Motor Ensembles Research Articles

Simulation studies of self-organization of microtubules and molecular motors

We perform Monte Carlo type simulation studies of self-organization of microtubules interacting with molecular motors. We model microtubules as stiff polar rods of equal length exhibiting anisotropic diffusion in the plane. The molecular motors are implicitly introduced by specifying certain probabilistic collision rules resulting in realignment of the rods. This approximation of the complicated microtubule-motor interaction by a simple instant collision allows us to bypass the "computational bottlenecks" associated with the details of the diffusion and the dynamics of motors and the reorientation of microtubules. Consequently, we are able to perform simulations of large ensembles of microtubules and motors on a very large time scale. This simple model reproduces all important phenomenology observed in in vitro experiments: Formation of vortices for low motor density and raylike asters and bundles for higher motor density.

In vitro assays of molecular motors - impact of motor-surface interactions

In many types of biophysical studies of both single molecules and ensembles of molecular motors the motors are adsorbed to artificial surfaces. Some of the most important assay systems of this type (in vitro motility assays and related single molecule techniques) will be briefly described together with an account of breakthroughs in the understanding of actomyosin function that have resulted from their use. A poorly characterized, but potentially important, entity in these studies is the mechanism of motor adsorption to surfaces and the effects of motor surface interactions on experimental results. A better understanding of these phenomena is also important for the development of commercially viable nanotechnological applications powered by molecular motors. Here, we will consider several aspects of motor surface interactions with a particular focus on heavy meromyosin (HMM) from skeletal muscle. These aspects will be related to heavy meromyosin structure and relevant parts of the vast literature on protein-surface interactions for non-motor proteins. An overview of methods for studying motor-surface interactions will also be given. The information is used as a basis for further development of a model for HMM-surface interactions and is discussed in relation to experiments where nanopatterning has been employed for in vitro reconstruction of actomyosin order. The challenges and potentials of this approach in biophysical studies, compared to the use of self-assembly of biological components into supramolecular protein aggregates (e.g. myosin filaments) will be considered. Finally, this review will consider the implications for further developments of motor-powered lab-on-a-chip devices.

Convergence of a Numerical Method for Solving Discontinuous Fokker–Planck Equations

In studies of molecular motors, the stochastic motion is modeled using the Langevin equation. If we consider an ensemble of motors, the probability density is governed by the corresponding Fokker–Planck equation. Average quantities, such as average velocity, effective diffusion coefficient, and randomness parameter, can be calculated from the probability density. A numerical method was previously developed to solve Fokker–Planck equations [H. Wang, C. Peskin, and T. Elston, J. Theoret. Biol., 221 (2003), pp. 491–511]. It preserves detailed balance, which ensures that if the system is forced to an equilibrium, the numerical solution will be the same as the Boltzmann distribution. Here we study the convergence of this numerical method when the potential has a finite number of discontinuities at half numerical grid points. We prove that this numerical method is stable and is consistent with the differential equation. Based on the consistency analysis, we propose a modified version of this numerical method to eliminate the first order error term caused by the discontinuity. We also show that in the presence of discontinuities, detailed balance is a necessary condition for converging to the correct solution. This explains why the central difference method converges to a wrong solution.

Kinetic Mechanism of Human Myosin IIIA

Myosin IIIA is specifically expressed in photoreceptors and cochlea and is important for the phototransduction and hearing processes. In addition, myosin IIIA contains a unique N-terminal kinase domain and C-terminal tail actin-binding motif. We examined the kinetic properties of baculovirus expressed human myosin IIIA containing the kinase, motor, and two IQ domains. The maximum actin-activated ATPase rate is relatively slow (k(cat) = 0.77 +/- 0.08 s(-1)), and high actin concentrations are required to fully activate the ATPase rate (K(ATPase) = 34 +/- 11 microm). However, actin co-sedimentation assays suggest that myosin III has a relatively high steady-state affinity for actin in the presence of ATP (K(actin) approximately 7 microm). The rate of ATP binding to the motor domain is quite slow both in the presence and absence of actin (K(1)k(+2) = 0.020 and 0.001 microm(-1).s(-1), respectively). The rate of actin-activated phosphate release is more than 100-fold faster (85 s(-1)) than the k(cat), whereas ADP release in the presence of actin follows a two-step mechanism (7.0 and 0.6 s(-1)). Thus, our data suggest a transition between two actomyosin-ADP states is the rate-limiting step in the actomyosin III ATPase cycle. Our data also suggest the myosin III motor spends a large fraction of its cycle in an actomyosin ADP state that has an intermediate affinity for actin (K(d) approximately 5 microm). The long lived actomyosin-ADP state may be important for the ability of myosin III to function as a cellular transporter and actin cross-linker in the actin bundles of sensory cells.

Force generation in small ensembles of Brownian motors

The motility of certain gram-negative bacteria is mediated by retraction of type IV pili surface filaments, which are essential for infectivity. The retraction is powered by a strong molecular motor protein, PilT, producing very high forces that can exceed 150 pN. The molecular details of the motor mechanism are still largely unknown, while other features have been identified, such as the ring-shaped protein structure of the PilT motor. The surprisingly high forces generated by the PilT system motivate a model investigation of the generation of large forces in molecular motors. We propose a simple model, involving a small ensemble of motor subunits interacting through the deformations on a circular backbone with finite stiffness. The model describes the motor subunits in terms of diffusing particles in an asymmetric, time-dependent binding potential (flashing ratchet potential), roughly corresponding to the ATP hydrolysis cycle. We compute force-velocity relations in a subset of the parameter space and explore how the maximum force (stall force) is determined by stiffness, binding strength, ensemble size, and degree of asymmetry. We identify two qualitatively different regimes of operation depending on the relation between ensemble size and asymmetry. In the transition between these two regimes, the stall force depends nonlinearly on the number of motor subunits. Compared to its constituents without interactions, we find higher efficiency and qualitatively different force-velocity relations. The model captures several of the qualitative features obtained in experiments on pilus retraction forces, such as roughly constant velocity at low applied forces and insensitivity in the stall force to changes in the ATP concentration.

Model of Chromosome Motility in Drosophila Embryos: Adaptation of a General Mechanism for Rapid Mitosis

During mitosis, ensembles of dynamic MTs and motors exert forces that coordinate chromosome segregation. Typically, chromosomes align at the metaphase spindle equator where they oscillate along the pole-pole axis before disjoining and moving poleward during anaphase A, but spindles in different cell types display differences in MT dynamicity, in the amplitude of chromosome oscillations and in rates of chromatid-to-pole motion. Drosophila embryonic mitotic spindles, for example, display remarkably dynamic MTs, barely detectable metaphase chromosome oscillations, and a rapid rate of “flux-pacman-dependent” anaphase chromatid-to-pole motility. Here we develop a force-balance model that describes Drosophila embryo chromosome motility in terms of a balance of forces acting on kinetochores and kMTs that is generated by multiple polymer ratchets and mitotic motors coupled to tension-dependent kMT dynamics. The model shows that i), multiple MTs displaying high dynamic instability can drive steady and rapid chromosome motion; ii), chromosome motility during metaphase and anaphase A can be described by a single mechanism; iii), high kinetochore dynein activity is deployed to dampen metaphase oscillations, to augment the basic flux-pacman mechanism, and to drive rapid anaphase A; iv), modulation of the MT rescue frequency by the kinetochore-associated kinesin-13 depolymerase promotes metaphase chromosome oscillations; and v), this basic mechanism can be adapted to a broad range of spindles.

Nonlinear dynamics and self-organization of rotary molecular motor ensembles

The coupled nonlinear dynamics of rotary motor ensembles and an embedding fluid medium are simulated on the basis of Navier-Stokes equations. The model description considers space-dependent rotation of motors and a spatially inhomogeneous motor distribution as well as spatial fluctuations in molecular properties. The mutual influence of motor rotations and fluid dynamics as well as spatial inhomogeneities of their molecular properties are self-consistently included. Space-time simulations visualize the complex interplay between the fluid dynamics and motor rotation and show that the dynamic coupling to the environment via the velocity field as well as the spatial inhomogeneity and distribution of rotors determines the overall behavior of the motor complex. Depending on the initial molecular distribution, rotation frequency, and fluid dynamics a spatial self-organization or a chaotic behavior may arise.

DNA Molecular Motor Driven Micromechanical Cantilever Arrays

The unique ability of living systems to translate biochemical reactions into mechanical work has inspired the design of synthetic DNA motors which generate nanoscale motion via controlled conformational change. However, while Nature has evolved intricate mechanisms to convert molecular shape change into specific micrometer-scale mechanical cellular responses, the integration of artificial DNA motors with mechanical devices presents a major challenge. Here we report the direct integration between an ensemble of DNA motors and an array of microfabricated silicon cantilevers. The forces exerted by the precise duplex to nonclassical i-motif conformational change were probed via differential measurements using an in-situ reference cantilever coated with a nonspecific sequence of DNA. Fueled by the addition of protons, the open to close stroke of the motor induced 32 +/- 3 mN/m compressive surface stress, which corresponds to a single motor force of approximately 11 pN/m, an order of magnitude larger than previous classical hybridization studies. Furthermore, the surface-tethered conformational change was found to be highly reversible, in contrast to classical DNA motors which typically suffer rapid system poisoning. The direction and amplitude of motor-induced cantilever motion was tuneable via control of buffer pH and ionic strength, indicating that electrostatic forces play an important role in stress generation. Hybrid devices which directly harness the multiple accessible conformational states of dynamic oligonucleotides and aptamers, translating biochemical energy into micromechanical work, present a radical new approach to the construction of "smart" nanoscale machinery and mechano-biosensors.

The kinesin Eg5 drives poleward microtubule flux in Xenopus laevis egg extract spindles

Although mitotic and meiotic spindles maintain a steady-state length during metaphase, their antiparallel microtubules slide toward spindle poles at a constant rate. This “poleward flux” of microtubules occurs in many organisms and may provide part of the force for chromosome segregation. We use quantitative image analysis to examine the role of the kinesin Eg5 in poleward flux in metaphase Xenopus laevis egg extract spindles. Pharmacological inhibition of Eg5 results in a dose–responsive slowing of flux, and biochemical depletion of Eg5 significantly decreases the flux rate. Our results suggest that ensembles of nonprocessive Eg5 motors drive flux in metaphase Xenopus extract spindles.

Computational model of dynein-dependent self-organization of microtubule asters.

Polar arrays of microtubules play many important roles in the cell. Normally, such arrays are organized by a centrosome anchoring the minus ends of the microtubules, while the plus ends extend to the cell periphery. However, ensembles of molecular motors and microtubules also demonstrate the ability to self-organize into polar arrays. We use quantitative modeling to analyze the self-organization of microtubule asters and the aggregation of motor-driven pigment granules in fragments of fish melanophore cells. The model is based on the observation that microtubules are immobile and treadmilling, and on the experimental evidence that cytoplasmic dynein motors associated with granules have the ability to nucleate MTs and attenuate their minus-end dynamics. The model explains the observed sequence of events as follows. Initially, pigment granules driven by cytoplasmic dynein motors aggregate to local clusters of microtubule minus ends. The pigment aggregates then nucleate microtubules with plus ends growing toward the fragment boundary, while the minus ends stay transiently in the aggregates. Microtubules emerging from one aggregate compete with any aggregates they encounter leading to the gradual formation of a single aggregate. Simultaneously, a positive feedback mechanism drives the formation of a single MT aster--a single loose aggregate leads to focused MT nucleation and hence a tighter aggregate which stabilizes MT minus ends more effectively leading to aster formation. We translate the model assumptions based on experimental measurements into mathematical equations. The model analysis and computer simulations successfully reproduce the observed pathways of pigment aggregation and microtubule aster self-organization. We test the model predictions by observing the self-organization in fragments of various sizes and in bi-lobed fragments. The model provides stringent constraints on rates and concentrations describing microtubule and motor dynamics, and sheds light on the role of polymer dynamics and polymer-motor interactions in cytoskeletal organization.

Cell division.

In creating the mitotic spindle and the contractile ring, natural selection has engineered fascinating precision machines whose movements depend upon forces generated by ensembles of cytoskeletal proteins. These machines segregate chromosomes and divide the cell with high fidelity. Current research on the mechanisms and regulation of spindle morphogenesis, chromosome motility and cytokinesis emphasizes how ensembles of dynamic cytoskeletal polymers and multiple motors cooperate to generate the forces that guide the cell through mitosis and cytokinesis.

Fluctuations in sliding motion generated by independent and random actions of protein motors

We consider theoretical fluctuations in the in vitro sliding movement of individual cytoskeletal filaments generated by an ensemble of protein motors whose actions are assumed to be statistically independent and random. We show that the mean square deviation of the sliding distances of a filament for a given period of time around their average is proportional to the inverse of the filament length. This result provides a basis for an experimental test of the general assumption on the independent and random actions of protein motors.

Thermodynamics and Kinetics of a Molecular Motor Ensemble

If, contrary to conventional models of muscle, it is assumed that molecular forces equilibrate among rather than within molecular motors, an equation of state and an expression for energy output can be obtained for a near-equilibrium, coworking ensemble of molecular motors. These equations predict clear, testable relationships between motor structure, motor biochemistry, and ensemble motor function, and we discuss these relationships in the context of various experimental studies. In this model, net work by molecular motors is performed with the relaxation of a near-equilibrium intermediate step in a motor-catalyzed reaction. The free energy available for work is localized to this step, and the rate at which this free energy is transferred to work is accelerated by the free energy of a motor-catalyzed reaction. This thermodynamic model implicitly deals with a motile cell system as a dynamic network (not a rigid lattice) of molecular motors within which the mechanochemistry of one motor influences and is influenced by the mechanochemistry of other motors in the ensemble.

The actomyosin engine

Two recent advances have extended our knowledge of the actomyosin interaction considerably. The first of these is the determination of the crystal structures of the actin monomer and of the myosin head. The structures of these proteins were fit into the quaternary protein complexes of the actin filament, and of the actin filament decorated by the myosin head, using data from fiber diffraction or electron microscopy as constraints. The atomic models of these protein complexes have led to new hypotheses concerning the large conformational changes that must occur in these proteins in order to generate force. The second advance has been the measurements of the forces and displacements of single motor proteins, in particular, of a single myosin molecule interacting with a single actin filament. These data lead to more unambiguous interpretations than did previous mechanical data, which were obtained using ensembles of motors that worked asynchronously on their filaments. This approach brings to the motor field the equivalent of what patch clamp techniques brought to the study of membrane channels: the ability to look at the function of a single molecule.