Session X48: Focus Session: Statistical Physics of Active Systems Away From Detailed Balance: Cytoskeleton, Flagella, and All That

The Role of Dynein in Microtubule Mechanics

Experiments in Lele's group have shown that microtubules severed by laser ablation do not straighten, as would be expected from the large bending moments along their lengths. Instead, segments near newly created minus ends typically increased in curvature following severing, while segments near new microtubule plus ends depolymerize before any observable change in shape. However, in dynein-inhibited cells, segments near the cut straightened rapidly following severing. These observations suggest that microtubules are subject to significant tangential forces, and that lateral motion of the microtubule is primarily opposed by frictional rather than elastic forces. To interpret the experimental results, we have developed a numerical model for intracellular microtubule mechanics, accounting for dynein-generated forces on the microtubules. We have supplemented the Kirchoff model for an elastic filament with the stochastic growth and collapse of microtubules, and by a model for dynein generated forces. I will present simulations of the dynamics of individual microtubules that show how motor forces result in the localization of short-wavelength buckles near the cell periphery. Our results suggest that microtubule shapes in vivo reflect a dynamic force balance, where bending moments are opposed by dynein-motor forces that include a large effective friction from the stochastic binding and unbinding of the motors. Simulations of the motion of the centrosome are consistent with a mechanism for centrosome centering driven by pulling forces exerted by dynein motors. I will explain how tension on the centrosome can be reconciled with buckled filaments near the cell periphery.   

Processivity and collectivity of molecular motors pulling membrane tubes

In every cell, directed transport involves proteins that convert chemical energy into mechanical work. Molecular motors responsible for this vital task are mostly too weak to carry biological cargo by themselves and some cannot even take more than a single step before unbinding from their cytoskeletal track. By acting collectively, however, they can muster the required forces. In this talk, we discuss interactions among motors and their collective effects on the extraction of membrane nanotubes. Via a force balance coupled to binding kinetics, we sketch the phase diagram of tube formation. Three regimes are identified: (1) tip clustering, in the sense that the driving force is concentrated at the tip of the tube, (2) viscous extraction, in which motors axially drag membrane, and (3) hybrid extraction, such that tip clustering and axial drag are equally important. Comparison with experiments indicates that synthetic membranes mostly fall into regime (1), while biological membranes tend to fall into regime (2). Our model suggests a unifying picture of tube extraction by both processive and nonprocessive motors.   

Carbon nanotubes as mechanical probes of equilibrium and non-equilibrium cytoskeletal networks

Networks of filamentous proteins underlie the mechanics of cells. The activity of motor proteins typically creates strong fluctuations that drive the system out of equilibrium. Understanding the behavior of such networks requires probes that ideally span the characteristic length-scales, from nanometers to micrometers. Single-walled carbon nanotubes (SWNTs) are nanometer-diameter filaments with micrometer length and tunable bending stiffness. On a Brownian energy scale they have persistence lengths of about 20-100 micrometers and show significant thermal fluctuations on the cellular scale of a few microns. Diffusive motion and local bending dynamics of SWNTs embedded in an active polymeric network reflect forces and fluctuations of the embedding medium. We study the motion of individual SWNTs in equilibrium and non-equilibrium networks by near infrared fluorescence microscopy. We show that SWNTs reptate in the network. We will discuss the possibility of using SWNTs as multi-scale probes relating their local dynamic behavior to the viscoelastic properties of the surrounding network.   

The metaphase and anaphase dynamics is dominated by the physical and mechanical properties of both microtubules and chromatin

One of the most interesting problems in biophysics involves the physical separation of chromosomes and the mechanical properties of both microtubules (MT's) and chromatin. This process involves the polymers MT's and chromatin, each of which has unique physical properties that have been determined extensively in vitro. Of further interest for physicists is the out-of-equilibrium nature of this process involving several force generators from motor proteins and MT depolymerization. We follow the dynamics of spindle pole bodies and centromeres of yeast cells during mitosis in three-dimensions at high spatial resolution. Using this novel approach, we are able to observe spindle oscillations during metaphase, and the three-dimensional dynamics of spindle elongation and chromosome separation during anaphase. With these data, we can separate the dynamics caused by MT depolymerization from those caused by the motors. This allows us to determine the depolymerization rate of the kinetochore MT's in vivo. Furthermore, we determine the temporal profile of the chromatin extension during anaphase we combine with the known force-extension curve of chromatin in vitro, to infer the expected force-velocity curve of the collective motors in vivo, which has never been measured in vivo or in vitro.   

Effective temperature and spontaneous collective motion of active matter

Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the cytoskeleton in which constituents are driven by energy-consuming motors. Although systems driven by small-step motors are described by an effective temperature and are thus quiescent, at higher order in step size, both homogeneous and inhomogeneous, flowing and oscillating behavior emerges. Motors that respond with a negative susceptibility to imposed forces lead to an apparent negative temperature system in which beautiful structures form resembling the asters seen in cell division.   

An active matter analysis of intracellular Active Transport

Tens of thousands of fluorescence-based trajectories at nm resolution have been analyzed, regarding active transport along microtubules in living cells. The following picture emerges. Directed motion to pre-determined locations is certainly an attractive idea, but cannot be pre-programmed as to do so would sacrifice adaptability. The polarity of microtubules is inadequate to identify these directions in cells, and no other mechanism is currently known. We conclude that molecular motors carry cargo through disordered intracellular microtubule networks in a statistical way, with loud cellular ``noise'' both in directionality and speed. Programmed random walks describe how local 1D active transport traverses crowded cellular space efficiently, rapidly, minimizing the energy waste that would result from redundant activity. The mechanism of statistical regulation is not yet understood, however.   

Constant torque in flagellar bacterial motors optimizes space exploration

Experiments indicate that the torque provided by the bacterial rotary motor is approximately constant over a large range of angular speeds. Constant torque implies that the power spent in active motion is proportional to the instantaneous bacterial speed, if the relation between angular speed and swimming speed is linear. Here we show that a constant torque maximizes the volume of the region explored by a bacterium in a resource-depleted medium. Given that nutrients in the ocean are often concentrated in separate, ephemeral patches, we propose that the observed constancy of the torque may be a trait evolved to maximize bacterial survival in the ocean. We also discuss the dependence of the explored volume with the particular features of the bacterial propulsion mechanism.   

Interplay between motor contractility and mechanical stability of active biopolymer networks

The mechanical properties of cells are regulated in part by internal stresses generated actively by molecular motors in the cytoskeletal filamentous actin network. On a larger scale, collective motor activity allows the cell to contract the surrounding extracellular matrix, consisting also of biopolymer networks. Experiments show that such active contractility dramatically affects the networks' elasticity, both in reconstituted intracellular F-actin networks with myosin motors as well as in extracellular gels with contractile cells. We provide insight into this remarkable behavior with a model for the mechanics of contractile disordered networks consisting of simple straight fibers with linear bending and stretching elasticity. We find that these networks exhibit a low-connectivity rigidity threshold governed by fiber-bending elasticity and a high-connectivity threshold that controls a crossover between bending and stretching dominated network elasticity. Owing to their low connectivity, typical biopolymer networks fall below this upper threshold and their mechanical stability thus relies on the fibers' bending rigidity. The macroscopic elasticity of such networks is governed by soft fiber bending deformations. However, we find that motor-generated contractile forces can ``pull out'' these soft bending modes, thereby inducing a crossover to a mechanically more stable regime governed by stiff fiber stretching modes. Using scaling arguments and mean field theory, we show that this transition---induced by motor contractility---can be understood from the stress-dependence of the mechanical stability thresholds. These results suggest a physical principle by which active contractility can control biopolymer network mechanics, even when the fiber constituents are linear elastic elements.   

Propulsion of microorganisms by a helical flagellum

Many bacteria (e.g. \textit{E. coli} and \textit{Salmonella}) swim by rotating rigid helical flagella, which are typically several $\mu$m long and 0.4 $\mu$m in diameter. We investigate this propulsion in laboratory measurements on macroscopic rotating helices (typical diameter, 12 mm) in a fluid with viscosity $10^{5}$ times that of water; thus the Reynolds number in the experiments is much less than unity, just as for bacteria. We measure the propulsive force and torque generated by a rotating flagellum, and the drag force on a translating flagellum; thus we can determine all elements of the propulsion matrices along the axial direction. We also compute force, torque and drag using the regularized Stokeslets method of Cortez et al. (2005). Our experimental and numerical results are in excellent agreement. However, these results differ significantly from the predictions of resistive force theories developed by Gray and Hancock (1953) and Lighthill (1975). The difference between our measurements and resistive force theory is especially large for helices with small pitch/diameter ratios, which is the regime of many bacteria.   

Energy Consumption of Actively Beating Flagella

Motile cilia and flagella are important for propelling cells or driving fluid over tissues. The microtubule-based core in these organelles, the axoneme, has a nearly universal ``9+2'' arrangement of 9 outer doublet microtubules assembled around two singlet microtubules in the center. Thousands of molecular motor proteins are attached to the doublets and walk on neighboring outer doublets. The motors convert the chemical energy of ATP hydrolysis into sliding motion between adjacent doublet microtubules, resulting in precisely regulated oscillatory beating. Using demembranated sea urchin sperm flagella as an experimental platform, we simultaneously monitor the axoneme's consumption of ATP and its beating dynamics while key parameters, such as solution viscosity and ATP concentration, are varied. Insights into motor cooperativity during beating and energetic consequences of hydrodynamic interactions will be presented.   

A particle based computational model for eukaryotic flagella

The structure of the eukaryotic flagella is very complex and the exact mechanisms responsible for flagellar beating are not clearly understood. Here we present a minimal model to study flagellar beating in two dimensions, which demonstrates that regular beating with a well defined characteristic frequency can arise spontaneously in the absence of external control. In this model, the flagella is represented by two stiff filaments clamped on a surface, on which model ``molecular motors'' take directed steps on one of the filaments and thereby apply a local force. The fluid medium is simulated using Multiparticle Collision dynamics (MPC), which is a particle based method for hydrodynamic simulations. Within a certain range of motor concentrations, large amplitude periodic oscillations with a well defined frequency are observed; other qualitatively different beating patterns arise outside of this range. We present a phase diagram that characterizes the beating behaviour as a function of relevant parameters such as filament length, motor density on the filament and motor velocity.   

Force generation in a regrowing eukaryotic flagellum

Flagella are whip-like organelles with a complex internal structure, the axoneme, highly conserved across eukaryotic species. The highly regulated activity of motor proteins arranged along the axoneme moves the flagellum in the surrounding fluid, generating forces that can be used for swimming or fluid propulsion. Although our understanding of the general mechanism behind flagellar motion is well established, the details of its implementation in a real axoneme is still poorly understood. Here we explore the inner working of the eukaryotic flagellum using a uniflagellated mutant of the unicellular green alga {\it Chlamydomonas reinhardtii} to investigate in detail the force and power generated by a moving flagellum during axonemal regrowth after deflagellation. These experiments will contribute to our understanding of the inner working of the eukaryotic flagellum.