Kinesin-1-transported liposomes prefer to go straight in 3D microtubule intersections by a mechanism shared by other molecular motors.

Kinesin-1 ensembles maneuver vesicular cargoes through the three-dimensional (3D) intracellular microtubule (MT) network. To define how such cargoes navigate MT intersections, we first determined how many kinesins from an ensemble on a lipid-based cargo simultaneously engage a MT, and then determined the directional outcomes (straight, turn, terminate) for liposome cargoes at perpendicular MT intersections. Run lengths of 350-nm diameter liposomes decorated with up to 20, constitutively active, truncated kinesin-1 KIF5B (K543) were longer than single motor transported cargo, suggesting multiple motor engagement. However, detachment forces of lipid-coated beads with ~20 kinesins, measured using an optical trap, showed no more than three simultaneously engaged motors, with a single engaged kinesin predominating, indicating anticooperative MT binding. At two-dimensional (2D) and 3D in vitro MT intersections, liposomes frequently paused (~2 s), suggesting kinesins simultaneously bind both MTs and engage in a tug-of-war. Liposomes showed no directional outcome bias in 2D (1.1 straight:turn ratio) but preferentially went straight (1.8 straight:turn ratio) in 3D intersections. To explain these data, we developed a mathematical model of liposome transport incorporating the known mechanochemistry of kinesins, which diffuse on the liposome surface, and have stiff tails in both compression and extension that impact how motors engage the intersecting MTs. Our model predicts the ~3 engaged motor limit observed in the optical trap and the bias toward going straight in 3D intersections. The striking similarity of these results to our previous study of liposome transport by myosin Va suggests a "universal" mechanism by which cargoes navigate 3D intersections.

Modeling thick filament activation suggests a molecular basis for force depression.

Multiscale models aiming to connect muscle's molecular and cellular function have been difficult to develop, in part due to a lack of self-consistent multiscale data. To address this gap, we measured the force response from single, skinned rabbit psoas muscle fibers to ramp shortenings and step stretches performed on the plateau region of the force-length relationship. We isolated myosin from the same muscles and, under similar conditions, performed single-molecule and ensemble measurements of myosin's ATP-dependent interaction with actin using laser trapping and in vitro motility assays. We fit the fiber data by developing a partial differential equation model that includes thick filament activation, whereby an increase in force on the thick filament pulls myosin out of an inhibited state. The model also includes a series elastic element and a parallel elastic element. This parallel elastic element models a titin-actin interaction proposed to account for the increase in isometric force after stretch (residual force enhancement). By optimizing the model fit to a subset of our fiber measurements, we specified seven unknown parameters. The model then successfully predicted the remainder of our fiber measurements and also our molecular measurements from the laser trap and in vitro motility. The success of the model suggests that our multiscale data are self-consistent and can serve as a testbed for other multiscale models. Moreover, the model captures the decrease in isometric force observed in our muscle fibers after active shortening (force depression), suggesting a molecular mechanism for force depression, whereby a parallel elastic element combines with thick filament activation to decrease the number of cycling cross-bridges.

"Spatial Relationships Matter: Kinesin-1 Molecular Motors Transport Liposome Cargo Through 3D Microtubule Intersections In Vitro".

Kinesin-1 ensembles maneuver vesicular cargoes through intersections in the 3-dimensional (3D) intracellular microtubule (MT) network. To characterize directional outcomes (straight, turn, terminate) at MT intersections, we challenge 350 nm fluid-like liposomes transported by ~10 constitutively active, truncated kinesin-1 KIF5B (K543) with perpendicular 2-dimensional (2D) and 3D intersections in vitro. Liposomes frequently pause at 2D and 3D intersections (~2s), suggesting that motor teams can simultaneously engage each MT and undergo a tug-of-war. Once resolved, the directional outcomes at 2D MT intersections have a straight to turn ratio of 1.1; whereas at 3D MT intersections, liposomes more frequently go straight (straight to turn ratio of 1.8), highlighting that spatial relationships at intersections bias directional outcomes. Using 3D super-resolution microscopy (STORM), we define the gap between intersecting MTs and the liposome azimuthal approach angle heading into the intersection. We develop an in silico model in which kinesin-1 motors diffuse on the liposome surface, simultaneously engage the intersecting MTs, generate forces and detach from MTs governed by the motors' mechanochemical cycle, and undergo a tug-of-war with the winning team determining the directional outcome in 3D. The model predicts that 1-3 motors typically engage the MT, consistent with optical trapping measurements. Modeled liposomes also predominantly go straight through 3D intersections over a range of intersection gaps and liposome approach angles, even when obstructed by the crossing MT. Our observations and modeling offer mechanistic insights into how cells might tune the MT cytoskeleton, cargo, and motors to modulate cargo transport.

Modeling Thick Filament Activation Suggests a Molecular Basis for Force Depression.

Multiscale models aiming to connect muscle's molecular and cellular function have been difficult to develop, in part, due to a lack of self-consistent multiscale data. To address this gap, we measured the force response from single skinned rabbit psoas muscle fibers to ramp shortenings and step stretches performed on the plateau region of the force-length relationship. We isolated myosin from the same muscles and, under similar conditions, performed single molecule and ensemble measurements of myosin's ATP-dependent interaction with actin using laser trapping and in vitro motility assays. We fit the fiber data by developing a partial differential equation model that includes thick filament activation, whereby an increase in force on the thick filament pulls myosin out of an inhibited state. The model also includes a series elastic element and a parallel elastic element. This parallel elastic element models a titin-actin interaction proposed to account for the increase in isometric force following stretch (residual force enhancement). By optimizing the model fit to a subset of our fiber measurements, we specified seven unknown parameters. The model then successfully predicted the remainder of our fiber measurements and also our molecular measurements from the laser trap and in vitro motility. The success of the model suggests that our multiscale data are self-consistent and can serve as a testbed for other multiscale models. Moreover, the model captures the decrease in isometric force observed in our muscle fibers after active shortening (force depression), suggesting a molecular mechanism for force depression, whereby a parallel elastic element combines with thick filament activation to decrease the number of cycling cross-bridges.

Metabolic cost for isometric force scales nonlinearly and predicts how humans distribute forces across limbs.

Muscles consume metabolic energy for force production and movement. A mathematical model of metabolic energy cost will be useful in predicting instantaneous costs during human exercise and in computing effort-minimizing movements via simulations. Previous in vivo data-derived models usually assumed either zero or linearly increasing cost with force, but a nonlinear relation could have significant metabolic or behavioural implications. Here, we show that metabolic cost scales nonlinearly with joint torque with an exponent of about 1.64, using calorimetric measurements of isometric squats. We then demonstrate that this metabolic nonlinearity is reflected in human behaviour: minimizing this nonlinear cost predicts how humans share forces between limbs in additional experiments involving arms and legs. This shows the utility of the nonlinear energy cost in predictive models and its generalizability across limbs. Finally, we show mathematical evidence that the same nonlinear metabolic objective may underlie force sharing at the muscle level.

Force-velocity and tension transient measurements from Drosophila jump muscle reveal the necessity of both weakly-bound cross-bridges and series elasticity in models of muscle contraction.

Muscle contraction is a fundamental biological process where molecular interactions between the myosin molecular motor and actin filaments result in contraction of a whole muscle, a process spanning size scales differing in eight orders of magnitude. Since unique behavior is observed at every scale in between these two extremes, to fully understand muscle function it is vital to develop multi-scale models. Based on simulations of classic measurements of muscle heat generation as a function of work, and shortening rate as a function of applied force, we hypothesize that a model based on molecular measurements must be modified to include a weakly-bound interaction between myosin and actin in order to fit measurements at the muscle fiber or whole muscle scales. This hypothesis is further supported by the model's need for a weakly-bound state in order to qualitatively reproduce the force response that occurs when a muscle fiber is rapidly stretched a small distance. We tested this hypothesis by measuring steady-state force as a function of shortening velocity, and the force transient caused by a rapid length step in Drosophila jump muscle fibers. Then, by performing global parameter optimization, we quantitatively compared the predictions of two mathematical models, one lacking a weakly-bound state and one with a weakly-bound state, to these measurements. Both models could reproduce our force-velocity measurements, but only the model with a weakly-bound state could reproduce our force transient measurements. However, neither model could concurrently fit both measurements. We find that only a model that includes weakly-bound cross-bridges with force-dependent detachment and an elastic element in series with the cross-bridges is able to fit both of our measurements. This result suggests that the force response after stretch is not a reflection of distinct steps in the cross-bridge cycle, but rather arises from the interaction of cross-bridges with a series elastic element. Additionally, the model suggests that the curvature of the force-velocity relationship arises from a combination of the force-dependence of weakly- and strongly-bound cross-bridges. Overall, this work presents a minimal cross-bridge model that has predictive power at the fiber level.

Myosin Va transport of liposomes in three-dimensional actin networks is modulated by actin filament density, position, and polarity.

The cell's dense 3D actin filament network presents numerous challenges to vesicular transport by teams of myosin Va (MyoVa) molecular motors. These teams must navigate their cargo through diverse actin structures ranging from Arp2/3-branched lamellipodial networks to the dense, unbranched cortical networks. To define how actin filament network organization affects MyoVa cargo transport, we created two different 3D actin networks in vitro. One network was comprised of randomly oriented, unbranched actin filaments; the other was comprised of Arp2/3-branched actin filaments, which effectively polarized the network by aligning the actin filament plus-ends. Within both networks, we defined each actin filament's 3D spatial position using superresolution stochastic optical reconstruction microscopy (STORM) and its polarity by observing the movement of single fluorescent reporter MyoVa. We then characterized the 3D trajectories of fluorescent, 350-nm fluid-like liposomes transported by MyoVa teams (∼10 motors) moving within each of the two networks. Compared with the unbranched network, we observed more liposomes with directed and fewer with stationary motion on the Arp2/3-branched network. This suggests that the modes of liposome transport by MyoVa motors are influenced by changes in the local actin filament polarity alignment within the network. This mechanism was supported by an in silico 3D model that provides a broader platform to understand how cellular regulation of the actin cytoskeletal architecture may fine tune MyoVa-based intracellular cargo transport.

The load dependence of muscle's force-velocity curve is modulated by alternative myosin converter domains.

The hyperbolic shape of the muscle force-velocity relationship (FVR) is characteristic of all muscle fiber types. The degree of curvature of the hyperbola varies between muscle fiber types and is thought to be set by force-dependent properties of different myosin isoforms. However, the structural elements in myosin and the mechanism that determines force dependence are unresolved. We tested our hypothesis that the myosin converter domain plays a critical role in the force-velocity relationship (FVR) mechanism. Drosophila contains a single myosin heavy chain gene with five converters encoded by alternative exons. We measured FVR properties of Drosophila jump muscle fibers from five transgenic lines each expressing a single converter. Consistent with our hypothesis, we observed up to 2.4-fold alterations in FVR curvature. Maximum shortening velocity (v0) and optimal velocity for maximum power generation were also altered, but isometric tension and maximum power generation were unaltered. Converter 11a, normally found in the indirect flight muscle (IFM), imparted the highest FVR curvature and v0, whereas converter 11d, found in larval body wall muscle, imparted the most linear FVR and slowest v0. Jump distance strongly correlated with increasing FVR curvature and v0, meaning flies expressing the converter from the IFM jumped farther than flies expressing the native jump muscle converter. Fitting our data with Huxley's two-state model and a biophysically based four-state model suggest a testable hypothesis that the converter sets muscle type FVR curvature by influencing the detachment rate of negatively strained myosin via changes in the force dependence of product release.

Acidosis affects muscle contraction by slowing the rates myosin attaches to and detaches from actin.

The loss of muscle force and power during fatigue from intense contractile activity is associated with, and likely caused by, elevated levels of phosphate ([Formula: see text]) and hydrogen ions (decreased pH). To understand how these deficits in muscle performance occur at the molecular level, we used direct measurements of mini-ensembles of myosin generating force in the laser trap assay at pH 7.4 and 6.5. The data are consistent with a mechanochemical model in which a decrease in pH reduces myosin's detachment from actin (by slowing ADP release), increases non-productive myosin binding (by detached myosin rebinding without a powerstroke), and reduces myosin's attachment to actin (by slowing the weak-to-strong binding transition). Additional support of this mechanism is found by incorporating it into a branched pathway model for the effects of [Formula: see text] on myosin's interaction with actin. Including pH-dependence in one additional parameter (acceleration of [Formula: see text]-induced detachment), the model reproduces experimental measurements at high and low pH, and variable [Formula: see text], from the single molecule to large ensemble levels. Furthermore, when scaled up, the model predicts force-velocity relationships that are consistent with muscle fiber measurements. The model suggests that reducing pH has two opposing effects, a decrease in attachment favoring a decrease in muscle force and a decrease in detachment favoring an increase in muscle force. Depending on experimental details, the addition of [Formula: see text] can strengthen one or the other effect, resulting in either synergistic or antagonistic effects. This detailed molecular description suggests a molecular basis for contractile failure during muscle fatigue.

Myosin Va molecular motors manoeuvre liposome cargo through suspended actin filament intersections in vitro.

Intracellular cargo transport relies on myosin Va molecular motor ensembles to travel along the cell's three-dimensional (3D) highway of actin filaments. At actin filament intersections, the intersecting filament is a structural barrier to and an alternate track for directed cargo transport. Here we use 3D super-resolution fluorescence imaging to determine the directional outcome (that is, continues straight, turns or terminates) for an ∼10 motor ensemble transporting a 350 nm lipid-bound cargo that encounters a suspended 3D actin filament intersection in vitro. Motor-cargo complexes that interact with the intersecting filament go straight through the intersection 62% of the time, nearly twice that for turning. To explain this, we develop an in silico model, supported by optical trapping data, suggesting that the motors' diffusive movements on the vesicle surface and the extent of their engagement with the two intersecting actin tracks biases the motor-cargo complex on average to go straight through the intersection.

Mechanosensitive Adhesion Explains Stepping Motility in Amoeboid Cells.

Cells employing amoeboid motility exhibit repetitive cycles of rapid expansion and contraction and apply coordinated traction forces to their environment. Although aspects of this process are well studied, it is unclear how the cell controls the coordination of cell length changes with adhesion to the surface. Here, we develop a simple model to mechanistically explain the emergence of periodic changes in length and spatiotemporal dynamics of traction forces measured in chemotaxing unicellular amoeba, Dictyostelium discoideum. In contrast to the biochemical mechanisms that have been implicated in the coordination of some cellular processes, we show that many features of amoeboid locomotion emerge from a simple mechanochemical model. The mechanism for interaction with the environment in Dictyostelium is unknown and thus, we explore different cell-environment interaction models to reveal that mechanosensitive adhesions are necessary to reproduce the spatiotemporal adhesion patterns. In this modeling framework, we find that the other motility modes, such as smooth gliding, arise naturally with variations in the physical properties of the surface. Thus, our work highlights the prominent role of biomechanics in determining the emergent features of amoeboid locomotion.

The molecular basis of thin filament activation: from single molecule to muscle.

For muscles to effectively power locomotion, trillions of myosin molecules must rapidly attach and detach from the actin thin filament. This is accomplished by precise regulation of the availability of the myosin binding sites on actin (i.e. activation). Both calcium (Ca++) and myosin binding contribute to activation, but both mechanisms are simultaneously active during contraction, making their relative contributions difficult to determine. Further complicating the process, myosin binding accelerates the attachment rate of neighboring myosin molecules, adding a cooperative element to the activation process. To de-convolve these two effects, we directly determined the effect of Ca++ on the rate of attachment of a single myosin molecule to a single regulated actin thin filament, and separately determined the distance over which myosin binding increases the attachment rate of neighboring molecules. Ca++ alone increases myosin's attachment rate ~50-fold, while myosin binding accelerates attachment of neighboring molecules 400 nm along the actin thin filament.

Direct Measurements of Local Coupling between Myosin Molecules Are Consistent with a Model of Muscle Activation.

Muscle contracts due to ATP-dependent interactions of myosin motors with thin filaments composed of the proteins actin, troponin, and tropomyosin. Contraction is initiated when calcium binds to troponin, which changes conformation and displaces tropomyosin, a filamentous protein that wraps around the actin filament, thereby exposing myosin binding sites on actin. Myosin motors interact with each other indirectly via tropomyosin, since myosin binding to actin locally displaces tropomyosin and thereby facilitates binding of nearby myosin. Defining and modeling this local coupling between myosin motors is an open problem in muscle modeling and, more broadly, a requirement to understanding the connection between muscle contraction at the molecular and macro scale. It is challenging to directly observe this coupling, and such measurements have only recently been made. Analysis of these data suggests that two myosin heads are required to activate the thin filament. This result contrasts with a theoretical model, which reproduces several indirect measurements of coupling between myosin, that assumes a single myosin head can activate the thin filament. To understand this apparent discrepancy, we incorporated the model into stochastic simulations of the experiments, which generated simulated data that were then analyzed identically to the experimental measurements. By varying a single parameter, good agreement between simulation and experiment was established. The conclusion that two myosin molecules are required to activate the thin filament arises from an assumption, made during data analysis, that the intensity of the fluorescent tags attached to myosin varies depending on experimental condition. We provide an alternative explanation that reconciles theory and experiment without assuming that the intensity of the fluorescent tags varies.

Effects of cardiac Myosin binding protein-C on actin motility are explained with a drag-activation-competition model.

Although mutations in cardiac myosin binding protein-C (cMyBP-C) cause heart disease, its role in muscle contraction is not well understood. A mechanism remains elusive partly because the protein can have multiple effects, such as dual biphasic activation and inhibition observed in actin motility assays. Here we develop a mathematical model for the interaction of cMyBP-C with the contractile proteins actin and myosin and the regulatory protein tropomyosin. We use this model to show that a drag-activation-competition mechanism accurately describes actin motility measurements, while models lacking either drag or competition do not. These results suggest that complex effects can arise simply from cMyBP-C binding to actin.

Muscle activation described with a differential equation model for large ensembles of locally coupled molecular motors.

Molecular motors, by turning chemical energy into mechanical work, are responsible for active cellular processes. Often groups of these motors work together to perform their biological role. Motors in an ensemble are coupled and exhibit complex emergent behavior. Although large motor ensembles can be modeled with partial differential equations (PDEs) by assuming that molecules function independently of their neighbors, this assumption is violated when motors are coupled locally. It is therefore unclear how to describe the ensemble behavior of the locally coupled motors responsible for biological processes such as calcium-dependent skeletal muscle activation. Here we develop a theory to describe locally coupled motor ensembles and apply the theory to skeletal muscle activation. The central idea is that a muscle filament can be divided into two phases: an active and an inactive phase. Dynamic changes in the relative size of these phases are described by a set of linear ordinary differential equations (ODEs). As the dynamics of the active phase are described by PDEs, muscle activation is governed by a set of coupled ODEs and PDEs, building on previous PDE models. With comparison to Monte Carlo simulations, we demonstrate that the theory captures the behavior of locally coupled ensembles. The theory also plausibly describes and predicts muscle experiments from molecular to whole muscle scales, suggesting that a micro- to macroscale muscle model is within reach.

Direct observation of phosphate inhibiting the force-generating capacity of a miniensemble of Myosin molecules.

Elevated levels of phosphate (Pi) reduce isometric force, providing support for the notion that the release of Pi from myosin is closely associated with the generation of muscular force. Pi is thought to rebind to actomyosin in an ADP-bound state and reverse the force-generating steps, including the rotation of the lever arm (i.e., the powerstroke). Despite extensive study, this mechanism remains controversial, in part because it fails to explain the effects of Pi on isometric ATPase and unloaded shortening velocity. To gain new insight into this process, we determined the effect of Pi on the force-generating capacity of a small ensemble of myosin (∼12 myosin heads) using a three-bead laser trap assay. In the absence of Pi, myosin pulled the actin filament out of the laser trap an average distance of 54 ± 4 nm, translating into an average peak force of 1.2 pN. By contrast, in the presence of 30 mM Pi, myosin generated only enough force to displace the actin filament by 13 ± 1 nm, generating just 0.2 pN of force. The elevated Pi also caused a >65% reduction in binding-event lifetime, suggesting that Pi induces premature detachment from a strongly bound state. Definitive evidence of a Pi-induced powerstroke reversal was not observed, therefore we determined if a branched kinetic model in which Pi induces detachment from a strongly bound, postpowerstroke state could explain these observations. The model was able to accurately reproduce not only the data presented here, but also the effects of Pi on both isometric ATPase in muscle fibers and actin filament velocity in a motility assay. The ability of the model to capture the findings presented here as well as previous findings suggests that Pi-induced inhibition of force may proceed along a kinetic pathway different from that of force generation.

Mechanical coupling between myosin molecules causes differences between ensemble and single-molecule measurements.

In contracting muscle, individual myosin molecules function as part of a large ensemble, hydrolyzing ATP to power the relative sliding of actin filaments. The technological advances that have enabled direct observation and manipulation of single molecules, including recent experiments that have explored myosin's force-dependent properties, provide detailed insight into the kinetics of myosin's mechanochemical interaction with actin. However, it has been difficult to reconcile these single-molecule observations with the behavior of myosin in an ensemble. Here, using a combination of simulations and theory, we show that the kinetic mechanism derived from single-molecule experiments describes ensemble behavior; but the connection between single molecule and ensemble is complex. In particular, even in the absence of external force, internal forces generated between myosin molecules in a large ensemble accelerate ADP release and increase how far actin moves during a single myosin attachment. These myosin-induced changes in strong binding lifetime and attachment distance cause measurable properties, such as actin speed in the motility assay, to vary depending on the number of myosin molecules interacting with an actin filament. This ensemble-size effect challenges the simple detachment limited model of motility, because even when motility speed is limited by ADP release, increasing attachment rate can increase motility speed.

Actin cap associated focal adhesions and their distinct role in cellular mechanosensing.

The ability for cells to sense and adapt to different physical microenvironments plays a critical role in development, immune responses, and cancer metastasis. Here we identify a small subset of focal adhesions that terminate fibers in the actin cap, a highly ordered filamentous actin structure that is anchored to the top of the nucleus by the LINC complexes; these differ from conventional focal adhesions in morphology, subcellular organization, movements, turnover dynamics, and response to biochemical stimuli. Actin cap associated focal adhesions (ACAFAs) dominate cell mechanosensing over a wide range of matrix stiffness, an ACAFA-specific function regulated by actomyosin contractility in the actin cap, while conventional focal adhesions are restrictively involved in mechanosensing for extremely soft substrates. These results establish the perinuclear actin cap and associated ACAFAs as major mediators of cellular mechanosensing and a critical element of the physical pathway that transduce mechanical cues all the way to the nucleus.

Phosphate enhances myosin-powered actin filament velocity under acidic conditions in a motility assay.

Elevated levels of inorganic phosphate (P(i)) are believed to inhibit muscular force by reversing myosin's force-generating step. These same levels of P(i) can also affect muscle velocity, but the molecular basis underlying these effects remains unclear. We directly examined the effect of P(i) (30 mM) on skeletal muscle myosin's ability to translocate actin (V(actin)) in an in vitro motility assay. Manipulation of the pH enabled us to probe rebinding of P(i) to myosin's ADP-bound state, while changing the ATP concentration probed rebinding to the rigor state. Surprisingly, the addition of P(i) significantly increased V(actin) at both pH 6.8 and 6.5, causing a doubling of V(actin) at pH 6.5. To probe the mechanisms underlying this increase in speed, we repeated these experiments while varying the ATP concentration. At pH 7.4, the effects of P(i) were highly ATP dependent, with P(i) slowing V(actin) at low ATP (<500 µM), but with a minor increase at 2 mM ATP. The P(i)-induced slowing of V(actin), evident at low ATP (pH 7.4), was minimized at pH 6.8 and completely reversed at pH 6.5. These data were accurately fit with a simple detachment-limited kinetic model of motility that incorporated a P(i)-induced prolongation of the rigor state, which accounted for the slowing of V(actin) at low ATP, and a P(i)-induced detachment from a strongly bound post-power-stroke state, which accounted for the increase in V(actin) at high ATP. These findings suggest that P(i) differentially affects myosin function: enhancing velocity, if it rebinds to the ADP-bound state, while slowing velocity, if it binds to the rigor state.