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.

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.

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.

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.

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.

Bacteria that glide with helical tracks.

Many bacteria glide smoothly on surfaces, despite having no discernable propulsive organelles on their surface. Recent experiments with Myxococcus xanthus and Flavobacterium johnsoniae show that both of these distantly related bacterial species glide using proteins that move in helical tracks, albeit with significantly different motility mechanisms. Both species utilize proton-motive force for movement. Although the motors that power gliding in M. xanthus have been identified, the F. johnsoniae motors remain to be discovered.

Small scale membrane mechanics.

Large scale changes to lipid bilayer shapes are well represented by the Helfrich model. However, there are membrane processes that take place at smaller length scales that this model cannot address. In this work, we present a one-dimensional continuum model that captures the mechanics of the lipid bilayer membrane at the length scale of the lipids themselves. The model is developed using the Cosserat theory of surfaces with lipid orientation, or 'tilt', as the fundamental degree of freedom. The Helfrich model can be recovered as a special case when the curvatures are small and the lipid tilt is everywhere zero. We use the tilt model to study local membrane deformations in response to a protein inclusion. Parameter estimates and boundary conditions are obtained from a coarse-grained molecular model using dissipative particle dynamics (DPD) to capture the same phenomenon. The continuum model is able to reproduce the membrane bending, stretch and lipid tilt as seen in the DPD model. The lipid tilt angle relaxes to the bulk tilt angle within 5-6 nm from the protein inclusion. Importantly, for large tilt gradients induced by the proteins, the tilt energy contribution is larger than the bending energy contribution. Thus, the continuum model of tilt accurately captures behaviors at length scales shorter than the membrane thickness.

Lipid tubule growth by osmotic pressure.

We present here a procedure for growing lipid tubules in vitro. This method allows us to grow tubules of consistent shape and structure, and thus can be a useful tool for nano-engineering applications. There are three stages during the tubule growth process: initiation, elongation and termination. Balancing the forces that act on the tubule head shows that the growth of tubules during the elongation phase depends on the balance between osmotic pressure and the viscous drag exerted on the membrane from the substrate and the external fluid. Using a combination of mathematical modelling and experiment, we identify the key forces that control tubule growth during the elongation phase.

Interaction between surface shape and intra-surface viscous flow on lipid membranes.

The theory of intra-surface viscous flow on lipid bilayers is developed by combining the equations for flow on a curved surface with those that describe the elastic resistance of the bilayer to flexure. The model is derived directly from balance laws and augments an alternative formulation based on a variational principle. Conditions holding along an edge of the membrane are emphasized, and the coupling between flow and membrane shape is simulated numerically.

On the mysterious propulsion of Synechococcus.

We propose a model for the self-propulsion of the marine bacterium Synechococcus utilizing a continuous looped helical track analogous to that found in Myxobacteria [1]. In our model cargo-carrying protein motors, driven by proton-motive force, move along a continuous looped helical track. The movement of the cargo creates surface distortions in the form of small amplitude traveling ridges along the S-layer above the helical track. The resulting fluid motion adjacent to the helical ribbon provides the propulsive thrust. A variation on the helical rotor model of [1] allows the motors to be anchored to the peptidoglycan layer, where they drive rotation of the track creating traveling helical waves along the S-layer. We derive expressions relating the swimming speed to the amplitude, wavelength, and velocity of the surface waves induced by the helical rotor, and show that they fall in reasonable ranges to explain the velocity and rotation rate of swimming Synechococcus.

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.

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.

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.

Coupling translocation with nucleic acid unwinding by NS3 helicase.

We present a semiquantitative model for translocation and unwinding activities of monomeric nonstructural protein 3 (NS3) helicase. The model is based on structural, biochemical, and single-molecule measurements. The model predicts that the NS3 helicase actively unwinds duplex by reducing more than 50% the free energy that stabilizes base pairing/stacking. The unwinding activity slows the movement of the helicase in a sequence-dependent manner, lowering the average unwinding efficiency to less than 1 bp per ATP cycle. When bound with ATP, the NS3 helicase can display significant translocational diffusion. This increases displacement fluctuations of the helicase, decreases the average unwinding efficiency, and enhances the sequence dependence. Also, interactions between the helicase and the duplex stabilize the helicase at the junction, facilitating the helicase's unwinding activity while preventing it from dissociating. In the presence of translocational diffusion during active unwinding, the dissociation rate of the helicase also exhibits sequence dependence. Based on unwinding velocity fluctuations measured from single-molecule experiments, we estimate the diffusion rate to be on the order of 10 s(-1). The generic features of coupling single-stranded nucleic acid translocation with duplex unwinding presented in this work may apply generally to a class of helicases.

Experimentalist meets theoretician: a tale of two scientific cultures.

This is the story of how a small team of experimentalists and theoreticians collaborated to develop a theoretical model for vesicle formation during endocytosis. In telling our story, we hope to distil some general conclusions about the purpose and value of theoretical models and how best to navigate collaborations between experimentalists and theoreticians. We encountered challenges in building and publishing our model, but through our experiences we gained insight into how such collaborations can be profitably conducted. We also developed opinions about how theoretical models should be evaluated by peer reviewers and editors. During the evolution of our theoretical model, we educated each other, organized our thoughts and our data, developed a conceptual framework for understanding the mechanochemistry of endocytosis, and generated testable hypotheses that stimulated new experiments.

Mechanochemistry of a viral DNA packaging motor.

The pentameric ATPase motor gp16 packages double-stranded DNA into the bacteriophage phi29 virus capsid. On the basis of the results of single-molecule experimental studies, we propose a push and roll mechanism to explain how the packaging motor translocates the DNA in bursts of four 2.5 bp power strokes, while rotating the DNA. In this mechanism, each power stroke accompanies P(i) release after ATP hydrolysis. Since the high-resolution structure of the gp16 motor is not available, we borrowed characterized features from the P4 RNA packaging motor in bacteriophage phi12. For each power stroke, a lumenal lever from a single subunit is electrostatically steered to the DNA backbone. The lever then pushes sterically, orthogonal to the backbone axis, such that the right-handed DNA helix is translocated and rotated in a left-handed direction. The electrostatic association allows tight coupling between the lever and the DNA and prevents DNA from slipping back. The lever affinity for DNA decreases towards the end of the power stroke and the DNA rolls to the lever on the next subunit. Each power stroke facilitates ATP hydrolysis in the next catalytic site by inserting an Arg -finger into the site, as captured in phi12-P4. At the end of every four power strokes, ADP release happens slowly, so the cycle pauses constituting a dwell phase during which four ATPs are loaded into the catalytic sites. The next burst phase of four power strokes starts once spontaneous ATP hydrolysis takes place in the fifth site without insertion of an Arg finger. The push and roll model provides a new perspective on how a multimeric ATPase transports DNA, and it might apply to other ring motors as well.

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.

Mechanochemical crosstalk during endocytic vesicle formation.

Membrane curvature has emerged as a key regulatory factor in endocytic vesicle formation. From a theoretical perspective, we summarize recent progress in understanding how membrane curvature and biochemical pathways are coupled and orchestrated during the coherent process of endocytic vesicle formation. We mainly focus on clathrin-mediated and actin-mediated endocytosis in yeast and in mammalian cells. We further speculate on how mechanochemical feedback could modulate other membrane-remodeling processes.

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.