Focal adhesions are controlled by microtubules through local contractility regulation.

Microtubules regulate cell polarity and migration via local activation of focal adhesion turnover, but the mechanism of this process is insufficiently understood. Molecular complexes containing KANK family proteins connect microtubules with talin, the major component of focal adhesions. Here, local optogenetic activation of KANK1-mediated microtubule/talin linkage promoted microtubule targeting to an individual focal adhesion and subsequent withdrawal, resulting in focal adhesion centripetal sliding and rapid disassembly. This sliding is preceded by a local increase of traction force due to accumulation of myosin-II and actin in the proximity of the focal adhesion. Knockdown of the Rho activator GEF-H1 prevented development of traction force and abolished sliding and disassembly of focal adhesions upon KANK1 activation. Other players participating in microtubule-driven, KANK-dependent focal adhesion disassembly include kinases ROCK, PAK, and FAK, as well as microtubules/focal adhesion-associated proteins kinesin-1, APC, and αTAT. Based on these data, we develop a mathematical model for a microtubule-driven focal adhesion disruption involving local GEF-H1/RhoA/ROCK-dependent activation of contractility, which is consistent with experimental data.

Recycling of the actin monomer pool limits the lifetime of network turnover.

Intracellular organization is largely mediated by actin turnover. Cellular actin networks continuously assemble and disassemble, while maintaining their overall appearance. This behavior, called "dynamic steady state," allows cells to sense and adapt to their environment. However, how structural stability can be maintained during the constant turnover of a limited actin monomer pool is poorly understood. To answer this question, we developed an experimental system where polystyrene beads are propelled by an actin comet in a microwell containing a limited amount of components. We used the speed and the size of the actin comet tails to evaluate the system's monomer consumption and its lifetime. We established the relative contribution of actin assembly, disassembly, and recycling for a bead movement over tens of hours. Recycling mediated by cyclase-associated protein (CAP) is the key step in allowing the reuse of monomers for multiple assembly cycles. ATP supply and protein aging are also factors that limit the lifetime of actin turnover. This work reveals the balancing mechanism for long-term network assembly with a limited amount of building blocks.

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.

The spatial arrangement of chromosomes during prometaphase facilitates spindle assembly.

Error-free chromosome segregation requires stable attachment of sister kinetochores to the opposite spindle poles (amphitelic attachment). Exactly how amphitelic attachments are achieved during spindle assembly remains elusive. We employed photoactivatable GFP and high-resolution live-cell confocal microscopy to visualize complete 3D movements of individual kinetochores throughout mitosis in nontransformed human cells. Combined with electron microscopy, molecular perturbations, and immunofluorescence analyses, this approach reveals unexpected details of chromosome behavior. Our data demonstrate that unstable lateral interactions between kinetochores and microtubules dominate during early prometaphase. These transient interactions lead to the reproducible arrangement of chromosomes in an equatorial ring on the surface of the nascent spindle. A computational model predicts that this toroidal distribution of chromosomes exposes kinetochores to a high density of microtubules which facilitates subsequent formation of amphitelic attachments. Thus, spindle formation involves a previously overlooked stage of chromosome prepositioning which promotes formation of amphitelic attachments.

Friction patterns guide actin network contraction.

The shape of cells is the outcome of the balance of inner forces produced by the actomyosin network and the resistive forces produced by cell adhesion to their environment. The specific contributions of contractile, anchoring and friction forces to network deformation rate and orientation are difficult to disentangle in living cells where they influence each other. Here, we reconstituted contractile actomyosin networks in vitro to study specifically the role of the friction forces between the network and its anchoring substrate. To modulate the magnitude and spatial distribution of friction forces, we used glass or lipids surface micropatterning to control the initial shape of the network. We adapted the concentration of Nucleating Promoting Factor on each surface to induce the assembly of actin networks of similar densities and compare the deformation of the network toward the centroid of the pattern shape upon myosin-induced contraction. We found that actin network deformation was faster and more coordinated on lipid bilayers than on glass, showing the resistance of friction to network contraction. To further study the role of the spatial distribution of these friction forces, we designed heterogeneous micropatterns made of glass and lipids. The deformation upon contraction was no longer symmetric but biased toward the region of higher friction. Furthermore, we showed that the pattern of friction could robustly drive network contraction and dominate the contribution of asymmetric distributions of myosins. Therefore, we demonstrate that during contraction, both the active and resistive forces are essential to direct the actin network deformation.

Crawling, waving, inch worming, dilating, and pivoting mechanics of migrating cells: Lessons from Ken Jacobson.

Research on the locomotion of single cells on hard, flat surfaces brought insight into the mechanisms of leading-edge protrusion, spatially graded adhesion, front-rear coordination, and how intracellular and traction forces are harnessed to execute various maneuvers. Here, we highlight how, by studying a variety of cell types, shapes, and movements, Ken Jacobson and his collaborators made several discoveries that triggered the mechanistic understanding of cell motility. We then review the recent advancements and current perspectives in this field.

CellDynaMo-stochastic reaction-diffusion-dynamics model: Application to search-and-capture process of mitotic spindle assembly.

We introduce a Stochastic Reaction-Diffusion-Dynamics Model (SRDDM) for simulations of cellular mechanochemical processes with high spatial and temporal resolution. The SRDDM is mapped into the CellDynaMo package, which couples the spatially inhomogeneous reaction-diffusion master equation to account for biochemical reactions and molecular transport within the Langevin Dynamics (LD) framework to describe dynamic mechanical processes. This computational infrastructure allows the simulation of hours of molecular machine dynamics in reasonable wall-clock time. We apply SRDDM to test performance of the Search-and-Capture of mitotic spindle assembly by simulating, in three spatial dimensions, dynamic instability of elastic microtubules anchored in two centrosomes, movement and deformations of geometrically realistic centromeres with flexible kinetochores and chromosome arms. Furthermore, the SRDDM describes the mechanics and kinetics of Ndc80 linkers mediating transient attachments of microtubules to the chromosomal kinetochores. The rates of these attachments and detachments depend upon phosphorylation states of the Ndc80 linkers, which are regulated in the model by explicitly accounting for the reactions of Aurora A and B kinase enzymes undergoing restricted diffusion. We find that there is an optimal rate of microtubule-kinetochore detachments which maximizes the accuracy of the chromosome connections, that adding chromosome arms to kinetochores improve the accuracy by slowing down chromosome movements, that Aurora A and kinetochore deformations have a small positive effect on the attachment accuracy, and that thermal fluctuations of the microtubules increase the rates of kinetochore capture and also improve the accuracy of spindle assembly.

Interplay between Brownian motion and cross-linking controls bundling dynamics in actin networks.

Morphology changes in cross-linked actin networks are important in cell motility, division, and cargo transport. Here, we study the transition from a weakly cross-linked network of actin filaments to a heavily cross-linked network of actin bundles through microscopic Brownian dynamics simulations. We show that this transition occurs in two stages: first, a composite bundle network of small and highly aligned bundles evolves from cross-linking of individual filaments and, second, small bundles coalesce into the clustered bundle state. We demonstrate that Brownian motion speeds up the first stage of this process at a faster rate than the second. We quantify the time to reach the composite bundle state and show that it strongly increases as the mesh size increases only when the concentration of cross-links is small and that it remains roughly constant if we decrease the relative ratio of cross-linkers as we increase the actin concentration. Finally, we examine the dependence of the bundling timescale on filament length, finding that shorter filaments bundle faster because they diffuse faster.

Experiment, theory, and the keratocyte: An ode to a simple model for cell motility.

Lamellipodial locomotion of fish keratocytes is one of the simplest examples of actin-based motility. In the last four decades, fruitful collaborations between experimentalists and theorists have resulted in a detailed mechanistic understanding of the self-organized lamellipodial engine powering keratocyte motility. Here we review the mechanical mechanisms underlying keratocyte migration, highlighting the interplay between modeling and experiments that led to insights regarding the dynamics of actin network organization, cell shape, and self-polarization. We discuss how to apply lessons learnt from keratocytes to understand cell migration in more complex, physiological contexts.

Stress fibres are embedded in a contractile cortical network.

Contractile actomyosin networks are responsible for the production of intracellular forces. There is increasing evidence that bundles of actin filaments form interconnected and interconvertible structures with the rest of the network. In this study, we explored the mechanical impact of these interconnections on the production and distribution of traction forces throughout the cell. By using a combination of hydrogel micropatterning, traction force microscopy and laser photoablation, we measured the relaxation of traction forces in response to local photoablations. Our experimental results and modelling of the mechanical response of the network revealed that bundles were fully embedded along their entire length in a continuous and contractile network of cortical filaments. Moreover, the propagation of the contraction of these bundles throughout the entire cell was dependent on this embedding. In addition, these bundles appeared to originate from the alignment and coalescence of thin and unattached cortical actin filaments from the surrounding mesh.

Infection-generated electric field in gut epithelium drives bidirectional migration of macrophages.

Many bacterial pathogens hijack macrophages to egress from the port of entry to the lymphatic drainage and/or bloodstream, causing dissemination of life-threatening infections. However, the underlying mechanisms are not well understood. Here, we report that Salmonella infection generates directional electric fields (EFs) in the follicle-associated epithelium of mouse cecum. In vitro application of an EF, mimicking the infection-generated electric field (IGEF), induces directional migration of primary mouse macrophages to the anode, which is reversed to the cathode upon Salmonella infection. This infection-dependent directional switch is independent of the Salmonella pathogenicity island 1 (SPI-1) type III secretion system. The switch is accompanied by a reduction of sialic acids on glycosylated surface components during phagocytosis of bacteria, which is absent in macrophages challenged by microspheres. Moreover, enzymatic cleavage of terminally exposed sialic acids reduces macrophage surface negativity and severely impairs directional migration of macrophages in response to an EF. Based on these findings, we propose that macrophages are attracted to the site of infection by a combination of chemotaxis and galvanotaxis; after phagocytosis of bacteria, surface electrical properties of the macrophage change, and galvanotaxis directs the cells away from the site of infection.

Simulations of dynamically cross-linked actin networks: Morphology, rheology, and hydrodynamic interactions.

Cross-linked actin networks are the primary component of the cell cytoskeleton and have been the subject of numerous experimental and modeling studies. While these studies have demonstrated that the networks are viscoelastic materials, evolving from elastic solids on short timescales to viscous fluids on long ones, questions remain about the duration of each asymptotic regime, the role of the surrounding fluid, and the behavior of the networks on intermediate timescales. Here we perform detailed simulations of passively cross-linked non-Brownian actin networks to quantify the principal timescales involved in the elastoviscous behavior, study the role of nonlocal hydrodynamic interactions, and parameterize continuum models from discrete stochastic simulations. To do this, we extend our recent computational framework for semiflexible filament suspensions, which is based on nonlocal slender body theory, to actin networks with dynamic cross linkers and finite filament lifetime. We introduce a model where the cross linkers are elastic springs with sticky ends stochastically binding to and unbinding from the elastic filaments, which randomly turn over at a characteristic rate. We show that, depending on the parameters, the network evolves to a steady state morphology that is either an isotropic actin mesh or a mesh with embedded actin bundles. For different degrees of bundling, we numerically apply small-amplitude oscillatory shear deformation to extract three timescales from networks of hundreds of filaments and cross linkers. We analyze the dependence of these timescales, which range from the order of hundredths of a second to the actin turnover time of several seconds, on the dynamic nature of the links, solvent viscosity, and filament bending stiffness. We show that the network is mostly elastic on the short time scale, with the elasticity coming mainly from the cross links, and viscous on the long time scale, with the effective viscosity originating primarily from stretching and breaking of the cross links. We show that the influence of nonlocal hydrodynamic interactions depends on the network morphology: for homogeneous meshworks, nonlocal hydrodynamics gives only a small correction to the viscous behavior, but for bundled networks it both hinders the formation of bundles and significantly lowers the resistance to shear once bundles are formed. We use our results to construct three-timescale generalized Maxwell models of the networks.

Mechanical Torque Promotes Bipolarity of the Mitotic Spindle Through Multi-centrosomal Clustering.

Intracellular forces shape cellular organization and function. One example is the mitotic spindle, a cellular machine consisting of multiple chromosomes and centrosomes which interact via dynamic microtubule filaments and motor proteins, resulting in complicated spatially dependent forces. For a cell to divide properly, it is important for the spindle to be bipolar, with chromosomes at the center and multiple centrosomes clustered into two 'poles' at opposite sides of the chromosomes. Experimental observations show that in unhealthy cells, the spindle can take on a variety of patterns. What forces drive each of these patterns? It is known that attraction between centrosomes is key to bipolarity, but what prevents the centrosomes from collapsing into a monopolar configuration? Here, we explore the hypothesis that torque rotating chromosome arms into orientations perpendicular to the centrosome-centromere vector promotes spindle bipolarity. To test this hypothesis, we construct a pairwise-interaction model of the spindle. On a continuum version of the model, an integro-PDE system, we perform linear stability analysis and construct numerical solutions which display a variety of spatial patterns. We also simulate a discrete particle model resulting in a phase diagram that confirms that the spindle bipolarity emerges most robustly with torque. Altogether, our results suggest that rotational forces may play an important role in dictating spindle patterning.

F-actin bending facilitates net actomyosin contraction By inhibiting expansion with plus-end-located myosin motors.

Contraction of actomyosin networks underpins important cellular processes including motility and division. The mechanical origin of actomyosin contraction is not fully-understood. We investigate whether contraction arises on the scale of individual filaments, without needing to invoke network-scale interactions. We derive discrete force-balance and continuum partial differential equations for two symmetric, semi-flexible actin filaments with an attached myosin motor. Assuming the system exists within a homogeneous background material, our method enables computation of the stress tensor, providing a measure of contractility. After deriving the model, we use a combination of asymptotic analysis and numerical solutions to show how F-actin bending facilitates contraction on the scale of two filaments. Rigid filaments exhibit polarity-reversal symmetry as the motor travels from the minus to plus-ends, such that contractile and expansive components cancel. Filament bending induces a geometric asymmetry that brings the filaments closer to parallel as a myosin motor approaches their plus-ends, decreasing the effective spring force opposing motor motion. The reduced spring force enables the motor to move faster close to filament plus-ends, which reduces expansive stress and gives rise to net contraction. Bending-induced geometric asymmetry provides both new understanding of actomyosin contraction mechanics, and a hypothesis that can be tested in experiments.

Protein friction and filament bending facilitate contraction of disordered actomyosin networks.

We use mathematical modeling and computation to investigate how protein friction facilitates contraction of disordered actomyosin networks. We simulate two-dimensional networks using an agent-based model, consisting of a system of force-balance equations for myosin motor proteins and semiflexible actin filaments. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use repeated simulations of disordered networks to confirm that both protein friction and actin filament bending are required for contraction. We then use simulations of elementary two-filament systems to show that filament bending flexibility can facilitate contraction on the microscopic scale. Finally, we show that actin filament turnover is necessary to sustain contraction and prevent filament aggregation. Simulations with and without turnover also exhibit contractile pulses. However, these pulses are aperiodic, suggesting that periodic pulsation can only arise because of additional regulatory mechanisms or more complex mechanical behavior.

Computational estimates of mechanical constraints on cell migration through the extracellular matrix.

Cell migration through a three-dimensional (3D) extracellular matrix (ECM) underlies important physiological phenomena and is based on a variety of mechanical strategies depending on the cell type and the properties of the ECM. By using computer simulations of the cell's mid-plane, we investigate two such migration mechanisms-'push-pull' (forming a finger-like protrusion, adhering to an ECM node, and pulling the cell body forward) and 'rear-squeezing' (pushing the cell body through the ECM by contracting the cell cortex and ECM at the cell rear). We present a computational model that accounts for both elastic deformation and forces of the ECM, an active cell cortex and nucleus, and for hydrodynamic forces and flow of the extracellular fluid, cytoplasm, and nucleoplasm. We find that relations between three mechanical parameters-the cortex's contractile force, nuclear elasticity, and ECM rigidity-determine the effectiveness of cell migration through the dense ECM. The cell can migrate persistently even if its cortical contraction cannot deform a near-rigid ECM, but then the contraction of the cortex has to be able to sufficiently deform the nucleus. The cell can also migrate even if it fails to deform a stiff nucleus, but then it has to be able to sufficiently deform the ECM. Simulation results show that nuclear stiffness limits the cell migration more than the ECM rigidity. Simulations show the rear-squeezing mechanism of motility results in more robust migration with larger cell displacements than those with the push-pull mechanism over a range of parameter values. Additionally, results show that the rear-squeezing mechanism is aided by hydrodynamics through a pressure gradient.

Mechanics of Multicentrosomal Clustering in Bipolar Mitotic Spindles.

To segregate chromosomes in mitosis, cells assemble a mitotic spindle, a molecular machine with centrosomes at two opposing cell poles and chromosomes at the equator. Microtubules and molecular motors connect the poles to kinetochores, specialized protein assemblies on the centromere regions of the chromosomes. Bipolarity of the spindle is crucial for the proper cell division, and two centrosomes in animal cells naturally become two spindle poles. Cancer cells are often multicentrosomal, yet they are able to assemble bipolar spindles by clustering centrosomes into two spindle poles. Mechanisms of this clustering are debated. In this study, we computationally screen effective forces between 1) centrosomes, 2) centrosomes and kinetochores, 3) centrosomes and chromosome arms, and 4) centrosomes and cell cortex to understand mechanics that determines three-dimensional spindle architecture. To do this, we use the stochastic Monte Carlo search for stable mechanical equilibria in the effective energy landscape of the spindle. We find that the following conditions have to be met to robustly assemble the bipolar spindle in a multicentrosomal cell: 1) the strengths of centrosomes' attraction to each other and to the cell cortex have to be proportional to each other and 2) the strengths of centrosomes' attraction to kinetochores and repulsion from the chromosome arms have to be proportional to each other. We also find that three other spindle configurations emerge if these conditions are not met: 1) collapsed, 2) monopolar, and 3) multipolar spindles, and the computational screen reveals mechanical conditions for these abnormal spindles.

Mechanical positioning of multiple nuclei in muscle cells.

Many types of large cells have multiple nuclei. In skeletal muscle fibers, the nuclei are distributed along the cell to maximize their internuclear distances. This myonuclear positioning is crucial for cell function. Although microtubules, microtubule associated proteins, and motors have been implicated, mechanisms responsible for myonuclear positioning remain unclear. We used a combination of rough interacting particle and detailed agent-based modeling to examine computationally the hypothesis that a force balance generated by microtubules positions the muscle nuclei. Rather than assuming the nature and identity of the forces, we simulated various types of forces between the pairs of nuclei and between the nuclei and cell boundary to position the myonuclei according to the laws of mechanics. We started with a large number of potential interacting particle models and computationally screened these models for their ability to fit biological data on nuclear positions in hundreds of Drosophila larval muscle cells. This reverse engineering approach resulted in a small number of feasible models, the one with the best fit suggests that the nuclei repel each other and the cell boundary with forces that decrease with distance. The model makes nontrivial predictions about the increased nuclear density near the cell poles, the zigzag patterns of the nuclear positions in wider cells, and about correlations between the cell width and elongated nuclear shapes, all of which we confirm by image analysis of the biological data. We support the predictions of the interacting particle model with simulations of an agent-based mechanical model. Taken together, our data suggest that microtubules growing from nuclear envelopes push on the neighboring nuclei and the cell boundaries, which is sufficient to establish the nearly-uniform nuclear spreading observed in muscle fibers.

Bidirectional sliding of two parallel microtubules generated by multiple identical motors.

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of this study is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.

Microtubule Dynamics, Kinesin-1 Sliding, and Dynein Action Drive Growth of Cell Processes.

Recent experimental studies of the role of microtubule sliding in neurite outgrowth suggested a qualitative model, according to which kinesin-1 motors push the minus-end-out microtubules against the cell membrane and generate the early cell processes. At the later stage, dynein takes over the sliding, expels the minus-end-out microtubules from the neurites, and pulls in the plus-end-out microtubules that continue to elongate the nascent axon. This model leaves unanswered a number of questions: why is dynein unable to generate the processes alone, whereas kinesin-1 can? What is the role of microtubule dynamics in process initiation and growth? Can the model correctly predict the rates of process growth in control and dynein-inhibited cases? What triggers the transition from kinesin-driven to dynein-driven sliding? To answer these questions, we combine computational modeling of a network of elastic dynamic microtubules and kinesin-1 and dynein motors with measurements of the process growth kinetics and pharmacological perturbations in Drosophila S2 cells. The results verify quantitatively the qualitative model of the microtubule polarity sorting and suggest that dynein-powered elongation is effective only when the processes are longer than a threshold length, which explains why kinesin-1 alone, but not dynein, is sufficient for the process growth. Furthermore, we show that the mechanism of process elongation depends critically on microtubule dynamic instability. Both modeling and experimental measurements show, surprisingly, that dynein inhibition accelerates the process extension. We discuss implications of the model for the general problems of cell polarization, cytoskeletal polarity emergence, and cell process protrusion.