Cells sense the environment's mechanical stiffness to control their own shape, migration, and fate. To better understand stiffness sensing, we constructed a stochastic model of the "motor-clutch" force transmission system, where molecular clutches link F-actin to the substrate and mechanically resist myosin-driven F-actin retrograde flow. The model predicts two distinct regimes: (i) "frictional slippage," with fast retrograde flow and low traction forces on stiff substrates and (ii) oscillatory "load-and-fail" dynamics, with slower retrograde flow and higher traction forces on soft substrates. We experimentally confirmed these model predictions in embryonic chick forebrain neurons by measuring the nanoscale dynamics of single-growth-cone filopodia. Furthermore, we experimentally observed a model-predicted switch in F-actin dynamics around an elastic modulus of 1 kilopascal. Thus, a motor-clutch system inherently senses and responds to the mechanical stiffness of the local environment.

We developed a stochastic model of microtubule (MT) assembly dynamics that estimates tubulin-tubulin bond energies, mechanical energy stored in the lattice dimers, and the size of the tubulin-GTP cap at MT tips. First, a simple assembly͞disassembly state model was used to screen possible combinations of lateral bond energy (⌬G Lat) and longitudinal bond energy (⌬GLong) plus the free energy of immobilizing a dimer in the MT lattice (⌬G S) for rates of MT growth and shortening measured experimentally. This analysis predicts ⌬G Lat in the range of ؊3.2 to ؊5.7 kBT and ⌬GLong plus ⌬GS in the range of ؊6.8 to ؊9.4 kBT. Based on these estimates, the energy of conformational stress for a single tubulin-GDP dimer in the lattice is 2.1-2.5 k BT. Second, we studied how tubulin-GTP cap size fluctuates with different hydrolysis rules and show that a mechanism of directly coupling subunit addition to hydrolysis fails to support MT growth, whereas a finite hydrolysis rate allows growth. By adding rules to mimic the mechanical constraints present at the MT tip, the model generates tubulin-GTP caps similar in size to experimental estimates. Finally, by combining assembly͞ disassembly and cap dynamics, we generate MT dynamic instability with rates and transition frequencies similar to those measured experimentally. Our model serves as a platform to examine GTPcap dynamics and allows predictions of how MT-associated proteins and other effectors alter the energetics of MT assembly.M icrotubule (MT) dynamic instability plays a critical role in chromosome movement and separation during mitosis. MTs grow, shorten, and transition between these states at rates governed by the presence of various MT effectors, such as divalent cations, MT-associated proteins (MAPs), and drugs such as Taxol (1).MTs are composed of heterodimer subunits of ␣-and ␤-tubulin. A guanine nucleotide (GTP or GDP) is positioned on the ␤ monomer opposite the interface between the two monomers, where it is hydrolyzable (if it is GTP) and exchangeable. The ␤ monomer end of the dimer faces the (ϩ) end of the MT, which is the end of more active dynamics and kinetochore attachment during cell division. The ␣ monomer faces the (Ϫ) end of the MT, which originates at the centrosome. MTs are thought to transition from a state of growth to a state of rapid shortening, termed a ''catastrophe'', when the tubulin-GTP ''cap'' is stochastically lost from the tip of the MT. The tubulin-GTP cap must exist because it has been demonstrated that tubulin-GTP subunits are added to the ends during assembly. A tubulin-GTP cap, however, has not been detected in experiments with porcine brain tubulin, and therefore must be small. Experiments suggest that the cap must be less than Ϸ200 dimers (2-4). Presumably, the transition from rapid shortening to growth, termed ''rescue,'' occurs when the tubulin-GTP cap is reestablished.Interactions at the surfaces of adjacent dimers occur through discrete lateral and longitudinal noncovalent bonds. Thermodynamic studies of MT assembly are unable to qua...

Microtubule assembly is vital for many fundamental cellular processes. Current models for microtubule assembly kinetics assume that the subunit disassociation rate from a microtubule tip is independent of free subunit concentration. Using Total-Internal-Reflection-Fluorescence (TIRF) microscopy and a laser tweezers assay to measure in vitro microtubule assembly with nanometer resolution, we find that the subunit dissociation rate from a microtubule tip increases as the free subunit concentration increases. These data are consistent with a two-dimensional model for microtubule assembly, and are explained by a shift in microtubule tip structure from a relatively blunt shape at low free concentrations to relatively tapered at high free concentrations. Because both the association and the dissociation rates increase at higher free subunit concentrations, we find that the kinetics of microtubule assembly are an order-of-magnitude higher than currently estimated in the literature.

Microtubule self-assembly is largely governed by the chemical kinetics and thermodynamics of tubulin-tubulin interactions. An important aspect of microtubule assembly is that hydrolysis of the beta-tubulin-associated GTP promotes protofilament curling. Protofilament curling presumably drives the transition from tip structures associated with growth (sheetlike projections and blunt ends) to those associated with shortening (rams' horns and frayed ends), and transitions between these structures have been proposed to be important for growth-shortening transitions. However, previous models for microtubule dynamic instability have not considered such structures or mechanics explicitly. Here we present a three-dimensional model that explicitly incorporates mechanical stress and strain within the microtubule lattice. First, we found that the model recapitulates three-dimensional tip structures and rates of assembly and disassembly for microtubules grown under standard conditions, and we propose that taxol may stabilize microtubule growth by reducing flexural rigidity. Second, in contrast to recent suggestions, it was determined that sheetlike tips are more likely to undergo catastrophe than blunt tips. Third, partial uncapping of the tubulin-GTP cap provides a possible mechanism for microtubule pause events. Finally, simulations of the binding and structural effects of XMAP215 produced the experimentally observed growth and shortening rates, and tip structure.

Cell migration, which is central to many biological processes including wound healing and cancer progression, is sensitive to environmental stiffness, and many cell types exhibit a stiffness optimum, at which migration is maximal. Here we present a cell migration simulator that predicts a stiffness optimum that can be shifted by altering the number of active molecular motors and clutches. This prediction is verified experimentally by comparing cell traction and F-actin retrograde flow for two cell types with differing amounts of active motors and clutches: embryonic chick forebrain neurons (ECFNs; optimum ∼1 kPa) and U251 glioma cells (optimum ∼100 kPa). In addition, the model predicts, and experiments confirm, that the stiffness optimum of U251 glioma cell migration, morphology and F-actin retrograde flow rate can be shifted to lower stiffness by simultaneous drug inhibition of myosin II motors and integrin-mediated adhesions.

Summary During mitosis, sister chromatids congress to the spindle equator and are subsequently segregated via attachment to dynamic kinetochore microtubule (kMT) plus-ends. A major question is how kMT plus-end assembly is spatially regulated to achieve chromosome congression. Here we find in budding yeast that the widely-conserved kinesin-5 sliding motor proteins, Cin8p and Kip1p, mediate chromosome congression by suppressing kMT plus-end assembly of longer kMTs. Of the two, Cin8p is the major effector and its activity requires a functional motor domain. In contrast, the depolymerizing kinesin-8 motor Kip3p plays a minor role in spatial regulation of yeast kMT assembly. Our analysis identified a model where kinesin-5 motors bind to kMTs, move to kMT plus ends, and upon arrival at a growing plus-end promote net kMT plus-end disassembly. In conclusion, we find that length-dependent control of net kMT assembly by kinesin-5 motors yields a simple and stable self-organizing mechanism for chromosome congression.

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.

SUMMARY Protein concentration gradients encode spatial information across cells and tissues, and often depend on spatially localized protein synthesis. Here, we report that a different mechanism underlies the MEX-5 gradient. MEX-5 is an RNA-binding protein that becomes distributed in a cytoplasmic gradient along the anterior-to-posterior axis of the 1-cell C. elegans embryo. We demonstrate that the MEX-5 gradient is a direct consequence of an underlying gradient in MEX-5 diffusion. The MEX-5 diffusion gradient arises when the PAR-1 kinase stimulates the release of MEX-5 from slow-diffusive, RNA-containing complexes in the posterior cytoplasm. PAR-1 directly phosphorylates MEX-5 and is antagonized by the spatially-uniform phosphatase PP2A. Mathematical modeling and in vivo observations demonstrate that spatially-segregated phosphorylation and dephosphorylation reactions are sufficient to generate stable protein concentration gradients in the cytoplasm. The principles demonstrated here apply to any spatially-segregated modification cycle that affects protein diffusion, and do not require protein synthesis or degradation.