Force versus axial deflection of pipette-aspirated closed membranes

The axial deformation of a pipette-pressurized fluid membrane bag produces minuscule yet well-defined, reproducible forces. The stiffness of this ultrasensitive force transducer is tunable and largely independent of the constitutive membrane behavior. Based on a rigorous variational treatment, we present both numerical as well as approximate analytical solutions for the force-deflection relation of this unique biophysical force probe. Our numerical results predict a measurably nonlinear force-deflection behavior at moderate-to-large deformations, which we confirm experimentally using red blood cells. Furthermore, considering nearly spherical membrane shapes and enforcing proper boundary conditions, we derive an analytical solution valid at small deformations. In this linear regime the pressurized membrane bag behaves like a Hookean spring, with a spring constant that is significantly larger than previously published for the biomembrane force probe.     

Nano- to microscale dynamics of P-selectin detachment from leukocyte interfaces. II. Tether flow terminated by P-selectin dissociation from PSGL-1

We have used a biomembrane force probe decorated with P-selectin to form point attachments with PSGL-1 receptors on a human neutrophil (PMN) in a calcium-containing medium and then to quantify the forces experienced by the attachment during retraction of the PMN at fixed speed. From first touch to final detachment, the typical force history exhibited the following sequence of events: i), an initial linear-elastic displacement of the PMN surface, ii), an abrupt crossover to viscoplastic flow that signaled membrane separation from the interior cytoskeleton and the beginning of a membrane tether, and iii), the final detachment from the probe tip most often by one precipitous step of P-selectin:PSGL-1 dissociation. Analyzing the initial elastic response and membrane unbinding from the cytoskeleton in our companion article I, we focus in this article on the regime of tether extrusion that nearly always occurred before release of the extracellular adhesion bond at pulling speeds > or =1 microm/s. The force during tether growth appeared to approach a plateau at long times. Examined over a large range of pulling speeds up to 150 microm/s, the plateau force exhibited a significant shear thinning as indicated by a weak power-law dependence on pulling speed, f(infinity) = 60 pN(nu(pull)/microm/s)(0.25). Using this shear-thinning response to describe the viscous element in a nonlinear Maxwell-like fluid model, we show that a weak serial-elastic component with a stiffness of approximately 0.07 pN/nm provides good agreement with the time course of the tether force approach to the plateau under constant pulling speed.     

Biomembrane force probe investigation of RNA dissociation

Investigations into the energy pathways of biomolecular interactions by use of dynamic force spectroscopy are limited by the range of loading rates accessible with a single technique. In the work discussed in this paper, this range has been extended for a previously studied system by using the biomembrane force probe (BFP). This work builds on our previous single-molecule atomic force microscopy (AFM) study of the dissociation of a bulge-motif-containing RNA complex. The disparity observed, at high loading rates, between the dissociation of a 12-base pair complex with and without a central three-base pair bulge was not observed at low rates. This suggests that the two species share a similar outer barrier to dissociation and that inclusion of the bulge motif creates an additional barrier at a distance closer to the bound state. Experiments performed in different buffer environments yielded similar results. The results, when combined with those of previous studies, suggest that the shared outer barrier to dissociation is that due to a rearrangement and fraying of the ends of the helix.     

Physicochemical properties of blue fluorescent protein determined via molecular dynamics simulation

Blue fluorescent protein (BFP) is a mutant of green fluorescent protein (GFP), where the chromophore has been modified to shift the emitted fluorescence into the blue spectral region. In this study, MD calculations were performed with the GROMACS simulation package and AMBER force field to investigate the dependence of BFPs physicochemical properties on temperature and applied pressure. The MD approach enabled us to calculate the compressibility of protein itself, separately from the nontrivial contribution of the hydration shell, which is difficult to achieve experimentally. The computed compressibility of BFP (3.94 x10(-5) MPa(-1)) is in agreement with experimental values of globular proteins. The center-of-mass diffusion coefficient of BFP and its dependence on temperature and pressure, which plays an important role in its application as a probe for intracellular liquid viscosity measurement, was calculated and found to be in good agreement with photobleaching recovery experimental data. We have shown that decreased temperature as well as applied pressure increases the water viscosity, but the concomitant decrease of the BFP diffusion coefficient behaves differently from Stokes-Einstein formula. It is shown that the number of hydrogen bonds around the protein grows with pressure, which explains the aforementioned deviation. Pressure also reduces root mean square (RMS) fluctuations, especially those of the most flexible residues situated in the loops. The analysis of the RMS fluctuations of the backbone Calpha atoms also reveals that the most rigid part of BFP is the center of the beta-barrel, in accord with temperature B factors obtained from the Protein Data Bank.     

Network dynamics of the nonlinear power-law relaxation of cell cortex

Living cells are known to exhibit universal power-law rheological behaviors, but their underlying biomechanical principles are still not fully understood. Here, we present a network dynamics picture to decipher the nonlinear power-law relaxation of cortical cytoskeleton. Under step strains, we present a scaling relation between instantaneous differential stiffness and external stress as a result of chain reorientation. Then, during the relaxation, we show how the scaling law theoretically originates from an exponential form of cortical disorder, with the scaling exponent decreased by the imposed strain or crosslinker density in the nonlinear regime. We attribute this exponent variation to the molecular realignment along the stretch direction or the transition of network structure from in-series to in-parallel modes, both solidifying the network toward our one-dimensional theoretical limit. In addition, the rebinding of crosslinkers is found to be crucial for moderating the relaxation speed under small strains. Together with the disorder nature, we demonstrate that the structural effects of networks provide a unified interpretation for the nonlinear power-law relaxation of cell cortex, and may help to understand cell mechanics from the molecular scale.     

Nonlinear stiffness--force relationships in whole mammalian skeletal muscles

Small, sinusoidal length changes were superimposed on isometric contractions of fast- and slow-twitch mouse muscles, which were stimulated maximally via their nerves. Stiffness increased with increasing frequency of sinusoidal stimulation, but the relative time course of force and stiffness changes during twitch, tetanic, or partially fused contractions was quite invariant over a range of frequencies in both muscles. Typically, stiffness increases more rapidly than force during contraction and decreases less rapidly during relaxation. This pattern was observed at various temperatures and with various numbers of stimuli. It can be described by a nonlinear relation between stiffness and force with some hysteresis. The presence in the muscle of parallel and series elastic elements, whose stiffness varies with force, may account for the nonlinear relation. This nonlinearity can be used to relate the patterns for summation of force and stiffness observed with brief trains of stimuli under a variety of conditions.     

Drastic change of local stiffness distribution correlating to cell migration in living fibroblasts.

Sequential images of the local stiffness distribution of living fibroblasts (NIH3T3) were captured under a culture condition using scanning probe microscopy in a force modulation mode. We found a clear relation between cell migration and local stiffness distribution on the cell: When cells were stationary at one position, the stiffness distribution of their cellular surface was quite stable. On the other hand, once the cells started to move, the stiffness in their nuclear regions drastically decreased. Possible explanations for the correlation between the cell migration and the cell stiffness are proposed.     

Catastrophic Regime Shift in Water Reservoirs and S?o Paulo Water Supply Crisis

The relation between rainfall and water accumulated in reservoirs comprises nonlinear feedbacks. Here we show that they may generate alternative equilibrium regimes, one of high water-volume, the other of low water-volume. Reservoirs can be seen as socio-environmental systems at risk of regime shifts, characteristic of tipping point transitions. We analyze data from stored water, rainfall, and water inflow and outflow in the main reservoir serving the metropolitan area of São Paulo, Brazil, by means of indicators of critical regime shifts, and find a strong signal of a transition. We furthermore build a mathematical model that gives a mechanistic view of the dynamics and demonstrates that alternative stable states are an expected property of water reservoirs. We also build a stochastic version of this model that fits well to the data. These results highlight the broader aspect that reservoir management must account for their intrinsic bistability, and should benefit from dynamical systems theory. Our case study illustrates the catastrophic consequences of failing to do so.     

Euler buckling and nonlinear kinking of double-stranded DNA

The bending stiffness of double-stranded DNA (dsDNA) at high curvatures is fundamental to its biological activity, yet this regime has been difficult to probe experimentally, and literature results have not been consistent. We created a ‘molecular vise’ in which base-pairing interactions generated a compressive force on sub-persistence length segments of dsDNA. Short dsDNA strands (<41 base pairs) resisted this force and remained straight; longer strands became bent, a phenomenon called ‘Euler buckling’. We monitored the buckling transition via Förster Resonance Energy Transfer (FRET) between appended fluorophores. For low-to-moderate concentrations of monovalent salt (up to ∼150 mM), our results are in quantitative agreement with the worm-like chain (WLC) model of DNA elasticity, without the need to invoke any ‘kinked’ states. Greater concentrations of monovalent salts or 1 mM Mg²⁺ induced an apparent softening of the dsDNA, which was best accounted for by a kink in the region of highest curvature. We tested the effects of all single-nucleotide mismatches on the DNA bending. Remarkably, the propensity to kink correlated with the thermodynamic destabilization of the mismatched DNA relative the perfectly complementary strand, suggesting that the kinked state is locally melted. The molecular vise is exquisitely sensitive to the sequence-dependent linear and nonlinear elastic properties of dsDNA.     

Hydrodynamic effects in fast AFM single-molecule force measurements

Atomic force microscopy (AFM) allows the critical forces that unfold single proteins and rupture individual receptor–ligand bonds to be measured. To derive the shape of the energy landscape, the dynamic strength of the system is probed at different force loading rates. This is usually achieved by varying the pulling speed between a few nm/s and a few m/s, although for a more complete investigation of the kinetic properties higher speeds are desirable. Above 10 m/s, the hydrodynamic drag force acting on the AFM cantilever reaches the same order of magnitude as the molecular forces. This has limited the maximum pulling speed in AFM single-molecule force spectroscopy experiments. Here, we present an approach for considering these hydrodynamic effects, thereby allowing a correct evaluation of AFM force measurements recorded over an extended range of pulling speeds (and thus loading rates). To support and illustrate our theoretical considerations, we experimentally evaluated the mechanical unfolding of a multi-domain protein recorded at 30 m/s pulling speed.Abbrevations AFM atomic force micrcoscopy - pN piconewton - BR bacteriorhodopsin - DFS dynamic force spectroscopy - Ig27 immunoglobulin 27 - If27-8 immunoglobulin 27 octameric construct - BFP biomembrane force probe     

Nano- to microscale dynamics of P-selectin detachment from leukocyte interfaces. I. Membrane separation from the cytoskeleton

We have used a biomembrane force probe decorated with P-selectin to form point attachments with PSGL-1 receptors on a human neutrophil (PMN) in a calcium-containing medium and then to quantify the forces experienced by the attachment during retraction of the PMN at fixed speed. From first touch to final detachment, the typical force history exhibited the following sequence of events: i), an initial linear-elastic displacement of the PMN surface, ii), an abrupt crossover to viscoplastic flow that signaled membrane separation from the interior cytoskeleton and the beginning of a membrane tether, and iii), the final detachment from the probe tip by usually one precipitous step of P-selectin:PSGL-1 dissociation. In this first article I, we focus on the initial elastic response and its termination by membrane separation from the cytoskeleton, initiating tether formation. Quantifying membrane unbinding forces for rates of loading (force/time) in the elastic regime from 240 pN/s to 38,000 pN/s, we discovered that the force distributions agreed well with the theory for kinetically limited failure of a weak bond. The kinetic rate for membrane unbinding was found to increase as an exponential function of the pulling force, characterized by an e-fold scale in force of approximately 17 pN and a preexponential factor, or apparent unstressed off rate, of approximately 1/s. The rheological properties of tether growth subsequent to the membrane unbinding events are presented in a companion article II.     

Mechanical properties of external confinement modulate the rounding dynamics of cells

Many studies have demonstrated that mitotic cells can round up against external impediments. However, how the stiffness of external confinement affects the dynamics of rounding force/pressure and cell volume remains largely unknown. Here, we develop a theoretical framework to study the rounding of adherent cells confined between a substrate and a cantilever. We show that the rounding force and pressure increase exclusively with the effective confinement on the cell, which is related to the cantilever stiffness and the separation between cantilever and substrate. Remarkably, an increase of cantilever stiffness from 0.001 to 1 N/m can lead to a 100-fold change in rounding force. This model also predicts an active role of confinement stiffness in regulating the dynamics of cell volume and hydrostatic pressure. We find that the dynamic changes of cellular volume and hydrostatic pressure after osmotic shocks are opposite if the cantilever is soft, whereas the dynamic changes of cellular volume and pressure are the same if the cantilever is stiff. Taken together, this work demonstrates that confinement stiffness appears as a critical regulator in regulating the dynamics of rounding force and pressure. Our findings also indicate that the difference in cantilever stiffness need to be considered when comparing the measured rounding force and pressure from various experiments.     

Conditional heteroscedasticity as a leading indicator of ecological regime shifts

Regime shifts are massive, often irreversible, rearrangements of nonlinear ecological processes that occur when systems pass critical transition points. Ecological regime shifts sometimes have severe consequences for human well-being, including eutrophication in lakes, desertification, and species extinctions. Theoretical and laboratory evidence suggests that statistical anomalies may be detectable leading indicators of regime shifts in ecological time series, making it possible to foresee and potentially avert incipient regime shifts. Conditional heteroscedasticity is persistent variance characteristic of time series with clustered volatility. Here, we analyze conditional heteroscedasticity as a potential leading indicator of regime shifts in ecological time series. We evaluate conditional heteroscedasticity by using ecological models with and without four types of critical transition. On approaching transition points, all time series contain significant conditional heteroscedasticity. This signal is detected hundreds of time steps in advance of the regime shift. Time series without regime shifts do not have significant conditional heteroscedasticity. Because probability values are easily associated with tests for conditional heteroscedasticity, detection of false positives in time series without regime shifts is minimized. This property reduces the need for a reference system to compare with the perturbed system.     

Observing force-regulated conformational changes and ligand dissociation from a single integrin on cells

As adhesion molecules, integrins connect a cell to its environment and transduce signals across the membrane. Their different functional states correspond to distinct conformations. Using a biomembrane force probe, we observed real-time reversible switches between bent and extended conformations of a single integrin, α_Lβ₂, on the surface of a living cell by measuring its nanometer-scale headpiece displacements, bending and unbending frequencies, and molecular stiffness changes. We determined the stabilities of these conformations, their dynamic equilibrium, speeds and rates of conformational changes, and the impact of divalent cations and tensile forces. We quantified how initial and subsequent conformations of α_Lβ₂ regulate the force-dependent kinetics of dissociation from intercellular adhesion molecule 1. Our findings provide new insights into how integrins function as nanomachines to precisely control cell adhesion and signaling.     

Characterizing Cell Adhesion by Using Micropipette Aspiration

We have developed a technique to directly quantify cell-substrate adhesion force using micropipette aspiration. The micropipette is positioned perpendicular to the surface of an adherent cell and a constant-rate aspiration pressure is applied. Since the micropipette diameter and the aspiration pressure are our control parameters, we have direct knowledge of the aspiration force, whereas the cell behavior is monitored either in brightfield or interference reflection microscopy. This setup thus allows us to explore a range of geometric parameters, such as projected cell area, adhesion area, or pipette size, as well as dynamical parameters such as the loading rate. We find that cell detachment is a well-defined event occurring at a critical aspiration pressure, and that the detachment force scales with the cell adhesion area (for a given micropipette diameter and loading rate), which defines a critical stress. Taking into account the cell adhesion area, intrinsic parameters of the adhesion bonds, and the loading rate, a minimal model provides an expression for the critical stress that helps rationalize our experimental results.     

Condensation prevails over B-A transition in the structure of DNA at low humidity

B-A transition and DNA condensation are processes regulated by base sequence and water activity. The constraints imposed by interhelical interactions in condensation compromise the observation of the mechanism by which B and A base-stacking modes influence the global state of the molecule. We used a single-molecule approach to prevent aggregation and mechanical force to control the intramolecular chain association involved in condensation. Force-extension experiments with optical tweezers revealed that DNA stretches as B-DNA under ethanol and spermine concentrations that favor the A-form. Moreover, we found no contour-length change compatible with a cooperative transition between the A and B forms within the intrinsic-force regime. Experiments performed at constant force in the entropic-force regime with magnetic tweezers similarly did not show a bistable contraction of the molecules that could be attributed to the B-A transition when the physiological buffer was replaced by a water-ethanol mixture. A total, stepwise collapse was found instead, which is characteristic of DNA condensation. Therefore, a low-humidity-induced change from the B- to the A-form base-stacking alone does not lead to a contour-length shortening. These results support a mechanism for the B-A transition in which low-humidity conditions locally change the base-stacking arrangement and globally induce DNA condensation, an effect that may eventually stabilize a molecular contour-length reduction.     

Enforced unbinding of biomembranes whose mutual adhesion is mediated by a specific interaction

The enforced unbinding of a biomembrane from a rigid substrate whose adhesion is mediated by a specific interaction has been studied theoretically. We argue that the unbinding takes place via motion of the adhesion rim. We account for the release of the elastic energy (stored in the membrane curvature) in the binding-unbinding process and obtain the dissociation constant at the rim. We further deduce an equation of motion for the rim. The solution exhibits an initial phase describing a slow motion followed by a regime of rapid motion and finally breaking of the adhesion. We show that the unbinding force depends on the rate of force application as F(*) approximately equal (F')(beta). When small forces are applied, beta=1/2, while in the case of large forces, beta=1/3.     

Mechanical equilibrium of thick, hollow, liquid membrane cylinders.

The mechanical equilibrium of bilayer membrane cylinders is analyzed. The analysis is motivated by the observation that mechanically formed membrane strands (tethers) can support significant axial loads and that the tether radius varies inversely with the axial force. Previously, thin shell theory has been used to analyze the tether formation process, but this approach is inadequate for describing and predicting the equilibrium state of the tether itself. In the present work the membrane is modeled as two adjacent, thick, anisotropic liquid shells. The analysis predicts an inverse relationship between axial force and tether radius, which is consistent with experimental observation. The area expansivity modulus and bending stiffness of the tether membrane are calculated using previously measured values of tether radii. These calculated values are consistent with values of membrane properties measured previously. Application of the analysis to precise measurements of the relationship between tether radius and axial force will provide a novel method for determining the mechanical properties of biomembrane.     

Transport Regulation of Two-Dimensional Receptor-Ligand Association

The impact of flow disturbances on platelet adhesion is complex and incompletely understood. At the molecular scale, platelet glycoprotein Ibα (GPIbα) must associate with the von Willebrand factor A1 domain (VWF-A1) with a rapid on-rate under high hemodynamic forces, as occurs in arterial thrombosis, where various transport mechanisms are at work. Here, we theoretically modeled the coupled transport-reaction process of the two-dimensional (2D) receptor-ligand association kinetics in a biomembrane force probe to explicitly account for the effects of molecular length, confinement stiffness, medium viscosity, surface curvature, and separation distance. We experimentally verified the theoretical approach by visualizing association and dissociation of individual VWF-A1-GPIbα bonds in a real-time thermal fluctuation assay. The apparent on-rate, reciprocal of the average time intervals between sequential bonds, decreased with the increasing gap distance between A1- and GPIbα-bearing surfaces with an 80-nm threshold (beyond which bond formation became prohibitive) identified as the combined contour length of the receptor and ligand molecules. The biomembrane force probe spring constant and diffusivity of the protein-bearing beads also significantly influenced the apparent on-rate, in accordance with the proposed transport mechanisms. The global agreement between the experimental data and the model predictions supports the hypothesis that receptor-ligand association behaves distinctly in the transport- and reaction-limited scenarios. To our knowledge, our results represent the first detailed quantification of physical regulation of the 2D on-rate that allows platelets to sense and respond to local changes in their hemodynamic environment. In addition, they provide an approach for determining the intrinsic kinetic parameters that employs simultaneous experimental measurements and theoretical modeling of bond association in a single assay. The 2D intrinsic forward rate for VWF-A1-GPIbα association was determined from the measurements to be (3.5 ± 0.67) × 10⁻⁴μm² s⁻¹.