Growth velocities of branched actin networks

On the generation of force required for actin-based motility

The fundamental question of how forces are generated in a motile cell, a lamellipodium, and a comet tail is the subject of this note. It is now well established that cellular motility results from the polymerization of actin, the most abundant protein in eukaryotic cells, into an interconnected set of filaments. We portray this process in a continuum mechanics framework, claiming that polymerization promotes a mechanical swelling in a narrow zone around the nucleation loci, which ultimately results in cellular or bacterial motility. To this aim, a new paradigm in continuum multi-physics has been designed, departing from the well-known theory of Larché-Cahn chemo-transport-mechanics. In this note, we set up the theory of network growth and compare the outcomes of numerical simulations with experimental evidence.

Synthetic control of actin polymerization and symmetry breaking in active protocells

Nonlinear biomolecular interactions on membranes drive membrane remodeling crucial for biological processes including chemotaxis, cytokinesis, and endocytosis. The complexity of biomolecular interactions, their redundancy, and the importance of spatiotemporal context in membrane organization impede understanding of the physical principles governing membrane mechanics. Developing a minimal in vitro system that mimics molecular signaling and membrane remodeling while maintaining physiological fidelity poses a major challenge. Inspired by chemotaxis, we reconstructed chemically regulated actin polymerization inside vesicles, guiding membrane self-organization. An external, undirected chemical input induced directed actin polymerization and membrane deformation uncorrelated with upstream biochemical cues, suggesting symmetry breaking. A biophysical model incorporating actin dynamics and membrane mechanics proposes that uneven actin distributions cause nonlinear membrane deformations, consistent with experimental findings. This protocellular system illuminates the interplay between actin dynamics and membrane shape during symmetry breaking, offering insights into chemotaxis and other cell biological processes.

Nascent adhesions shorten the period of lamellipodium protrusion through the Brownian ratchet mechanism

Directional cell migration is driven by the conversion of oscillating edge motion into lasting periods of leading edge protrusion. Actin polymerization against the membrane and adhesions control edge motion, but the exact mechanisms that determine protrusion period remain elusive. We addressed this by developing a computational model in which polymerization of actin filaments against a deformable membrane and variable adhesion dynamics support edge motion. Consistent with previous reports, our model showed that actin polymerization and adhesion lifetime power protrusion velocity. However, increasing adhesion lifetime decreased the protrusion period. Measurements of adhesion lifetime and edge motion in migrating cells confirmed that adhesion lifetime is associated with and promotes protrusion velocity, but decreased duration. Our model showed that adhesions' control of protrusion persistence originates from the Brownian ratchet mechanism for actin filament polymerization. With longer adhesion lifetime or increased-adhesion density, the proportion of actin filaments tethered to the substrate increased, maintaining filaments against the cell membrane. The reduced filament-membrane distance generated pushing force for high edge velocity, but limited further polymerization needed for protrusion duration. We propose a mechanism for cell edge protrusion in which adhesion strength regulates actin filament polymerization to control the periods of leading edge protrusion.

Synthetic control of actin polymerization and symmetry breaking in active protocells

Non-linear biomolecular interactions on the membranes drive membrane remodeling that underlies fundamental biological processes including chemotaxis, cytokinesis, and endocytosis. The multitude of biomolecules, the redundancy in their interactions, and the importance of spatiotemporal context in membrane organization hampers understanding the physical principles governing membrane mechanics. A minimal, in vitro system that models the functional interactions between molecular signaling and membrane remodeling, while remaining faithful to cellular physiology and geometry is powerful yet remains unachieved. Here, inspired by the biophysical processes underpinning chemotaxis, we reconstituted externally-controlled actin polymerization inside giant unilamellar vesicles, guiding self-organization on the membrane. We show that applying undirected external chemical inputs to this system results in directed actin polymerization and membrane deformation that are uncorrelated with upstream biochemical cues, indicating symmetry breaking. A biophysical model of the dynamics and mechanics of both actin polymerization and membrane shape suggests that inhomogeneous distributions of actin generate membrane shape deformations in a non-linear fashion, a prediction consistent with experimental measurements and subsequent local perturbations. The active protocellular system demonstrates the interplay between actin dynamics and membrane shape in a symmetry breaking context that is relevant to chemotaxis and a suite of other biological processes.

From WRC to Arp2/3: Collective molecular mechanisms of branched actin network assembly

Branched actin networks have emerged as major force-generating structures driving the protrusions in various distinct cell types and processes, ranging from lamellipodia operating in mesenchymal and epithelial cell migration or tails pushing intracellular pathogens and vesicles to developing spine heads on neurons. Many key molecular features are conserved among all those Arp2/3 complex-containing, branched actin networks. Here, we will review recent progress in our molecular understanding of the core biochemical machinery driving branched actin nucleation, from the generation of filament primers to Arp2/3 activator recruitment, regulation and turnover. Due to the wealth of information on distinct, Arp2/3 network-containing structures, we are largely focusing-in an exemplary fashion-on canonical lamellipodia of mesenchymal cells, which are regulated by Rac GTPases, their downstream effector WAVE Regulatory Complex and its target Arp2/3 complex. Novel insight additionally confirms that WAVE and Arp2/3 complexes regulate or are themselves tuned by additional prominent actin regulatory factors, including Ena/VASP family members and heterodimeric capping protein. Finally, we are considering recent insights into effects exerted by mechanical force, both at the branched network and individual actin regulator level.

The molecular mechanism of load adaptation by branched actin networks

Branched actin networks are self-assembling molecular motors that move biological membranes and drive many important cellular processes, including phagocytosis, endocytosis, and pseudopod protrusion. When confronted with opposing forces, the growth rate of these networks slows and their density increases, but the stoichiometry of key components does not change. The molecular mechanisms governing this force response are not well understood, so we used single-molecule imaging and AFM cantilever deflection to measure how applied forces affect each step in branched actin network assembly. Although load forces are observed to increase the density of growing filaments, we find that they actually decrease the rate of filament nucleation due to inhibitory interactions between actin filament ends and nucleation promoting factors. The force-induced increase in network density turns out to result from an exponential drop in the rate constant that governs filament capping. The force dependence of filament capping matches that of filament elongation and can be explained by expanding Brownian Ratchet theory to cover both processes. We tested a key prediction of this expanded theory by measuring the force-dependent activity of engineered capping protein variants and found that increasing the size of the capping protein increases its sensitivity to applied forces. In summary, we find that Brownian Ratchets underlie not only the ability of growing actin filaments to generate force but also the ability of branched actin networks to adapt their architecture to changing loads.

Granger-causal inference of the lamellipodial actin regulator hierarchy by live cell imaging without perturbation

Many cell regulatory systems implicate nonlinearity and redundancy among components. The regulatory network governing lamellipodial and lamellar actin structures is prototypical of such a system, containing tens of actin-nucleating and -modulating molecules with functional overlap and feedback loops. Due to instantaneous and long-term compensation, phenotyping the system response to perturbation provides limited information on the roles the targeted component plays in the unperturbed system. Accordingly, how individual actin regulators contribute to lamellipodial dynamics remains ambiguous. Here, we present a perturbation-free reconstruction of cause-effect relations among actin regulators by applying Granger-causal inference to constitutive image fluctuations that indicate regulator recruitment as a proxy for activity. Our analysis identifies distinct zones of actin regulator activation and of causal effects on filament assembly and delineates actin-dependent and actin-independent regulator roles in controlling edge motion. We propose that edge motion is driven by assembly of two independently operating actin filament systems.

Effective Force Generation During Mammalian Cell Migration Under Different Molecular and Physical Mechanisms

We have developed much understanding of actin-driven cell migration and the forces that propel cell motility. However, fewer studies focused on estimating the effective forces generated by migrating cells. Since cells in vivo are exposed to complex physical environments with various barriers, understanding the forces generated by cells will provide insights into how cells manage to navigate challenging environments. In this work, we use theoretical models to discuss actin-driven and water-driven cell migration and the effect of cell shapes on force generation. The results show that the effective force generated by actin-driven cell migration is proportional to the rate of actin polymerization and the strength of focal adhesion; the energy source comes from the actin polymerization against the actin network pressure. The effective force generated by water-driven cell migration is proportional to the rate of active solute flux and the coefficient of external hydraulic resistance; the energy sources come from active solute pumping against the solute concentration gradient. The model further predicts that the actin network distribution is mechanosensitive and the presence of globular actin helps to establish a biphasic cell velocity in the strength of focal adhesion. The cell velocity and effective force generation also depend on the cell shape through the intracellular actin flow field.

Modeling the dynamic growth and branching of actin filaments

As an essential component of the cytoskeleton, actin filaments play a key role in a variety of cellular physiological activities. To better understand the function of actin filaments, which are a special kind of polymer chain, researchers have started to focus on the Brownian dynamics of polymers. Currently, to study the dynamics of polymers, classical explicit bead-spring models and finite-element methods (FEMs) have both been broadly used. However, compared to bead-spring models, FEMs can address the mechanical properties of actin filaments and actin networks with more detail and better accuracy. However, current FEMs do not consider the dynamic assembly of actin into an actin filament network. Here, we extend the traditional FEM and present a new framework of the FEM based on the co-rotational grid method, which allows us to simulate the dynamic growth and branching of actin filaments. Several examples are studied. The proposed numerical model is capable of capturing the dynamic assembly of actin filaments.

Leading edge maintenance in migrating cells is an emergent property of branched actin network growth

Animal cell migration is predominantly driven by the coordinated, yet stochastic, polymerization of thousands of nanometer-scale actin filaments across micron-scale cell leading edges. It remains unclear how such inherently noisy processes generate robust cellular behavior. We employed high-speed imaging of migrating neutrophil-like HL-60 cells to explore the fine-scale shape fluctuations that emerge and relax throughout the process of leading edge maintenance. We then developed a minimal stochastic model of the leading edge that reproduces this stable relaxation behavior. Remarkably, we find lamellipodial stability naturally emerges from the interplay between branched actin network growth and leading edge shape – with no additional feedback required – based on a synergy between membrane-proximal branching and lateral spreading of filaments. These results thus demonstrate a novel biological noise-suppression mechanism based entirely on system geometry. Furthermore, our model suggests that the Arp2/3-mediated ~70–80° branching angle optimally smooths lamellipodial shape, addressing its long-mysterious conservation from protists to mammals.

In every human cell, there are tens of millions of proteins which work together to control everything from the cell’s shape to its behavior. One of the most abundant proteins is actin, which organizes itself into filaments that mechanically support the cell and help it to move.

These filaments are very dynamic, with individual actin molecules constantly being added or removed. This allows the cell to build large structures with distinct shapes and properties. Many motile cells, for example, have a structure called a lamellipodium which protrudes at their ‘leading edge’ and pushes them forward. The lamellipodium has a very robust shape that does not vary much between different cell types, or change significantly as cells migrate. But how the tens of thousands of actin molecules inside the lamellipodium organize themselves into this large, stable structure is not fully understood.

To investigate, Garner and Theriot used high-speed video microscopy to track the shape of human cells cultured in the laboratory. As the cells crawled along a glass surface, their leading edge undulated like strings being plucked on a guitar. A computer simulation showed that these ripples can be caused by filaments randomly adding and removing actin molecules.

While these random movements could destabilize the structure of the leading edge, the simulation suggests that another aspect of actin filament growth smooths out any fluctuations in the lamellipodium’s shape. Actin networks in the lamellipodium have a branched configuration, with new strands emerging off each other at an angle like branches in a tree. Garner and Theriot found that the specific angle in which new filaments are added smooths out the lamellipodium’s shape, which may explain why this geometry has persisted throughout evolution.

These findings suggest that the way in which actin filaments join together helps to maintain the shape of large cellular structures. In the future, scientists could use this design principle to build molecular machines that can self-organize into microstructures. These engineered constructs could be used to modulate the activity of living cells that have been damaged by disease.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

Actin-Membrane Release Initiates Cell Protrusions

Despite the well-established role of actin polymerization as a driving mechanism for cell protrusion, upregulated actin polymerization alone does not initiate protrusions. Using a combination of theoretical modeling and quantitative live-cell imaging experiments, we show that local depletion of actin-membrane links is needed for protrusion initiation. Specifically, we show that the actin-membrane linker ezrin is depleted prior to protrusion onset and that perturbation of ezrin's affinity for actin modulates protrusion frequency and efficiency. We also show how actin-membrane release works in concert with actin polymerization, leading to a comprehensive model for actin-driven shape changes. Actin-membrane release plays a similar role in protrusions driven by intracellular pressure. Thus, our findings suggest that protrusion initiation might be governed by a universal regulatory mechanism, whereas the mechanism of force generation determines the shape and expansion properties of the protrusion.

Lamellipodium tip actin barbed ends serve as a force sensor

Cells change direction of migration by sensing rigidity of environment and traction force, yet its underlying mechanism is unclear. Here, we show that tip actin barbed ends serve as an active "force sensor" at the leading edge. We established a method to visualize intracellular single-molecule fluorescent actin through an elastic culture substrate. We found that immediately after cell edge stretch, actin assembly increased specifically at the lamellipodium tip. The rate of actin assembly increased with increasing stretch speed. Furthermore, tip actin polymerization remained elevated at the subsequent hold step, which was accompanied by a decrease in the load on the tip barbed ends. Stretch-induced tip actin polymerization was still observed without either the WAVE complex or Ena/VASP proteins. The observed relationships between forces and tip actin polymerization are consistent with a force-velocity relationship as predicted by the Brownian ratchet mechanism. Stretch caused extra membrane protrusion with respect to the stretched substrate and increased local tip polymerization by >5% of total cellular actin in 30 s. Our data reveal that augmentation of lamellipodium tip actin assembly is directly coupled to the load decrease, which may serve as a force sensor for directed cell protrusion.

Building a dendritic actin filament network branch by branch: models of filament orientation pattern and force generation in lamellipodia

We review mathematical and computational models of the structure, dynamics, and force generation properties of dendritic actin networks. These models have been motivated by the dendritic nucleation model, which provided a mechanistic picture of how the actin cytoskeleton system powers cell motility. We describe how they aimed to explain the self-organization of the branched network into a bimodal distribution of filament orientations peaked at 35° and - 35° with respect to the direction of membrane protrusion, as well as other patterns. Concave and convex force-velocity relationships were derived, depending on network organization, filament, and membrane elasticity and accounting for actin polymerization at the barbed end as a Brownian ratchet. This review also describes models that considered the kinetics and transport of actin and diffuse regulators and mechanical coupling to a substrate, together with explicit modeling of dendritic networks.

Mechanotransduction by the Actin Cytoskeleton: Converting Mechanical Stimuli into Biochemical Signals

Force transmission through the actin cytoskeleton plays a central role in cell movements, shape change, and internal organization. Dynamic reorganization of actin filaments by an array of specialized binding proteins creates biochemically and architecturally distinct structures, many of which are finely tuned to exert or resist mechanical loads. The molecular complexity of the actin cytoskeleton continues to be revealed by detailed biochemical assays, and the architectural diversity and dynamics of actin structures are being uncovered by advances in super-resolution fluorescence microscopy and electron microscopy. However, our understanding of how mechanical forces feed back on cytoskeletal architecture and actin-binding protein organization is comparatively limited. In this review, we discuss recent work investigating how mechanical forces applied to cytoskeletal proteins are transduced into biochemical signals. We explore multiple mechanisms for mechanical signal transduction, including the mechanosensitive behavior of actin-binding proteins, the effect of mechanical force on actin filament dynamics, and the influence of mechanical forces on the structure of single actin filaments. The emerging picture is one in which the actin cytoskeleton is defined not only by the set of proteins that constitute a network but also by the constant interplay of mechanical forces and biochemistry.

MEDYAN: Mechanochemical Simulations of Contraction and Polarity Alignment in Actomyosin Networks

Active matter systems, and in particular the cell cytoskeleton, exhibit complex mechanochemical dynamics that are still not well understood. While prior computational models of cytoskeletal dynamics have lead to many conceptual insights, an important niche still needs to be filled with a high-resolution structural modeling framework, which includes a minimally-complete set of cytoskeletal chemistries, stochastically treats reaction and diffusion processes in three spatial dimensions, accurately and efficiently describes mechanical deformations of the filamentous network under stresses generated by molecular motors, and deeply couples mechanics and chemistry at high spatial resolution. To address this need, we propose a novel reactive coarse-grained force field, as well as a publicly available software package, named the Mechanochemical Dynamics of Active Networks (MEDYAN), for simulating active network evolution and dynamics (available at www.medyan.org). This model can be used to study the non-linear, far from equilibrium processes in active matter systems, in particular, comprised of interacting semi-flexible polymers embedded in a solution with complex reaction-diffusion processes. In this work, we applied MEDYAN to investigate a contractile actomyosin network consisting of actin filaments, alpha-actinin cross-linking proteins, and non-muscle myosin IIA mini-filaments. We found that these systems undergo a switch-like transition in simulations from a random network to ordered, bundled structures when cross-linker concentration is increased above a threshold value, inducing contraction driven by myosin II mini-filaments. Our simulations also show how myosin II mini-filaments, in tandem with cross-linkers, can produce a range of actin filament polarity distributions and alignment, which is crucially dependent on the rate of actin filament turnover and the actin filament’s resulting super-diffusive behavior in the actomyosin-cross-linker system. We discuss the biological implications of these findings for the arc formation in lamellipodium-to-lamellum architectural remodeling. Lastly, our simulations produce force-dependent accumulation of myosin II, which is thought to be responsible for their mechanosensation ability, also spontaneously generating myosin II concentration gradients in the solution phase of the simulation volume.

Active matter systems have the distinct ability to convert energy from their surroundings into mechanical work, which gives rise to them having highly dynamic properties. Modeling active matter systems and capturing their complex behavior has been a great challenge in past years due to the many coupled interactions between their constituent parts, including not only distinct chemical and mechanical properties, but also feedback between them. One of the most intriguing biological active matter systems is the cell cytoskeleton, which can dynamically respond to chemical and mechanical cues to control cell structure and shape, playing a central role in many higher-order cellular processes. To model these systems and reproduce their behavior, we present a new modeling approach which combines the chemical, mechanical, and molecular transport aspects of active matter systems, all represented with equivalent complexity, while also allowing for various forms of mechanochemical feedback. This modeling approach, named MEDYAN, and software implementation is flexible so that a wide range of active matter systems can be simulated with a high level of detail, and ultimately can help to describe active matter phenomena, and in particular, the dynamics of the cell cytoskeleton. In this work, we have used MEDYAN to simulate a cytoskeletal network consisting of actin filaments, cross-linking proteins, and myosin II molecular motors. We found that these systems show rich dynamical behaviors, undergoing alignment and bundling transitions, with an emergent contractility, as the concentrations of myosin II and cross-linking proteins, as well as actin filament turnover rates, are varied.

Self-propulsion and interactions of catalytic particles in a chemically active medium

Enzymatic "machines," such as catalytic rods or colloids, can self-propel and interact by generating gradients of their substrates. We theoretically investigate the behaviors of such machines in a chemically active environment where their catalytic substrates are continuously synthesized and destroyed, as occurs in living cells. We show how the kinetic properties of the medium modulate self-propulsion and pairwise interactions between machines, with the latter controlled by a tunable characteristic interaction range analogous to the Debye screening length in an electrolytic solution. Finally, we discuss the effective force arising between interacting machines and possible biological applications, such as partitioning of bacterial plasmids.

Branching influences force-velocity curves and length fluctuations in actin networks

We investigate collective dynamics of branched actin networks growing against a rigid movable wall constrained by a resistive force. Computing the force velocity relations, we show that the stall force of such networks depends not only on the average number of filaments touching the wall, but also on the amount of fluctuation of the leading edge of the network. These differences arise due to differences in the network architecture, namely, distance between two adjacent branching points and the initial distance of the starting filament from the wall, with their relative magnitudes influencing the nature of the force velocity curves (convex versus concave). We also show that the introduction of branching results in nonmonotonic diffusion constant, a quantity that measures the growth in length fluctuation of the leading edge of the network, as a function of externally applied force. Together our results demonstrate how the collective dynamics of a branched network differs from that of a parallel filament network.

Communication: Dominance of extreme statistics in a prototype many-body Brownian ratchet

Many forms of cell motility rely on Brownian ratchet mechanisms that involve multiple stochastic processes. We present a computational and theoretical study of the nonequilibrium statistical dynamics of such a many-body ratchet, in the specific form of a growing polymer gel that pushes a diffusing obstacle. We find that oft-neglected correlations among constituent filaments impact steady-state kinetics and significantly deplete the gel's density within molecular distances of its leading edge. These behaviors are captured quantitatively by a self-consistent theory for extreme fluctuations in filaments' spatial distribution.

Active mechanics and dynamics of cell spreading on elastic substrates

The spreading area of cells has been shown to play a central role in the determination of cell fate and tissue morphogenesis; however, a clear understanding of how spread cell area is determined is still lacking. The observation that cell area and force generally increase with substrate rigidity suggests that cell area is dictated mechanically, by means of a force-balance between the cell and the substrate. A simple mechanical model, corroborated by experimental measurements of cell area and force is presented to analyze the temporal force balance between the cell and the substrate during spreading. The cell is modeled as a thin elastic disc that is actively pulled by lamellipodia protrusions at the cell front. The essential molecular mechanisms of the motor activity at the cell front, including, actin polymerization, adhesion kinetics, and the actin retrograde flow, are accounted for and used to predict the dynamics of cell spreading on elastic substrates; simple, closed-form expressions for the evolution of cell size and force are derived. Time-resolved, traction force microscopy, combined with measurements of cell area are performed to investigate the simultaneous variations of cell size and force. We find that cell area and force increase simultaneously during spreading but the force develops with an apparent delay relative to the increase in cell area. We demonstrate that this may reflect the strain-stiffening property of the cytoskeleton. We further demonstrate that the radial cell force is a concave function of spreading speed and that this may reflect the strengthening of cell-substrate adhesions during spreading.

Multiscale modeling of cell shape from the actin cytoskeleton

The actin cytoskeleton is a dynamic structure that constantly undergoes complex reorganization events during many cellular processes. Mathematical models and simulations are powerful tools that can provide insight into the physical mechanisms underlying these processes and make predictions that can be experimentally tested. Representation of the interactions of the actin filaments with the plasma membrane and the movement of the plasma membrane for computation remains a challenge. Here, we provide an overview of the different modeling approaches used to study cytoskeletal dynamics and highlight the differential geometry approach that we have used to implement the interactions between the plasma membrane and the cytoskeleton. Using cell spreading as an example, we demonstrate how this approach is able to successfully capture in simulations, experimentally observed behavior. We provide a perspective on how the differential geometry approach can be used for other biological processes.

Control of actin-based motility through localized actin binding

A wide variety of cell biological and biomimetic systems use actin polymerization to drive motility. It has been suggested that an object such as a bacterium can propel itself by self-assembling a high concentration of actin behind it, if it is repelled by actin. However, it is also known that it is essential for the moving object to bind actin. Therefore, a key question is how the actin tail can propel an object when it both binds and repels the object. We present a physically consistent Brownian dynamics model for actin-based motility that includes the minimal components of the dendritic nucleation model and allows for both attractive and repulsive interactions between actin and a moveable disc. We find that the concentration gradient of filamentous actin generated by polymerization is sufficient to propel the object, even with moderately strong binding interactions. Additionally, actin binding can act as a biophysical cap, and may directly control motility through modulation of network growth. Overall, this mechanism is robust in that it can drive motility against a load up to a stall pressure that depends on the Young's modulus of the actin network and can explain several aspects of actin-based motility.

Molecular transport modulates the adaptive response of branched actin networks to an external force

Actin networks are an integral part of the cytoskeleton of eukaryotic cells and play an essential role in determining cellular shape and movement. Understanding the underlying mechanism of actin network assembly is of fundamental importance. We developed in this work a minimal motility model and performed stochastic simulations to study mechanical regulation of the growth dynamics of lamellipodia-like branched actin networks, characterized by various force-velocity relations. In such networks, the treadmilling process leads to a concentration gradient of G-actin, and thus G-actin transport is essential to effective actin network assembly. We first explore how capping protein modulates force-velocity relations and then discuss how actin transport due to diffusion and facilitated transport such as advective flow tunes the growth dynamics of the branched actin network. Our work demonstrates the important role of molecular transport in determining the adaptive response of the actin network to an external force.

Comparison of the force exerted by hippocampal and DRG growth cones

Mechanical properties such as force generation are fundamental for neuronal motility, development and regeneration. We used optical tweezers to compare the force exerted by growth cones (GCs) of neurons from the Peripheral Nervous System (PNS), such as Dorsal Root Ganglia (DRG) neurons, and from the Central Nervous System (CNS) such as hippocampal neurons. Developing GCs from dissociated DRG and hippocampal neurons were obtained from P1-P2 and P10-P12 rats. Comparing their morphology, we observed that the area of GCs of hippocampal neurons was 8-10 µm² and did not vary between P1-P2 and P10-P12 rats, but GCs of DRG neurons were larger and their area increased from P1-P2 to P10-P12 by 2-4 times. The force exerted by DRG filopodia was in the order of 1-2 pN and never exceeded 5 pN, while hippocampal filopodia exerted a larger force, often in the order of 5 pN. Hippocampal and DRG lamellipodia exerted lateral forces up to 20 pN, but lamellipodia of DRG neurons could exert a vertical force larger than that of hippocampal neurons. Force-velocity relationships (Fv) in both types of neurons had the same qualitative behaviour, consistent with a common autocatalytic model of force generation. These results indicate that molecular mechanisms of force generation of GC from CNS and PNS neurons are similar but the amplitude of generated force is influenced by their cytoskeletal properties.

There is more than one way to model an elephant. Experiment-driven modeling of the actin cytoskeleton

Mathematical modeling has established its value for investigating the interplay of biochemical and mechanical mechanisms underlying actin-based motility. Because of the complex nature of actin dynamics and its regulation, many of these models are phenomenological or conceptual, providing a general understanding of the physics at play. But the wealth of carefully measured kinetic data on the interactions of many of the players in actin biochemistry cries out for the creation of more detailed and accurate models that could permit investigators to dissect interdependent roles of individual molecular components. Moreover, no human mind can assimilate all of the mechanisms underlying complex protein networks; so an additional benefit of a detailed kinetic model is that the numerous binding proteins, signaling mechanisms, and biochemical reactions can be computationally organized in a fully explicit, accessible, visualizable, and reusable structure. In this review, we will focus on how comprehensive and adaptable modeling allows investigators to explain experimental observations and develop testable hypotheses on the intracellular dynamics of the actin cytoskeleton.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Unifying autocatalytic and zeroth-order branching models for growing actin networks

The directed polymerization of actin networks is an essential element of many biological processes, including cell migration. Different theoretical models considering the interplay between the underlying processes of polymerization, capping, and branching have resulted in conflicting predictions. One of the main reasons for this discrepancy is the assumption of a branching reaction that is either first order (autocatalytic) or zeroth order in the number of existing filaments. Here we introduce a unifying framework from which the two established scenarios emerge as limiting cases for low and high filament numbers. A smooth transition between the two cases is found at intermediate conditions. We also derive a threshold for the capping rate above which autocatalytic growth is predicted at sufficiently low filament number. Below the threshold, zeroth-order characteristics are predicted to dominate the dynamics of the network for all accessible filament numbers. Together, these mechanisms allow cells to grow stable actin networks over a large range of different conditions.

Crawling of a driven adherent membrane

We discuss motions of an elastic N × M membrane model whose constituents can bind reversibly with strength ε to adhesive sites of a flat substrate. One of the edges of the membrane ("front") is driven in one direction at rate constant p by N stochastically treadmilling short parallel lines ("cortex"). The main conclusions derived from Monte Carlo studies of this model are the following: (a) Since the polymerizing cortex pushes only the leading edge of the membrane, the major part of the membranes is dragged behind. Therefore, the locomotion of the membrane can be described by frictional sliding processes which are asymmetrically distributed between front and rear of the membrane. A signature of this asymmetry is the difference between the life times of adhesion bonds at front and rear, τ(1) and τ(M), respectively, where τ(1) ≫ τ(M). (b) There are four characteristic times for the membrane motion: The first time, T(0) ~ τ(M) ~ e(aε), is the resting time where the displacement of the membrane is practically zero. The second time, T(p) ~ τ(1) ~ M, is the friction time which characterizes the time between two consecutive ruptures of adhesion bonds at the front, and which signalizes the onset of drift ("protrusion") at the leading edge. The third time, T(r) ~ M(γ(ε)) (γ > 1), characterizes the "retraction" of the trailing edge, which is the retarded response to the pulling leading edge. The fourth time, T(L) ~ M(2), is the growth time for fluctuation of the end-to-end distance. (c) The separation of time scales, T(r)/T(p) ~ M(γ(ε) - 1), leads to stretched fluctuations of the end-to-end distance, which are considered as stochastic cycles of protrusion and retraction on the time scale of T(L). (d) The drift velocity v obeys anomalous scaling, v/p~f(p(1/γ(ε))M), where f(z) ~ const. for small drag pM ≪ 1, and f(z) ~ z(-γ(ε)) for pM ≫ 1, which implies v~M(-γ(ε)). These results may also turn out to be useful for the (more difficult) problem of understanding the protrusion-retraction cycle of crawling biological cells. We compare our model and our results to previous two-particle theories for membrane protrusion and to known stochastic friction models.

Branching and capping determine the force-velocity relationships of branching actin networks

A branching actin network is the major engine that drives cell motility. A measure of the effectiveness of an engine is the velocity the engine is able to produce at a given resistance-the force-velocity relationship. Concave force-velocity relationships consist of a force-insensitive region, indicative of an adaptive response. In contrast, convex force-velocity relationships would reflect a passive response. Even in in vitro experiments, branching actin networks can exhibit both concave and convex force-velocity curves. However, the exact mechanism that can explain both force-velocity curves is not yet known. We carried out an agent-based stochastic simulation to explore such a mechanism. We discovered an emergent behavior of a branching actin network: Upon resistance, it remodels itself by increasing the number of filaments growing in contact with the load. The remodeling is favored by branching events and limited by capping. The force-velocity relationship hinges on the relative time-scale between the intrinsic kinetics of the branching actin network and the loading. Shortly after encountering resistance (∼seconds), the force-velocity relationship of the actin network is always convex, as it does not have enough time to remodel itself. A concave force-velocity relationship requires network remodeling at longer time-scales (∼tens of seconds to minutes) and the faster branching event relative to capping. Furthermore, our model explains the observed hysteresis in the force-velocity relationship of actin networks. Our model thus establishes a unified mechanism that can account for both convex and concave force-velocity relationships observed in branching actin networks.

Mesoscopic model of actin-based propulsion

Two theoretical models dominate current understanding of actin-based propulsion: microscopic polymerization ratchet model predicts that growing and writhing actin filaments generate forces and movements, while macroscopic elastic propulsion model suggests that deformation and stress of growing actin gel are responsible for the propulsion. We examine both experimentally and computationally the 2D movement of ellipsoidal beads propelled by actin tails and show that neither of the two models can explain the observed bistability of the orientation of the beads. To explain the data, we develop a 2D hybrid mesoscopic model by reconciling these two models such that individual actin filaments undergoing nucleation, elongation, attachment, detachment and capping are embedded into the boundary of a node-spring viscoelastic network representing the macroscopic actin gel. Stochastic simulations of this ‘in silico’ actin network show that the combined effects of the macroscopic elastic deformation and microscopic ratchets can explain the observed bistable orientation of the actin-propelled ellipsoidal beads. To test the theory further, we analyze observed distribution of the curvatures of the trajectories and show that the hybrid model's predictions fit the data. Finally, we demonstrate that the model can explain both concave-up and concave-down force-velocity relations for growing actin networks depending on the characteristic time scale and network recoil. To summarize, we propose that both microscopic polymerization ratchets and macroscopic stresses of the deformable actin network are responsible for the force and movement generation.

There are two major ideas about how actin networks generate force against an obstacle: one is that the force comes directly from the elongation and bending of individual actin filaments against the surface of the obstacle; the other is that a growing actin gel can build up stress around the obstacle to squeeze it forward. Neither of the two models can explain why actin-propelled ellipsoidal beads move with equal bias toward long- and short-axes. We propose a hybrid model by combining those two ideas so that individual actin filaments are embedded into the boundary of a deformable actin gel. Simulations of this model show that the combined effects of pushing from individual filaments and squeezing from the actin network explain the observed bi-orientation of ellipsoidal beads as well as the curvature of trajectories of spherical beads and the force-velocity relation of actin networks.

How actin network dynamics control the onset of actin-based motility

Cells use their dynamic actin network to control their mechanics and motility. These networks are made of branched actin filaments generated by the Arp2/3 complex. Here we study under which conditions the microscopic organization of branched actin networks builds up a sufficient stress to trigger sustained motility. In our experimental setup, dynamic actin networks or "gels" are grown on a hard bead in a controlled minimal protein system containing actin monomers, profilin, the Arp2/3 complex and capping protein. We vary protein concentrations and follow experimentally and through simulations the shape and mechanical properties of the actin gel growing around beads. Actin gel morphology is controlled by elementary steps including "primer" contact, growth of the network, entanglement, mechanical interaction and force production. We show that varying the biochemical orchestration of these steps can lead to the loss of network cohesion and the lack of effective force production. We propose a predictive phase diagram of actin gel fate as a function of protein concentrations. This work unveils how, in growing actin networks, a tight biochemical and physical coupling smoothens initial primer-caused heterogeneities and governs force buildup and cell motility.

The role of membrane stiffness and actin turnover on the force exerted by DRG lamellipodia

We used optical tweezers to analyze the effect of jasplakinolide and cyclodextrin on the force exerted by lamellipodia from developing growth cones (GCs) of isolated dorsal root ganglia (DRG) neurons. We found that 25 nM of jasplakinolide, which is known to inhibit actin filament turnover, reduced both the maximal exerted force and maximal velocity during lamellipodia leading-edge protrusion. By using atomic force microscopy, we verified that cyclodextrin, which is known to remove cholesterol from membranes, decreased the membrane stiffness of DRG neurons. Lamellipodia treated with 2.5 mM of cyclodextrin exerted a larger force, and their leading edge could advance with a higher velocity. Neither jasplakinolide nor cyclodextrin affected force or velocity during lamellipodia retraction. The amplitude and frequency of elementary jumps underlying force generation were reduced by jasplakinolide but not by cyclodextrin. The action of both drugs at the used concentration was fully reversible. These results support the notion that membrane stiffness provides a selective pressure that shapes force generation, and confirm the pivotal role of actin turnover during protrusion.

Actin network growth under load

Many processes in eukaryotic cells, including the crawling motion of the whole cell, rely on the growth of branched actin networks from surfaces. In addition to their well-known role in generating propulsive forces, actin networks can also sustain substantial pulling loads thanks to their persistent attachment to the surface from which they grow. The simultaneous network elongation and surface attachment inevitably generate a force that opposes network growth. Here, we study the local dynamics of a growing actin network, accounting for simultaneous network elongation and surface attachment, and show that there exist several dynamical regimes that depend on both network elasticity and the kinetic parameters of actin polymerization. We characterize this in terms of a phase diagram and provide a connection between mesoscopic theories and the microscopic dynamics of an actin network at a surface. Our framework predicts the onset of instabilities that lead to the local detachment of the network and translate to oscillatory behavior and waves, as observed in many cellular phenomena and in vitro systems involving actin network growth, such as the saltatory dynamics of actin-propelled oil drops.

Actin filament curvature biases branching direction

Mechanical cues affect many important biological processes in metazoan cells, such as migration, proliferation, and differentiation. Such cues are thought to be detected by specialized mechanosensing molecules linked to the cytoskeleton, an intracellular network of protein filaments that provide mechanical rigidity to the cell and drive cellular shape change. The most abundant such filament, actin, forms branched networks nucleated by the actin-related protein (Arp) 2/3 complex that support or induce membrane protrusions and display adaptive behavior in response to compressive forces. Here we show that filamentous actin serves in a mechanosensitive capacity itself, by biasing the location of actin branch nucleation in response to filament bending. Using an in vitro assay to measure branching from curved sections of immobilized actin filaments, we observed preferential branch formation by the Arp2/3 complex on the convex face of the curved filament. To explain this behavior, we propose a fluctuation gating model in which filament binding or branch nucleation by Arp2/3 occur only when a sufficiently large, transient, local curvature fluctuation causes a favorable conformational change in the filament, and we show with Monte Carlo simulations that this model can quantitatively account for our experimental data. We also show how the branching bias can reinforce actin networks in response to compressive forces. These results demonstrate how filament curvature can alter the interaction of cytoskeletal filaments with regulatory proteins, suggesting that direct mechanotransduction by actin may serve as a general mechanism for organizing the cytoskeleton in response to force.

Modeling morphodynamic phenotypes and dynamic regimes of cell motion

Many cellular processes and signaling pathways converge onto cell morphology and cell motion, which share important components. The mechanisms used for propulsion could also be responsible for shape changes, if they are capable of generating the rich observed variety of dynamic regimes. Additionally, the analysis of cell shape changes in space and time promises insight into the state of the cytoskeleton and signaling pathways controlling it. While this has been obvious for some time by now, little effort has been made to systematically and quantitatively explore this source of information. First pioneering experimental work revealed morphodynamic phenotypes which can be associated with dynamic regimes like oscillations and excitability. Here, we review the current state of modeling of morphodynamic phenotypes, the experimental results and discuss the ideas on the mechanisms driving shape changes which are suggested by modeling.

Mechanochemical model of cell migration on substrates of varying stiffness

Cells propel themselves along a substrate by organizing structures at the leading edge called lamellipodia that contain the actin network, myosin, integrin, and other proteins. In this article, we describe a quantitative model that couples the response of stretch-sensitive proteins in the lamellipodia to the dynamics of the actin cytoskeleton, therefore allowing the cell to respond to different substrate stiffnesses. Using this model, we predict the various phases of dynamics possible, including continuous protrusion, unstable retractions leading to ruffling, and periodic protrusion-retraction cycles. We explain the necessary conditions for each type of migratory behavior to occur. In particular, we show that, for periodic protrusion-retraction cycles to occur, the stiffness of the substrate must be high, the myosin-dependent maturation rate of nascent to focal adhesions must be high, and the myosin-independent integrin activation rate must be low. In addition, we also predict the dynamics expected at a given substrate stiffness, leading to a quantitative explanation of experimental data that showed that periodic protrusion-retraction cycles disappear when cells are placed on soft substrates. We also suggest experiments with downregulating α actinin and/or talin and upregulating p130Cas and make predictions on what types of migratory dynamics will be observed.

How does the antagonism between capping and anti-capping proteins affect actin network dynamics?

Actin-based cell motility is essential to many biological processes. We built a simplified, three-dimensional computational model and subsequently performed stochastic simulations to study the growth dynamics of lamellipodia-like branched networks. In this work, we shed light on the antagonism between capping and anti-capping proteins in regulating actin dynamics in the filamentous network. We discuss detailed mechanisms by which capping and anti-capping proteins affect the protrusion speed of the actin network and the rate of nucleation of filaments. We computed a phase diagram showing the regimes of motility enhancement and inhibition by these proteins. Our work shows that the effects of capping and anti-capping proteins are mainly transmitted by modulation of the filamentous network density and local availability of monomeric actin. We discovered that the combination of the capping/anti-capping regulatory network with nucleation-promoting proteins introduces robustness and redundancy in cell motility machinery, allowing the cell to easily achieve maximal protrusion speeds under a broader set of conditions. Finally, we discuss distributions of filament lengths under various conditions and speculate on their potential implication for the emergence of filopodia from the lamellipodial network.

Cell motility: the integrating role of the plasma membrane

The plasma membrane is of central importance in the motility process. It defines the boundary separating the intracellular and extracellular environments, and mediates the interactions between a motile cell and its environment. Furthermore, the membrane serves as a dynamic platform for localization of various components which actively participate in all aspects of the motility process, including force generation, adhesion, signaling, and regulation. Membrane transport between internal membranes and the plasma membrane, and in particular polarized membrane transport, facilitates continuous reorganization of the plasma membrane and is thought to be involved in maintaining polarity and recycling of essential components in some motile cell types. Beyond its biochemical composition, the mechanical characteristics of the plasma membrane and, in particular, membrane tension are of central importance in cell motility; membrane tension affects the rates of all the processes which involve membrane deformation including edge extension, endocytosis, and exocytosis. Most importantly, the mechanical characteristics of the membrane and its biochemical composition are tightly intertwined; membrane tension and local curvature are largely determined by the biochemical composition of the membrane and the biochemical reactions taking place; at the same time, curvature and tension affect the localization of components and reaction rates. This review focuses on this dynamic interplay and the feedbacks between the biochemical and biophysical characteristics of the membrane and their effects on cell movement. New insight on these will be crucial for understanding the motility process.

Cytoskeletal actin networks in motile cells are critically self-organized systems synchronized by mechanical interactions

Growing networks of actin fibers are able to organize into compact, stiff two-dimensional structures inside lamellipodia of crawling cells. We put forward the hypothesis that the growing actin network is a critically self-organized system, in which long-range mechanical stresses arising from the interaction with the plasma membrane provide the selective pressure leading to organization. We show that a simple model based only on this principle reproduces the stochastic nature of lamellipodia protrusion (growth periods alternating with fast retractions) and several of the features observed in experiments: a growth velocity initially insensitive to the external force; the capability of the network to organize its orientation; a load-history-dependent growth velocity. Our model predicts that the spectrum of the time series of the height of a growing lamellipodium decays with the inverse of the frequency. This behavior is a well-known signature of self-organized criticality and is confirmed by unique optical tweezer measurements performed in vivo on neuronal growth cones.

Keratocyte lamellipodial protrusion is characterized by a concave force-velocity relation

We report on the characterization of actin driven lamellipodial protrusion forces and velocities in keratocytes. A vertically mounted glass fiber acted as a flexible barrier positioned in front of migrating keratocytes with parallel phase contrast microscopy. A laser beam was coupled into the fiber and allowed detecting the position of the fiber by a segmented photodiode. Calibration of the fiber was carried out with the thermal oscillation method. Deflection and force signals were measured during lamellipodial protrusion. Velocity was constant during initial contact whereas loading force increased until finally the cell was stalled at higher forces. Stall forces were on the order of 2.9 ± 0.6 nN, which corresponds to a stall pressure of 2.7 ± 1.6 nN/μm(2). Assuming a density of actin filaments of 240 filaments per μm, we can estimate a stall force per actin filament of 1.7 ± 0.8 pN. To check for adaption of the cell against an external force, we let the cell push toward the glass fiber several times. On the timescale of the experiment (∼1 min), however, the cell did not adapt to previous loading events.

Force-velocity measurements of a few growing actin filaments

The authors propose a new mechanism for actin-based force generation based on results using chains of actin-grafted magnetic colloids.

The polymerization of actin in filaments generates forces that play a pivotal role in many cellular processes. We introduce a novel technique to determine the force-velocity relation when a few independent anchored filaments grow between magnetic colloidal particles. When a magnetic field is applied, the colloidal particles assemble into chains under controlled loading or spacing. As the filaments elongate, the beads separate, allowing the force-velocity curve to be precisely measured. In the widely accepted Brownian ratchet model, the transduced force is associated with the slowing down of the on-rate polymerization. Unexpectedly, in our experiments, filaments are shown to grow at the same rate as when they are free in solution. However, as they elongate, filaments are more confined in the interspace between beads. Higher repulsive forces result from this higher confinement, which is associated with a lower entropy. In this mechanism, the production of force is not controlled by the polymerization rate, but is a consequence of the restriction of filaments' orientational fluctuations at their attachment point.

Actin self-assembles into filaments, and this produces forces that deform cell membranes in a large number of motile processes. While physical measurements have been performed of the force produced by growth of either a single filament or a large intricate array of filaments organized in an active macroscopic gel, these measurements don't provide a clear picture of how force is produced by the assembly of each filament within a complex structure. The present study explores a situation between these two extremes by measuring the force produced by the assembly of a small number of filaments. We developed a method in which actin filaments grow from the surface of magnetic beads that are aligned by a controlled magnetic field. The distance between beads in a chain-like arrangement increases with time when the force is kept constant. We observe that the growth of actin filaments is not affected by the load, in contrast to the widely accepted “Brownian ratchet model.” Instead, our results suggest that the surface opposite growing filaments imposes restrictions on the rotational fluctuations of a filament at its free hinge anchoring point, inducing a repulsive force. The confinement of filaments increases as they grow, and this in turn increases the repulsive force developed by their growth. This entropy-based mechanism may operate during motile processes when actin networks are loosely organized.

Signaling network triggers and membrane physical properties control the actin cytoskeleton-driven isotropic phase of cell spreading

Cell spreading is regulated by signaling from the integrin receptors that activate intracellular signaling pathways to control actin filament regulatory proteins. We developed a hybrid model of whole-cell spreading in which we modeled the integrin signaling network as ordinary differential equations in multiple compartments, and cell spreading as a three-dimensional stochastic model. The computed activity of the signaling network, represented as time-dependent activity levels of the actin filament regulatory proteins, is used to drive the filament dynamics. We analyzed the hybrid model to understand the role of signaling during the isotropic phase of fibroblasts spreading on fibronectin-coated surfaces. Simulations showed that the isotropic phase of spreading depends on integrin signaling to initiate spreading but not to maintain the spreading dynamics. Simulations predicted that signal flow in the absence of Cdc42 or WASP would reduce the spreading rate but would not affect the shape evolution of the spreading cell. These predictions were verified experimentally. Computational analyses showed that the rate of spreading and the evolution of cell shape are largely controlled by the membrane surface load and membrane bending rigidity, and changing information flow through the integrin signaling network has little effect. Overall, the plasma membrane acts as a damper such that only ∼5% of the actin dynamics capability is needed for isotropic spreading. Thus, the biophysical properties of the plasma membrane can condense varying levels of signaling network activities into a single cohesive macroscopic cellular behavior.

Robust organizational principles of protrusive biopolymer networks in migrating living cells

Cell migration is associated with the dynamic protrusion of a thin actin-based cytoskeletal extension at the cell front, which has been shown to consist of two different substructures, the leading lamellipodium and the subsequent lamellum. While the formation of the lamellipodium is increasingly well understood, organizational principles underlying the emergence of the lamellum are just beginning to be unraveled. We report here on a 1D mathematical model which describes the reaction-diffusion processes of a polarized actin network in steady state, and reproduces essential characteristics of the lamellipodium-lamellum system. We observe a steep gradient in filament lengths at the protruding edge, a local depolymerization maximum a few microns behind the edge, as well as a differential dominance of the network destabilizer ADF/cofilin and the stabilizer tropomyosin. We identify simple and robust organizational principles giving rise to the derived network characteristics, uncoupled from the specifics of any molecular implementation, and thus plausibly valid across cell types. An analysis of network length dependence on physico-chemical system parameters implies that to limit array treadmilling to cellular dimensions, network growth has to be truncated by mechanisms other than aging-induced depolymerization, e.g., by myosin-associated network dissociation at the transition to the cell body. Our work contributes to the analytical understanding of the cytoskeletal extension's bisection into lamellipodium and lamellum and sheds light on how cells organize their molecular machinery to achieve motility.

Mechanics and regulation of cell shape during the cell cycle

Many cell types undergo dramatic changes in shape throughout the cell cycle. For individual cells, a tight control of cell shape is crucial during cell division, but also in interphase, for example during cell migration. Moreover, cell cycle-related cell shape changes have been shown to be important for tissue morphogenesis in a number of developmental contexts. Cell shape is the physical result of cellular mechanical properties and of the forces exerted on the cell. An understanding of the causes and repercussions of cell shape changes thus requires knowledge of both the molecular regulation of cellular mechanics and how specific changes in cell mechanics in turn effect global shape changes. In this chapter, we provide an overview of the current knowledge on the control of cell morphology, both in terms of general cell mechanics and specifically during the cell cycle.

Observation and kinematic description of long actin tracks induced by spherical beads

We report an in vitro study comparing the growth of long actin tails induced by spherical beads coated with the verprolin central acidic domain of the polymerization enzyme N-WASP to that induced by Listeria monocytogenes in similar cellular extracts. The tracks behind the beads show characteristic differences in shape and curvature from those left by the bacteria, which have an elongated shape and a similar polymerization-inducing enzyme distributed only on the rear surface of the cell. The experimental tracks are simulated using a generalized kinematic model, which incorporates three modes of bead rotation with respect to the tail. The results show that the trajectories of spherical beads are mechanically deterministic rather than random, as suggested by stochastic models. Assessment of the bead rotation and its mechanistic basis offers insights into the biological function of actin-based motility.

Leading-edge-gel coupling in lamellipodium motion

We present a model for actin-based motility that combines the dynamics of the semiflexible region at the leading edge of the lamellipodium with actomyosin gel properties in the bulk described by the theory of active polar gels. We calculate the velocity of the lamellipodium determined by the interaction of the gel and adhesion with forces in the semiflexible region. The stationary concave force-velocity relation of the model reproduces experimental results. We suggest that it is determined by retrograde flow at small forces and gel formation and retrograde flow at large ones. The variety of dynamic regimes of the semiflexible region reproducing experimentally observed morphodynamics is conserved when we couple the leading edge to the gel.

Modeling the formation of in vitro filopodia

Filopodia are bundles of actin filaments that extend out ahead of the leading edge of a crawling cell to probe its upcoming environment. In vitro experiments (Vignjevic et al. in J Cell Biol 160:951-962, 2003) have determined the minimal ingredients required for the formation of filopodia from the dendritic-like morphology of the leading edge. We model these experiments using kinetic aggregation equations for the density of growing bundle tips. In mean field, we determine the bundle size distribution to be broad for bundle sizes smaller than a characteristic bundle size above which the distribution decays exponentially. Two-dimensional simulations incorporating both bundling and cross-linking measure a bundle size distribution that agrees qualitatively with mean field. The simulations also demonstrate a nonmonotonicity in the radial extent of the dendritic region as a function of capping protein concentration, as was observed in experiments, due to the interplay between percolation and the ratcheting of growing filaments off a spherical obstacle.