Thermal motion and forced migration of colloidal particles generate hydrostatic pressure in solvent

A colloidal solution of ferrite particles in an osmometer has been used to demonstrate that the property that propels water across the semipermeable membrane is the decrease in hydrostatic pressure in the water of the solution. A magnetic field gradient directed so as to force the ferrite particles away from the semipermeable membrane of the osmometer and toward the free surface of the solution enhanced the colloidal osmotic pressure. The enhancement of this pressure was always exactly equal to the augmentation of the pressure as measured by the outward force of the particles, against the area of the free surface. Contrariwise, directing the magnetic field gradient so as to force the ferrite particles away from the free surface and toward the semipermeable membrane diminished the colloidal osmotic pressure of the solution. For a sufficiently forceful field gradient, the initial colloidal osmotic pressure could be negative, followed by an equilibrium pressure approaching zero regardless of the force of the particles against the membrane. Thus, the osmotic pressure of a solution is to be attributed to the pressure in the solvent generated in opposition to the pressure of the solute particles caused by their interaction with the free surface (Brownian motion and/or an external field force), or by their viscous shear when they migrate through the solvent, or both.     

Calculation of absolute protein-ligand binding free energy from computer simulations

A general methodology for calculating the equilibrium binding constant of a flexible ligand to a protein receptor is formulated on the basis of potentials of mean force. The overall process is decomposed into several stages that can be computed separately: the free ligand in the bulk is first restrained into the conformation it adopts in the bound state, position, and orientation by applying biasing potentials, then it is translated into the binding site, where it is released completely. The conformational restraining potential is based on the root-mean-square deviation of the peptide coordinates relative to its average conformation in the bound complex. Free energy contributions from each stage are calculated by means of free energy perturbation potential of mean force techniques by using appropriate order parameters. The present approach avoids the need to decouple the ligand from its surrounding (bulk solvent and receptor protein) as is traditionally performed in the double-decoupling scheme. It is believed that the present formulation will be particularly useful when the solvation free energy of the ligand is very large. As an application, the equilibrium binding constant of the phosphotyrosine peptide pYEEI to the Src homology 2 domain of human Lck has been calculated. The results are in good agreement with experimental values.     

Analysis of a Simple Prototypal Muscle Model Near to and Far from Equilibrium

Accurate calculations of force generated and of ATP flux, in steady isotonic contractions, are made on a simple, two-state muscle model of the sliding filament type. The objective is to illustrate the proper formulation and use of a complete and self-consistent molecular model of muscle. Otherwise, the model is not meant to be realistic. Calculations were made near equilibrium, including linear and quadratic terms in a power series expansion, and arbitrarily far from equilibrium by direct numerical solution of the appropriate differential equation in the probability of cross-bridge attachment. The results obtained from the two different methods agree where they overlap near equilibrium. The efficiency of free energy conversion is emphasized, and the relation to linear irreversible thermodynamics is pointed out.     

Deconvolution of dynamic mechanical networks

Time-resolved single-molecule biophysical experiments yield data that contain a wealth of dynamic information, in addition to the equilibrium distributions derived from histograms of the time series. In typical force spectroscopic setups the molecule is connected via linkers to a readout device, forming a mechanically coupled dynamic network. Deconvolution of equilibrium distributions, filtering out the influence of the linkers, is a straightforward and common practice. We have developed an analogous dynamic deconvolution theory for the more challenging task of extracting kinetic properties of individual components in networks of arbitrary complexity and topology. Our method determines the intrinsic linear response functions of a given object in the network, describing the power spectrum of conformational fluctuations. The practicality of our approach is demonstrated for the particular case of a protein linked via DNA handles to two optically trapped beads at constant stretching force, which we mimic through Brownian dynamics simulations. Each well in the protein free energy landscape (corresponding to folded, unfolded, or possibly intermediate states) will have its own characteristic equilibrium fluctuations. The associated linear response function is rich in physical content, because it depends both on the shape of the well and its diffusivity-a measure of the internal friction arising from such processes as the transient breaking and reformation of bonds in the protein structure. Starting from the autocorrelation functions of the equilibrium bead fluctuations measured in this force clamp setup, we show how an experimentalist can accurately extract the state-dependent protein diffusivity using a straightforward two-step procedure.     

Experiments on osmosis with magnetic fluid

Experiments on a ferromagnetic colloidal fluid at equilibrium showed equality between magnetic and osmotic force; this result identifies solute pressure against the free surface as the cause of the negative solvent pressure. Except for water of hydration, there is no other osmotic interaction between solute and solvent.     

Exploring the energy landscape of GFP by single-molecule mechanical experiments

We use single-molecule force spectroscopy to drive single GFP molecules from the native state through their complex energy landscape into the completely unfolded state. Unlike many smaller proteins, mechanical GFP unfolding proceeds by means of two subsequent intermediate states. The transition from the native state to the first intermediate state occurs near thermal equilibrium at approximately 35 pN and is characterized by detachment of a seven-residue N-terminal alpha-helix from the beta barrel. We measure the equilibrium free energy cost associated with this transition as 22 k(B)T. Detachment of this small alpha-helix completely destabilizes GFP thermodynamically even though the beta-barrel is still intact and can bear load. Mechanical stability of the protein on the millisecond timescale, however, is determined by the activation barrier of unfolding the beta-barrel out of this thermodynamically unstable intermediate state. High bandwidth, time-resolved measurements of the cantilever relaxation phase upon unfolding of the beta-barrel revealed a second metastable mechanical intermediate with one complete beta-strand detached from the barrel. Quantitative analysis of force distributions and lifetimes lead to a detailed picture of the complex mechanical unfolding pathway through a rough energy landscape.     

What drives the translocation of stiff chains?

We study the dynamics of the passage of a stiff chain through a pore into a cell containing particles that bind reversibly to it. Using Brownian molecular dynamics simulations we investigate the mean first-passage time as a function of the length of the chain inside for different concentrations of binding particles. As a consequence of the interactions with these particles, the chain experiences a net force along its length whose calculated value from the simulations accounts for the velocity at which it enters the cell. This force can in turn be obtained from the solution of a generalized diffusion equation incorporating an effective Langmuir adsorption free energy for the chain plus binding particles. These results suggest a role of binding particles in the translocation process that is in general quite different from that of a Brownian ratchet. Furthermore, nonequilibrium effects contribute significantly to the dynamics; e.g., the chain often enters the cell faster than particle binding can be saturated, resulting in a force several times smaller than the equilibrium value.     

A two-state kinetic model for the unfolding of single molecules by mechanical force

We investigate the work dissipated during the irreversible unfolding of single molecules by mechanical force, using the simplest model necessary to represent experimental data. The model consists of two levels (folded and unfolded states) separated by an intermediate barrier. We compute the probability distribution for the dissipated work and give analytical expressions for the average and variance of the distribution. To first order, the amount of dissipated work is directly proportional to the rate of application of force (the loading rate) and to the relaxation time of the molecule. The model yields estimates for parameters that characterize the unfolding kinetics under force in agreement with those obtained in recent experimental results. We obtain a general equation for the minimum number of repeated experiments needed to obtain an equilibrium free energy, to within k(B)T, from nonequilibrium experiments by using the Jarzynski formula. The number of irreversible experiments grows exponentially with the ratio of the average dissipated work, W(dis) to k(B)T.     

Normal free thyroxine in critical nonthyroidal illnesses measured by ultrafiltration of undiluted serum and equilibrium dialysis

Considerable controversy exists concerning the assessment of thyroidal state in critically ill patients with decreased serum T4 and T3 concentrations, in part because serum free T4 values are often low in such patients no matter what method of measurement is used. We developed an ultrafiltration method to measure free T4 and free T3 in undiluted serum and compared the results with those obtained using a standard equilibrium dialysis method to measure free T4 and T3. In 30 consecutive intensive care unit (ICU) patients, serum free T4 values were similar to or higher than those in 12 normal subjects by both methods in most patients and were clearly distinguishable from those in hypothyroid patients. The serum total T4 concentrations in these patients ranged from 12.9-131.3 nmol/L (mean, 68.2; normal mean, 115.8). Free T4 by equilibrium dialysis was highly correlated with free T4 by ultrafiltration in the ICU group (r = 0.91; P less than 0.001). Serum free T3 levels, however, whether measured by equilibrium dialysis or ultrafiltration, were decreased in the ICU patients, confirming other reports of lowered free T3 in critically ill clinically euthyroid patients. Our findings suggest that the use of equilibrium dialysis of undiluted serum or ultrafiltration to measure serum free T4 concentrations will distinguish euthyroid hypothyroxinemic ICU patients from those with hypothyroidism.     

Kinetic equilibrium of forces and molecular events in muscle contraction

In biomolecular systems, the mechanical transfer of free energy occurs with both high efficiency and high speed. It is shown here that such a transfer can be achieved only if the participating free-energy-storing elements exhibit opposing relationships between their content of free energy and the force they exert in the transfer direction. A kinetic equilibrium of forces (KEF) results, in which the transfer of free energy is mediated essentially by thermal molecular motion. On the basis of present evidence, KEF is used as a guiding principle in developing a mechanical model of the crossbridge cycle in muscle contraction. The model allows the basic features of molecular events to be visualized in terms of plausible structures. Real understanding of the process will require identification of the elements that perform the functions described here. Besides chemomechanical energy transduction, KEF may have a role in other biomolecular processes in which free energy is transferred mechanically over large distances.     

Measurement of free testosterone in normal women and women with androgen deficiency: comparison of methods

Androgen deficiency in women is increasingly recognized as a new clinical syndrome and has raised our awareness of the importance of accurate and well-validated measurements of serum free testosterone (T) concentrations in women. Therefore, we compared serum free T levels measured by equilibrium dialysis to those measured by a direct RIA (analog method) and to those calculated from the law of mass action (requires the measurement of total T and SHBG). We also calculated the free androgen index, 100 x T/SHBG, as a simple index known to correlate with free T. Subjects were 147 women with variable androgen and estrogen statuses. All were studied three times in 1 month and included women 1) with regular menses (estrogen positive, T positive), 2) more than 50 yr old and not receiving estrogen (estrogen negative, T positive), 3) receiving estrogen (estrogen positive, T negative), and 4) with severe androgen deficiency secondary to hypopituitarism (estrogen negative, T negative). Calculated values for free T using the laws of mass action correlated well with those obtained from equilibrium dialysis (r = 0.99; P < 0.0001). However, the agreement depended strongly on the specific assays used for total T and SHBG. In contrast, the direct RIA method had unacceptably high systematic bias and random variability and did not correlate as well with equilibrium dialysis values (r = 0.81; P < 0.0001). In addition, the lower limit of detection was higher for the direct RIA than for equilibrium dialysis or calculated free T. Free androgen index correlates well with free T by equilibrium dialysis (r = 0.93; P < 0.0001), but is a unitless number without reference to the physical reality of free T. We conclude that the mass action equation and equilibrium dialysis are the preferred methods for use in diagnosing androgen deficiency in women.     

Mechanism of the heparin-induced increase in the concentration of free thyroxine in plasma

The iv administration of heparin causes an increase in the plasma free T4 concentration, as determined by equilibrium dialysis. The mechanism and physiological consequences of this action of heparin are unknown. To explore the possibility that the heparin-induced increase in plasma free T4 is an in vitro artifact due to generation of FFA during equilibrium dialysis, we studied plasma samples from 10 subjects treated with iv heparin. In plasma from 4 of these subjects, free T4 concentrations measured by equilibrium dialysis did not increase above baseline values after heparin administration. In incubations performed in parallel with the equilibrium dialysis measurements, FFA concentrations in these plasma samples were found to increase, but in no subject did they exceed 2.5 meq/L after incubation. In contrast, in plasma from the other 6 subjects treated with heparin, free T4 concentrations rose markedly (by 130-520%) above baseline values after heparin administration. In all of these postheparin plasma samples, FFA concentrations were less than 2.8 meq/L before incubation, but rose during incubation by 80-270% to more than 3.8 meq/L. Treatment of these plasma samples with protamine to inhibit lipoprotein lipase and with specific antiserum to inhibit hepatic triglyceride lipase before equilibrium dialysis or incubation prevented, in parallel, the heparin-induced increases in FFA and free T4 concentrations. From these findings we conclude that the heparin-induced increase in free T4 is usually an in vitro artifact, and that most subjects receiving heparin have a normal plasma free T4 concentration in vivo. We also conclude that this in vitro artifact may account for many of the findings that led to the postulate of an inhibitor of T4 binding to plasma and intracellular proteins in heparin-treated patients and perhaps in patients with nonthyroid illness as well.     

Molecular buoyancy and osmotic equilibrium

Measurements of osmotic equilibrium of colloidal solutions, ranging from 170,000 to 20,000 in molecular weight, show that the negative buoyancy of the solute molecules is additive to the osmotic pressure. Seen together with earlier measurements of positive buoyancy of oil suspensions, these data confirm that the osmotic interaction between solute and solvent at equilibrium is purely a force-coupling at the free surface.     

Influence of cellular energy metabolism on contractions of porcine carotid artery smooth muscle

A number of cellular metabolites, including inorganic phosphate and ADP, have been proposed to regulate the contractions of smooth muscle. Hypothesizing that one of these would have a greater influence than the others, parallel experiments using tissue mechanics and (31)P-NMR allowed comparison of several metabolic components with the generation of force in porcine carotid artery smooth muscle during long-term contractions. P(i), ADP, ATP, PCr, free energy, pH, and free Mg(2+) were determined from phosphate spectra during a control-hypoxia-postcontrol sequence generated during K(+) stimulation by replacement of oxygen with nitrogen using either pyruvate or glucose as substrate. Both pH and free Mg(2+) were significantly lower in control pyruvate-supplied tissues than in glucose-supplied tissues. Mechanical experiments following the same protocol produced variations in force. The pyruvate series produced the greater range of mechanical and metabolic changes. Linear and logarithmic regression analysis found the order of correlation with force to be highest for P(i), followed by pH, free energy, PCr, ATP, ADP, and free Mg(2+). The results are consistent with models for the regulation of myosin ATPase by free phosphate inhibition. The results are inconsistent with models of ADP as a regulator of smooth muscle force. Perturbations which alter intracellular phosphate, such as creatine loading, may produce side effects on the contractions of vascular smooth muscle.     

Membrane shape as a reporter for applied forces

Recent advances have enabled 3-dimensional reconstructions of biological structures in vivo, ranging in size and complexity from single proteins to multicellular structures. In particular, tomography and confocal microscopy have been exploited to capture detailed 3-dimensional conformations of membranes in cellular processes ranging from viral budding and organelle maintenance to phagocytosis. Despite the wealth of membrane structures available, there is as yet no generic, quantitative method for their interpretation. We propose that by modeling these observed biomembrane shapes as fluid lipid bilayers in mechanical equilibrium, the externally applied forces as well as the pressure, tension, and spontaneous curvature can be computed directly from the shape alone. To illustrate the potential power of this technique, we apply an axial force with optical tweezers to vesicles and explicitly demonstrate that the applied force is equal to the force computed from the membrane conformation.     

Construction of effective free energy landscape from single-molecule time series

A scheme for extracting an effective free energy landscape from single-molecule time series is presented. This procedure uniquely identifies a non-Gaussian distribution of the observable associated with each local equilibrium state (LES). Both the number of LESs and the shape of the non-Gaussian distributions depend on the time scale of observation. By assessing how often the system visits and resides in a chosen LES and escapes from one LES to another (with checking whether the local detailed balance is satisfied), our scheme naturally leads to an effective free energy landscape whose topography depends on in which time scale the system experiences the underlying landscape. For example, two metastable states are unified as one if the time scale of observation is longer than the escape time scale for which the system can visit mutually these two states. As an illustrative example, we present the application of extracting the effective free energy landscapes from time series of the end-to-end distance of a three-color, 46-bead model protein. It indicates that the time scales to attain the local equilibrium tend to be longer in the unfolded state than those in the compact collapsed state.     

Effect of gravity on osmotic equilibria

Experiments with colloidal suspensions of oil and iron oxide at equilibrium show that the buoyancy of the suspended particles is additive to the osmotic pressure and that the relation remains with the free surface and not with the membrane. The experiments contribute to the general concept that osmotic pressure is caused by the dispersal pressure of solute molecules and that the osmotic interaction with the water at equilibrium is due solely to a coupling at the free surface.     

Calculation of protein-ligand binding free energy by using a polarizable potential

The binding of charged ligands benzamidine and diazamidine to trypsin was investigated by using a polarizable potential energy function and explicit-water molecular dynamics simulations. The binding free energies were computed from the difference between the free energies of decoupling the ligand from water and protein environments. Both the absolute and the relative free energies from the perturbation simulations agree with experimental measurements to within 0.5 kcal.mol(-1). Comparison of free-energy components sampled from different thermodynamic paths indicates that electrostatics is the main driving force behind benzamidine recognition of trypsin. The contribution of electronic polarization to binding appears to be crucial. By computing the free-energy contribution caused by the polarization between the ligand and its surroundings, we found that polarization has the opposite effect in dissimilar environments. Although polarization favors ligand solvation in water, it weakens the protein-ligand attraction by screening the electrostatic interaction between trypsin and benzamidine. We also examined the relative binding free energies of a benzamidine analog diazamidine to trypsin. The changes in free energy on benzamidine-diazamidine substitution were tens of kilocalories in both water and trypsin environments; however, the change in the total binding free energy is <2 kcal.mol(-1) because of cancellation, consistent with the experimental results. Overall, our results suggest that the use of a polarizable force field, given adequate sampling, is capable of achieving chemical accuracy in molecular simulations of protein-ligand recognition.     

Dynamic force sensing of filamin revealed in single-molecule experiments

Mechanical forces are important signals for cell response and development, but detailed molecular mechanisms of force sensing are largely unexplored. The cytoskeletal protein filamin is a key connecting element between the cytoskeleton and transmembrane complexes such as integrins or the von Willebrand receptor glycoprotein Ib. Here, we show using single-molecule mechanical measurements that the recently reported Ig domain pair 20–21 of human filamin A acts as an autoinhibited force-activatable mechanosensor. We developed a mechanical single-molecule competition assay that allows online observation of binding events of target peptides in solution to the strained domain pair. We find that filamin force sensing is a highly dynamic process occurring in rapid equilibrium that increases the affinity to the target peptides by up to a factor of 17 between 2 and 5 pN. The equilibrium mechanism we find here can offer a general scheme for cellular force sensing.     

Stepping down backwards as a means of detecting biomechanical differences between healthy older and younger adults

BACKGROUND AND AIMS: Changes in sensory- motor systems that occur with age result in a decrease in postural equilibrium, which has been linked with an increased risk of falling in the elderly. Stepping down backwards from a step perturbs dynamic postural equilibrium, thus offering an opportunity to analyze the biomechanical parameters underlying the control of balance. The aim of this study was to analyze modifications in motor patterns used by older adult subjects to control equilibrium under the environmental constraint of a backward stepping-down movement. METHODS: Ten healthy young adult and 10 healthy older adult subjects with no previous history of falls stepped down backward from a stable position on a force plate 7 cm high, at a spontaneous velocity. Each subject performed five trials, and the mean of all trials was used for subsequent analyses. An ANOVA was performed, with temporal parameters defining the phases of the stepping-down backward movement, center of mass velocity, vertical ground reaction force, impulse, and slope as dependent variables, and subject group as independent variable. RESULTS: Older adult subjects had a longer total movement duration, a longer phase of anticipatory postural adjustments, and a longer weight-transfer phase than young adult subjects (p<0.05). In contrast, older adults had a shorter relative swing phase than the young adults (p<0.05) and a lower center of mass velocity, impulse and slope (p<0.05) than young adults. CONCLUSIONS: To counterbalance the perturbation of postural equilibrium created by the backward stepping-down movement, older adults decreased the intensity of ground reaction forces and spent correspondingly more time in double-support phases than young adults.