Direct Measurements of Drag Forces in C.?elegans Crawling Locomotion

With a simple and versatile microcantilever-based force measurement technique, we have probed the drag forces involved in Caenorhabditis elegans locomotion. As a worm crawls on an agar surface, we found that substrate viscoelasticity introduces nonlinearities in the force-velocity relationships, yielding nonconstant drag coefficients that are not captured by original resistive force theory. A major contributing factor to these nonlinearities is the formation of a shallow groove on the agar surface. We measured both the adhesion forces that cause the worm’s body to settle into the agar and the resulting dynamics of groove formation. Furthermore, we quantified the locomotive forces produced by C. elegans undulatory motions on a wet viscoelastic agar surface. We show that an extension of resistive force theory is able to use the dynamics of a nematode’s body shape along with the measured drag coefficients to predict the forces generated by a crawling nematode.     

Undulatory locomotion of Caenorhabditis elegans on wet surfaces

The physical and biomechanical principles that govern undulatory movement on wet surfaces have important applications in physiology, physics, and engineering. The nematode Caenorhabditis elegans, with its highly stereotypical and functionally distinct sinusoidal locomotory gaits, is an excellent system in which to dissect these properties. Measurements of the main forces governing the C. elegans crawling gait on lubricated surfaces have been scarce, primarily due to difficulties in estimating the physical features at the nematode-gel interface. Using kinematic data and a hydrodynamic model based on lubrication theory, we calculate both the surface drag forces and the nematode's bending force while crawling on the surface of agar gels within a preexisting groove. We find that the normal and tangential surface drag coefficients during crawling are ～222 and 22, respectively, and the drag coefficient ratio is ～10. During crawling, the calculated internal bending force is time-periodic and spatially complex, exhibiting a phase lag with respect to the nematode's body bending curvature. This phase lag is largely due to viscous drag forces, which are higher during crawling as compared to swimming in an aqueous buffer solution. The spatial patterns of bending force generated during either swimming or crawling correlate well with previously described gait-specific features of calcium signals in muscle. Further, our analysis indicates that one may be able to control the motility gait of C. elegans by judiciously adjusting the magnitude of the surface drag coefficients.     

Mechanics of Undulatory Swimming in a Frictional Fluid

The sandfish lizard (Scincus scincus) swims within granular media (sand) using axial body undulations to propel itself without the use of limbs. In previous work we predicted average swimming speed by developing a numerical simulation that incorporated experimentally measured biological kinematics into a multibody sandfish model. The model was coupled to an experimentally validated soft sphere discrete element method simulation of the granular medium. In this paper, we use the simulation to study the detailed mechanics of undulatory swimming in a “granular frictional fluid” and compare the predictions to our previously developed resistive force theory (RFT) which models sand-swimming using empirically determined granular drag laws. The simulation reveals that the forward speed of the center of mass (CoM) oscillates about its average speed in antiphase with head drag. The coupling between overall body motion and body deformation results in a non-trivial pattern in the magnitude of lateral displacement of the segments along the body. The actuator torque and segment power are maximal near the center of the body and decrease to zero toward the head and the tail. Approximately 30% of the net swimming power is dissipated in head drag. The power consumption is proportional to the frequency in the biologically relevant range, which confirms that frictional forces dominate during sand-swimming by the sandfish. Comparison of the segmental forces measured in simulation with the force on a laterally oscillating rod reveals that a granular hysteresis effect causes the overestimation of the body thrust forces in the RFT. Our models provide detailed testable predictions for biological locomotion in a granular environment.     

A computational fluid dynamics (CFD) based method for assessing the hydrodynamic impact of animal borne data loggers on host marine mammals

Animal‐borne data loggers (ABDLs) or “tags” are regularly used to elucidate animal ecology and physiology, but current literature highlights the need to assess associated deleterious impacts including increased resistive force to motion. Previous studies have used computational fluid dynamics (CFD) to estimate this impact, but many suffer limitations (e.g., inaccurate turbulence modeling, neglecting boundary layer transition, neglecting added mass effects, and analyzing the ABDL in isolation from the animal). A novel CFD‐based method is presented in which a “tag impact envelope” is defined utilizing simulations with and without transition modeling to define upper and lower drag limits, respectively, and added mass coefficients are found via simulations with sinusoidally varying inlet velocity, with modified Navier‐Stokes conservation of momentum equations enforcing a shift to the animal's noninertial reference frame. The method generates coefficients for calculating total resistive force for any velocity and acceleration combination, and is validated against theory for a prolate spheroid. An example case shows ABDL drag impact on a harp seal of 11.21%–16.24%, with negligible influence on added mass. By considering the effects of added mass and boundary layer transition, the approach presented is an enhancement to the CFD‐based ABDL impact assessment methods previously applied by researchers.     

In situ measurements of hydrodynamic forces imposed on Chondrus crispus Stackhouse

Among the hydrodynamic forces experienced by intertidal organisms, drag and the impingement force are thought to have the greatest effect on macroalgae. These forces are modified by biotic factors such as algal morphology, reconfiguration, and the presence of a canopy. However, much of what is known about the hydrodynamics of macroalgae has been garnered from low-velocity laboratory flume studies. Few field studies have measured drag and none have directly measured the effects of the canopy on force. To examine in situ hydrodynamic forces imposed on the turf forming macroalga Chondrus crispus, compact digital force sensors were developed that measure and record the 3-dimensional force imposed on a macroalga without disturbing the surrounding canopy. Sensors were positioned within natural Chondrus beds and the effects of the canopy, algal morphology, and sea state on in situ hydrodynamic force were examined. Additionally, the predictions of a new model for drag on flexible macroalgae were tested by simultaneously measuring force and water velocity. Digital force recordings indicated that Chondrus only experience drag; lift and impingement force were negligible in all combinations of factors. Canopies significantly reduced drag by 15-65%. Morphology and size also influenced drag, such that lower forces were imposed on small planar algae than large arborescent individuals. Further, planar algae experienced low drag in all combinations of sea and canopy state, indicating that these individuals may not be as susceptible to wave disturbance as arborescent individuals. Overall, these data indicate that the ability for Chondrus to grow large, arborescent individuals is dependent on the drag reducing properties of the canopy, while more hydrodynamically harsh habitats may be accessible to planar morphologies. Additionally, these data suggest that drag models for canopy forming macroalgae must incorporate the effects of the canopy to predict drag accurately in situ.     

Micromechanics of the Vertebrate Meiotic Spindle Examined by Stretching along the Pole-to-Pole Axis

The meiotic spindle is a bipolar molecular machine that is designed to segregate duplicated chromosomes toward the opposite poles of the cell. The size and shape of the spindle are considered to be maintained by a balance of forces produced by molecular motors and microtubule assembly dynamics. Several studies have probed how mechanical perturbations of the force balance affect the spindle structure. However, the spindle’s response to a stretching force acting at the spindle pole and along its long axis, i.e., the direction in which chromosomes are segregated, has not been examined. Here, we describe a method to apply a stretching force to the metaphase spindle assembled in Xenopus egg extracts and measure the relationship between the force and the three-dimensional deformation of the spindle. We found that the spindle behaves as a Zener-type viscoelastic body when forces are applied at the spindle pole, generating a restoring force for several minutes. In addition, both the volume of the spindle and the tubulin density are conserved under the stretching force. These results provide insight into how the spindle size is maintained at metaphase.     

Scaling of lunge feeding in rorqual whales: An integrated model of engulfment duration

Rorqual whales (Balaenopteridae) obtain their food by lunge feeding, a dynamic process that involves the intermittent engulfment and filtering of large amounts of water and prey. During a lunge, whales accelerate to high speed and open their mouth wide, thereby exposing a highly distensible buccal cavity to the flow and facilitating its inflation. Unsteady hydrodynamic models suggest that the muscles associated with the ventral groove blubber undergo eccentric contraction in order to stiffen and control the inflation of the buccal cavity; in doing so the engulfed water mass is accelerated forward as the whale’s body slows down. Although the basic mechanics of lunge feeding are relatively well known, the scaling of this process remains poorly understood, particularly with regards to its duration (from mouth opening to closure). Here we formulate a new theory of engulfment time which integrates prey escape behavior with the mechanics of the whale’s body, including lunge speed and acceleration, gape angle dynamics, and the controlled inflation of the buccal cavity. Given that the complex interaction between these factors must be highly coordinated in order to maximize engulfment volume, the proposed formulation rests on the scenario of Synchronized Engulfment, whereby the filling of the cavity (posterior to the temporomandibular joint) coincides with the moment of maximum gape. When formulated specifically for large rorquals feeding on krill, our analysis predicts that engulfment time increases with body size, but in amounts dictated by the specifics of krill escape and avoidance kinematics. The predictions generated by the model are corroborated by limited empirical data on a species-specific basis, particularly for humpback and blue whales chasing krill. A sensitivity analysis applied to all possible sized fin whales also suggests that engulfment duration and lunge speed will increase intra-specifically with body size under a wide range of predator-prey scenarios. This study provides the theoretical framework required to estimate the scaling of the mass-specific drag being generated during engulfment, as well as the energy expenditures incurred.     

Getting a grip on the intertidal: flow microhabitat and substratum type determine the dislodgement of the crab Pachygrapsus crassipes (Randall) on rocky shores and in estuaries

Hydrodynamic forces can affect survival as well as limit the movement of motile benthic animals. An animal's danger of dislodgement depends on the hydrodynamic forces it experiences in its microhabitat relative to the force required to dislodge it (tenacity) from the substratum. We measured water flow and substratum characteristics in two different habitats of the shore crab Pachygrapsus crassipes: a wave-swept rocky shore and an intertidal mudflat. The maximum water velocities and accelerations in the microhabitats of the crabs at the wave-swept site were three times and two times greater, respectively, than at the mudflat site. In the laboratory, we measured the tenacity of crabs of various sizes on different substrata, and also measured their drag, lift and added-mass coefficients. Using these data, we calculated the flow conditions under which crabs would be overturned or sheared off the substratum in their two habitats. The net horizontal force (drag plus acceleration reaction) required to dislodge a crab on a rugose rock substratum was an order of magnitude greater than on smooth rock and two orders of magnitude greater than on mud. Our calculations indicate that, under non-storm conditions, crabs will not be dislodged from the substratum in either the mudflat or the wave-swept habitat when grasping the substratum with maximum tenacity. Moving crabs have lower tenacity and our calculations predict that hydrodynamic forces will restrict the mobility of large crabs more than that of small ones on smooth, but not on rugose rock.     

Membrane-based actuation for high-speed single molecule force spectroscopy studies using AFM

Atomic force microscopy (AFM)-based dynamic force spectroscopy of single molecular interactions involves characterizing unbinding/unfolding force distributions over a range of pulling speeds. Owing to their size and stiffness, AFM cantilevers are adversely affected by hydrodynamic forces, especially at pulling speeds >10 μm/s, when the viscous drag becomes comparable to the unbinding/unfolding forces. To circumvent these adverse effects, we have fabricated polymer-based membranes capable of actuating commercial AFM cantilevers at speeds ≥100 μm/s with minimal viscous drag effects. We have used FLUENT^®, a computational fluid dynamics (CFD) software, to simulate high-speed pulling and fast actuation of AFM cantilevers and membranes in different experimental configurations. The simulation results support the experimental findings on a variety of commercial AFM cantilevers and predict significant reduction in drag forces when membrane actuators are used. Unbinding force experiments involving human antibodies using these membranes demonstrate that it is possible to achieve bond loading rates ≥10⁶ pN/s, an order of magnitude greater than that reported with commercial AFM cantilevers and systems.     

A modified blade element theory for estimation of forces generated by a beetle-mimicking flapping wing system

We present an unsteady blade element theory (BET) model to estimate the aerodynamic forces produced by a freely flying beetle and a beetle-mimicking flapping wing system. Added mass and rotational forces are included to accommodate the unsteady force. In addition to the aerodynamic forces needed to accurately estimate the time history of the forces, the inertial forces of the wings are also calculated. All of the force components are considered based on the full three-dimensional (3D) motion of the wing. The result obtained by the present BET model is validated with the data which were presented in a reference paper. The difference between the averages of the estimated forces (lift and drag) and the measured forces in the reference is about 5.7%. The BET model is also used to estimate the force produced by a freely flying beetle and a beetle-mimicking flapping wing system. The wing kinematics used in the BET calculation of a real beetle and the flapping wing system are captured using high-speed cameras. The results show that the average estimated vertical force of the beetle is reasonably close to the weight of the beetle, and the average estimated thrust of the beetle-mimicking flapping wing system is in good agreement with the measured value. Our results show that the unsteady lift and drag coefficients measured by Dickinson et al are still useful for relatively higher Reynolds number cases, and the proposed BET can be a good way to estimate the force produced by a flapping wing system.     

Single-molecule force spectroscopy reveals a stepwise unfolding of Caenorhabditis elegans giant protein kinase domains

Myofibril assembly and disassembly are complex processes that regulate overall muscle mass. Titin kinase has been implicated as an initiating catalyst in signaling pathways that ultimately result in myofibril growth. In titin, the kinase domain is in an ideal position to sense mechanical strain that occurs during muscle activity. The enzyme is negatively regulated by intramolecular interactions occurring between the kinase catalytic core and autoinhibitory/regulatory region. Molecular dynamics simulations suggest that human titin kinase acts as a force sensor. However, the precise mechanism(s) resulting in the conformational changes that relieve the kinase of this autoinhibition are unknown. Here we measured the mechanical properties of the kinase domain and flanking Ig/Fn domains of the Caenorhabditis elegans titin-like proteins twitchin and TTN-1 using single-molecule atomic force microscopy. Our results show that these kinase domains have significant mechanical resistance, unfolding at forces similar to those for Ig/Fn β-sandwich domains (30-150 pN). Further, our atomic force microscopy data is consistent with molecular dynamic simulations, which show that these kinases unfold in a stepwise fashion, first an unwinding of the autoinhibitory region, followed by a two-step unfolding of the catalytic core. These data support the hypothesis that titin kinase may function as an effective force sensor.     

The forces that position a mitotic spindle asymmetrically are tethered until after the time of spindle assembly

Regulation of the mitotic spindle's position is important for cells to divide asymmetrically. Here, we use Caenorhabditis elegans embryos to provide the first analysis of the temporal regulation of forces that asymmetrically position a mitotic spindle. We find that asymmetric pulling forces, regulated by cortical PAR proteins, begin to act as early as prophase and prometaphase, even before the spindle forms and shifts to a posterior position. The spindle does not shift asymmetrically during these early phases due to a tethering force, mediated by astral microtubules that reach the anterior cell cortex. We show that this tether is normally released after spindle assembly and independently of anaphase entry. Monitoring microtubule dynamics by photobleaching segments of microtubules during anaphase revealed that spindle microtubules do not undergo significant poleward flux in C. elegans. Together with the known absence of anaphase A, these data suggest that the major forces contributing to chromosome separation during anaphase originate outside the spindle. We propose that the forces positioning the mitotic spindle asymmetrically are tethered until after the time of spindle assembly and that these same forces are used later to drive chromosome segregation at anaphase.     

A Force Balance Model of Early Spindle Pole Separation in Drosophila Embryos

The formation and function of the mitotic spindle depends upon force generation by multiple molecular motors and by the dynamics of microtubules, but how these force-generating mechanisms relate to one another is unclear. To address this issue we have modeled the separation of spindle poles as a function of time during the early stages of spindle morphogenesis in Drosophila embryos. We propose that the outward forces that drive the separation of the spindle poles depend upon forces exerted by cortical dynein and by microtubule polymerization, and that these forces are antagonized by a C-terminal kinesin, Ncd, which generates an inward force on the poles. We computed the sum of the forces generated by dynein, microtubule polymerization, and Ncd, as a function of the extent of spindle pole separation and solved an equation relating the rate of pole separation to the net force. As a result, we obtained graphs of the time course of spindle pole separation during interphase and prophase that display a reasonable fit to the experimental data for wild-type and motor-inhibited embryos. Among the novel contributions of the model are an explanation of pole separation after simultaneous loss of Ncd and dynein function, and the prediction of a large value for the effective centrosomal drag that is needed to fit the experimental data. The results demonstrate the utility of force balance models for explaining certain mitotic movements because they explain semiquantitatively how the force generators drive a rapid initial burst of pole separation when the net force is great, how pole separation slows down as the force decreases, and how a stable separation of the spindle poles characteristic of the prophase steady state is achieved when the force reaches zero.     

Drag of suction cup tags on swimming animals: Modeling and measurement

Bio‐logging tags are widely used to study the behavior and movements of marine mammals with the tacit assumption of little impact to the animal. However, tags on fast‐swimming animals generate substantial hydrodynamic forces potentially affecting behavior and energetics adversely, or promoting early removal of the tag. In this work, hydrodynamic loading of three novel tag housing designs are compared over a range of swimming speeds using computational fluid dynamics (CFD). Results from CFD simulation were verified using tag models in a water flume with close agreement. Drag forces were reduced by minimizing geometric disruptions to the flow around the housing, while lift forces were reduced by minimizing the frontal cross‐sectional area of the housing and holding the tag close to the attachment surface. Hydrodynamic tag design resulted in an experimentally measured 60% drag force reduction in 5.6 m/s flow. For all housing designs, off‐axis flow increased the magnitude of the force on the tag. Experimental work with a common dolphin (Delphinus delphis) cadaver indicates that the suction cups used to attach the types of tags described here provide sufficient attachment force to resist failure to predicted forces at swimming speeds of up to 10 m/s.     

Influence of surface penetration on measured fluid force on a hand model

The purpose of this study was to quantify the effect of wave drag due to surface penetration on drag and lift forces (C_d and C_l) acting on a hand model. The values of C_d and C_l had been acquired to gain the hydrodynamic characteristics of the swimmer's hand and predict force on the swimmer's hand. These values have also been used to benchmark computational fluid dynamics analysis. Because the previous studies used a hand/forearm model which penetrated the water's surface, the values of C_d and C_l include the effect of the surface wave on the model. Wave formation causes pressure differences between the frontal and rear sides of a surface-penetrating model as a result of depressions and elevations in the water's surface. This may be considered as wave drag due to surface penetration. Fluid forces due to wave drag on the forearm should not be included in the measured C_d and C_l of a swimmer's hand that does not sweep near the water's surface. Two hand/forearm models are compared, one with the hand rigidly connected to the forearm. The other model was constructed to isolate the fluid forces acting on the hand from the influence of wave drag on the forearm. The measurements showed that the effect of wave drag on the hand model caused large increases in the values of C_d, up to 46–98% with lesser increases in C_l of 2–12% depending on the hand orientation. The present study provides an improved method to determine the values of C_d and C_l that eliminates the effect of wave drag on a hand/forearm model by isolating the measurement of fluid forces on the forearm of the hand/forearm model in order to separately acquire the forces on the hand.     

Impact of chemical manipulation of tarsal liquids on attachment in the Colorado potato beetle, Leptinotarsa decemlineata

Insect tarsal attachment forces are thought to be influenced by the viscosity and surface tension of a thin film of adhesive liquid (wet adhesion). In beetles, this fluid has been shown to be composed mainly of lipophilic substances that are similar to the cuticular lipids. In this study we investigate whether and how the chemical composition of footprint lipids affects attachment forces in the Colorado potato beetle, Leptinotarsa decemlineata. After application of standardised mixtures of synthetic n-alkanes or alkenes, or a concentrated hydrocarbon extract to the surface of the elytra, we tested the beetles’ attachment performance using a beam force transducer. The results show that only the unsaturated components, but not the straight-chained alkanes reduced friction forces, confirming that attachment performance is influenced by the chemical composition of the adhesive secretion. We estimated the volume of footprint droplets and calculated a mean thickness of the liquid layer of 0.04 μm. The measured friction exceeded the viscous and capillary force expected for a film of this thickness. Therefore, alternative mechanisms (i.e. shear-thinning and solid-like behaviour) for the generation of attachment forces and their dependence on the chemical composition of the liquid are discussed.     

Unsteady Motion, Finite Reynolds Numbers, and Wall Effect on Vorticella convallaria Contribute Contraction Force Greater than the Stokes Drag

Contraction of Vorticella convallaria, a sessile ciliated protozoan, is completed within a few milliseconds and results in a retraction of its cell body toward the substratum by coiling its stalk. Previous studies have modeled the cell body as a sphere and assumed a drag force that satisfies Stokes' law. However, the contraction-induced flow of the medium is transient and bounded by the substrate, and the maximum Reynolds number is larger than unity. Thus, calculations of contractile force from the drag force are incomplete. In this study, we analyzed fluid flow during contraction by the particle tracking velocimetry and computational fluid dynamics simulations to estimate the contractile force. Particle paths show that the induced flow is limited by the substrate. Simulation-based force estimates suggest that the combined effect of the flow unsteadiness, the finite Reynolds number, and the substrate comprises 35% of the total force. The work done in the early stage of contraction and the maximum power output are similar regardless of the medium viscosity. These results suggest that, during the initial development of force, V. convallaria uses a common mechanism for performing mechanical work irrespective of viscous loading conditions.     

The coordination of spindle‐positioning forces during the asymmetric division of the Caenorhabditis elegans zygote

In Caenorhabditis elegans zygote, astral microtubules generate forces essential to position the mitotic spindle, by pushing against and pulling from the cortex. Measuring microtubule dynamics there, we revealed the presence of two populations, corresponding to pulling and pushing events. It offers a unique opportunity to study, under physiological conditions, the variations of both spindle‐positioning forces along space and time. We propose a threefold control of pulling force, by polarity, spindle position and mitotic progression. We showed that the sole anteroposterior asymmetry in dynein on‐rate, encoding pulling force imbalance, is sufficient to cause posterior spindle displacement. The positional regulation, reflecting the number of microtubule contacts in the posterior‐most region, reinforces this imbalance only in late anaphase. Furthermore, we exhibited the first direct proof that dynein processivity increases along mitosis. It reflects the temporal control of pulling forces, which strengthens at anaphase onset following mitotic progression and independently from chromatid separation. In contrast, the pushing force remains constant and symmetric and contributes to maintaining the spindle at the cell centre during metaphase.     

Force measurements of Myosin II waves at the yolk surface during Drosophila dorsal closure

The mechanical properties and the forces involved during tissue morphogenesis have been the focus of much research in the last years. Absolute values of forces during tissue closure events have not yet been measured. This is also true for a common force-producing mechanism involving Myosin II waves that results in pulsed cell surface contractions. Our patented magnetic tweezer, CAARMA, integrated into a spinning disk confocal microscope, provides a powerful explorative tool for quantitatively measuring forces during tissue morphogenesis. Here, we used this tool to quantify the in vivo force production of Myosin II waves that we observed at the dorsal surface of the yolk cell in stage 13 Drosophila melanogaster embryos. In addition to providing for the first time to our knowledge quantitative values on an active Myosin-driven force, we elucidated the dynamics of the Myosin II waves by measuring their periodicity in both absence and presence of external perturbations, and we characterized the mechanical properties of the dorsal yolk cell surface.     

Flexibility of the Bacterial Chaperone Trigger Factor in Microsecond-Timescale Molecular Dynamics Simulations

The bacterial chaperone trigger factor (TF) is the first chaperone to be encountered by a nascent protein chain as it emerges from the ribosome exit tunnel. Experimental results suggest that TF possesses considerable conformational flexibility, and in an attempt to provide an atomic-level view of this flexibility, we have performed independent 1.5-μs molecular dynamics simulations of TF in explicit solvent using two different simulation force fields (OPLS-AA/L and AMBER ff99SB-ILDN). Both simulations indicate that TF possesses tremendous flexibility, with huge excursions from the crystallographic conformation caused by reorientations of the protein’s constituent domains; both simulations also predict the formation of extensive contacts between TF’s PPIase domain and the Arm 1 domain that is involved in nascent-chain binding. In the OPLS simulation, however, TF rapidly settles into a very compact conformation that persists for at least 1 μs, whereas in the AMBER simulation, it remains highly dynamic; additional simulations in which the two force fields were swapped suggest that these differences are at least partly attributable to sampling issues. The simulation results provide potential rationalizations of a number of experimental observations regarding TF’s conformational behavior and have implications for using simulations to model TF’s function on translating ribosomes.