Design principles for robust vesiculation in clathrin-mediated endocytosis.

A critical step in cellular-trafficking pathways is the budding of membranes by protein coats, which recent experiments have demonstrated can be inhibited by elevated membrane tension. The robustness of processes like clathrin-mediated endocytosis (CME) across a diverse range of organisms and mechanical environments suggests that the protein machinery in this process has evolved to take advantage of some set of physical design principles to ensure robust vesiculation against opposing forces like membrane tension. Using a theoretical model for membrane mechanics and membrane protein interaction, we have systematically investigated the influence of membrane rigidity, curvature induced by the protein coat, area covered by the protein coat, membrane tension, and force from actin polymerization on bud formation. Under low tension, the membrane smoothly evolves from a flat to budded morphology as the coat area or spontaneous curvature increases, whereas the membrane remains essentially flat at high tensions. At intermediate, physiologically relevant, tensions, the membrane undergoes a "snap-through instability" in which small changes in the coat area, spontaneous curvature or membrane tension cause the membrane to "snap" from an open, U-shape to a closed bud. This instability can be smoothed out by increasing the bending rigidity of the coat, allowing for successful budding at higher membrane tensions. Additionally, applied force from actin polymerization can bypass the instability by inducing a smooth transition from an open to a closed bud. Finally, a combination of increased coat rigidity and force from actin polymerization enables robust vesiculation even at high membrane tensions.

Paradoxical signaling regulates structural plasticity in dendritic spines.

Transient spine enlargement (3- to 5-min timescale) is an important event associated with the structural plasticity of dendritic spines. Many of the molecular mechanisms associated with transient spine enlargement have been identified experimentally. Here, we use a systems biology approach to construct a mathematical model of biochemical signaling and actin-mediated transient spine expansion in response to calcium influx caused by NMDA receptor activation. We have identified that a key feature of this signaling network is the paradoxical signaling loop. Paradoxical components act bifunctionally in signaling networks, and their role is to control both the activation and the inhibition of a desired response function (protein activity or spine volume). Using ordinary differential equation (ODE)-based modeling, we show that the dynamics of different regulators of transient spine expansion, including calmodulin-dependent protein kinase II (CaMKII), RhoA, and Cdc42, and the spine volume can be described using paradoxical signaling loops. Our model is able to capture the experimentally observed dynamics of transient spine volume. Furthermore, we show that actin remodeling events provide a robustness to spine volume dynamics. We also generate experimentally testable predictions about the role of different components and parameters of the network on spine dynamics.

Mechanics of torque generation in the bacterial flagellar motor.

The bacterial flagellar motor (BFM) is responsible for driving bacterial locomotion and chemotaxis, fundamental processes in pathogenesis and biofilm formation. In the BFM, torque is generated at the interface between transmembrane proteins (stators) and a rotor. It is well established that the passage of ions down a transmembrane gradient through the stator complex provides the energy for torque generation. However, the physics involved in this energy conversion remain poorly understood. Here we propose a mechanically specific model for torque generation in the BFM. In particular, we identify roles for two fundamental forces involved in torque generation: electrostatic and steric. We propose that electrostatic forces serve to position the stator, whereas steric forces comprise the actual "power stroke." Specifically, we propose that ion-induced conformational changes about a proline "hinge" residue in a stator α-helix are directly responsible for generating the power stroke. Our model predictions fit well with recent experiments on a single-stator motor. The proposed model provides a mechanical explanation for several fundamental properties of the flagellar motor, including torque-speed and speed-ion motive force relationships, backstepping, variation in step sizes, and the effects of key mutations in the stator.

The Limiting Speed of the Bacterial Flagellar Motor.

Recent experiments on the bacterial flagellar motor have shown that the structure of this nanomachine, which drives locomotion in a wide range of bacterial species, is more dynamic than previously believed. Specifically, the number of active torque-generating complexes (stators) was shown to vary across applied loads. This finding brings under scrutiny the experimental evidence reporting that limiting (zero-torque) speed is independent of the number of active stators. In this study, we propose that, contrary to previous assumptions, the maximum speed of the motor increases as additional stators are recruited. This result arises from our assumption that stators disengage from the motor for a significant portion of their mechanochemical cycles at low loads. We show that this assumption is consistent with current experimental evidence in chimeric motors, as well as with the requirement that a processive motor driving a large load via an elastic linkage must have a high duty ratio.

Evolution of patterns on Conus shells.

The pigmentation patterns of shells in the genus Conus can be generated by a neural-network model of the mantle. We fit model parameters to the shell pigmentation patterns of 19 living Conus species for which a well resolved phylogeny is available. We infer the evolutionary history of these parameters and use these results to infer the pigmentation patterns of ancestral species. The methods we use allow us to characterize the evolutionary history of a neural network, an organ that cannot be preserved in the fossil record. These results are also notable because the inferred patterns of ancestral species sometimes lie outside the range of patterns of their living descendants, and illustrate how development imposes constraints on the evolution of complex phenotypes.

Protein-induced membrane curvature alters local membrane tension.

Adsorption of proteins onto membranes can alter the local membrane curvature. This phenomenon has been observed in biological processes such as endocytosis, tubulation, and vesiculation. However, it is not clear how the local surface properties of the membrane, such as membrane tension, change in response to protein adsorption. In this article, we show that the partial differential equations arising from classical elastic model of lipid membranes, which account for simultaneous changes in shape and membrane tension due to protein adsorption in a local region, cannot be solved for nonaxisymmetric geometries using straightforward numerical techniques; instead, a viscous-elastic formulation is necessary to fully describe the system. Therefore, we develop a viscous-elastic model for inhomogeneous membranes of the Helfrich type. Using the newly available viscous-elastic model, we find that the lipids flow to accommodate changes in membrane curvature during protein adsorption. We show that, at the end of protein adsorption process, the system sustains a residual local tension to balance the difference between the actual mean curvature and the imposed spontaneous curvature. We also show that this change in membrane tension can have a functional impact such as altered response to pulling forces in the presence of proteins.

Dual biochemical oscillators may control cellular reversals in Myxococcus xanthus.

Myxococcus xanthus is a Gram-negative, soil-dwelling bacterium that glides on surfaces, reversing direction approximately once every 6 min. Motility in M. xanthus is governed by the Che-like Frz pathway and the Ras-like Mgl pathway, which together cause the cell to oscillate back and forth. Previously, Igoshin et al. (2004) suggested that the cellular oscillations are caused by cyclic changes in concentration of active Frz proteins that govern motility. In this study, we present a computational model that integrates both the Frz and Mgl pathways, and whose downstream components can be read as motor activity governing cellular reversals. This model faithfully reproduces wildtype and mutant behaviors by simulating individual protein knockouts. In addition, the model can be used to examine the impact of contact stimuli on cellular reversals. The basic model construction relies on the presence of two nested feedback circuits, which prompted us to reexamine the behavior of M. xanthus cells. We performed experiments to test the model, and this cell analysis challenges previous assumptions of 30 to 60 min reversal periods in frzCD, frzF, frzE, and frzZ mutants. We demonstrate that this average reversal period is an artifact of the method employed to record reversal data, and that in the absence of signal from the Frz pathway, Mgl components can occasionally reverse the cell near wildtype periodicity, but frz- cells are otherwise in a long nonoscillating state.

Myxobacteria gliding motility requires cytoskeleton rotation powered by proton motive force.

Myxococcus xanthus is a Gram-negative bacterium that glides over surfaces without the aid of flagella. Two motility systems are used for locomotion: social-motility, powered by the retraction of type IV pili, and adventurous (A)-motility, powered by unknown mechanism(s). We have shown that AgmU, an A-motility protein, is part of a multiprotein complex that spans the inner membrane and periplasm of M. xanthus. In this paper, we present evidence that periplasmic AgmU decorates a looped continuous helix that rotates clockwise as cells glide forward, reversing its rotation when cells reverse polarity. Inhibitor studies showed that the AgmU helix rotation is driven by proton motive force (PMF) and depends on actin-like MreB cytoskeletal filaments. The AgmU motility complex was found to interact with MotAB homologs. Our data are consistent with a mechanochemical model in which PMF-driven motors, similar to bacterial flagella stator complexes, run along an endless looped helical track, driving rotation of the track; deformation of the cell surface by the AgmU-associated proteins creates pressure waves in the slime, pushing cells forward.

A small post-translocation energy bias aids nucleotide selection in T7 RNA polymerase transcription.

The RNA polymerase (RNAP) of bacteriophage T7 is a single subunit enzyme that can transcribe DNA to RNA in the absence of additional protein factors. In this work, we present a model of T7 RNAP translocation during elongation. Based on structural information and experimental data from single-molecule force measurements, we show that a small component of facilitated translocation or power stroke coexists with the Brownian-ratchet-driven motions, and plays a crucial role in nucleotide selection at pre-insertion. The facilitated translocation is carried out by the conserved Tyr(639) that moves its side chain into the active site, pushing aside the 3'-end of the RNA, and forming a locally stabilized post-translocation intermediate. Pre-insertion of an incoming nucleotide into this stabilized intermediate state ensures that Tyr(639) closely participates in selecting correct nucleotides. A similar translocation mechanism has been suggested for multi-subunit RNAPs involving the bridge-helix bending. Nevertheless, the bent bridge-helix sterically prohibits nucleotide binding in the post-transolocation intermediate analog; moreover, the analog is not stabilized unless an inhibitory protein factor binds to the enzyme. Using our scheme, we also compared the efficiencies of different strategies for nucleotide selection, and examined effects of facilitated translocation on forward tracking.

The mechanochemistry of endocytosis.

Endocytic vesicle formation is a complex process that couples sequential protein recruitment and lipid modifications with dramatic shape transformations of the plasma membrane. Although individual molecular players have been studied intensively, how they all fit into a coherent picture of endocytosis remains unclear. That is, how the proper temporal and spatial coordination of endocytic events is achieved and what drives vesicle scission are not known. Drawing upon detailed knowledge from experiments in yeast, we develop the first integrated mechanochemical model that quantitatively recapitulates the temporal and spatial progression of endocytic events leading to vesicle scission. The central idea is that membrane curvature is coupled to the accompanying biochemical reactions. This coupling ensures that the process is robust and culminates in an interfacial force that pinches off the vesicle. Calculated phase diagrams reproduce endocytic mutant phenotypes observed in experiments and predict unique testable endocytic phenotypes in yeast and mammalian cells. The combination of experiments and theory in this work suggest a unified mechanism for endocytic vesicle formation across eukaryotes.

The neural origins of shell structure and pattern in aquatic mollusks.

We present a model to explain how the neurosecretory system of aquatic mollusks generates their diversity of shell structures and pigmentation patterns. The anatomical and physiological basis of this model sets it apart from other models used to explain shape and pattern. The model reproduces most known shell shapes and patterns and accurately predicts how the pattern alters in response to environmental disruption and subsequent repair. Finally, we connect the model to a larger class of neural models.

Large-scale simulations of fluctuating biological membranes.

We present a simple, and physically motivated, coarse-grained model of a lipid bilayer, suited for micron scale computer simulations. Each approximately 25 nm(2) patch of bilayer is represented by a spherical particle. Mimicking forces of hydrophobic association, multiparticle interactions suppress the exposure of each sphere's equator to its implicit solvent surroundings. The requirement of high equatorial density stabilizes two-dimensional structures without necessitating crystalline order, allowing us to match both the elasticity and fluidity of natural lipid membranes. We illustrate the model's versatility and realism by characterizing a membrane's response to a prodding nanorod.

Motor-substrate interactions in mycoplasma motility explains non-Arrhenius temperature dependence.

Mycoplasmas exhibit a novel, substrate-dependent gliding motility that is driven by approximately 400 "leg" proteins. The legs interact with the substrate and transmit the forces generated by an assembly of ATPase motors. The velocity of the cell increases linearly by nearly 10-fold over a narrow temperature range of 10-40 degrees C. This corresponds to an Arrhenius factor that decreases from approximately 45 k(B)T at 10 degrees C to approximately 10 k(B)T at 40 degrees C. On the other hand, load-velocity curves at different temperatures extrapolate to nearly the same stall force, suggesting a temperature-insensitive force-generation mechanism near stall. In this article, we propose a leg-substrate interaction mechanism that explains the intriguing temperature sensitivity of this motility. The large Arrhenius factor at low temperature comes about from the addition of many smaller energy barriers arising from many substrate-binding sites at the distal end of the leg protein. The Arrhenius dependence attenuates at high temperature due to two factors: 1), the reduced effective multiplicity of energy barriers intrinsic to the multiple-site binding mechanism; and 2), the temperature-sensitive weakly facilitated leg release that curtails the power stroke. The model suggests an explanation for the similar steep, sub-Arrhenius temperature-velocity curves observed in many molecular motors, such as kinesin and myosin, wherein the temperature behavior is dominated not by the catalytic biochemistry, but by the motor-substrate interaction.

Rotary protein motors.

Three protein motors have been unambiguously identified as rotary engines: the bacterial flagellar motor and the two motors that constitute ATP synthase (F(0)F(1) ATPase). Of these, the bacterial flagellar motor and F(0) motors derive their energy from a transmembrane ion-motive force, whereas the F(1) motor is driven by ATP hydrolysis. Here, we review the current understanding of how these protein motors convert their energy supply into a rotary torque.

Voltage and frequency dependence of prestin-associated charge transfer.

Membrane protein prestin is a critical component of the motor complex that generates forces and dimensional changes in cells in response to changes in the cell membrane potential. In its native cochlear outer hair cell, prestin is crucial to the amplification and frequency selectivity of the mammalian ear up to frequencies of tens of kHz. Other cells transfected with prestin acquire voltage-dependent properties similar to those of the native cell. The protein performance is critically dependent on chloride ions, and intrinsic protein charges also play a role. We propose an electro-diffusion model to reveal the frequency and voltage dependence of electric charge transfer by prestin. The movement of the combined charge (i.e., anion and protein charges) across the membrane is described with a Fokker-Planck equation coupled to a kinetic equation that describes the binding of chloride ions to prestin. We found a voltage- and frequency-dependent phase shift between the transferred charge and the applied electric field that determines capacitive and resistive components of the transferred charge. The phase shift monotonically decreases from zero to -90 degrees as a function of frequency. The capacitive component as a function of voltage is bell-shaped, and decreases with frequency. The resistive component is bell-shaped for both voltage and frequency. The capacitive and resistive components are similar to experimental measurements of charge transfer at high frequencies. The revealed nature of the transferred charge can help reconcile the high-frequency electrical and mechanical observations associated with prestin, and it is important for further analysis of the structure and function of this protein.

Q & A. [interview].

George Oster is Professor of Biophysics, University of California, Berkeley. He received his B.S. at the U.S. Merchant Marine Academy and his Ph.D. at Columbia University. He began his career in biophysics as a postdoc at the Weizmann Institute under Aharon Katchalsky, where his research involved membrane biophysics and irreversible thermodynamics. His concern for environmental issues led him into population biology, which shaded into evolutionary biology and thence to developmental biology, cell biology and, most recently, protein motors and bacterial motility and pattern formation. His tools are mathematics, physics and computer simulation. He is currently a faculty member in the Departments of Molecular and Cellular Biology and the College of Natural Resources at Berkeley.

Polymer motors: pushing out the front and pulling up the back.

Mechanical work in cells is performed by specialized motor proteins that operate in a continuous mechanochemical cycle. Less complex, but still efficient, 'one-shot' motors evolved based on the assembly and disassembly of polymers. We review the mechanisms of pushing and pulling by actin and microtubule filaments and the organizational principles of actin networks. We show how these polymer force generators are used for the propulsion of intracellular pathogens, protrusion of lamellipodia and mitotic movements. We discuss several examples of cellular forces generated by the assembly and disassembly of polymer gels.

How myxobacteria glide.

BACKGROUND: Many microorganisms, including myxobacteria, cyanobacteria, and flexibacteria, move by gliding. Although gliding always describes a slow surface-associated translocation in the direction of the cell's long axis, it can result from two very different propulsion mechanisms: social (S) motility and adventurous (A) motility. The force for S motility is generated by retraction of type 4 pili. A motility may be associated with the extrusion of slime, but evidence has been lacking, and how force might be generated has remained an enigma. Recently, nozzle-like structures were discovered in cyanobacteria from which slime emanated at the same rate at which the bacteria moved. This strongly implicates slime extrusion as a propulsion mechanism for gliding. RESULTS: Here we show that similar but smaller nozzle-like structures are found in Myxococcus xanthus and that they are clustered at both cell poles, where one might expect propulsive organelles. Furthermore, light and electron microscopical observations show that slime is secreted in ribbons from the ends of cells. To test whether the slime propulsion hypothesis is physically reasonable, we construct a mathematical model of the slime nozzle to see if it can generate a force sufficient to propel M. xanthus at the observed velocities. The model assumes that the hydration of slime, a cationic polyelectrolyte, is the force-generating mechanism. CONCLUSIONS: The discovery of nozzle-like organelles in various gliding bacteria suggests their role in prokaryotic gliding. Our calculations and our observations of slime trails demonstrate that slime extrusion from such nozzles can account for most of the observed properties of A motile gliding.

Mechanochemistry of t7 DNA helicase.

The bacteriophage T7 helicase is a ring-shaped hexameric motor protein that unwinds double-stranded DNA during DNA replication and recombination. To accomplish this it couples energy from the nucleotide hydrolysis cycle to translocate along one of the DNA strands. Here, we combine computational biology with new biochemical measurements to infer the following properties of the T7 helicase: (1) all hexameric subunits are catalytic; (2) the mechanical movement along the DNA strand is driven by the binding transition of nucleotide into the catalytic site; (3) hydrolysis is coordinated between adjacent subunits that bind DNA; (4) the hydrolysis step changes the affinity of a subunit for DNA allowing passage of DNA from one subunit to the next. We construct a numerical optimization scheme to analyze transient and steady-state biochemical measurements to determine the rate constants for the hydrolysis cycle and determine the flux distribution through the reaction network. We find that, under physiological and experimental conditions, there is no dominant pathway; rather there is a distribution of pathways that varies with the ambient conditions. Our analysis methods provide a systematic procedure to study kinetic pathways of multi-subunit, multi-state cooperative enzymes.

How to make a spiral bacterium.

The motility of some kinds of bacteria depends on their spiral form, as does the virulence of certain pathogenic species. We propose a novel mechanism for the development of spiral shape in bacteria and the supercoiling of chains ('filaments') of many cells. Recently discovered actin-like proteins lying just under the cell wall form fibers that play a role in maintaining cell shape. Some species have a single actin-like fiber helically wrapped around the cell, while others have two fibers wrapped in the same direction. Here, we show that if these fibers elongate more slowly than growth lengthens the cell, the cell both twists and bends, taking on a spiral shape. We tested this mechanism using a mathematical model of expanding fiber-wound structures and via experiments that measure the shape changes of elongating physical models. Comparison of the model with in vivo experiments on stationary phase Caulobacter crescentus filaments provide the first evidence that mechanical stretching of cytoskeletal fibers influences cell morphology. Any hydraulic cylinder can spiral by this mechanism if it is reinforced by stretch-resistant fibers wrapped helically in the same direction, or shortened by contractile elements. This might be useful in the design of man-made actuators.