A master equation approach to actin polymerization applied to endocytosis in yeast.

We present a Master Equation approach to calculating polymerization dynamics and force generation by branched actin networks at membranes. The method treats the time evolution of the F-actin distribution in three dimensions, with branching included as a directional spreading term. It is validated by comparison with stochastic simulations of force generation by actin polymerization at obstacles coated with actin "nucleation promoting factors" (NPFs). The method is then used to treat the dynamics of actin polymerization and force generation during endocytosis in yeast, using a model in which NPFs form a ring around the endocytic site, centered by a spot of molecules attaching the actin network strongly to the membrane. We find that a spontaneous actin filament nucleation mechanism is required for adequate forces to drive the process, that partial inhibition of branching and polymerization lead to different characteristic responses, and that a limited range of polymerization-rate values provide effective invagination and obtain correct predictions for the effects of mutations in the active regions of the NPFs.

Local Turgor Pressure Reduction via Channel Clustering.

The primary drivers of yeast endocytosis are actin polymerization and curvature-generating proteins, such as clathrin and BAR domain proteins. Previous work has indicated that these factors may not be capable of generating the forces necessary to overcome turgor pressure. Thus local reduction of the turgor pressure, via localized accumulation or activation of solute channels, might facilitate endocytosis. The possible reduction in turgor pressure was calculated numerically, by solving the diffusion equation through a Legendre polynomial expansion. It was found that for a region of increased permeability having radius 45 nm, as few as 60 channels with a spacing of 10 nm could locally decrease the turgor pressure by 50%. We identified a key dimensionless parameter, p = P1a/D, where P1 is the increased permeability, a is the radius of the permeable region, and D is the solute diffusion coefficient. When p > 0.44, the turgor pressure is locally reduced by >50%. An approximate analytic theory was used to generate explicit formulas for the turgor pressure reduction in terms of key parameters. These findings may also be relevant to plants, where the mechanisms that allow endocytosis to proceed despite high turgor pressure are largely unknown.

Actin-Regulator Feedback Interactions during Endocytosis.

Endocytosis mediated by clathrin, a cellular process by which cells internalize membrane receptors and their extracellular ligands, is an important component of cell signaling regulation. Actin polymerization is involved in endocytosis in varying degrees depending on the cellular context. In yeast, clathrin-mediated endocytosis requires a pulse of polymerized actin and its regulators, which recruit and activate the Arp2/3 complex. In this article, we seek to identify the main protein-protein interactions that 1) cause actin and its regulators to appear in pulses, and 2) determine the effects of key mutations and drug treatments on actin and regulator assembly. We perform a joint modeling/experimental study of actin and regulator dynamics during endocytosis in the budding yeast Saccharomyces cerevisiae. We treat both a stochastic model that grows an explicit three-dimensional actin network, and a simpler two-variable Fitzhugh-Nagumo type model. The models include a negative-feedback interaction of F-actin onto the Arp2/3 regulators. Both models explain the pulse time courses and the effects of interventions on actin polymerization: the surprising increase in the peak F-actin count caused by reduced regulator branching activity, the increase in F-actin resulting from slowing of actin disassembly, and the increased Arp2/3 regulator lifetime resulting from latrunculin treatment. In addition, they predict that decreases in the regulator branching activity lead to increases in accumulation of regulators, and we confirmed this prediction with experiments on yeast harboring mutations in the Arp2/3 regulators, using quantitative fluorescence microscopy. Our experimental measurements suggest that the regulators act quasi-independently, in the sense that accumulation of a particular regulator is most strongly affected by mutations of that regulator, as opposed to the others.

Modeling large-scale dynamic processes in the cell: polarization, waves, and division.

The past decade has witnessed significant developments in molecular biology techniques, fluorescent labeling, and super-resolution microscopy, and together these advances have vastly increased our quantitative understanding of the cell. This detailed knowledge has concomitantly opened the door for biophysical modeling on a cellular scale. There have been comprehensive models produced describing many processes such as motility, transport, gene regulation, and chemotaxis. However, in this review we focus on a specific set of phenomena, namely cell polarization, F-actin waves, and cytokinesis. In each case, we compare and contrast various published models, highlight the relevant aspects of the biology, and provide a sense of the direction in which the field is moving.

How capping protein enhances actin filament growth and nucleation on biomimetic beads.

Capping protein (CP), which caps the growing ends of actin filaments, accelerates actin-based motility. Recent experiments on biomimetic beads have shown that CP also enhances the rate of actin filament nucleation. Proposed explanations for these phenomena include (i) the actin funneling hypothesis (AFH), in which the presence of CP increases the free-actin concentration, and (ii) the monomer gating model, in which CP binding to actin filament barbed ends makes more monomers available for filament nucleation. To establish how CP increases the rates of filament elongation and nucleation on biomimetic beads, we perform a quantitative modeling analysis of actin polymerization, using rate equations that include actin filament nucleation, polymerization and capping, as modified by monomer depletion near the surface of the bead. With one adjustable parameter, our simulation results match previously measured time courses of polymerized actin and filament number. The results support a version of the AFH where CP increases the local actin monomer concentration at the bead surface, but leaves the global free-actin concentration nearly constant. Because the rate of filament nucleation increases with the monomer concentration, the increased local monomer concentration enhances actin filament nucleation. We derive a closed-form formula for the characteristic CP concentration where the local free-actin concentration reaches half the bulk value, and find it to be comparable to the global Arp2/3 complex concentration. We also propose an experimental protocol for distinguishing branching nucleation of filaments from spontaneous nucleation.

Force generation by endocytic actin patches in budding yeast.

Membrane deformation during endocytosis in yeast is driven by local, templated assembly of a sequence of proteins including polymerized actin and curvature-generating coat proteins such as clathrin. Actin polymerization is required for successful endocytosis, but it is not known by what mechanisms actin polymerization generates the required pulling forces. To address this issue, we develop a simulation method in which the actin network at the protein patch is modeled as an active gel. The deformation of the gel is treated using a finite-element approach. We explore the effects and interplay of three different types of force driving invagination: 1), forces perpendicular to the membrane, generated by differences between actin polymerization rates at the edge of the patch and those at the center; 2), the inherent curvature of the coat-protein layer; and 3), forces parallel to the membrane that buckle the coat protein layer, generated by an actomyosin contractile ring. We find that with optimistic estimates for the stall stress of actin gel growth and the shear modulus of the actin gel, actin polymerization can generate almost enough force to overcome the turgor pressure. In combination with the other mechanisms, actin polymerization can the force over the critical value.

Feedback mechanisms in a mechanical model of cell polarization.

Directed cell migration requires a spatially polarized distribution of polymerized actin. We develop and treat a mechanical model of cell polarization based on polymerization and depolymerization of actin filaments at the two ends of a cell, modulated by forces at either end that are coupled by the cell membrane. We solve this model using both a simulation approach that treats filament nucleation, polymerization, and depolymerization stochastically, and a rate-equation approach based on key properties such as the number of filaments N and the number of polymerized subunits F at either end of the cell. The rate-equation approach agrees closely with the stochastic approach at steady state and, when appropriately generalized, also predicts the dynamic behavior accurately. The calculated transitions from symmetric to polarized states show that polarization is enhanced by a high free-actin concentration, a large pointed-end off-rate, a small barbed-end off-rate, and a small spontaneous nucleation rate. The rate-equation approach allows us to perform a linear-stability analysis to pin down the key interactions that drive the polarization. The polarization is driven by a positive-feedback loop having two interactions. First, an increase in F at one side of the cell lengthens the filaments and thus reduces the decay rate of N (increasing N); second, increasing N enhances F because the force per growing filament tip is reduced. We find that the transitions induced by changing system properties result from supercritical pitchfork bifurcations. The filament lifetime depends strongly on the average filament length, and this effect is crucial for obtaining polarization correctly.

Stress generation by myosin minifilaments in actin bundles.

Forces and stresses generated by the action of myosin minifilaments are analyzed in idealized computer-generated actin bundles, and compared to results for isotropic actin networks. The bundles are generated as random collections of actin filaments in two dimensions with constrained orientations, crosslinked and attached to two fixed walls. Myosin minifilaments are placed on actin filament pairs and allowed to move and deform the network so that it exerts forces on the walls. The vast majority of simulation runs end with contractile minifilament stress, because minifilaments rotate into energetically stable contractile configurations. This process is aided by the bending and stretching of actin filaments, which accomodate minifilament rotation. Stresses for bundles are greater than those for isotropic networks, and antiparallel filaments generate more tension than parallel filaments. The forces transmitted by the actin network to the walls of the simulation cell often exceed the tension in the minifilament itself.

Quantitative analysis of approaches to measure cooperative phosphate release in polymerized actin.

We use stochastic simulations that treat several experimental probes of actin dynamics to explore the extent to which phosphate dissociation in filamentous actin may be cooperative. Phosphate time-courses from polymerization and copolymerization experiments of ATP- and ADP-actin are studied, including the effects of variations in filament-number concentration as well as single-filament depolymerization time-courses. We find that highly cooperative models are consistent with the treated experimental data. We also find that some types of experiments that are believed to provide strong constraints on the cooperativity of actin hydrolysis models do not provide such constraints.

Molecular analysis of Arp2/3 complex activation in cells.

Many forms of cellular motility are driven by the growth of branched networks of actin filaments, which push against a membrane. In the dendritic nucleation model, Arp2/3 complex is critical, binding to the side of an existing mother filament, nucleating a new daughter filament, and thus creating a branch. Spatial and temporal regulation of Arp2/3 activity is critical for efficient generation of force and movement. A diverse collection of Arp2/3 regulatory proteins has been identified. They bind to and/or activate Arp2/3 complex via an acidic motif with a conserved tryptophan residue. We tested this model for Arp2/3 regulator function in vivo, by examining the roles of multiple Arp2/3 regulators in endocytosis in living yeast cells. We measured the molecular composition of the actin network in cells with mutations that removed the acidic motifs of the four Arp2/3 regulators previously shown to influence the proper function of the actin network. Unexpectedly, we did not find a simple or direct correlation between defects in patch assembly and movement and changes in the composition and dynamics of dendritic nucleation proteins. Taken together our data does not support the simple hypothesis that the primary role for Arp2/3 regulators is to recruit and activate Arp2/3. Rather our data suggests that these regulators may be playing more subtle roles in establishing functional networks in vivo.

Self-feedback in actin polymerization.

Polymerization of actin, which is crucial for functions such as cell migration, membrane ruffling, cytokinesis, and endocytosis, must be tightly regulated in order to preserve an adequate supply of free actin monomers to respond to changing external conditions. The paper will describe mechanisms by which F-actin feeds back on its own assembly, thus regulating itself. I will present the experimental evidence for such feedback terms, discuss their use in current models of actin dynamics in cells, and present preliminary calculations for the role of feedback in transient endocytic actin patches. These calculations suggest a partial homeostasis of F-actin, in which the F-actin peak height depends only weakly on the actin filament nucleation rate.

General mechanism of actomyosin contractility.

Stress generation by myosin minifilaments is analyzed via simulation of their motion in a random actin network. The stresses are overwhelmingly contractile because minifilament equilibrium positions having contractile stress have lower energy than those for expansive stress. Force chains lead to unexpectedly large stresses.

Dendritic actin filament nucleation causes traveling waves and patches.

The polymerization of actin via branching at a cell membrane containing nucleation-promoting factors is simulated using a stochastic-growth methodology. The polymerized-actin distribution displays three types of behavior: (a) traveling waves, (b) moving patches, and (c) random fluctuations. Increasing actin concentration causes a transition from patches to waves. The waves and patches move by a treadmilling mechanism not involving myosin II. The effects of downregulation of key proteins on actin wave behavior are evaluated.

Membrane bending by actin polymerization.

Actin polymerization provides driving force to aid several types of processes that involve pulling the plasma membrane into the cell, including phagocytosis, cellular entry of large viruses, and endocytosis. In endocytosis, actin polymerization is especially important under conditions of high membrane tension or high turgor pressure. Recent modeling efforts have shown how actin polymerization can give rise to a distribution of forces around the endocytic site, and explored how these forces affect the shape dynamics; experiments have revealed the structure of the endocytic machinery in increasing detail, and demonstrated key feedback interactions between actin assembly and membrane curvature. Here we provide a perspective on these findings and suggest avenues for future research.

Damped and persistent oscillations in a simple model of cell crawling.

A very simple, one-dimensional, discrete, autonomous model of cell crawling is proposed; the model involves only three or four coupled first-order differential equations. This form is sufficient to describe many general features of cell migration, including both steady forward motion and oscillatory progress. Closed-form expressions for crawling speeds and internal forces are obtained in terms of dimensionless parameters that characterize active intracellular processes and the passive mechanical properties of the cell. Two versions of the model are described: a basic cell model with simple elastic coupling between front and rear, which exhibits stable, steady forward crawling after initial transient oscillations have decayed, and a poroelastic model, which can exhibit oscillatory crawling in the steady state.

Actin dynamics: from nanoscale to microscale.

The dynamic nature of actin in cells manifests itself constantly. Polymerization near the cell edge is balanced by depolymerization in the interior, externally induced actin polymerization is followed by depolymerization, and spontaneous oscillations of actin at the cell periphery are frequently seen. I discuss how mathematical modeling relates quantitative measures of actin dynamics to the rates of underlying molecular level processes. The dynamic properties addressed include the rate of actin assembly at the leading edge of a moving cell, the disassembly rates of intracellular actin networks, the polymerization time course in externally stimulated cells, and spontaneous spatiotemporal patterns formed by actin. Although several aspects of actin assembly have been clarified by increasingly sophisticated models, our understanding of rapid actin disassembly is limited, and the origins of nonmonotonic features in externally stimulated actin polymerization remain unclear. Theory has generated several concrete, testable hypotheses for the origins of spontaneous actin waves and cell-edge oscillations. The development and use of more biomimetic systems applicable to the geometry of a cell will be key to obtaining a quantitative understanding of actin dynamics in cells.

Kinetic overshoot in actin network assembly induced jointly by branching and capping proteins.

We report an experimental study of the kinetics of actin assembly mediated by branching and capping proteins. Our findings confirm the recent prediction of a "branching explosion" occurring during polymerization. Fluorescence imaging shows a number of actin filaments with branches within a few minutes of polymerization, induced by the activated branching protein complex Arp2/3, but the number of visible branches decreases over time. The light-scattering intensity displays an overshoot as a function of time, which we attribute to the formation of highly branched clusters early in polymerization. Furthermore, the overshoot occurs over a limited range of the ratio of concentrations of branching and capping proteins, also consistent with the theoretical model. These results establish a natural link between the kinetic theory of actin assembly in vitro and the cytoskeletal structure and actin dynamics in motile cells.

Mathematical modeling of cell migration.

Mathematical modeling has become increasingly important in many areas of biology during the past two decades, and the area of cell migration and motility has seen significant contributions from a wide range of modeling approaches. In this chapter, we cover examples from the broad range of work in this area, emphasizing the models' biological significance and the relationships between them. We focus on three specific areas: cell protrusion, cell adhesion, and retraction/whole-cell models. At the end of this chapter, we provide our perspective on issues that future models and experiments should consider in order to advance the boundaries of this field.

Motion repulsion arises from stimulus statistics when analyzed with a clustering algorithm.

Motion repulsion is the perceived enlargement of the angle between the directions of motion of two transparently moving patterns. An explanation of this illusion has long been sought for in the neural circuitry of the brain. We show that motion repulsion already arises from the statistical properties of the motion transparency problem when analyzed with a clustering algorithm.