Force generation by cytoskeletal filament end-tracking proteins

Biophysics in tumor growth and progression: from single mechano-sensitive molecules to mechanomedicine

Evidence from physical sciences in oncology increasingly suggests that the interplay between the biophysical tumor microenvironment and genetic regulation has significant impact on tumor progression. Especially, tumor cells and the associated stromal cells not only alter their own cytoskeleton and physical properties but also remodel the microenvironment with anomalous physical properties. Together, these altered mechano-omics of tumor tissues and their constituents fundamentally shift the mechanotransduction paradigms in tumorous and stromal cells and activate oncogenic signaling within the neoplastic niche to facilitate tumor progression. However, current findings on tumor biophysics are limited, scattered, and often contradictory in multiple contexts. Systematic understanding of how biophysical cues influence tumor pathophysiology is still lacking. This review discusses recent different schools of findings in tumor biophysics that have arisen from multi-scale mechanobiology and the cutting-edge technologies. These findings range from the molecular and cellular to the whole tissue level and feature functional crosstalk between mechanotransduction and oncogenic signaling. We highlight the potential of these anomalous physical alterations as new therapeutic targets for cancer mechanomedicine. This framework reconciles opposing opinions in the field, proposes new directions for future cancer research, and conceptualizes novel mechanomedicine landscape to overcome the inherent shortcomings of conventional cancer diagnosis and therapies.

Integrative experimental/computational approach establishes active cellular protrusion as the primary driving force of phagocytic spreading by immune cells

The dynamic interplay between cell adhesion and protrusion is a critical determinant of many forms of cell motility. When modeling cell spreading on adhesive surfaces, traditional mathematical treatments often consider passive cell adhesion as the primary, if not exclusive, mechanistic driving force of this cellular motion. To better assess the contribution of active cytoskeletal protrusion to immune-cell spreading during phagocytosis, we here develop a computational framework that allows us to optionally investigate both purely adhesive spreading (“Brownian zipper hypothesis”) as well as protrusion-dominated spreading (“protrusive zipper hypothesis”). We model the cell as an axisymmetric body of highly viscous fluid surrounded by a cortex with uniform surface tension and incorporate as potential driving forces of cell spreading an attractive stress due to receptor-ligand binding and an outward normal stress representing cytoskeletal protrusion, both acting on the cell boundary. We leverage various model predictions against the results of a directly related experimental companion study of human neutrophil phagocytic spreading on substrates coated with different densities of antibodies. We find that the concept of adhesion-driven spreading is incompatible with experimental results such as the independence of the cell-spreading speed on the density of immobilized antibodies. In contrast, the protrusive zipper model agrees well with experimental findings and, when adapted to simulate cell spreading on discrete adhesion sites, it also reproduces the observed positive correlation between antibody density and maximum cell-substrate contact area. Together, our integrative experimental/computational approach shows that phagocytic spreading is driven by cellular protrusion, and that the extent of spreading is limited by the density of adhesion sites.

To accomplish many routine biological tasks, cells must rapidly spread over different types of surfaces. Here, we examine the biophysical underpinnings of immune cell spreading during phagocytosis, the process by which white blood cells such as neutrophils engulf pathogens or other foreign objects. Our computational framework models the case in which a human neutrophil spreads over a flat surface coated with antibodies, which we also test experimentally in a companion paper. Our primary purpose is to assess whether phagocytic spreading is actively driven by protrusive forces exerted by the cell, or passively by adhesive forces acting between receptors in the cell membrane and antibodies on the surface. By directly comparing our model predictions to experimental results, we demonstrate that phagocytic spreading is primarily driven by protrusion, but the extent of spreading is still limited by the availability of binding sites. Our findings improve the fundamental understanding of phagocytosis and may also pave the way for future investigations of the balance between adhesion and protrusion in other forms of cell spreading, such as wound healing or cancer cell migration.

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.

Ena/VASP proteins in cell edge protrusion, migration and adhesion

The tightly coordinated, spatiotemporal control of actin filament remodeling provides the basis of fundamental cellular processes, such as cell migration and adhesion. Specific protein assemblies, composed of various actin-binding proteins, are thought to operate in these processes to nucleate and elongate new filaments, arrange them into complex three-dimensional (3D) arrays and recycle them to replenish the actin monomer pool. Actin filament assembly is not only necessary to generate pushing forces against the leading edge membrane or to propel pathogens through the cytoplasm, but also coincides with the generation of stress fibers (SFs) and focal adhesions (FAs) that generate, transmit and sense mechanical tension. The only protein families known to date that directly enhance the elongation of actin filaments are formins and the family of Ena/VASP proteins. Their mechanisms of action, however, in enhancing processive filament elongation are distinct. The aim of this Review is to summarize our current knowledge on the molecular mechanisms of Ena/VASP-mediated actin filament assembly, and to discuss recent insights into the cell biological functions of Ena/VASP proteins in cell edge protrusion, migration and adhesion.

Materials for Smart Soft Actuator Systems

In contrast to conventional hard actuators, soft actuators offer many vivid advantages, such as improved flexibility, adaptability, and reconfigurability, which are intrinsic to living systems. These properties make them particularly promising for different applications, including soft electronics, surgery, drug delivery, artificial organs, or prosthesis. The additional degree of freedom for soft actuatoric devices can be provided through the use of intelligent materials, which are able to change their structure, macroscopic properties, and shape under the influence of external signals. The use of such intelligent materials allows a substantial reduction of a device's size, which enables a number of applications that cannot be realized by externally powered systems. This review aims to provide an overview of the properties of intelligent synthetic and living/natural materials used for the fabrication of soft robotic devices. We discuss basic physical/chemical properties of the main kinds of materials (elastomers, gels, shape memory polymers and gels, liquid crystalline elastomers, semicrystalline ferroelectric polymers, gels and hydrogels, other swelling polymers, materials with volume change during melting/crystallization, materials with tunable mechanical properties, and living and naturally derived materials), how they are related to actuation and soft robotic application, and effects of micro/macro structures on shape transformation, fabrication methods, and we highlight selected applications.

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.

Multi-Stimuli-Triggered Shape Transformation of Polymeric Filaments Derived from Dynamic Covalent Block Copolymers

Using dynamic polymers to achieve the morphology transformation of polymeric assemblies under different conditions is challenging. Herein, we reported diversiform shape transformation of multi-responsive polymer filaments, which were self-assembled by a new kind of amphiphilic block copolymer (PVEG-PVEA) possessing dynamic and reversible acylhydrazone bonds through reacting benzaldehyde-containing block copolymers poly(vinylbenzaldehyde)-b-poly(N-(4-vinylbenzyl)-N,N-diethylamine) (PVBA-PVEA) with acylhydrazine-modified oligoethylene glycol. It was found that the resulting amphiphilic and dynamic PVEG-PVEA was capable of hierarchically self-assembling into intriguing core-branched filaments in aqueous solution. Notably, the features of acylhydrazone bonds and PVEA block endow the filaments with multi-responsiveness including acid, base, and temperature, leading to the multiple morphological transformations under such stimuli. Moreover, the core-branched filaments would further transform into polymeric braided bundles driven by hydrogen-bonding interactions of amide bonds. It is noteworthy that both core-branched filaments and braided bundles made from polymers are quite rare. These diversiform polymeric assemblies and their morphological evolution were characterized by TEM, Cryo-TEM, SEM, and DLS. Finally, we used PVBA-PVEA as a platform to facilely prepare functional polymers, such as glycopolymers via the reaction of amino-containing sugars and aldehyde groups. The obtained glycopolymers self-assembled into glycofibers for the biomimicry of glycans via binding with lectins. These findings not only are conducive to understanding of the stimulated shape change process of dynamic polymeric assemblies in water but also provide a new method for the facile fabrication of smart and functional polymeric assemblies for different potential applications, such as biomimicry and targeted drug nanocarriers or delivery vehicles.

Thermodynamically consistent treatment of the growth of a biopolymer in the presence of a smooth obstacle interaction potential

We investigate the effect of filament-obstacle interactions on the force-velocity relation of growing biopolymers, via calculations explicitly treating obstacle diffusion and stochastic addition and subtraction of subunits. We first show that the instantaneous subunit on- and off-rates satisfy a rigorous thermodynamic relationship determined by the filament-obstacle interaction potential, which has been violated by several calculations in the literature. The instantaneous rates depend not only on the average force on the obstacle but also on the shape of the potential on the nanometer length scale. Basing obstacle-induced reduction of the on-rate entirely on the force, as previous work has often done, is thermodynamically inconsistent and can overestimate the stall force, sometimes by more than a factor of two. We perform simulations and analytic calculations of the force-velocity relation satisfying the thermodynamic relationship. The force-velocity relation can deviate strongly from the Brownian-Ratchet predictions. For shallow potential wells of depth ∼5k_{B}T, which might correspond to transient filament-membrane attachments, the velocity drops more rapidly than predicted by the Brownian-Ratchet model, in some cases by as much as a factor of 50 at an opposing force of only 1 pN. On the other hand, the zero-force velocity is much less affected than would be expected from naive use of the Boltzmann factor. Furthermore, the growth velocity has a surprisingly strong dependence on the obstacle diffusion coefficient even when the dimensionless diffusion coefficient is large. For deep potential wells, as might result from strong filament-membrane links, both the on- and off-rates are reduced significantly, slowing polymerization. Such potentials can sustain pulling forces while polymerizing but only if the attractive well is relatively flat over a region comparable to or greater than the monomer size. For double-well potentials, which have such a flat region, the slowing of polymerization by external pushing force is almost linear up to the stall force in some parameter ranges.

Pulling-force generation by ensembles of polymerizing actin filaments

The process by which actin polymerization generates pulling forces in cellular processes such as endocytosis is less well understood than pushing-force generation. To clarify the basic mechanisms of pulling-force generation, we perform stochastic polymerization simulations for a square array of polymerizing semiflexible actin filaments, having different interactions with the membrane. The filaments near the array center have a strong attractive component. Filament bending and actin-network elasticity are treated explicitly. We find that the outer filaments push on the membrane and the inner filaments pull, with a net balance of forces. The total calculated pulling force is maximized when the central filaments have a very deep potential well, and the outer filaments have no well. The steady-state force is unaffected by the gel rigidity, but equilibration takes longer for softer gels. The force distributions are flat over the pulling and pushing regions. Actin polymerization is enhanced by softening the gel or reducing the filament binding to the membrane. Filament-membrane detachment can occur for softer gels, even if the total binding energy of the filaments to the membrane is 100 [Formula: see text] or more. It propagates via a stress-concentration mechanism similar to that of a brittle crack in a solid, and the breaking stress is determined by a criterion similar to that of the 'Griffith' theory of crack propagation.

Modeling Cell Migration Mechanics

Cell migration is the physical movement of cells and is responsible for the extensive cellular invasion and metastasis that occur in high-grade tumors. Motivated by decades of direct observation of cell migration via light microscopy, theoretical models have emerged to capture various aspects of the fundamental physical phenomena underlying cell migration. Yet, the motility mechanisms actually used by tumor cells during invasion are still poorly understood, as is the role of cellular interactions with the extracellular environment. In this chapter, we review key physical principles of cytoskeletal self-assembly and force generation, membrane tension, biological adhesion, hydrostatic and osmotic pressures, and their integration in mathematical models of cell migration. With the goal of modeling-driven cancer therapy, we provide examples to guide oncologists and physical scientists in developing next-generation models to predict disease progression and treatment.

Network Contractility During Cytokinesis-from Molecular to Global Views

Cytokinesis is the last stage of cell division, which partitions the mother cell into two daughter cells. It requires the assembly and constriction of a contractile ring that consists of a filamentous contractile network of actin and myosin. Network contractility depends on network architecture, level of connectivity and myosin motor activity, but how exactly is the contractile ring network organized or interconnected and how much it depends on motor activity remains unclear. Moreover, the contractile ring is not an isolated entity; rather, it is integrated into the surrounding cortex. Therefore, the mechanical properties of the cell cortex and cortical behaviors are expected to impact contractile ring functioning. Due to the complexity of the process, experimental approaches have been coupled to theoretical modeling in order to advance its global understanding. While earlier coarse-grained descriptions attempted to provide an integrated view of the process, recent models have mostly focused on understanding the behavior of an isolated contractile ring. Here we provide an overview of the organization and dynamics of the actomyosin network during cytokinesis and discuss existing theoretical models in light of cortical behaviors and experimental evidence from several systems. Our view on what is missing in current models and should be tested in the future is provided.

Distinct VASP tetramers synergize in the processive elongation of individual actin filaments from clustered arrays

Ena/VASP proteins act as actin polymerases that drive the processive elongation of filament barbed ends in membrane protrusions or at the surface of bacterial pathogens. Based on previous analyses of fast and slow elongating VASP proteins by in vitro total internal reflection fluorescence microscopy (TIRFM) and kinetic and thermodynamic measurements, we established a kinetic model of Ena/VASP-mediated actin filament elongation. At steady state, it entails that tetrameric VASP uses one of its arms to processively track growing filament barbed ends while three G-actin-binding sites (GABs) on other arms are available to recruit and deliver monomers to the filament tip, suggesting that VASP operates as a single tetramer in solution or when clustered on a surface, albeit processivity and resistance toward capping protein (CP) differ dramatically between both conditions. Here, we tested the model by variation of the oligomerization state and by increase of the number of GABs on individual polypeptide chains. In excellent agreement with model predictions, we show that in solution the rates of filament elongation directly correlate with the number of free GABs. Strikingly, however, irrespective of the oligomerization state or presence of additional GABs, filament elongation on a surface invariably proceeded with the same rate as with the VASP tetramer, demonstrating that adjacent VASP molecules synergize in the elongation of a single filament. Additionally, we reveal that actin ATP hydrolysis is not required for VASP-mediated filament assembly. Finally, we show evidence for the requirement of VASP to form tetramers and provide an amended model of processive VASP-mediated actin assembly in clustered arrays.

Controlling amphiphilic copolymer self-assembly morphologies based on macrocycle/anion recognition and nucleotide-induced payload release

We report here a new approach to creating diversiform copolymer-derived self-assembly morphologies that relies on macrocycle/anion recognition in aqueous media. This approach exploits the anion binding features of a water-soluble form of the so-called 'Texas-sized' molecular box. When this tetracationic receptor is added to an aqueous solution of an amphiphilic copolymer bearing tethered carboxylate anion substituents, binding occurs to form a macrocycle/polymer complex. As the concentration of the box-like receptor increases, the relative hydrophilic fraction of the copolymer complex likewise increases. This leads to changes in the overall morphology of the self-assembled ensemble. The net result is an environmentally controllable system that mimics on a proof-of-concept level the structural evolution of organelles seen in living cells. The macrocycle/anion interactions respond in differing degrees to three key biological species, namely ATP, ADP, and AMP, which may be used as "inputs" to induce disassembly of these vehicles. As a result of this triggering and the nature of the morphological changes induced, the present copolymer system is capable of capturing and releasing in controlled manner various test payloads, including hydrophobic and hydrophilic fluorophores. The copolymer displays low inherent cytotoxicity as inferred from cell proliferation assays involving the HUVEC and HepG2 cell lines.

Statistics of actin-propelled trajectories in noisy environments

Actin polymerization is ubiquitously utilized to power the locomotion of eukaryotic cells and pathogenic bacteria in living systems. Inevitably, actin polymerization and depolymerization proceed in a fluctuating environment that renders the locomotion stochastic. Previously, we have introduced a deterministic model that manages to reproduce actin-propelled trajectories in experiments, but not to address fluctuations around them. To remedy this, here we supplement the deterministic model with noise terms. It enables us to compute the effects of fluctuating actin density and forces on the trajectories. Specifically, the mean-squared displacement (MSD) of the trajectories is computed and found to show a super-ballistic scaling with an exponent 3 in the early stage, followed by a crossover to a normal, diffusive scaling of exponent 1 in the late stage. For open-end trajectories such as straights and S-shaped curves, the time of crossover matches the decay time of orientational order of the velocities along trajectories, suggesting that it is the spreading of velocities that leads to the crossover. We show that the super-ballistic scaling of MSD arises from the initial, linearly increasing correlation of velocities, before time translational symmetry is established. When the spreading of velocities reaches a steady state in the long-time limit, short-range correlation then yields a diffusive scaling in MSD. In contrast, close-loop trajectories like circles exhibit localized periodic motion, which inhibits spreading. The initial super-ballistic scaling of MSD arises from velocity correlation that both linearly increases and oscillates in time. Finally, we find that the above statistical features of the trajectories transcend the nature of noises, be it additive or multiplicative, and generalize to other self-propelled systems that are not necessarily actin based.

Actoclampin (+)-end-tracking motors: How the pursuit of profilin's role(s) in actin-based motility twice led to the discovery of how cells crawl

The path to the discovery of the actoclampins began with efforts to define profilin's role in actin-based pathogen and endosome rocketing. That research identified a set of FPPPP-containing cargo proteins and FPPPP-binding proteins that are consistently stationed within the polymerization zone during episodes of active motility. The very same biophysical clues that forced us to abandon Brownian Ratchet models guided us to the Actoclampin Hypothesis, which asserts that every propulsive filament possesses a (+)-end-tracking motor that generates the forces cells need to crawl. Each actoclampin motor is a multi-arm oligomeric complex, employing one arm to recruit/deliver Profilin•Actin•ATP to a growth-site located at the (+)-end of the lagging subfilament, while a second arm maintains an affinity-modulated binding interaction with the extreme (+)-end of the other subfilament. The alternating actions of these arms define a true molecular motor, the processivity of which explains why propelling filaments maintain full possession of their cargo. The Actoclampin Hypothesis also suggests how the energetics of tracker interactions with the (+)-end determines whether a given actoclampin is a passive (low force-producing) or active (high force-producing) motor, the latter requiring the Gibbs free energy of ATP hydrolysis. Another aim of this review is to acknowledge an earlier notional model that emerged from efforts to comprehend profilin's pivotal role(s) in actin-based cell motility.

The flatness of Lamellipodia explained by the interaction between actin dynamics and membrane deformation

The crawling motility of many cell types relies on lamellipodia, flat protrusions spreading on flat substrates but (on cells in suspension) also growing into three-dimensional space. Lamellipodia consist of a plasma membrane wrapped around an oriented actin filament meshwork. It is well known that the actin density is controlled by coordinated polymerization, branching, and capping processes, but the mechanisms producing the small aspect ratios of lamellipodia (hundreds of nm thickness vs. several μm lateral and inward extension) remain unclear. The main hypothesis of this work is a strong influence of the local geometry of the plasma membrane on the actin dynamics. This is motivated by observations of co-localization of proteins with I-BAR domains (like IRSp53) with polymerization and branching agents along the membrane. The I-BAR domains are known to bind to the membrane and to prefer and promote membrane curvature. This hypothesis is translated into a stochastic mathematical model where branching and capping rates, and polymerization speeds depend on the local membrane geometry and branching directions are influenced by the principal curvature directions. This requires the knowledge of the deformation of the membrane, being described in a quasi-stationary approximation by minimization of a modified Helfrich energy, subject to the actin filaments acting as obstacles. Simulations with this model predict pieces of flat lamellipodia without any prescribed geometric restrictions.

Determinants of maximal force transmission in a motor-clutch model of cell traction in a compliant microenvironment

The mechanical stiffness of a cell's environment exerts a strong, but variable, influence on cell behavior and fate. For example, different cell types cultured on compliant substrates have opposite trends of cell migration and traction as a function of substrate stiffness. Here, we describe how a motor-clutch model of cell traction, which exhibits a maximum in traction force with respect to substrate stiffness, may provide a mechanistic basis for understanding how cells are tuned to sense the stiffness of specific microenvironments. We find that the optimal stiffness is generally more sensitive to clutch parameters than to motor parameters, but that single parameter changes are generally only effective over a small range of values. By contrast, dual parameter changes, such as coordinately increasing the numbers of both motors and clutches offer a larger dynamic range for tuning the optimum. The model exhibits distinct regimes: at high substrate stiffness, clutches quickly build force and fail (so-called frictional slippage), whereas at low substrate stiffness, clutches fail spontaneously before the motors can load the substrate appreciably (a second regime of frictional slippage). Between the two extremes, we find the maximum traction force, which occurs when the substrate load-and-fail cycle time equals the expected time for all clutches to bind. At this stiffness, clutches are used to their fullest extent, and motors are therefore resisted to their fullest extent. The analysis suggests that coordinate parameter shifts, such as increasing the numbers of motors and clutches, could underlie tumor progression and collective cell migration.

Formin mDia1 senses and generates mechanical forces on actin filaments

Cytoskeleton assembly is instrumental in the regulation of biological functions by physical forces. In a number of key cellular processes, actin filaments elongated by formins such as mDia are subject to mechanical tension, yet how mechanical forces modulate the assembly of actin filaments is an open question. Here, using the viscous drag of a microfluidic flow, we apply calibrated piconewton pulling forces to individual actin filaments that are being elongated at their barbed end by surface-anchored mDia1 proteins. We show that mDia1 is mechanosensitive and that the elongation rate of filaments is increased up to two-fold by the application of a pulling force. We also show that mDia1 is able to track a depolymerizing barbed end in spite of an opposing pulling force, which means that mDia1 can efficiently put actin filaments under mechanical tension. Our findings suggest that formin function in cells is tightly coupled to the mechanical activity of other machineries.

Plasma membrane--cortical cytoskeleton interactions: a cell biology approach with biophysical considerations

From a biophysical standpoint, the interface between the cell membrane and the cytoskeleton is an intriguing site where a "two-dimensional fluid" interacts with an exceedingly complex three-dimensional protein meshwork. The membrane is a key regulator of the cytoskeleton, which not only provides docking sites for cytoskeletal elements through transmembrane proteins, lipid binding-based, and electrostatic interactions, but also serves as the source of the signaling events and molecules that control cytoskeletal organization and remolding. Conversely, the cytoskeleton is a key determinant of the biophysical and biochemical properties of the membrane, including its shape, tension, movement, composition, as well as the mobility, partitioning, and recycling of its constituents. From a cell biological standpoint, the membrane-cytoskeleton interplay underlies--as a central executor and/or regulator--a multitude of complex processes including chemical and mechanical signal transduction, motility/migration, endo-/exo-/phagocytosis, and other forms of membrane traffic, cell-cell, and cell-matrix adhesion. The aim of this article is to provide an overview of the tight structural and functional coupling between the membrane and the cytoskeleton. As biophysical approaches, both theoretical and experimental, proved to be instrumental for our understanding of the membrane/cytoskeleton interplay, this review will "oscillate" between the cell biological phenomena and the corresponding biophysical principles and considerations. After describing the types of connections between the membrane and the cytoskeleton, we will focus on a few key physical parameters and processes (force generation, curvature, tension, and surface charge) and will discuss how these contribute to a variety of fundamental cell biological functions.

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.

Processive acceleration of actin barbed-end assembly by N-WASP

Clustered N-WASP binds directly to actin-filament barbed ends and can either slow individual filament growth or processively accelerate the assembly of bundled actin filaments. This novel Arp2/3-independent mechanism of N-WASP likely plays a role in invadopodia and podosome formation, in which both N-WASP and actin filaments are tightly clustered.

Neuronal Wiskott–Aldrich syndrome protein (N-WASP)–activated actin polymerization drives extension of invadopodia and podosomes into the basement layer. In addition to activating Arp2/3, N-WASP binds actin-filament barbed ends, and both N-WASP and barbed ends are tightly clustered in these invasive structures. We use nanofibers coated with N-WASP WWCA domains as model cell surfaces and single-actin-filament imaging to determine how clustered N-WASP affects Arp2/3-independent barbed-end assembly. Individual barbed ends captured by WWCA domains grow at or below their diffusion-limited assembly rate. At high filament densities, however, overlapping filaments form buckles between their nanofiber tethers and myosin attachment points. These buckles grew ∼3.4-fold faster than the diffusion-limited rate of unattached barbed ends. N-WASP constructs with and without the native polyproline (PP) region show similar rate enhancements in the absence of profilin, but profilin slows barbed-end acceleration from constructs containing the PP region. Increasing Mg²⁺ to enhance filament bundling increases the frequency of filament buckle formation, consistent with a requirement of accelerated assembly on barbed-end bundling. We propose that this novel N-WASP assembly activity provides an Arp2/3-independent force that drives nascent filament bundles into the basement layer during cell invasion.

Master equation-based analysis of a motor-clutch model for cell traction force

Microenvironmental mechanics play an important role in determining the morphology, traction, migration, proliferation, and differentiation of cells. A stochastic motor-clutch model has been proposed to describe this stiffness sensitivity. In this work, we present a master equation-based ordinary differential equation (ODE) description of the motor-clutch model, from which we derive an analytical expression to for a cell's optimum stiffness (i.e. the stiffness at which the traction force is maximal). This analytical expression provides insight into the requirements for stiffness sensing by establishing fundamental relationships between the key controlling cell-specific parameters. We find that the fundamental controlling parameters are the numbers of motors and clutches (constrained to be nearly equal), and the time scale of the on-off kinetics of the clutches (constrained to favor clutch binding over clutch unbinding). Both the ODE solution and the analytical expression show good agreement with Monte Carlo motor-clutch output, and reduce computation time by several orders of magnitude, which potentially enables long time scale behaviors (hours-days) to be studied computationally in an efficient manner. The ODE solution and the analytical expression may be incorporated into larger scale models of cellular behavior to bridge the gap from molecular time scales to cellular and tissue time scales.

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.

Actin cortex mechanics and cellular morphogenesis

The cortex is a thin, crosslinked actin network lying immediately beneath the plasma membrane of animal cells. Myosin motors exert contractile forces in the meshwork. Because the cortex is attached to the cell membrane, it plays a central role in cell shape control. The proteic constituents of the cortex undergo rapid turnover, making the cortex both mechanically rigid and highly plastic, two properties essential to its function. The cortex has recently attracted increasing attention and its functions in cellular processes such as cytokinesis, cell migration, and embryogenesis are progressively being dissected. In this review, we summarize current knowledge on the structural organization, composition, and mechanics of the actin cortex, focusing on the link between molecular processes and macroscopic physical properties. We also highlight consequences of cortex dysfunction in disease.

Cytokinesis in animal cells

Cytokinesis, the final step in cell division, partitions the contents of a single cell into two. In animal cells, cytokinesis occurs through cortical remodeling orchestrated by the anaphase spindle. Cytokinesis relies on a tight interplay between signaling and cellular mechanics and has attracted the attention of both biologists and physicists for more than a century. In this review, we provide an overview of four topics in animal cell cytokinesis: (a) signaling between the anaphase spindle and cortex, (b) the mechanics of cortical remodeling, (c) abscission, and (d) regulation of cytokinesis by the cell cycle machinery. We report on recent progress in these areas and highlight some of the outstanding questions that these findings bring into focus.

Actin filament attachments for sustained motility in vitro are maintained by filament bundling

We reconstructed cellular motility in vitro from individual proteins to investigate how actin filaments are organized at the leading edge. Using total internal reflection fluorescence microscopy of actin filaments, we tested how profilin, Arp2/3, and capping protein (CP) function together to propel thin glass nanofibers or beads coated with N-WASP WCA domains. Thin nanofibers produced wide comet tails that showed more structural variation in actin filament organization than did bead substrates. During sustained motility, physiological concentrations of Mg(2+) generated actin filament bundles that processively attached to the nanofiber. Reduction of total Mg(2+) abolished particle motility and actin attachment to the particle surface without affecting actin polymerization, Arp2/3 nucleation, or filament capping. Analysis of similar motility of microspheres showed that loss of filament bundling did not affect actin shell formation or symmetry breaking but eliminated sustained attachments between the comet tail and the particle surface. Addition of Mg(2+), Lys-Lys(2+), or fascin restored both comet tail attachment and sustained particle motility in low Mg(2+) buffers. TIRF microscopic analysis of filaments captured by WCA-coated beads in the absence of Arp2/3, profilin, and CP showed that filament bundling by polycation or fascin addition increased barbed end capture by WCA domains. We propose a model in which CP directs barbed ends toward the leading edge and polycation-induced filament bundling sustains processive barbed end attachment to the leading edge.

Blurred line between chemotactic chase and phagocytic consumption: an immunophysical single-cell perspective

An innate immune cell can sense a pathogen, either from a distance by recognizing chemoattractant stimuli or by direct physical contact. The pathogen is subsequently neutralized, which usually occurs through its phagocytic internalization. By investigating chemotaxis and phagocytosis from an immunophysical single-cell perspective, it now appears that the demarcation between these two processes is less distinct than originally thought. Several lines of evidence support this notion. First, chemotactic stimulation does not cease at the moment of initial contact between the cell and the pathogenic target. Second, even when classical chemotaxis of neutrophils is suppressed, the early cell response to contact with typical chemoattractant targets, such as zymosan, fungal spores or chemokine-coated particles, can still involve morphological attributes of chemotaxis. Recognizing that the changing morphology of motile cells is inextricably linked to physical cell behavior, this Commentary focuses on the mechanical aspects of the early response of innate immune cells to chemotactic and phagocytic stimuli. On the basis of this perspective, we propose that the combined study of chemotaxis and phagocytosis will, potentially, not only advance our grasp of the mechanisms underlying immune-cell motility but also open new lines of research that will promote a deeper understanding of the innate recognition of pathogens.

Choosing orientation: influence of cargo geometry and ActA polarization on actin comet tails

Networks of polymerizing actin filaments can propel intracellular pathogens and drive movement of artificial particles in reconstituted systems. While biochemical mechanisms activating actin network assembly have been well characterized, it remains unclear how particle geometry and large-scale force balance affect emergent properties of movement. We reconstituted actin-based motility using ellipsoidal beads resembling the geometry of Listeria monocytogenes. Beads coated uniformly with the L. monocytogenes ActA protein migrated equally well in either of two distinct orientations, with their long axes parallel or perpendicular to the direction of motion, while intermediate orientations were unstable. When beads were coated with a fluid lipid bilayer rendering ActA laterally mobile, beads predominantly migrated with their long axes parallel to the direction of motion, mimicking the orientation of motile L. monocytogenes. Generating an accurate biophysical model to account for our observations required the combination of elastic-propulsion and tethered-ratchet actin-polymerization theories. Our results indicate that the characteristic orientation of L. monocytogenes must be due to polarized ActA rather than intrinsic actin network forces. Furthermore, viscoelastic stresses, forces, and torques produced by individual actin filaments and lateral movement of molecular complexes must all be incorporated to correctly predict large-scale behavior in the actin-based movement of nonspherical particles.

Cofilin cooperates with fascin to disassemble filopodial actin filaments

Cells use a large repertoire of proteins to remodel the actin cytoskeleton. Depending on the proteins involved, F-actin is organized in specialized protrusions such as lamellipodia or filopodia, which serve diverse functions in cell migration and sensing. Although factors responsible for directed filament assembly in filopodia have been extensively characterized, the mechanisms of filament disassembly in these structures are mostly unknown. We investigated how the actin-depolymerizing factor cofilin-1 affects the dynamics of fascincrosslinked actin filaments in vitro and in live cells. By multicolor total internal reflection fluorescence microscopy and fluorimetric assays, we found that cofilin-mediated severing is enhanced in fascin-crosslinked bundles compared with isolated filaments, and that fascin and cofilin act synergistically in filament severing. Immunolabeling experiments demonstrated for the first time that besides its known localization in lamellipodia and membrane ruffles, endogenous cofilin can also accumulate in the tips and shafts of filopodia. Live-cell imaging of fluorescently tagged proteins revealed that cofilin is specifically targeted to filopodia upon stalling of protrusion and during their retraction. Subsequent electron tomography established filopodial actin filament and/or bundle fragmentation to precisely correlate with cofilin accumulation. These results identify a new mechanism of filopodium disassembly involving both fascin and cofilin.

THE BROWNIAN RATCHET REVISITED: DIFFUSION FORMALISM, POLYMER-BARRIER ATTRACTIONS, AND MULTIPLE FILAMENTOUS BUNDLE GROWTH

Actin polymerization driven stochastic movement of the bacteria Listeria monocytogenes is often measured using single-particle tracking (SPT) methodology and analyzed in terms of statistics. Experimental results suggested a dynamic association between the growing actin filaments and the propelled bacteria. Based on an alternative mathematical formalism for a Brownian ratchet (BR), we introduce such an attractive interaction into the one-dimensional BR model and show that its effect is equivalent to an external resistant force on the bacterium. Such a force significantly reduces the Brownian motion of a driven bacterium, and accentuates the stepping due to polymerization. We then consider the growth, with and without a barrier, of a filamentous bundle consisting of N identical filaments. It is shown that the bundle grows with a similar rate as a single filament in the absence of a load, but can oppose N times the external force under the stalling condition. A set of relationships describing the velocity of the bacterium movement (Vz) and its apparent diffusivity (Dz) as functions of the resistant force (F) and the number of filaments in a bundle (N) are obtained. The theoretical study suggests methods for data analysis in future experiments with applied external resistant force.

The role of actin bundling proteins in the assembly of filopodia in epithelial cells

The goal of this review is to highlight how emerging new models of filopodia assembly, which include tissue specific actin-bundling proteins, could provide more comprehensive representations of filopodia assembly that would describe more adequately and effectively the complexity and plasticity of epithelial cells.  This review also describes how the true diversity of actin bundling proteins must be considered to predict the far-reaching significance and versatile functions of filopodia in epithelial cells.

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.

A theoretical analysis of filament length fluctuations in actin and other polymers

Control of the structure and dynamics of the actin cytoskeleton is essential for cell motility and for maintaining the structural integrity of cells. Central to understanding the control of these features is an understanding of the dynamics of actin filaments, first as isolated filaments, then as integrated networks, and finally as networks containing higher-order structures such as bundles, stress fibers and acto-myosin complexes. It is known experimentally that single filaments can exhibit large fluctuations, but a detailed understanding of the transient dynamics involved is still lacking. Here we first study stochastic models of a general system involving two-monomer types that can be analyzed completely, and then we report stochastic simulations on the complete actin model with three monomer types. We systematically examine the transient behavior of filament length dynamics so as to gain a better understanding of the time scales involved in reaching a steady state. We predict the lifetime of a cap of one monomer type and obtain the mean and variance of the survival time of a cap at the filament end, which together determine the filament length fluctuations.

Molecular mechanism of Ena/VASP-mediated actin-filament elongation

Ena/VASP proteins are implicated in a variety of fundamental cellular processes including axon guidance and cell migration. In vitro, they enhance elongation of actin filaments, but at rates differing in nearly an order of magnitude according to species, raising questions about the molecular determinants of rate control. Chimeras from fast and slow elongating VASP proteins were generated and their ability to promote actin polymerization and to bind G-actin was assessed. By in vitro TIRF microscopy as well as thermodynamic and kinetic analyses, we show that the velocity of VASP-mediated filament elongation depends on G-actin recruitment by the WASP homology 2 motif. Comparison of the experimentally observed elongation rates with a quantitative mathematical model moreover revealed that Ena/VASP-mediated filament elongation displays a saturation dependence on the actin monomer concentration, implying that Ena/VASP proteins, independent of species, are fully saturated with actin in vivo and generally act as potent filament elongators. Moreover, our data showed that spontaneous addition of monomers does not occur during processive VASP-mediated filament elongation on surfaces, suggesting that most filament formation in cells is actively controlled.

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.

A microscopic formulation for the actin-driven motion of listeria in curved paths

Using a generalized Brownian ratchet model that accounts for the interactions of actin filaments with the surface of Listeria mediated by proteins like ActA and Arp2/3, we have developed a microscopic model for the movement of Listeria. Specifically, we show that a net torque can be generated within the comet tail, causing the bacteria to spin about its long axis, which in conjunction with spatially varying polymerization at the surface leads to motions of bacteria in curved paths that include circles, sinusoidal-like curves, translating figure eights, and serpentine shapes, as observed in recent experiments. A key ingredient in our formulation is the coupling between the motion of Listeria and the force-dependent rate of filament growth. For this reason, a numerical scheme was developed to determine the kinematic parameters of motion and stress distribution among filaments in a self-consistent manner. We find that a 5-15% variation in polymerization rates can lead to radii of curvatures of the order of 4-20 microm, measured in experiments. In a similar way, our results also show that most of the observed trajectories can be produced by a very low degree of correlation, <10%, among filament orientations. Since small fluctuations in polymerization rate, as well as filament orientation, can easily be induced by various factors, our findings here provide a reasonable explanation for why Listeria can travel along totally different paths under seemingly identical experimental conditions. Besides trajectories, stress distributions corresponding to different polymerization profiles are also presented. We have found that although some actin filaments generate propelling forces that push the bacteria forward, others can exert forces opposing the movement of Listeria, consistent with recent experimental observations.

Control of actin filament treadmilling in cell motility

Recent advances in structural, biochemical, biophysical, and live cell imaging approaches have furthered our understanding of the molecular mechanisms by which regulated assembly dynamics of actin filaments drive motile processes. Attention is focused on lamellipodium protrusion, powered by the turnover of a branched filament array. ATP hydrolysis on actin is the key reaction that allows filament treadmilling. It regulates barbed-end dynamics and length fluctuations at steady state and specifies the functional interaction of actin with essential regulatory proteins such as profilin and ADF/cofilin. ATP hydrolysis on actin and Arp2/3 acts as a timer, regulating the assembly and disassembly of the branched array to generate tropomyosin-mediated heterogeneity in the structure and dynamics of the lamellipodial network. The detailed molecular mechanisms of ATP hydrolysis/Pi release on F-actin remain elusive, as well as the mechanism of filament branching with Arp2/3 complex or that of the formin-driven processive actin assembly. Novel biophysical methods involving single-molecule measurements should foster progress in these crucial issues.

Force generation in lamellipodia is a probabilistic process with fast growth and retraction events

Polymerization of actin filaments is the primary source of motility in lamellipodia and it is controlled by a variety of regulatory proteins. The underlying molecular mechanisms are only partially understood and a precise determination of dynamical properties of force generation is necessary. Using optical tweezers, we have measured with millisecond (ms) temporal resolution and picoNewton (pN) sensitivity the force-velocity (Fv) relationship and the power dissipated by lamellipodia of dorsal root ganglia neurons. When force and velocity are averaged over 3-5 s, the Fv relationships can be flat. On a finer timescale, random occurrence of fast growth and subsecond retractions become predominant. The maximal power dissipated by lamellipodia over a silica bead with a diameter of 1 microm is 10(-16) W. Our results clarify the dynamical properties of force generation: i), force generation is a probabilistic process; ii), underlying biological events have a bandwidth up to at least 10 Hz; and iii), fast growth of lamellipodia leading edge alternates with local retractions.

Crowding effects on association reactions at membranes

The effect of macromolecular crowding on the binding of ligands to a receptor near membranes is studied using Brownian dynamics simulations. The receptor is modeled as a reactive patch on a hard surface and the ligands and crowding agents are modeled as spheres that interact via a steep repulsive interaction potential. When a ligand collides with the patch, it reacts with probability p(rxn). The association rate constant (k(infinity)) can be decomposed into contributions from diffusion-limited (k(D)) and reaction-limited (k(R)) rates, i.e., 1/k(infinity) = 1/k(D) + 1/k(R). The simulations show that k(D) is a nonmonotonic function of the volume fraction of crowding agents for receptors of small sizes. k(R) is always an increasing function of the volume fraction of crowding agents, and the association rate constant k(infinity) determined from both contributions has a qualitatively different dependence on the macromolecular crowding for high and low values of the reaction probability p(rxn). The simulation results are used to predict the velocity of the membrane protrusion driven by actin filament elongation. Based on the simple model where the protrusive force on the membrane is generated by the intercalation of actin monomers between the membrane and actin filament ends, we predict that crowding increases the local concentration of actin monomers near the filament ends and hence accelerates the membrane protrusion.

Effects of molecular-scale processes on observable growth properties of actin networks

The properties of actin network growth against a flat obstacle are studied using several different sets of molecular-level assumptions regarding filament growth and nucleation. These assumptions are incorporated into a multifilament methodology which treats both the distribution of filament orientations and bending of filaments. Three single-filament force-generation mechanisms in the literature are compared within this framework. Each mechanism is treated using two different filament nucleation modes, namely, spontaneous nucleation and branching off pre-existing filaments. We find that the shape of the force-velocity relation depends mainly on the ratio of the thermodynamic and mechanical stall forces of the filaments. If the thermodynamic stall force greatly exceeds the mechanical stall force, the velocity drops abruptly to zero when the mechanical stall force is reached; otherwise, it goes more gradually to zero. In addition, branching nucleation gives a steeper increase in the filament number with opposing force than spontaneous nucleation does. Finally, the zero-force velocity of the obstacle as a function of the detachment and capping rates differs significantly between the different single-filament growth mechanisms. Experiments are proposed to use these differences to discriminate between the network growth models.

Force-induced myofibroblast differentiation through collagen receptors is dependent on mammalian diaphanous (mDia)

The development of fibrosis promotes the differentiation of myofibroblasts, pro-fibrotic cells, which contribute to tissue dysfunction. Myofibroblast differentiation is dependent on actin assembly, which in response to force, is mediated by various actin-binding proteins including the mammalian Diaphanous-related formins (mDia). We examined the role of mDia in the mechano-sensing pathway that leads to force-induced expression of alpha-smooth muscle actin (SMA), a marker and critical determinant of myofibroblast differentiation. In cells treated with siRNA to knockdown mDia and then subjected to tensile force using collagen-coated magnetite beads attached to beta1 integrins, actin assembly was inhibited at bead contact sites. Force-induced nuclear translocation of MRTF-A, a transcriptional co-activator of SMA, was reduced 50% by mDia knockdown. The expression of the transcriptional co-activator of SMA, serum response factor, was reduced by 50% after siRNA knockdown of mDia or by 100% in cells transfected with catalytically inactive mDia. Force-induced activation of the SMA promoter and SMA expression were blocked by knockdown of siRNA of mDia. In anchored collagen gel assays to measure myofibroblast-mediated contraction, knockdown of mDia reduced contraction by 50%. We conclude that mDia plays an important role in the development of force-induced transcriptional activation of SMA and myofibroblast differentiation.