformation and displaces tropomyosin, a filamentous protein that wraps around the actin filament, thereby exposing myosin binding sites on actin. Myosin motors interact with each other indirectly via tropomyosin, since myosin binding to actin locally displaces tropomyosin and thereby facilitates binding of nearby myosin. Defining and modeling this local coupling between myosin motors is an open problem in muscle modeling and, more broadly, a requirement to understanding the connection between muscle contraction at the molecular and macro scale. It is challenging to directly observe this coupling, and such measurements have only recently been made. Analysis of these data suggests that two myosin heads are required to activate the thin filament. This result contrasts with a theoretical model, which reproduces several indirect measurements of coupling between myosin, that assumes a single myosin head can activate the thin filament. To understand this apparent discrepancy, we incorporated the model into stochastic simulations of the experiments, which generated simulated data that were then analyzed identically to the experimental measurements. By varying a single parameter, good agreement between simulation and experiment was established. The conclusion that two myosin molecules are required to activate the thin filament arises from an assumption, made during data analysis, that the intensity of the fluorescent tags attached to myosin varies depending on experimental condition. We provide an alternative explanation that reconciles theory and experiment without assuming that the intensity of the fluorescent tags varies.     

Mechanical Distortion of Single Actin Filaments Induced by External Force: Detection by Fluorescence Imaging

Actin is a major component of the cytoskeleton that transmits mechanical stress in both muscle and nonmuscle cells. As the first step toward developing a “bio-nano strain gauge” that would be able to report the mechanical stress imposed on an actin filament, we quantitatively examined the fluorescence intensity of dyes attached to single actin filaments under various tensile forces (5-20 pN). Tensile force was applied via two optically trapped plastic beads covalently coated with chemically modified heavy meromyosin molecules that were attached to both end regions of an actin filament. As a result, we found that the fluorescence intensity of an actin filament, where 20% of monomers were labeled with tetramethylrhodamine (TMR)-5-maleimide at Cys³⁷⁴ and the filamentous structure was stabilized with nonfluorescent phalloidin, decreased by ∼6% per 10 pN of the applied force, whereas the fluorescence intensity of an actin filament labeled with either BODIPY TMR cadaverin-iodoacetamide at Cys³⁷⁴ or rhodamine-phalloidin showed only an ∼2% decrease per 10 pN of the applied force. On the other hand, spectroscopic measurements of actin solutions showed that the fluorescence intensity of TMR-actin increased 1.65-fold upon polymerization (G-F transformation), whereas that of BODIPY-actin increased only 1.06-fold. These results indicate that the external force distorts the filament structure, such that the microenvironment around Cys³⁷⁴ approaches that in G-actin. We thus conclude that the fluorescent dye incorporated into an appropriate site of actin can report the mechanical distortion of the binding site, which is a necessary condition for the bio-nano strain gauge.     

A Periodic Pattern of Evolutionarily Conserved Basic and Acidic Residues Constitutes the Binding Interface of Actin-Tropomyosin

Actin filament cytoskeletal and muscle functions are regulated by actin binding proteins using a variety of mechanisms. A universal actin filament regulator is the protein tropomyosin, which binds end-to-end along the length of the filament. The actin-tropomyosin filament structure is unknown, but there are atomic models in different regulatory states based on electron microscopy reconstructions, computational modeling of actin-tropomyosin, and docking of atomic resolution structures of tropomyosin to actin filament models. Here, we have tested models of the actin-tropomyosin interface in the “closed state” where tropomyosin binds to actin in the absence of myosin or troponin. Using mutagenesis coupled with functional analyses, we determined residues of actin and tropomyosin required for complex formation. The sites of mutations in tropomyosin were based on an evolutionary analysis and revealed a pattern of basic and acidic residues in the first halves of the periodic repeats (periods) in tropomyosin. In periods P1, P4, and P6, basic residues are most important for actin affinity, in contrast to periods P2, P3, P5, and P7, where both basic and acidic residues or predominantly acidic residues contribute to actin affinity. Hydrophobic interactions were found to be relatively less important for actin binding. We mutated actin residues in subdomains 1 and 3 (Asp²⁵-Glu³³⁴-Lys³²⁶-Lys³²⁸) that are poised to make electrostatic interactions with the residues in the repeating motif on tropomyosin in the models. Tropomyosin failed to bind mutant actin filaments. Our mutagenesis studies provide the first experimental support for the atomic models of the actin-tropomyosin interface.     

Movement of tropomyosin during regulation of vertebrate skeletal muscle: a simple physical model.

Using data obtained from (a) X-ray diffraction patterns of vertebrate skeletal muscle (b) three dimensional reconstructions from electron micrographs of in vitro aggregates of the thin filament proteins in the switched “on” and “off” positions (c) the analysis of the sequence of tropomyosin, a simple model can be proposed which may explain the geometrical arrangement of actin, tropomyosin and troponin during regulation. The “cooperative” behaviour exhibited by the thin filament can also be explained in terms of this arrangement.     

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.     

Differential Effects of Caldesmon on the Intermediate Conformational States of Polymerizing Actin

The actin-binding protein caldesmon (CaD) reversibly inhibits smooth muscle contraction. In non-muscle cells, a shorter CaD isoform co-exists with microfilaments in the stress fibers at the quiescent state, but the phosphorylated CaD is found at the leading edge of migrating cells where dynamic actin filament remodeling occurs. We have studied the effect of a C-terminal fragment of CaD (H32K) on the kinetics of the in vitro actin polymerization by monitoring the fluorescence of pyrene-labeled actin. Addition of H32K or its phosphorylated form either attenuated or accelerated the pyrene emission enhancement, depending on whether it was added at the early or the late phase of actin polymerization. However, the CaD fragment had no effect on the yield of sedimentable actin, nor did it affect the actin ATPase activity. Our findings can be explained by a model in which nascent actin filaments undergo a maturation process that involves at least two intermediate conformational states. If present at early stages of actin polymerization, CaD stabilizes one of the intermediate states and blocks the subsequent filament maturation. Addition of CaD at a later phase accelerates F-actin formation. The fact that CaD is capable of inhibiting actin filament maturation provides a novel function for CaD and suggests an active role in the dynamic reorganization of the actin cytoskeleton.     

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.     

The regulatory action of α-actinin on actin filaments is enhanced by cofilin

We have used fluorescence recovery after photobleaching to study the effect of muscle α-actinin on the structure of actin filaments in dilute solutions. Unexpectedly we found that α-actinin partitioned filaments into two types: those with a high mobility and those with low mobility. We have determined that the high mobility (smaller sized) population is too large to be simple monomeric actin:α-actinin complexes. Although it is known that cofilin encourages the transformation of α-actinin:actin gels into large meshworks of inter-digitating actin filament bundles (Maciver et al. 1991), we have found that the presence of cofilin also increases the cross-linking of actin filaments by α-actinin and hypothesize that this is due to cofilin’s ability to alter the filament twist. This effectively makes more potential α-actinin binding sites per unit of actin filament. As expected from previous work, this effect was more marked at pH 6.5 than at pH 8.0. Both effects are likely to operate in cells to deny other actin-binding proteins access to binding these particular filaments and may explain how very different actin cytoskeletal structures may co-exist in the same cell at the same time.     

Arabidopsis AtADF1 is Functionally Affected by Mutations on Actin Binding Sites

The plant actin depolymerizing factor (ADF) binds to both monomeric and filamentous actin, and is directly involved in the depolymerization of actin filaments. To better understand the actin binding sites of the Arabidopsis thaliana L. AtADF1, we generated mutants of AtADF1 and investigated their functions in vitro and in vivo. Analysis of mutants harboring amino acid substitutions revealed that charged residues (Arg98 and Lys100) located at the α‐helix 3 and forming an actin binding site together with the N‐terminus are essential for both G‐ and F‐actin binding. The basic residues on the β‐strand 5 (K82/A) and the α‐helix 4 (R135/A, R137/A) form another actin binding site that is important for F‐actin binding. Using transient expression of CFP‐tagged AtADF1 mutant proteins in onion (Allium cepa) peel epidermal cells and transgenic Arabidopsis thaliana L. plants overexpressing these mutants, we analyzed how these mutant proteins regulate actin organization and affect seedling growth. Our results show that the ADF mutants with a lower affinity for actin filament binding can still be functional, unless the affinity for actin monomers is also affected. The G‐actin binding activity of the ADF plays an essential role in actin binding, depolymerization of actin polymers, and therefore in the control of actin organization.     

A Mechanism for Actin Filament Severing by Malaria Parasite Actin Depolymerizing Factor 1 via a Low Affinity Binding Interface

Actin depolymerizing factor (ADF)/cofilins are essential regulators of actin turnover in eukaryotic cells. These multifunctional proteins facilitate both stabilization and severing of filamentous (F)-actin in a concentration-dependent manner. At high concentrations ADF/cofilins bind stably to F-actin longitudinally between two adjacent actin protomers forming what is called a decorative interaction. Low densities of ADF/cofilins, in contrast, result in the optimal severing of the filament. To date, how these two contrasting modalities are achieved by the same protein remains uncertain. Here, we define the proximate amino acids between the actin filament and the malaria parasite ADF/cofilin, PfADF1 from Plasmodium falciparum. PfADF1 is unique among ADF/cofilins in being able to sever F-actin but do so without stable filament binding. Using chemical cross-linking and mass spectrometry (XL-MS) combined with structure reconstruction we describe a previously overlooked binding interface on the actin filament targeted by PfADF1. This site is distinct from the known binding site that defines decoration. Furthermore, total internal reflection fluorescence (TIRF) microscopy imaging of single actin filaments confirms that this novel low affinity site is required for F-actin severing. Exploring beyond malaria parasites, selective blocking of the decoration site with human cofilin (HsCOF1) using cytochalasin D increases its severing rate. HsCOF1 may therefore also use a decoration-independent site for filament severing. Thus our data suggest that a second, low affinity actin-binding site may be universally used by ADF/cofilins for actin filament severing.     

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.