Mesoscopic model of actin-based propulsion

Physical principles and mechanisms of cell migration

Cell migration is critical in processes such as developmental biology, wound healing, immune response, and cancer invasion/metastasis. Understanding its regulation is essential for developing targeted therapies in regenerative medicine, cancer treatment and immune modulation. This review examines cell migration mechanisms, highlighting fundamental physical principles, key molecular components, and cellular behaviors, identifying existing gaps in current knowledge, and suggesting potential directions for future research.

Cellular and Nuclear Forces: An Overview

Biological cells sample their surrounding microenvironments using nanoscale force sensors on the cell surfaces. These surface-based force and stress sensors generate physical and chemical responses inside the cell. The inherently well-connected cytoskeleton and its physical contacts with the force elements on the nuclear membrane lead these physicochemical responses to cascade all the way inside the cell nucleus, physically altering the nuclear state. These physical alterations of the cell nucleus, through yet-unknown complex steps, elicit physical and functional responses from the chromatin in the form of altered gene expression profiles. This mechanism of force/stress sensing by the cell and then its nuclear response has been shown to play a vital role in maintaining robust cellular homeostasis, controlling gene expression profiles during developmental phases as well as cell differentiation. In the last few years, there has been appreciable progress toward the identification of the molecular players responsible for force sensing. However, the actual sensing mechanism of cell surface-bound force sensors and more importantly cascading of the signals, both physical (via cytosolic force sensing elements such as microtubule and actin framework) as well as chemical (cascade of biochemical signaling from cell surface to nuclear surface and further to the chromatin), inside the cell is poorly understood. In this chapter, we present a review of the currently known molecular players in cellular as well as nuclear force sensing repertoire and their possible mechanistic aspects. We also introduce various biophysical concepts and review some frequently used techniques that are used to describe the force/stress sensing and response of a cell. We hope that this will help in asking clearer questions and designing pointed experiments for better understanding of the force-dependent design principles of the cell surface, nuclear surface, and gene expression.

Energy-based modelling of single actin filament polymerization using bond graphs

Bond graphs provide an energy-based methodology for modelling complex systems hierarchically; at the moment, the method allows biological systems with both chemical and electrical subsystems to be modelled. Herein, the bond graph approach is extended to include chemomechanical transduction thus extending the range of biological systems to be modelled. Actin filament polymerization and force generation is used as an example of chemomechanical transduction, and it is shown that the TF (transformer) bond graph component provides a practical, and conceptually simple, alternative to the Brownian ratchet approach of Peskin, Odell, Oster and Mogilner. Furthermore, it is shown that the bond graph approach leads to the same equation as the Brownian ratchet approach in the simplest case. The approach is illustrated by showing that flexibility and non-normal incidence can be modelled by simply adding additional bond graph components and that compliance leads to non-convexity of the force-velocity curve. Energy flows are fundamental to life; for this reason, the energy-based approach is utilized to investigate the power transmission by the actin filament and its corresponding efficiency. The bond graph model is fitted to experimental data by adjusting the model physical parameters.

Rigidity in mechanical biological networks

Multicellular organisms generate complex morphologies required for their function. Organisms control these morphologies by tuning active forces and by altering the emergent 'material properties' of a tissue, i.e. the rheology of the tissue. In many cases, organisms take advantage of dramatic changes in the rheology that occur when the material undergoes a rigidity transition from a fluid-like or floppy state to a solid-like or rigid state. This transition in turn depends on internal parameters at the scale of cells and molecules. This review highlights recent theoretical work identifying the mechanisms that drive such transitions, so that biologists can look for these mechanisms in in vivo or in vitro systems. We discuss two main types of transition: a first-order rigidity transition that depends on the connectivity of small-scale structures, such as the number of contacts between cells or the number of branch points in a biopolymer network; and a second-order rigidity transition that depends on the geometry of small-scale structures, such as the shape of cells or the distance between crosslinks in a polymer network. We provide examples of each type of transition in model organisms and discuss methods for distinguishing between the mechanisms in future experiments.

Myosin-I synergizes with Arp2/3 complex to enhance the pushing forces of branched actin networks

Class I myosins (myosin-Is) colocalize with Arp2/3 complex-nucleated actin networks at sites of membrane protrusion and invagination, but the mechanisms by which myosin-I motor activity coordinates with branched actin assembly to generate force are unknown. We mimicked the interplay of these proteins using the "comet tail" bead motility assay, where branched actin networks are nucleated by the Arp2/3 complex on the surface of beads coated with myosin-I and nucleation-promoting factor. We observed that myosin-I increased bead movement efficiency by thinning actin networks without affecting growth rates. Myosin-I triggered symmetry breaking and comet tail formation in dense networks resistant to spontaneous fracturing. Even with arrested actin assembly, myosin-I alone could break the network. Computational modeling recapitulated these observations, suggesting myosin-I acts as a repulsive force shaping the network's architecture and boosting its force-generating capacity. We propose that myosin-I leverages its power stroke to amplify the forces generated by Arp2/3 complex-nucleated actin networks.

Mechanically induced topological transition of spectrin regulates its distribution in the mammalian cell cortex

The cell cortex is a dynamic assembly formed by the plasma membrane and underlying cytoskeleton. As the main determinant of cell shape, the cortex ensures its integrity during passive and active deformations by adapting cytoskeleton topologies through yet poorly understood mechanisms. The spectrin meshwork ensures such adaptation in erythrocytes and neurons by adopting different organizations. Erythrocytes rely on triangular-like lattices of spectrin tetramers, whereas in neurons they are organized in parallel, periodic arrays. Since spectrin is ubiquitously expressed, we exploited Expansion Microscopy to discover that, in fibroblasts, distinct meshwork densities co-exist. Through biophysical measurements and computational modeling, we show that the non-polarized spectrin meshwork, with the intervention of actomyosin, can dynamically transition into polarized clusters fenced by actin stress fibers that resemble periodic arrays as found in neurons. Clusters experience lower mechanical stress and turnover, despite displaying an extension close to the tetramer contour length. Our study sheds light on the adaptive properties of spectrin, which participates in the protection of the cell cortex by varying its densities in response to key mechanical features.

Myosin-I Synergizes with Arp2/3 Complex to Enhance Pushing Forces of Branched Actin Networks

Myosin-Is colocalize with Arp2/3 complex-nucleated actin networks at sites of membrane protrusion and invagination, but the mechanisms by which myosin-I motor activity coordinates with branched actin assembly to generate force are unknown. We mimicked the interplay of these proteins using the "comet tail" bead motility assay, where branched actin networks are nucleated by Arp2/3 complex on the surface of beads coated with myosin-I and the WCA domain of N-WASP. We observed that myosin-I increased bead movement efficiency by thinning actin networks without affecting growth rates. Remarkably, myosin-I triggered symmetry breaking and comet-tail formation in dense networks resistant to spontaneous fracturing. Even with arrested actin assembly, myosin-I alone could break the network. Computational modeling recapitulated these observations suggesting myosin-I acts as a repulsive force shaping the network's architecture and boosting its force-generating capacity. We propose that myosin-I leverages its power stroke to amplify the forces generated by Arp2/3 complex-nucleated actin networks.

Reconstitution of actin-based cellular processes: Why encapsulation changes the rules

While in vitro reconstitution of cellular processes is progressing rapidly, the encapsulation of biomimetic systems to reproduce the cellular environment is a major challenge. Here we review the difficulties, using reconstitution of processes dependent on actin polymerization as an example. Some of the problems are purely technical, due to the need for engineering strategies to encapsulate concentrated solutions in micrometer-sized compartments. However, other significant issues arise from the reduction of experimental volumes, which alters the chemical evolution of these non-equilibrium systems. Important parameters to consider for successful reconstitutions are the amount of each component, their consumption and renewal rates to guarantee their continuous availability.

Mechanically induced topological transition of spectrin regulates its distribution in the mammalian cortex

The cell cortex is a dynamic assembly that ensures cell integrity during passive deformation or active response by adapting cytoskeleton topologies with poorly understood mechanisms. The spectrin meshwork ensures such adaptation in erythrocytes and neurons. Erythrocytes rely on triangular-like lattices of spectrin tetramers, which in neurons are organized in periodic arrays. We exploited Expansion Microscopy to discover that these two distinct topologies can co-exist in other mammalian cells such as fibroblasts. We show through biophysical measurements and computational modeling that spectrin provides coverage of the cortex and, with the intervention of actomyosin, erythroid-like lattices can dynamically transition into condensates resembling neuron-like periodic arrays fenced by actin stress fibers. Spectrin condensates experience lower mechanical stress and turnover despite displaying an extension close to the contour length of the tetramer. Our study sheds light on the adaptive properties of spectrin, which ensures protection of the cortex by undergoing mechanically induced topological transitions.

A computational modeling of invadopodia protrusion into an extracellular matrix fiber network

Invadopodia are dynamic actin-rich membrane protrusions that have been implicated in cancer cell invasion and metastasis. In addition, invasiveness of cancer cells is strongly correlated with invadopodia formation, which are observed during extravasation and colonization of metastatic cancer cells at secondary sites. However, quantitative understanding of the interaction of invadopodia with extracellular matrix (ECM) is lacking, and how invadopodia protrusion speed is associated with the frequency of protrusion-retraction cycles remains unknown. Here, we present a computational framework for the characterization of invadopodia protrusions which allows two way interactions between intracellular branched actin network and ECM fibers network. We have applied this approach to predicting the invasiveness of cancer cells by computationally knocking out actin-crosslinking molecules, such as α-actinin, filamin and fascin. The resulting simulations reveal distinct invadopodia dynamics with cycles of protrusion and retraction. Specifically, we found that (1) increasing accumulation of MT1-MMP at tips of invadopodia as the duration of protrusive phase is increased, and (2) the movement of nucleus toward the leading edge of the cell becomes unstable as duration of the retractile phase (or myosin turnover time) is longer than 1 min.

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.

Cellular fate of deformable needle-shaped PLGA-PEG fibers

Deformability of micro/nanometer sized particles plays an important role in particle-cell interactions and thus becomes a key parameter in carrier design in biomedicine application such as drug delivery and vaccinology. Yet the influence of material's deformability on the cellular fate of the particles as well as physiology response of live cells are to be understood. Here we show the cellular fate of needle shaped (high aspect ratio ~25) PLGA-PEG copolymer fibers depending on their deformability. We found that all the fibers entered murine macrophage cells (RAW 264.7) via phagocytosis. While the fibers of high apparent Young's modulus (average value = 872 kPa) maintained their original shape upon phagocytosis, their counterparts of low apparent Young's modulus (average value = 56 kPa) curled in cells. The observed deformation of fibers of low apparent Young's modulus in cells coincided with abnormal intracellular actin translocation and absence of lysosome/phagosome fusion in macrophages, suggesting the important role of material mechanical properties and mechano-related cellular pathway in affecting cell physiology. STATEMENT OF SIGNIFICANCE: Particles are increasingly important in the field of biomedicine, especially when they are serving as drug carriers. Physical cues, such as mechanical properties, were shown to provide insight into their stability and influence on physiology inside the cell. In the current study, we managed to fabricate 5 types of needle shaped PLGA-PEG fibers with controlled Young's modulus. We found that hard fibers maintained their original shape upon phagocytosis, while soft fibers were curled by actin compressive force inside the cell, causing abnormal actin translocation and impediment of lysosome/phagosome fusion, suggesting the important role of material mechanical properties and mechano-related cellular pathway in affecting cell physiology.

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.

Efficient simulation of thermally fluctuating biopolymers immersed in fluids on 1-micron, 1-second scales

The combination of fluid-structure interactions with stochasticity, due to thermal fluctuations, remains a challenging problem in computational fluid dynamics. We develop an efficient scheme based on the stochastic immersed boundary method, Stokeslets, and multiple timestepping. We test our method for spherical particles and filaments under purely thermal and deterministic forces and find good agreement with theoretical predictions for Brownian Motion of a particle and equilibrium thermal undulations of a semi-flexible filament. As an initial application, we simulate bio-filaments with the properties of F-actin. We specifically study the average time for two nearby parallel filaments to bundle together. Interestingly, we find a two-fold acceleration in this time between simulations that account for long-range hydrodynamics compared to those that do not, suggesting that our method will reveal significant hydrodynamic effects in biological phenomena.

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.

Molecular mechanisms of force production in clathrin-mediated endocytosis

During clathrin-mediated endocytosis (CME), a flat patch of membrane is invaginated and pinched off to release a vesicle into the cytoplasm. In yeast CME, over 60 proteins-including a dynamic actin meshwork-self-assemble to deform the plasma membrane. Several models have been proposed for how actin and other molecules produce the forces necessary to overcome the mechanical barriers of membrane tension and turgor pressure, but the precise mechanisms and a full picture of their interplay are still not clear. In this review, we discuss the evidence for these force production models from a quantitative perspective and propose future directions for experimental and theoretical work that could clarify their various contributions.

Actin filaments growing against an elastic membrane: Effect of membrane tension

We study the force generation by a set of parallel actin filaments growing against an elastic membrane. The elastic membrane tries to stay flat and any deformation from this flat state, either caused by thermal fluctuations or due to protrusive polymerization force exerted by the filaments, costs energy. We study two lattice models to describe the membrane dynamics. In one case, the energy cost is assumed to be proportional to the absolute magnitude of the height gradient (gradient model) and in the other case it is proportional to the square of the height gradient (Gaussian model). For the gradient model we find that the membrane velocity is a nonmonotonic function of the elastic constant μ and reaches a peak at μ=μ^{*}. For μ<μ^{*} the system fails to reach a steady state and the membrane energy keeps increasing with time. For the Gaussian model, the system always reaches a steady state and the membrane velocity decreases monotonically with the elastic constant ν for all nonzero values of ν. Multiple filaments give rise to protrusions at different regions of the membrane and the elasticity of the membrane induces an effective attraction between the two protrusions in the Gaussian model which causes the protrusions to merge and a single wide protrusion is present in the system. In both the models, the relative time scale between the membrane and filament dynamics plays an important role in deciding whether the shape of elasticity-velocity curve is concave or convex. Our numerical simulations agree reasonably well with our analytical calculations.

Cell-cell adhesion interface: orthogonal and parallel forces from contraction, protrusion, and retraction

The epithelial lateral membrane plays a central role in the integration of intercellular signals and, by doing so, is a principal determinant in the emerging properties of epithelial tissues. Mechanical force, when applied to the lateral cell-cell interface, can modulate the strength of adhesion and influence intercellular dynamics. Yet the relationship between mechanical force and epithelial cell behavior is complex and not completely understood. This commentary aims to provide an investigative look at the usage of cellular forces at the epithelial cell-cell adhesion interface.

Computational modeling of single-cell mechanics and cytoskeletal mechanobiology

Cellular cytoskeletal mechanics plays a major role in many aspects of human health from organ development to wound healing, tissue homeostasis and cancer metastasis. We summarize the state-of-the-art techniques for mathematically modeling cellular stiffness and mechanics and the cytoskeletal components and factors that regulate them. We highlight key experiments that have assisted model parameterization and compare the advantages of different models that have been used to recapitulate these experiments. An overview of feed-forward mechanisms from signaling to cytoskeleton remodeling is provided, followed by a discussion of the rapidly growing niche of encapsulating feedback mechanisms from cytoskeletal and cell mechanics to signaling. We discuss broad areas of advancement that could accelerate research and understanding of cellular mechanobiology. A precise understanding of the molecular mechanisms that affect cell and tissue mechanics and function will underpin innovations in medical device technologies of the future. WIREs Syst Biol Med 2018, 10:e1407. doi: 10.1002/wsbm.1407 This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Cellular and Nuclear Forces: An Overview

Biological cells sample their surrounding microenvironments using nanoscale force sensors on the cell surfaces. These surface-based force and stress sensors generate physical and chemical responses inside the cell. The inherently well-connected cytoskeleton and its physical contacts with the force elements on the nuclear membrane lead these physicochemical responses to cascade all the way inside the cell nucleus, physically altering the nuclear state. These physical alterations of the cell nucleus, through yet-unknown complex steps elicit physical and functional response from the chromatin in the form of altered gene expression profiles. This mechanism of force/stress sensing by the cell and then its nuclear response has been shown to play a vital role in maintaining robust cellular homeostasis, controlling gene expression profiles during developmental phases as well as cell differentiation. Over the last few years, there has been appreciable progress toward identification of the molecular players responsible for force sensing. However, the actual sensing mechanism of cell surface bound force sensors and more importantly cascading of the signals, both physical (via cytosolic force sensing elements such as microtubule and actin framework) and chemical (cascade of biochemical signaling from cell surface to nuclear surface and further to the chromatin), inside the cell is poorly understood. In this chapter, we present a review of the currently known molecular players in cellular as well as nuclear force sensing repertoire and their possible mechanistic aspects. We also introduce various biophysical concepts that are used to describe the force/stress sensing and response of a cell. We hope this will help asking clearer questions and designing pointed experiments for better understanding of the force-dependent design principles of the cell surface, nuclear surface, and gene expression.

Comparison of cell migration mechanical strategies in three-dimensional matrices: a computational study

Cell migration on a two-dimensional flat surface has been extensively studied and is generally characterized by a front-protrusion-rear-contraction process. In a three-dimensional (3D) environment, on the other hand, cells adopt multiple migration strategies depending on the cell type and the properties of the extracellular matrix (ECM). By using computer simulations, we find that these migration strategies can be classified by various spatial-temporal dynamics of actin protrusion, actin-myosin contraction and actin-ECM adhesion. We demonstrate that if we include or exclude proteolysis of ECM, and vary adhesion dynamics and spatial distributions of protrusion, contraction and adhesion, our model can reproduce six experimentally observed motility modes: mesenchymal, chimneying, amoeboid, blebbing, finger-like protrusion and rear-squeezing cell locomotory behaviours. We further find that the mode of the cell motility evolves in response to the ECM density and adhesion detachment rate. The model makes non-trivial predictions about cell speed as a function of the adhesion strength, and ECM elasticity and mesh size.

Progress and perspectives in signal transduction, actin dynamics, and movement at the cell and tissue level: lessons from Dictyostelium

Movement of cells and tissues is a basic biological process that is used in development, wound repair, the immune response to bacterial invasion, tumour formation and metastasis, and the search for food and mates. While some cell movement is random, directed movement stimulated by extracellular signals is our focus here. This involves a sequence of steps in which cells first detect extracellular chemical and/or mechanical signals via membrane receptors that activate signal transduction cascades and produce intracellular signals. These intracellular signals control the motile machinery of the cell and thereby determine the spatial localization of the sites of force generation needed to produce directed motion. Understanding how force generation within cells and mechanical interactions with their surroundings, including other cells, are controlled in space and time to produce cell-level movement is a major challenge, and involves many issues that are amenable to mathematical modelling.

A variational approach to the growth dynamics of pre-stressed actin filament networks

In order to model the growth dynamics of elastic bodies with residual stresses a thermodynamically consistent approach is needed such that the cross-coupling between growth and mechanics can be correctly described. In the present work we apply a variational principle to the formulation of the interfacial growth dynamics of dendritic actin filament networks growing from biomimetic beads, an experimentally well studied system, where the buildup of residual stresses governs the network growth. We first introduce the material model for the network via a strain energy density for an isotropic weakly nonlinear elastic material and then derive consistently from this model the dynamic equations for the interfaces, i.e. for a polymerizing internal interface in contact with the bead and a depolymerizing external interface directed towards the solvent. We show that (i) this approach automatically preserves thermodynamic symmetry-properties, which is not the case for the often cited 'rubber-band-model' (Sekimoto et al 2004 Eur. Phys. J. E 13 247-59, Plastino et al 2004 Eur. Biophys. J. 33 310-20) and (ii) leads to a robust morphological instability of the treadmilling network interfaces. The nature of the instability depends on the interplay of the two dynamic interfaces. Depending on the biochemical conditions the network envelope evolves into a comet-like shape (i.e. the actin envelope thins out at one side and thickens on the opposite side of the bead) via a varicose instability or it breaks the symmetry via higher order zigzag modes. We conclude that morphological instabilities due to mechano-chemical coupling mechanisms and the presences of mechancial pre-stresses can play a major role in locally organizing the cytoskeleton of living cells.

Actin filaments growing against a barrier with fluctuating shape

We study force generation by a set of parallel actin filaments growing against a nonrigid obstacle, in the presence of an external load. The filaments polymerize by either moving the whole obstacle, with a large energy cost, or by causing local distortion in its shape which costs much less energy. The nonrigid obstacle also has local thermal fluctuations due to which its shape can change with time and we describe this using fluctuations in the height profile of a one-dimensional interface with Kardar-Parisi-Zhang dynamics. We find the shape fluctuations of the barrier strongly affect the force generation mechanism. The qualitative nature of the force-velocity curve is crucially determined by the relative time scale of filament and barrier dynamics. The height profile of the barrier also shows interesting variation with the external load. Our analytical calculations within mean-field theory show reasonable agreement with our simulation results.

Amplification of actin polymerization forces

The actin cytoskeleton drives many essential processes in vivo, using molecular motors and actin assembly as force generators. We discuss here the propagation of forces caused by actin polymerization, highlighting simple configurations where the force developed by the network can exceed the sum of the polymerization forces from all filaments.

Specification of Architecture and Function of Actin Structures by Actin Nucleation Factors

The actin cytoskeleton is essential for diverse processes in mammalian cells; these processes range from establishing cell polarity to powering cell migration to driving cytokinesis to positioning intracellular organelles. How these many functions are carried out in a spatiotemporally regulated manner in a single cytoplasm has been the subject of much study in the cytoskeleton field. Recent work has identified a host of actin nucleation factors that can build architecturally diverse actin structures. The biochemical properties of these factors, coupled with their cellular location, likely define the functional properties of actin structures. In this article, we describe how recent advances in cell biology and biochemistry have begun to elucidate the role of individual actin nucleation factors in generating distinct cellular structures. We also consider how the localization and orientation of actin nucleation factors, in addition to their kinetic properties, are critical to their ability to build a functional actin cytoskeleton.

CPI motif interaction is necessary for capping protein function in cells

Capping protein (CP) has critical roles in actin assembly in vivo and in vitro. CP binds with high affinity to the barbed end of actin filaments, blocking the addition and loss of actin subunits. Heretofore, models for actin assembly in cells generally assumed that CP is constitutively active, diffusing freely to find and cap barbed ends. However, CP can be regulated by binding of the 'capping protein interaction' (CPI) motif, found in a diverse and otherwise unrelated set of proteins that decreases, but does not abolish, the actin-capping activity of CP and promotes uncapping in biochemical experiments. Here, we report that CP localization and the ability of CP to function in cells requires interaction with a CPI-motif-containing protein. Our discovery shows that cells target and/or modulate the capping activity of CP via CPI motif interactions in order for CP to localize and function in cells.

Connecting local active forces to macroscopic stress in elastic media

In contrast with ordinary materials, living matter drives its own motion by generating active, out-of-equilibrium internal stresses. These stresses typically originate from localized active elements embedded in an elastic medium, such as molecular motors inside the cell or contractile cells in a tissue. While many large-scale phenomenological theories of such active media have been developed, a systematic understanding of the emergence of stress from the local force-generating elements is lacking. In this paper, we present a rigorous theoretical framework to study this relationship. We show that the medium's macroscopic active stress tensor is equal to the active elements' force dipole tensor per unit volume in both continuum and discrete linear homogeneous media of arbitrary geometries. This relationship is conserved on average in the presence of disorder, but can be violated in nonlinear elastic media. Such effects can lead to either a reinforcement or an attenuation of the active stresses, giving us a glimpse of the ways in which nature might harness microscopic forces to create active materials.

Bistable forespore engulfment in Bacillus subtilis by a zipper mechanism in absence of the cell wall

To survive starvation, the bacterium Bacillus subtilis forms durable spores. The initial step of sporulation is asymmetric cell division, leading to a large mother-cell and a small forespore compartment. After division is completed and the dividing septum is thinned, the mother cell engulfs the forespore in a slow process based on cell-wall degradation and synthesis. However, recently a new cell-wall independent mechanism was shown to significantly contribute, which can even lead to fast engulfment in 60 of the cases when the cell wall is completely removed. In this backup mechanism, strong ligand-receptor binding between mother-cell protein SpoIIIAH and forespore-protein SpoIIQ leads to zipper-like engulfment, but quantitative understanding is missing. In our work, we combined fluorescence image analysis and stochastic Langevin simulations of the fluctuating membrane to investigate the origin of fast bistable engulfment in absence of the cell wall. Our cell morphologies compare favorably with experimental time-lapse microscopy, with engulfment sensitive to the number of SpoIIQ-SpoIIIAH bonds in a threshold-like manner. By systematic exploration of model parameters, we predict regions of osmotic pressure and membrane-surface tension that produce successful engulfment. Indeed, decreasing the medium osmolarity in experiments prevents engulfment in line with our predictions. Forespore engulfment may thus not only be an ideal model system to study decision-making in single cells, but its biophysical principles are likely applicable to engulfment in other cell types, e.g. during phagocytosis in eukaryotes.

When the bacterium B. subtilis runs out of food, it undergoes a fundamental development process by which it forms durable spores. Sporulation is initiated by asymmetric cell division after which the larger mother cell engulfs the smaller forespore, followed by spore maturation and release. This survival strategy is so robust that engulfment even proceeds when cells are deprived of their protective cell wall. Under these severe perturbations, 60 of the mother cells still engulf their forespores in only 10 of the normal engulfment time, while the remaining 40 of mother cells withdraw from engulfment. This all-or-none outcome of engulfment suggests decision-making, which was recently also identified in other types of engulfment, e.g. during phagocytosis when immune cells engulf and destroy pathogens. Here, we developed a biophysical model to explain fast bistable forespore engulfment in absence of the cell wall and energy sources. Our discovered principles may prove very general, thus predicting key ingredients of successful engulfment across all kingdoms of life.

Actin dynamics, architecture, and mechanics in cell motility

Tight coupling between biochemical and mechanical properties of the actin cytoskeleton drives a large range of cellular processes including polarity establishment, morphogenesis, and motility. This is possible because actin filaments are semi-flexible polymers that, in conjunction with the molecular motor myosin, can act as biological active springs or "dashpots" (in laymen's terms, shock absorbers or fluidizers) able to exert or resist against force in a cellular environment. To modulate their mechanical properties, actin filaments can organize into a variety of architectures generating a diversity of cellular organizations including branched or crosslinked networks in the lamellipodium, parallel bundles in filopodia, and antiparallel structures in contractile fibers. In this review we describe the feedback loop between biochemical and mechanical properties of actin organization at the molecular level in vitro, then we integrate this knowledge into our current understanding of cellular actin organization and its physiological roles.

Methods for modeling cytoskeletal and DNA filaments

This review summarizes the models that researchers use to represent the conformations and dynamics of cytoskeletal and DNA filaments. It focuses on models that address individual filaments in continuous space. Conformation models include the freely jointed, Gaussian, angle-biased chain (ABC), and wormlike chain (WLC) models, of which the first three bend at discrete joints and the last bends continuously. Predictions from the WLC model generally agree well with experiment. Dynamics models include the Rouse, Zimm, stiff rod, dynamic WLC, and reptation models, of which the first four apply to isolated filaments and the last to entangled filaments. Experiments show that the dynamic WLC and reptation models are most accurate. They also show that biological filaments typically experience strong hydrodynamic coupling and/or constrained motion. Computer simulation methods that address filament dynamics typically compute filament segment velocities from local forces using the Langevin equation and then integrate these velocities with explicit or implicit methods; the former are more versatile and the latter are more efficient. Much remains to be discovered in biological filament modeling. In particular, filament dynamics in living cells are not well understood, and current computational methods are too slow and not sufficiently versatile. Although primarily a review, this paper also presents new statistical calculations for the ABC and WLC models. Additionally, it corrects several discrepancies in the literature about bending and torsional persistence length definitions, and their relations to flexural and torsional rigidities.

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.

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.