Estimation of cell Young's modulus of adherent cells probed by optical and magnetic tweezers: influence of cell thickness and bead immersion

A precise characterization of cell elastic properties is crucial for understanding the mechanisms by which cells sense mechanical stimuli and how these factors alter cellular functions. Optical and magnetic tweezers are micromanipulation techniques which are widely used for quantifying the stiffness of adherent cells from their response to an external force applied on a bead partially embedded within the cell cortex. However, the relationships between imposed external force and resulting bead translation or rotation obtained from these experimental techniques only characterize the apparent cell stiffness. Indeed, the value of the estimated apparent cell stiffness integrates the effect of different geometrical parameters, the most important being the bead embedding angle 2gamma, bead radius R, and cell height h. In this paper, a three-dimensional finite element analysis was used to compute the cell mechanical response to applied force in tweezer experiments and to explicit the correcting functions which have to be used in order to infer the intrinsic cell Young's modulus from the apparent elasticity modulus. Our analysis, performed for an extensive set of values of gamma, h, and R, shows that the most relevant parameters for computing the correcting functions are the embedding half angle gamma and the ratio h(u)/2R, where h(u) is the under bead cell thickness. This paper provides original analytical expressions of these correcting functions as well as the critical values of the cell thickness below which corrections of the apparent modulus are necessary to get an accurate value of cell Young's modulus. Moreover, considering these results and taking benefit of previous results obtained on the estimation of cell Young's modulus of adherent cells probed by magnetic twisting cytometry (MTC) (Ohayon, J., and Tracqui, P., 2005, Ann. Biomed. Eng., 33, pp. 131-141), we were able to clarify and to solve the still unexplained discrepancies reported between estimations of elasticity modulus performed on the same cell type and probed with MTC and optical tweezers (OT). More generally, this study may strengthen the applicability of optical and magnetic tweezers techniques by insuring a more precise estimation of the intrinsic cell Young's modulus (CYM).     

Rotation of bacterial flagella as driven by cytomembrane streaming

Theoretical estimates are given to check the possibility that flagellar rotation of bacteria is driven by viscous forces exerted from a streaming cytomembrane matrix to the basal structure of the flagellum.For different regimes of cytomembrane streaming, i.e. for circular, shearing and uniform linear motion of the membrane matrix past the basal ring of the flagellum, the velocity of streaming is computed that will yield the necessary mechanical torque for rotation of a helical flagellum in a watery medium.It is shown that in the range of surface viscosities determined for phospholipid monolayers the required velocities of cytomembrane streaming may be expected in the range of 3 μm/s to 60 μm/s. Since streaming velocities of the latter order of magnitude have been observed in protozoan membrane, efforts appear warranted to demonstrate experimentally the existence of cytomembrane streaming in bacteria and to characterize the surface viscosity of their cytomembrane.     

Study of the torque of the bacterial flagellar motor using a rotating electric field.

Bacterial flagella are driven by a rotary motor that is energized by an electrochemical ion gradient across the cell membrane. In this study the torque generated by the flagellar motor was measured in tethered cells of a smooth-swimming Escherichia coli strain by using rotating electric fields to determine the relationship between the torque and speed over a wide range. By measuring the electric current applied to the sample cell and combining the data obtained at different viscosities, the torque of the flagellar motor was estimated up to 55 Hz, and also at negative rotation rates. By this method we have found that the torque of the flagellar motor linearly decreases with rotation rate from negative through positive rate of rotation. In addition, the dependence of torque upon temperature was also investigated. We showed that torque at the high speeds encountered in swimming cells had a much steeper dependence on temperature that at the low speeds encountered in tethered cells. From these results, the activation energy of the proton transfer reaction in the torque-generating unit was calculated to be about 7.0 x 10(-20) J.     

Preparation of bead-tailed actin filaments: estimation of the torque produced by the sliding force in an in vitro motility assay.

By coating covalently the surface of a polystyrene bead (diameter = 1 micron) with gelsolin, we have succeeded in attaching the bead selectively at the barbed end of an actin filament and forming a 1:1 bead-actin filament complex. On a layer of heavy meromyosin on a nitrocellulose-coated coverglass, this bead-actin filament complex slid smoothly, trailing the bead at its end. Therefore we called this preparation "bead-tailed" actin filaments. The sliding velocity was indistinguishable from that of nonbeaded filaments. With use of this system, we tried to detect the axial rotation (rotation around the filament axis) in a sliding actin filament. Although a single bead at the tail end did not serve as the marker for the axial rotation, we occasionally found another bead bound to the tail bead. In this case, the orientation of the bead-aggregate could be followed continuously with a video monitor while the filament was sliding over heavy meromyosin. We observed that actin filaments slid over distances of many tens of micrometers without showing a complete turn of the bead-aggregates. On the basis of the calculation of rotational friction drag on the bead-aggregate, we estimate that the rotational component of the sliding force and the torque produced on a sliding actin filament (length < or = 10 microns) did not accumulate > 1 pN and 5 pN.nm, respectively. In the present system of randomly oriented heavy meromyosin lying on a nitrocellulose film without an external load.     

Effect of the cytoskeletal prestress on the mechanical impedance of cultured airway smooth muscle cells.

We investigated the effect of the cytoskeletal prestress (P) on the elastic and frictional properties of cultured human airway smooth muscle cells during oscillatory loading; P is preexisting tensile stress in the actin cytoskeleton generated by the cell contractile apparatus. We oscillated (0.1 Hz, 6 Pa peak to peak) small ferromagnetic beads bound to integrin receptors and computed the storage (elastic) modulus (G') and the loss (frictional) modulus (G") from the applied torque and the corresponding bead rotation. All measurements were done at baseline and after cells were treated with graded doses of either histamine (0.1, 1, 10 microM) or isoproterenol (0.01, 0.1, 1, 10 microM). Values for P for these concentrations were taken from a previous study (Wang et al., Am J Physiol Cell Physiol, in press). It was found that G' and G", as well as P, increased/decreased with increasing doses of histamine/isoproterenol. Both G' and G" exhibited linear dependences on P: G'(Pa) = 0.20P + 82 and G"(Pa) = 0.05P + 32. The dependence of G' on P is consistent with our previous findings and with the behavior of stress-supported structures. The dependence of G" on P is a novel finding. It could be attributed to a variety of mechanisms. Some of those mechanisms are discussed in detail. We concluded that, in addition to the central mechanisms by which stress-supported structures develop mechanical stresses, other mechanisms might need to be invoked to fully explain the observed dependence of the cell mechanical properties on the state of cell contractility.     

Control for small-speed lateral flight in a model insect

Controls required for small-speed lateral flight of a model insect were studied using techniques based on the linear theories of stability and control (the stability and control derivatives were computed by the method of computational fluid dynamics). The main results are as follows. (1) Two steady-state lateral motions can exist: one is a horizontal side translation with the body rolling to the same side of the translation by a small angle, and the other is a constant-rate yaw rotation (rotation about the vertical axis). (2) The side translation requires an anti-symmetrical change in the stroke amplitudes of the contralateral wings, and/or an anti-symmetrical change in the angles of attack of the contralateral wings, with the down- and upstroke angles of attack of a wing having equal change. The constant-rate yaw rotation requires an anti-symmetrical change in the angles of attack of the contralateral wings, with the down- and upstroke angles of attack of a wing having differential change. (3) For the control of the horizontal side translation, control input required for the steady-state motion has an opposite sign to that needed for initiating the motion. For example, to have a steady-state left side-translation, the insect needs to increase the stroke amplitude of the left wing and decrease that of the right wing to maintain the steady-state flight, but it needs an opposite change in stroke amplitude (decreasing the stroke amplitude of the left wing and increasing that of the right wing) to enter the flight.     

Viscoelasticity of the human red blood cell

We report here the first measurements of the complex modulus of the isolated red blood cell (RBC). Because the RBC is often larger than capillary diameter, important determinants of microcirculatory function are RBC deformability and its changes with pathologies, such as sickle cell disease and malaria. A functionalized ferrimagnetic microbead was attached to the membrane of healthy RBC and then subjected to an oscillatory magnetic field. The resulting torque caused cell deformation. From the oscillatory forcing and resulting bead motions, which were tracked optically, we computed elastic and frictional moduli, g' and g', respectively, from 0.1 to 100 Hz. The g' was nearly frequency independent and dominated the response at all but the highest frequencies measured. Over three frequency decades, g' increased as a power law with an exponent of 0.64, a result not predicted by any simple model. These data suggest that RBC relaxation times that have been reported previously, and any models that rest upon them, are artifactual; the artifact, we suggest, arises from forcing to an exponential fit data of limited temporal duration. A linear range of response was observed, but, as forcing amplitude increased, nonlinearities became clearly apparent. A finite element model suggests that membrane bending was localized to the vicinity of the bead and dominated membrane shear. While the mechanisms accounting for these RBC dynamics remain unclear, methods described here establish new avenues for the exploration of connections among the mechanical, chemical, and biological characteristics of the RBC in health and disease. storage modulus; loss modulus; magnetic twisting cytometry; erythrocyte; viscoelasticity; rheology     

Sensitive force technique to probe molecular adhesion and structural linkages at biological interfaces.

Adhesion and cytoskeletal structure are intimately related in biological cell function. Even with the vast amount of biological and biochemical data that exist, little is known at the molecular level about physical mechanisms involved in attachments between cells or about consequences of adhesion on the material structure. To expose physical actions at soft biological interfaces, we have combined an ultrasensitive transducer and reflection interference microscopy to image submicroscopic displacements of probe contact with a test surface under minuscule forces. The transducer is a cell-size membrane capsule pressurized by micropipette suction where displacement normal to the membrane under tension is proportional to the applied force. Pressure control of the tension tunes the sensitivity in operation over four orders of magnitude through a range of force from 0.01 pN up to the strength of covalent bonds (approximately 1000 pN)! As the surface probe, a microscopic bead is biochemically glued to the transducer with a densely-bound ligand that is indifferent to the test surface. Movements of the probe under applied force are resolved down to an accuracy of approximately 5 nm from the interference fringe pattern created by light reflected from the bead. With this arrangement, we show that local mechanical compliance of a cell surface can be measured at a displacement resolution set by structural fluctuations. When desired, a second ligand is bound sparsely to the probe for focal adhesion to specific receptors in the test surface. We demonstrate that monitoring fluctuations in probe position at low transducer stiffness enhances detection of molecular adhesion and activation of cytoskeletal structure.(ABSTRACT TRUNCATED AT 250 WORDS)     

Condylar translation and the function of the superficial masseter muscle in the rhesus monkey (M. mulatta).

The relationship between translation of the mandibular condyle during symmetrical mandibular rotation, i.e., symmetrical jaw depression and elevation, and the function of the superficial masseter muscle was examined in light of relative torque and the length-tension relationship for muscle. Lateral cephalograms of live adult rhesus monkeys (Macaca mulatta) were analyzed using two models: (1) Model A, normal symmetrical jaw rotation accompanied by condylar translation; and (2) Model B, mandibular rotation about an axis fixed at the position of the condyles during centric occlusion. The decrease in relative torque and the excursion of the superficial masseter at mouth-open positions are significantly greater in Model B than in Model A. Symmetrical rotation of the jaw about a fixed axis would result in a 35% greater loss of maximum producible tension at maximum gape than rotation associated with condylar translation. These results suggest that condylar translation during mandibular depression and elevation functions to minimize reduction in relative torque and excursion of superficial masseter muscle, thereby maintaining optimal potential for exerting maximum tension during jaw closure.     

Remodeling of the temporomandibular joint following mandibular distraction osteogenesis in the transverse dimension

Transverse mandibular distraction osteogenesis involves moving the osteotomized segments of the mandible in either a varus or valgus direction. This maneuver allows for widening of the bigonial distance or for a lateral shift of an asymmetric mandibular midline. During this process, a significant amount of torque is placed on the mandibular condyles, because they act as the pivot point for the mandibular translation. Although standard linear distraction osteogenesis induces transient, reversible changes in the temporomandibular joint, it is not known what effect the varus and valgus stresses of transverse distraction have on the temporomandibular joint. We therefore designed a study to document the temporomandibular joint changes following various degrees of transverse distraction.Bilateral transverse mandibular distraction was performed on 10 adult, female mongrel dogs using an external, multiplanar mandibular distraction device. The distraction protocol was as follows: (1) complete osteotomy at the angle of the mandible, (2) 5-day latency period, (3) distraction rate of 1 mm/day, (4) rhythm of one turn per day, (5) linear activation 16 to 30 mm bilaterally, and (6) 8-week consolidation period. A variety of varus and valgus distraction vectors were applied to the mandible only after 10 mm of initial linear distraction had been achieved. Posteroanterior and lateral cephalograms were performed throughout the entire process. Pre-distraction and post-consolidation computed tomographic scans were also performed. Changes in mandibular conformation, axis of rotation, temporomandibular joint structure, and glenoid fossa changes were directly assessed by evaluating the postmortem craniofacial skeleton. The findings were compared with those of normal, age-matched mongrel dog skulls.Significant remodeling changes were observed in the temporomandibular joints of all animals involved in the study. The mandibular condyles demonstrated varying degrees of flattening and erosion at all contact points with the craniofacial skeleton. In some cases, the condyle became part of the distraction regenerate process and was hypertrophied in all dimensions. The condyles were frequently displaced out of the glenoid fossa, particularly on the side in the direction of varus distraction. When the latter occurred, a new fossa was created on the undersurface of the zygomatic arch. Varying degrees of mandibular rotation in the sagittal plane were also observed, which led to abnormal torquing of the condyles in the coronal plane, depending on whether the axis of rotation occurred primarily around the condyle or around the distraction regenerate zone.In conclusion, transverse mandibular distraction is an effective means of producing a varus or valgus shift in the gonion relative to the midsagittal plane. However, unlike linear or angular mandibular distraction, transverse distraction has a multitude of nontransient effects on the temporomandibular joint. Therefore it must be emphasized that in clinical practice, transverse distraction should be used cautiously. One must also be aware that such a maneuver in distraction can have negative effects on the temporomandibular joint.     

Stretching short sequences of DNA with constant force axial optical tweezers

Single-molecule techniques for stretching DNA of contour lengths less than a kilobase are fraught with experimental difficulties. However, many interesting biological events such as histone binding and protein-mediated looping of DNA, occur on this length scale. In recent years, the mechanical properties of DNA have been shown to play a significant role in fundamental cellular processes like the packaging of DNA into compact nucleosomes and chromatin fibers. Clearly, it is then important to understand the mechanical properties of short stretches of DNA. In this paper, we provide a practical guide to a single-molecule optical tweezing technique that we have developed to study the mechanical behavior of DNA with contour lengths as short as a few hundred basepairs. The major hurdle in stretching short segments of DNA is that conventional optical tweezers are generally designed to apply force in a direction lateral to the stage (see Fig. 1). In this geometry, the angle between the bead and the coverslip, to which the DNA is tethered, becomes very steep for submicron length DNA. The axial position must now be accounted for, which can be a challenge, and, since the extension drags the microsphere closer to the coverslip, steric effects are enhanced. Furthermore, as a result of the asymmetry of the microspheres, lateral extensions will generate varying levels of torque due to rotation of the microsphere within the optical trap since the direction of the reactive force changes during the extension. Alternate methods for stretching submicron DNA run up against their own unique hurdles. For instance, a dual-beam optical trap is limited to stretching DNA of around a wavelength, at which point interference effects between the two traps and from light scattering between the microspheres begin to pose a significant problem. Replacing one of the traps with a micropipette would most likely suffer from similar challenges. While one could directly use the axial potential to stretch the DNA, an active feedback scheme would be needed to apply a constant force and the bandwidth of this will be quite limited, especially at low forces. We circumvent these fundamental problems by directly pulling the DNA away from the coverslip by using a constant force axial optical tweezers. This is achieved by trapping the bead in a linear region of the optical potential, where the optical force is constant-the strength of which can be tuned by adjusting the laser power. Trapping within the linear region also serves as an all optical force-clamp on the DNA that extends for nearly 350 nm in the axial direction. We simultaneously compensate for thermal and mechanical drift by finely adjusting the position of the stage so that a reference microsphere stuck to the coverslip remains at the same position and focus, allowing for a virtually limitless observation period.     

The response of the human periodontal ligament to torsional loading—I. Experimental methods

A test technique is described in which torsional loads are applied to human maxillary central incisors in vivo. The central axis of the incisor is located by a stereoscopic X-ray method and the tooth loading device is adjusted by means of a setting jig so that torque is applied about this central axis. The maximum torque which can be applied is ± 0·05 Nm and the maximum rotation of the tooth is ± 0·02 radian (± 1·2). A servo-control system allows one to apply any desired torque or deflection history. e.g. a creep or stress relaxation test or a cyclic load of any waveform.

Initial results obtained with this system are presented and show that the torque vs rotation response is initially linear but becomes highly nonlinear at higher torques. Creep tests and cyclic loading tests indicate that the periodontal ligament is viscoelastic in nature. Periods of cyclic loading separated by rest periods of 1–20 min show that the commonly observed decrease in tissue stiffness during the initial few load cycles is not a permanent effect in this in vivo test. Recovery to the initial stiffness takes place rapidly. e.g. 50 per cent recovery in 5 min at zero load.     

The effect of the point of application of anterior tibial loads on human knee kinematics

Coupled axial tibial rotation in response to an anterior tibial load has been used as a common diagnostic measurement and as a means to load the ligamentous structures during laboratory tests. However, the exact location of the point of application of these loads as well as the corresponding sensitivity of the coupled tibial rotation to this point can have an effect on the function of the soft tissues at the joint. Therefore, the purpose of this study was to determine the effects of four different points of application of the anterior tibial load on the anterior tibial translation and coupled axial tibial rotation. The four points include: (1) geometric point - midway between the collateral ligament insertion sites on the tibia, (2) clinical point - a position that attempts to simulate clinical diagnostic tests, (3) medial point - a position medial to the geometric point and (4) lateral point - a position lateral to the clinical point. A robotic/universal force-moment sensor testing system was used to apply the anterior tibial load at the four points of application and to record the resulting joint motion. Anterior tibial translation in response to an anterior tibial load of 100N was found not to vary between the four points of application of the anterior tibial load at all flexion angles examined. However, internal tibial rotation was found for the lateral point (13+/-10 degrees at 30 degrees of knee flexion) in all specimens and clinical point (8+/-10 degrees at 30 degrees of knee flexion) while external rotation resulted when the load was applied at the medial point (-8+/-7 degrees at 30 degrees of knee flexion). Both internal and external tibial rotations occurred throughout the range of flexion when the tibial load was applied at the geometric point. The results suggest that the clinical point should be used as the point of application of the anterior tibial load whenever clinical examinations are simulated and multi-degree-of-freedom joint and soft tissue function are examined.     

Temperature Dependences of Torque Generation and Membrane Voltage in the Bacterial Flagellar Motor

In their natural habitats bacteria are frequently exposed to sudden changes in temperature that have been shown to affect their swimming. With our believed to be new methods of rapid temperature control for single-molecule microscopy, we measured here the thermal response of the Na⁺-driven chimeric motor expressed in Escherichia coli cells. Motor torque at low load (0.35 μm bead) increased linearly with temperature, twofold between 15°C and 40°C, and torque at high load (1.0 μm bead) was independent of temperature, as reported for the H⁺-driven motor. Single cell membrane voltages were measured by fluorescence imaging and these were almost constant (∼120 mV) over the same temperature range. When the motor was heated above 40°C for 1–2 min the torque at high load dropped reversibly, recovering upon cooling below 40°C. This response was repeatable over as many as 10 heating cycles. Both increases and decreases in torque showed stepwise torque changes with unitary size ∼150 pN nm, close to the torque of a single stator at room temperature (∼180 pN nm), indicating that dynamic stator dissociation occurs at high temperature, with rebinding upon cooling. Our results suggest that the temperature-dependent assembly of stators is a general feature of flagellar motors.     

Behavior of Cells in Rotating Electric Fields with Account to Surface Charges and Cell Structures

The behavior of a single biological cell in a rotating electric field is investigated both theoretically and experimentally. The torque acting on the cell is calculated. The dependence of the torque on electric cell properties (the dielectric constants, the conductivities, and the surface charges of the cell components) and the field frequency is discussed. The dependence of the rotation velocity on the field frequency shows a typical resonance behavior. It is discussed in which manner the single rotation extrema are related to the different homogeneous cell compartments (cytoplasm, cell membrane, and cell wall). It is shown that the cell surface charge shifts the resonance frequency and influences the absolute value of rotation velocity.     

Curvature and torsion in growing actin networks

Intracellular pathogens such as Listeria monocytogenes and Rickettsia rickettsii move within a host cell by polymerizing a comet-tail of actin fibers that ultimately pushes the cell forward. This dense network of cross-linked actin polymers typically exhibits a striking curvature that causes bacteria to move in gently looping paths. Theoretically, tail curvature has been linked to details of motility by considering force and torque balances from a finite number of polymerizing filaments. Here we track beads coated with a prokaryotic activator of actin polymerization in three dimensions to directly quantify the curvature and torsion of bead motility paths. We find that bead paths are more likely to have low rather than high curvature at any given time. Furthermore, path curvature changes very slowly in time, with an autocorrelation decay time of 200 s. Paths with a small radius of curvature, therefore, remain so for an extended period resulting in loops when confined to two dimensions. When allowed to explore a three-dimensional (3D) space, path loops are less evident. Finally, we quantify the torsion in the bead paths and show that beads do not exhibit a significant left- or right-handed bias to their motion in 3D. These results suggest that paths of actin-propelled objects may be attributed to slow changes in curvature, possibly associated with filament debranching, rather than a fixed torque.     

Simultaneous measurement of bacterial flagellar rotation rate and swimming speed.

Swimming speeds and flagellar rotation rates of individual free-swimming Vibrio alginolyticus cells were measured simultaneously by laser dark-field microscopy at 25, 30, and 35 degrees C. A roughly linear relation between swimming speed and flagellar rotation rate was observed. The ratio of swimming speed to flagellar rotation rate was 0.113 microns, which indicated that a cell progressed by 7% of pitch of flagellar helix during one flagellar rotation. At each temperature, however, swimming speed had a tendency to saturate at high flagellar rotation rate. That is, the cell with a faster-rotating flagellum did not always swim faster. To analyze the bacterial motion, we proposed a model in which the torque characteristics of the flagellar motor were considered. The model could be analytically solved, and it qualitatively explained the experimental results. The discrepancy between the experimental and the calculated ratios of swimming speed to flagellar rotation rate was about 20%. The apparent saturation in swimming speed was considered to be caused by shorter flagella that rotated faster but produced less propelling force.     

What does inductance plethysmography really measure?

Inasmuch as it has been claimed that inductance plethysmography can measure cross-sectional area changes, we tested this assumption. We present experimental and computed relationships between self-inductance (L) of coils and areas (A) included inside for a coil with a well-defined side wavy pattern (triangular or sinusoidal) and for a real belt (Respitrace) placed on elliptical or rectangular configurations. The results are applied to the physiological field using measurements obtained from a computed tomography experiment. We demonstrate that the L-A relationships vary not only with shape or ellipticity of the cross section but also with the wavy pattern shape. This last parameter is critical because it is difficult to actually control. When the coil wavy pattern remains steady, there are some physiological situations where inductance plethysmography can more accurately estimate area changes: when the configuration shape is constant, the correspondence between delta L and delta A is almost linear with a shape-dependent sensitivity; when the configuration is nearly circular (ellipticity in the range 0.8-1), the relative error in delta A estimation is less than 5%.     

Energy Landscape for DNA Rotation and Sliding through a Phage Portal

Molecular motors involved in the packaging of DNA in tailed viruses are among the strongest known. The mechanism by which the motors operate has long been speculated to involve a coupling between rotation of the portal pore (the gate through which DNA passes upon its packaging or ejection), and translation of DNA. Recent experimental evidence rules out portal rotation with a substantial degree of certainty. We have created an atomistic model for the interaction between DNA and the portal of the bacteriophage SPP1, on the basis of cryo-electron microscopy images and of a recently solved crystal structure. A free energy surface describing the interaction is calculated using molecular dynamics simulations, and found to be inconsistent with a mechanism in which portal rotation drives DNA import. The low-energy pathways on the surface are used to advance a hypothesis on DNA import compatible with all available experiments. Additionally, temperature-dependent kinetic data are used to validate computed barriers to DNA ejection.     

Torque generated by the bacterial flagellar motor close to stall.

In earlier work in which electrorotation was used to apply external torque to tethered cells of the bacterium Escherichia coli, it was found that the torque required to force flagellar motors backward was considerably larger than the torque required to stop them. That is, there appeared to be substantial barrier to backward rotation. Here, we show that in most, possibly all, cases this barrier is an artifact due to angular variation of the torque applied by electrorotation, of the motor torque, or both; the motor torque appears to be independent to speed or to vary linearly with speed up to speeds of tens of Hertz, in either direction. However, motors often break catastrophically when driven backward, so backward rotation is not equivalent to forward rotation. Also, cells can rotate backward while stalled, either in randomly timed jumps of 180 degrees or very slowly and smoothly. When cells rotate slowly and smoothly backward, the motor takes several seconds to recover after electrorotation is stopped, suggesting that some form of reversible damage has occurred. These findings do not affect the interpretation of electrorotation experiments in which motors are driven rapidly forward.