Reconstructing potential energy functions from simulated force-induced unbinding processes

Atomistic simulation combined with analytic theory to study the response of the P-selectin/PSGL-1 complex to an external force

Steered molecular dynamics simulations are combined with analytic theory in order to gain insights into the properties of the P-selectin/PSGL-1 catch-slip bond at the atomistic level of detail. The simulations allow us to monitor the conformational changes in the P-selectin/PSGL-1 complex in response to an external force, while the theory provides a unified framework bridging the simulation data with experiment over 9 orders of magnitude. The theory predicts that the probability of bond dissociation by the catch mechanism is extremely low in the simulations; however, a few or even a single trajectory can be sufficient for characterization of the slip mechanism. Theoretical analysis of the simulation data shows that the bond responds to the force in a highly nonlinear way, with the bond stiffness changing considerably as a function of the force ramp rate. The Langevin description of the simulation provides spring constants of the proteins and the binding interaction and gives the friction coefficient associated with the receptor-ligand motion in water. The estimated relaxation time shows that the simple probabilistic description is accurate for the experimental regime and remains approximately valid for the high ramp rates used in simulations. The simulations establish that bond deformation occurs primarily within the P-selectin receptor region. The two interaction sites within the binding pocket dissociate sequentially, raising the possibility of observing these independent rupture events in experiment. The stronger interaction that determines the overall properties of the bond dissociates first, indicating that the experimental data indeed capture the main rupture event and not the secondary weaker site rupture. The main rupture event involves the interaction between the calcium ion of the receptor and the ligand residue FUC-623. It is followed by new interactions, supporting the sliding-rebinding behavior observed in the earlier simulation [ Lou, J. Zhu, C. Biophys. J. 2007 , 92 , 1471 - 1485 ]. The weaker binding site shows fewer interaction features, suggesting that the sliding-rebinding behavior may be determined by the unique properties of the calcium site. The agreement between simulation and experiment provided by the two-pathway and deformation models, each containing only four parameters, indicates that the essential physics of the catch-slip bond should be relatively simple and robust over a wide range of pulling regimes.

Using stochastic models calibrated from nanosecond nonequilibrium simulations to approximate mesoscale information

We demonstrate how the surrogate process approximation (SPA) method can be used to compute both the potential of mean force along a reaction coordinate and the associated diffusion coefficient using a relatively small number (10-20) of bidirectional nonequilibrium trajectories coming from a complex system. Our method provides confidence bands which take the variability of the initial configuration of the high-dimensional system, continuous nature of the work paths, and thermal fluctuations into account. Maximum-likelihood-type methods are used to estimate a stochastic differential equation (SDE) approximating the dynamics. For each observed time series, we estimate a new SDE resulting in a collection of SPA models. The physical significance of the collection of SPA models is discussed and methods for exploiting information in the population of estimated SPA models are demonstrated and suggested. Molecular dynamics simulations of potassium ion dynamics inside a gramicidin A channel are used to demonstrate the methodology, although SPA-type modeling has also proven useful in analyzing single-molecule experimental time series [J. Phys. Chem. B 113, 118 (2009)].

Computer simulations and theory of protein translocation

The translocation of proteins through pores is central to many biological phenomena, such as mitochondrial protein import, protein degradation, and delivery of protein toxins to their cytosolic targets. Because proteins typically have to pass through constrictions that are too narrow to accommodate folded structures, translocation must be coupled to protein unfolding. The simplest model that accounts for such co-translocational unfolding assumes that both translocation and unfolding are accomplished by pulling on the end of the polypeptide chain mechanically. In this Account, we describe theoretical studies and computer simulations of this model and discuss how the time scales of translocation depend on the pulling force and on the protein structure. Computationally, this is a difficult problem because biologically or experimentally relevant time scales of translocation are typically orders of magnitude slower than those accessible by fully atomistic simulations. For this reason, we explore one-dimensional free energy landscapes along suitably defined translocation coordinates and discuss various approaches to their computation. We argue that the free energy landscape of translocation is often bumpy because confinement partitions the protein's configuration space into distinct basins of attraction separated by large entropic barriers. Favorable protein-pore interactions and nonnative interactions within the protein further contribute to the complexity. Computer simulations and simple scaling estimates show that forces of just 2-6 pN are often sufficient to ensure transport of unstructured polypeptides, whereas much higher forces are typically needed to translocate folded protein domains. The unfolding mechanisms found from simulations of translocation are different from those observed in the much better understood case of atomic force microscopy (AFM) pulling studies, in which proteins are unraveled by stretching them between their N- and C-termini. In contrast to AFM experiments, single-molecule experimental studies of protein translocation have just started to emerge. We describe one example of a collaborative study, in which dwell times of beta-hairpin-forming peptides inside the alpha-hemolysin pore were both measured experimentally and estimated using computer simulations. Analysis of the simulated trajectories has explained the experimental finding that more stable hairpins take, on the average, longer to traverse the pore. Despite the insight we have gained, the general relationship between the structure of proteins and their resistance to mechanically driven co-translocational unfolding remains poorly understood. Future theoretical progress likely will be made in conjunction with single-molecule experiments and will require realistic models to account for specific protein-pore interactions and for solvent effects.

Quantifying multiscale noise sources in single-molecule time series

When analyzing single-molecule data, a low-dimensional set of system observables typically serves as the observational data. We calibrate stochastic dynamical models from time series that record such observables. Numerical techniques for quantifying noise from multiple time scales in a single trajectory, including experimental instrument and inherent thermal noise, are demonstrated. The techniques are applied to study time series coming from both simulations and experiments associated with the nonequilibrium mechanical unfolding of titin's I27 domain. The estimated models can be used for several purposes, (1) detect dynamical signatures of "rare events" by analyzing the effective diffusion and force as a function of the monitored observable, (2) quantify the influence that conformational degrees of freedom, which are typically difficult to directly monitor experimentally, have on the dynamics of the monitored observable, (3) quantitatively compare the inherent thermal noise to other noise sources, for example, instrument noise, variation induced by conformational heterogeneity, and so forth, (4) simulate random quantities associated with repeated experiments, and (5) apply pathwise, that is, trajectory-wise, hypothesis tests to assess the goodness-of-fit of the models and even detect conformational transitions in noisy signals. These items are all illustrated with several examples.

Study of the interaction between progesterone and beta-cyclodextrin by electrochemical techniques and steered molecular dynamics

The interaction of progesterone with beta-cyclodextrin (beta-CD) was studied by differential pulse polarography. The aim of the present work was to study the effect of beta-CD on the electrochemical behavior of progesterone in aqueous solution and also to analyze the molecular interactions involved in formation of the inclusion complex. The complex with stoichiometry of 1:1 was thermodynamically characterized. In addition, steered molecular dynamics (SMD) was used to investigate the energetic properties of formation of the inclusion complex along four different pathways (reaction coordinates), considering two possible orientations. From multiple trajectories along these pathways, the potentials of mean force for formation of the beta-CD progesterone inclusion complex were calculated. The energy analysis was in good agreement with the experimental results. In the beta-CD progesterone inclusion complex, a large portion of the steroid skeleton is included in the beta-CD cavity. The lowest energy was found when the D-ring of the guest molecule is located near the secondary hydroxyls of the beta-CD cavity. In the most probable orientation, one intermolecular hydrogen bond is formed between the O of the C-20 keto group of the progesterone and a secondary hydroxyl of the beta-CD.

How force might activate talin's vinculin binding sites: SMD reveals a structural mechanism

Upon cell adhesion, talin physically couples the cytoskeleton via integrins to the extracellular matrix, and subsequent vinculin recruitment is enhanced by locally applied tensile force. Since the vinculin binding (VB) sites are buried in the talin rod under equilibrium conditions, the structural mechanism of how vinculin binding to talin is force-activated remains unknown. Taken together with experimental data, a biphasic vinculin binding model, as derived from steered molecular dynamics, provides high resolution structural insights how tensile mechanical force applied to the talin rod fragment (residues 486–889 constituting helices H1–H12) might activate the VB sites. Fragmentation of the rod into three helix subbundles is prerequisite to the sequential exposure of VB helices to water. Finally, unfolding of a VB helix into a completely stretched polypeptide might inhibit further binding of vinculin. The first events in fracturing the H1–H12 rods of talin1 and talin2 in subbundles are similar. The proposed force-activated α-helix swapping mechanism by which vinculin binding sites in talin rods are exposed works distinctly different from that of other force-activated bonds, including catch bonds.

For cell survival, most eukaryotic cells need to be mechanically anchored to their environment. This is done by transmembrane proteins, including integrins, which externally bind to the extracellular matrix and on the cell interior to the contractile cytoskeleton via scaffolding proteins. One essential scaffolding protein is talin, which binds to integrins via its head and to the cytoskeletal filament f-actin via its rodlike tail. As cells apply tensile forces to newly formed adhesion sites, proteins that are part of such force-bearing networks get stretched and might change their structure and thus function. One of many proteins that are recruited to newly formed adhesions is vinculin, and vinculin recruitment is upregulated by tensile mechanical force—but how? Since talin's vinculin binding sites are buried in its native structure, we used steered molecular dynamics here to derive a high resolution structural model of how tensile mechanical forces might activate talin's vinculin binding sites. Once tensile forces break up the talin rod into helix subbundles, an event that we find here to constitute the main energy barrier, we propose how the strain-induced gradual exposure of the vinculin-binding helices finally allows for their activation and enables helix swapping with the vinculin head.

Analyzing different parameters of steered molecular dynamics for small membrane interacting molecules

The aim of this report is two-fold: First, to show the applicability of the Steered Molecular Dynamics (SMD) methodology for analyzing non-specific interactions governing the membrane affinity process of small biological molecules. Second, to point out a correlation between the system response and certain combinations of the SMD parameters (spring-elastic-constant and pulling-group). For these purposes, a simplified membrane model was used, modeled as a non-polar region limited by two polar aqueous media in a continuous dielectric representation. Polarization-induced effects at both interfaces were taken into account by the "electrostatic images" method. To perform SMD simulations a harmonic external force, representing a spring acting on a selected atom, forces the molecule to "break" its interaction with the surrounding environment by extracting it out of the membrane. With this approach, small molecules and peptides, with known affinity for the membrane environment, were studied: the zwitterionic tryptophan residue and a pentapeptide AcWLKLL. The SMD parameters, spring-elastic-constant and pulling-group, were varied and combined in order to analyze the systems responses in each case. It was observed that, the spring stiffness was crucial to reveal specific events that occur during the molecule behavior; hence, it was directly responsible for the sensitivity of this methodology. The pulling-group selected highly influenced on the reaction pathway, a fact that it was not observed with other parameters; consequently, force profiles are like the "fingerprints" of these induced pathways. The potential profile for the tryptophan was recovered from the SMD simulations being in good agreement with that estimated by an approximation method. With this rather simple model approach, SMD methodology has proven to be suitable for revealing the main interactions that govern the membrane affinity processes of small molecules and peptides.

Computational modeling of poly(alkylthiophene) conductive polymer insertion into phospholipid bilayers

We have previously demonstrated that some poly(alkylthiophenes) (PATs) are able to increase the electrical conductance of unsupported phospholipid bilayers and have hypothesized that this effect is due to the ability of some PAT side chains to permit stable insertion into the bilayer. We have further proposed the development of long-term intracellular electrodes based on that phenomenon. In this article, we apply molecular dynamics techniques to study the insertion of two model PATs into a patch of a lipid bilayer. Steered molecular dynamics is used to obtain potential trajectories of insertion, followed by umbrella sampling to determine the free-energy change upon insertion. Our results indicate that both branched-side-chain poly(3-(2-ethylhexyl)thiophene) (EHPT) and straight-side-chain poly(3-hexylthiophene) (HPT) are able to enter the bilayer but only EHPT can cross the center of the membrane and establish an electrical bridge. HPT penetrates the head groups but is not able to enter the alkyl tail phase. These findings support the feasibility of our electrode concept and raise questions regarding the mechanisms by which branched side chains grant PATs greater solubility in a lipid bilayer environment. The parameters and methods used in this study establish a novel framework for studying these and similar systems, and the results hold promise for the use of EHPT in biosensing and neural interfacing.

Computer simulations of the translocation and unfolding of a protein pulled mechanically through a pore

Protein degradation by ATP-dependent proteases and protein import into the mitochondrial matrix involve the unfolding of proteins upon their passing through narrow constrictions. It has been hypothesized that the cellular machinery accomplishes protein unfolding by pulling mechanically at one end of the polypeptide chain. Here, we use Langevin dynamics simulations of a minimalist off-lattice model to examine this hypothesis and to study the unfolding of a protein domain pulled mechanically through a long narrow pore. We compute the potential of mean force (PMF) experienced by the domain as a function of its displacement along the pore and identify the unfolding intermediates corresponding to the local minima of the PMF. The observed unfolding mechanism is different from that found when the two termini are pulled apart, as in single-molecule mechanical unfolding experiments. It depends on the pore diameter, the magnitude of the pulling force, and on whether the force is applied at the N- or the C-terminus of the chain. Consequently, the translocation time exhibits a pulling force dependence that is more complex than a simple exponential function expected on the basis of simple phenomenological models of translocation.

Steered Molecular Dynamics Studies of the Potential of Mean Force of a Na+ or K+ Ion in a Cyclic Peptide Nanotube

Potential of mean force (PMF) profiles of a single Na+ or K+ ion passing through a cyclic peptide nanotube, cyclo[-(D-Ala-Glu-D-Ala-Gln)2-], in water are calculated to provide insight into ion transport and to understand the conductance difference between these two ions. The PMF profiles are obtained by performing steered molecular dynamics (SMD) simulations that are based on the Jarzynski equality. The computed PMF profiles for both ions show barriers of around 2.4 kcal/mol at the channel entrances and exits and energy wells in the middle of the tube. The energy barriers, so-called dielectric energy barriers, arise due to the desolvation of water molecules when ions move across the nanotube, and the energy wells appear as a result of attractive interactions between the cations and negatively charged carbonyl oxygens on the backbone of the tube. We find more and deeper energy wells in the PMF profile for Na+ than for K+, which suggests that Na+ ions have a longer residence time inside the nanotube and that permeation of Na+ ions is reduced compared to K+ ions. Calculations of the radial distribution functions (RDF) between the ions and oxygens in the water molecules and in carbonyl groups on the tube and an investigation of the orientations of the carbonyl groups show that, in contrast with the dynamic carbonyl groups observed in the selectivity filter of the KcsA ion channel, the carbonyl groups in the cyclic peptide nanotube are relatively rigid, with only slight reorientation of the carbonyl groups as the cations pass through. The rigidity of the carbonyl groups in the cyclic peptide nanotube can be attributed to their role in hydrogen bonding, which is responsible for the tube structure. Comparison of the PMF profiles with the electrostatic energy profiles calculated from the Poisson-Boltzmann (PB) equation, a dielectric continuum model, reveals that the dielectric continuum model breaks down in the confined region within the tube that governs ion transport.

Mechanism and dynamics of cadherin adhesion

Cadherins are essential cell adhesion molecules involved in tissue morphogenesis and the maintenance of tissue architecture in adults. The adhesion and selectivity functions of cadherins are located in their extracellular regions. Biophysical studies show that the adhesive activity is not confined to a single interface. Instead, multiple cadherin domains contribute to binding. By contrast, the specificity-determining site maps to the N-terminal domains, which adhere by the reciprocal binding of Trp2 residues from opposing proteins. Structural cooperativity can transmit the effects of subtle structural changes or ligand binding over large distances in the protein. Increasingly, studies show that differential cadherin-mediated adhesion, rather than exclusive homophilic binding between identical cadherins, direct cell segregation and the organization of tissue interfaces during morphogenesis. Force measurements quantified both kinetic and strength differences between different classical cadherins that may underlie cell sorting behavior. Despite the complex adhesion mechanisms and differences in binding properties, cadherin-mediated cell adhesion is also regulated by many other biochemical processes. Elucidating the mechanisms by which cadherins organize cell junctions and tissue architecture requires not only quantitative, mechanistic investigations of cadherin function but also investigations of the biochemical and cellular processes that can modulate those functions.

Exploration of the conformational space of myosin recovery stroke via molecular dynamics

Muscle contractions are driven by cyclic conformational changes of myosin, whose molecular mechanisms of operation are being elucidated by recent advances in crystallographic studies and single molecule experiments. To complement such structural studies and consider the energetics of the conformational changes of myosin head, umbrella sampling molecular dynamics (MD) simulations were performed with the all-atom model of the scallop myosin sub-fragment 1 (S1) with a bound ATP in solution in explicit water using the crystallographic near-rigor and transition state conformations as two references. The constraints on RMSD reaction coordinates used for the umbrella sampling were found to steer the conformational changes efficiently, and relatively close correlations have been observed between the set of characteristic structural changes including the lever arm rotation and the closing of the nucleotide binding pocket. The lever arm angle and key residue interaction distances in the nucleotide binding pocket and the relay helix show gradual changes along the recovery stroke reaction coordinate, consistent with previous crystallographic and computational minimum energy studies. Thermal fluctuations, however, appear to make the switch-2 coordination of ATP more flexible than suggested by crystal structures. The local solvation environment of the fluorescence probe, Trp 507 (scallop numbering), also appears highly mobile in the presence of thermal fluctuations.

An asymptotic comparative analysis of the thermodynamics of non-covalent association

There is an ambiguity in the theoretical models for computing association constants, the key observable in a laboratory, of non-covalent associations. We show that three different models give unique result asymptotically in the limit of strong associate. For weak associations, the disagreement reflects the nature of ill-defined "associated complex'' which can be defined, among various ways, either geometrically or thermodynamically depending on measurement techniques. Furthermore, even when the free energy of association is unique, the corresponding entropy and enthalpy can still be different from different types of measurements--a surprising source of entropy-enthalpy compensation. This work provides a mathematical basis for modeling non-covalent association processes in biology.

Multiple-bond kinetics from single-molecule pulling experiments: evidence for multiple NCAM bonds

The kinetic parameters of single bonds between neural cell adhesion molecules were determined from atomic force microscope measurements of the forced dissociation of the homophilic protein-protein bonds. The analytical approach described provides a systematic procedure for obtaining rupture kinetics for single protein bonds from bond breakage frequency distributions obtained from single-molecule pulling experiments. For these studies, we used the neural cell adhesion molecule (NCAM), which was recently shown to form two independent protein bonds. The analysis of the bond rupture data at different loading rates, using the single-bond full microscopic model, indicates that the breakage frequency distribution is most sensitive to the distance to the transition state and least sensitive to the molecular spring constant. The analysis of bond failure data, however, motivates the use of a double-bond microscopic model that requires an additional kinetic parameter. This double-bond microscopic model assumes two independent NCAM-NCAM bonds, and more accurately describes the breakage frequency distribution, particularly at high loading rates. This finding agrees with recent surface-force measurements, which showed that NCAM forms two spatially distinct bonds between opposed proteins.

Topography of the free-energy landscape probed via mechanical unfolding of proteins

Single-molecule experiments in which proteins are unfolded by applying mechanical stretching forces generally force unfolding to proceed along a reaction coordinate that is different from that in chemical or thermal denaturation. Here we simulate the mechanical unfolding and refolding of a minimalist off-lattice model of the protein ubiquitin to explore in detail the slice of the multidimensional free-energy landscape that is accessible via mechanical pulling experiments. We find that while the free-energy profile along typical "chemical" reaction coordinates may exhibit two minima, corresponding to the native and denatured states, the free energy G(z) is typically a monotonic function of the mechanical coordinate z equal to the protein extension. Application of a stretching force along z tilts the free-energy landscape resulting in a bistable (or multistable) free energy G(z)-fz probed in mechanical unfolding experiments. We construct a two-dimensional free-energy surface as a function of both chemical and mechanical reaction coordinates and examine the coupling between the two. We further study the refolding trajectories after the protein has been prestretched by a large force, as well as the mechanical unfolding trajectories in the presence of a large stretching force. We demonstrate that the stretching forces required to destabilize the native state thermodynamically are larger than those expected on the basis of previous experimental estimates of G(z). This finding is consistent with the recent experimental studies, indicating that proteins may refold even in the presence of a substantial stretching force. Finally, we show that for certain temperatures the free energy of a polyprotein chain consisting of multiple domains is a linear function of the chain extension. We propose that the recently observed "slow phase" in the refolding of proteins under mechanical tension may be viewed as downhill diffusion in such a linear potential.

Unfolding a linker between helical repeats

In many multi-repeat proteins, linkers between repeats have little secondary structure and place few constraints on folding or unfolding. However, the large family of spectrin-like proteins, including alpha-actinin, spectrin, and dystrophin, share three-helix bundle, spectrin repeats that appear in crystal structures to be linked by long helices. All of these proteins are regularly subjected to mechanical stress. Recent single molecule atomic force microscopy (AFM) experiments demonstrate not only forced unfolding but also simultaneous unfolding of tandem repeats at finite frequency, which suggests that the contiguous helix between spectrin repeats can propagate a cooperative helix-to-coil transition. Here, we address what happens atomistically to the linker under stress by steered molecular dynamics simulations of tandem spectrin repeats in explicit water. The results for alpha-actinin repeats reveal rate-dependent pathways, with one pathway showing that the linker between repeats unfolds, which may explain the single-repeat unfolding pathway observed in AFM experiments. A second pathway preserves the structural integrity of the linker, which explains the tandem-repeat unfolding event. Unfolding of the linker begins with a splay distortion of proximal loops away from hydrophobic contacts with the linker. This is followed by linker destabilization and unwinding with increased hydration of the backbone. The end result is an unfolded helix that mechanically decouples tandem repeats. Molecularly detailed insights obtained here aid in understanding the mechanical coupling of domain stability in spectrin family proteins.

Theoretical determination of the strength of soft noncovalent molecular bonds

The strength of a simple soft bond under a constant loading rate is studied theoretically. There is a scaling regime where rebinding is negligible and the rupture force scales as const+ [ln (kv) ](2/3) , where kv is the loading rate. The scaling regime is smaller for weaker bonds and broader for stronger bonds. For a loading rate beyond the upper limit of the scaling regime, the bond rupture is deterministic and the thermal effects are negligible. For a loading rate below the lower limit of the scaling regime, the contribution from rebinding becomes important and there is no simple scaling relation between the rupture force and the loading rate. Our theory takes the effect of rebinding in the calculation, therefore we find good agreement between theory and simulation even below the scaling regime.

Molecular dynamics simulations of duplex stretching reveal the importance of entropy in determining the biomechanical properties of DNA

Advances in nanomanipulation techniques have made it possible to measure the response of an individual biomolecule to a force applied in the laboratory. Experiments that stretch a single molecule of duplex DNA have been difficult to interpret theoretically, particularly as the major changes in molecular structure caused by the force cannot be measured. In principle, computer simulation can calculate these conformational changes in atomic level detail, but to date such studies have failed to reproduce the experimental data due to the computational expense of the calculations. Here we show that a combination of molecular modeling and statistical physics can be used successfully to understand the stretching behavior of DNA. Our simulations provide new information about the dynamics of DNA denaturation under force in atomic level detail and also show the importance of entropy in determining biomechanical properties in general.

Force history dependence of receptor-ligand dissociation

Receptor-ligand bonds that mediate cell adhesion are often subjected to forces that regulate their dissociation via modulating off-rates. Off-rates control how long receptor-ligand bonds last and how much force they withstand. One should therefore be able to determine off-rates from either bond lifetime or unbinding force measurements. However, substantial discrepancies exist between the force dependence of off-rates derived from the two types of measurements even for the same interactions, e.g., selectins dissociating from their ligands, which mediate the tethering and rolling of leukocytes on vascular surfaces during inflammation and immune surveillance. We used atomic force microscopy to measure survival times of P-selectin dissociating from P-selectin glycoprotein ligand 1 or from an antibody in both bond lifetime and unbinding force experiments. By a new method of data analysis, we showed that the discrepancies resulted from the assumption that off-rates were functions of force only. The off-rates derived from forced dissociation data depended not only on force but also on the history of force application. This finding provides a new paradigm for understanding how force regulates receptor-ligand interactions.

Simulation of the mechanical unfolding of ubiquitin: Probing different unfolding reaction coordinates by changing the pulling geometry

Motivated by the recent experimental atomic force microscopy (AFM) measurements of the mechanical unfolding of proteins pulled in different directions [D. J. Brockwell et al., Nat. Struct. Biol. 10, 731 (2003); M. Carrion-Vazquez et al., ibid 10, 738 (2003)] we have computed the unfolding free energy profiles for the ubiquitin domain when it is stretched between its (A) N and C termini, (B) Lys48 and C terminus, (C) Lys11 and C terminus, and (D) N terminus and Lys63. Our results for cases (A) and (B) are in good agreement with the experimental unfolding forces measured for the N-C and Lys48-C linked polyubiquitin, in particular, indicating a considerably lower unfolding force in the latter case. Mechanical unfolding in case (A) involves longitudinal shearing of the terminal parallel strands while in case (C) the same strands are "unzipped" by the force. The computed unfolding forces in case (C) are found to be very low, less than 50 pN for pulling rates typical of AFM experiments. The unfolding free energy barrier found in case (C) is approximately 13 kcal/mol, which corresponds to a zero-force unfolding rate constant that is comparable to the rate of chemical unfolding extrapolated to zero denaturant concentration. The unfolding barrier calculated in case (A) in the limit of zero force is much higher, suggesting that mechanical unfolding in this case follows a pathway that is different from that of thermal/chemical denaturation.

Theoretical analysis of dynamic force spectroscopy experiments on ligand–receptor complexes

The forced rupture of single chemical bonds in biomolecular compounds (e.g. ligand-receptor systems) as observed in dynamic force spectroscopy experiments is addressed. Under the assumption that the probability of bond rupture depends only on the instantaneously acting force, a data collapse onto a single master curve is predicted. For rupture data obtained experimentally by dynamic AFM force spectroscopy of a ligand-receptor bond between a DNA and a regulatory protein we do not find such a collapse. We conclude that the above mentioned, generally accepted assumption is not satisfied and we discuss possible explanations.

Single molecular mechanics of a cholesterol-bearing pullulan nanogel at the hydrophobic interfaces

The study of nanogel (hydrogel nanoparticle) has intensified in the last decade due to the enormous potential applications in biomimetics, biosensors, artificial muscles and drug delivery (or release) systems. Cholesterol-bearing pullulan (CHP) is composed of hydrophilic pullulan backbone and partly substituted hydrophobic cholesterol, and is capable of forming a stable hydrogel nanoparticle in aqueous solution due to the self-assembly of hydrophobic cholesterol moieties. The conformation of CHP changes dramatically at the hydrophobic interfaces. In order to understand the interfacial responses of CHP, the interaction forces of CHP nanogels to the hydrophobic HOPG (highly orientated pyrolytic graphite) or carbon-coated surfaces were measured using atomic force microscope. The freely jointed china model for CHP molecular elasticity was applied to the force-extension curves and debonding force-pull-off distance in order to estimate the contour lengths and the segment lengths of the CHP molecules. The segment length of CHP chains in aqueous solution was estimated 0.32+/-0.19 nm showing a very flexible chain. From our analysis of the dynamic force measurements, the debonding forces were shown to depend on the applied loading forces. The zero kinetic off-rate K(off)(0) and the transition state x(b) were estimated to be 1.1 x 10(-3)s(-1) and 2.9A, respectively.

Tuning the Mechanical Stability of Fibronectin Type III Modules through Sequence Variations

Cells can switch the functional states of extracellular matrix proteins by stretching them while exerting mechanical force. Using steered molecular dynamics, we investigated how the mechanical stability of FnIII modules from the cell adhesion protein fibronectin is affected by natural variations in their amino acid sequences. Despite remarkably similar tertiary structures, FnIII modules share low sequence homology. Conversely, the sequence homology for the same FnIII module across multiple species is notably higher, suggesting that sequence variability is functionally significant. Our studies find that the mechanical stability of FnIII modules can be tuned through substitutions of just a few key amino acids by altering access of water molecules to hydrogen bonds that break early in the unfolding pathway. Furthermore, the FnIII hierarchy of mechanical unfolding can be changed by environmental conditions, such as pH for FnIII10, or by forming complexes with other molecules, such as heparin binding to FnIII13.

Force spectroscopy with a small dithering of AFM tip: a method of direct and continuous measurement of the spring constant of single molecules and molecular complexes

A new method of direct and continuous measurement of the spring constant of single molecule or molecular complex is elaborated. To that end the standard force spectroscopy technique with functionalized tips and samples is combined with a small dithering of the tip. The change of the dithering amplitude as a function of the pulling force is measured to extract the spring constant of the complex. The potentialities of this method are illustrated for the experiments with single bovine serum albumin-its polyclonal antibody (Ab-BSA) and fibrinogen-fibrinogen complexes.

Molecular interaction studies of hemostasis: fibrinogen ligand-human platelet receptor interactions

The interactions between fibrinogen ligands and platelet receptor alpha(IIb)beta(3) were studied under physiological conditions by atomic force microscopy (AFM). Two linear peptide sequences in fibrinogen, RGD and HHLGGAKQAGDV, play central roles in the regulation of hemostasis and thrombosis by facilitating adhesion and aggregation of platelets. In order to measure the interactions (i.e., debonding force), oligopeptides, GSSSGaaa, where aaa is -RGDSPA or -HHLGGAKQAGDV, were synthesized and grafted on to the surface of AFM probe tips. The interaction forces between a peptide-modified AFM probe tip and platelet surface were determined from pN to nN levels using AFM force measurements. Our results show that the zero kinetic off-rate, K(off)(0), for RGDSPA is significantly smaller than that for HHLGGAKQAGDV, under the consideration of flexible receptor surfaces. From our analysis, the K(off)(0), the single molecular binding energy E(b), and the transition state x(b), were extracted from the data, and estimated to be 1.53s(-1), -2.64x10(-20)J and 1.03A for the RGD-alpha(IIb)beta(3) system, and 47.58s(-1), 2.67x10(-20), 1.09A for the HHLGGAKQAGDV-alpha(IIb)beta(3) system, respectively.

Kinetics from nonequilibrium single-molecule pulling experiments

Mechanical forces exerted by laser tweezers or atomic force microscopes can be used to drive rare transitions in single molecules, such as unfolding of a protein or dissociation of a ligand. The phenomenological description of pulling experiments based on Bell's expression for the force-induced rupture rate is found to be inadequate when tested against computer simulations of a simple microscopic model of the dynamics. We introduce a new approach of comparable complexity to extract more accurate kinetic information about the molecular events from pulling experiments. Our procedure is based on the analysis of a simple stochastic model of pulling with a harmonic spring and encompasses the phenomenological approach, reducing to it in the appropriate limit. Our approach is tested against computer simulations of a multimodule titin model with anharmonic linkers and then an illustrative application is made to the forced unfolding of I27 subunits of the protein titin. Our procedure to extract kinetic information from pulling experiments is simple to implement and should prove useful in the analysis of experiments on a variety of systems.

Nonspecific interaction forces at water-membrane interface by forced molecular dynamics simulations

Nonspecific interactions are the main driving forces for the behavior of molecules with great affinity for biologic membranes. To investigate not only the molecular details of these interactions but to estimate their magnitude as well, the theoretical method of Forced Molecular Dynamics Simulations, based on the Atomic Force Spectroscopy experimental technique, was applied. In this approach, an additional one-dimensional elastic force, representing the cantilever probe, was incorporated to the force field of a Molecular Dynamics computational program. This force represents a spring fixed on one end to a selected atom of the molecule; the other end of the spring is displaced at constant velocity to pull the molecule out of the membrane. The force experimented by the molecule due to the spring, is proportional to the spring elongation relative to its equilibrium position. This value is registered during the entire simulation, and its maximum value will determine the molecule-membrane interaction force. Nonexplicit medium simulations were carried out. Polar and apolar media were considered according to their polarizability degree and a specific dielectric constant value was assigned. In this approach, the membrane was considered as the apolar region limited by two flat surfaces with a polar aqueous medium. The potential energy discontinuity at the interfaces was smoothed by considering the polarization-induced effects using the image method. The results of this methodology are presented using a small system, a single Alanine amino acid model, which enables extended simulations in a microsecond time scale. The confinement of this amino acid at the interface reduces its degrees of freedom and forces it to adopt one of the six defined conformations. A correlation between these stable structures at the water-membrane interface and the interaction force value was determined.

Pathways and intermediates in forced unfolding of spectrin repeats

Spectrin repeats are triple-helical coiled-coil domains found in many proteins that are regularly subjected to mechanical stress. We used atomic force microscopy technique and steered molecular dynamics simulations to study the behavior of a wild-type spectrin repeat and two mutants. The experiments indicate that spectrin repeats can form stable unfolding intermediates when subjected to external forces. In the simulations the unfolding proceeded via a variety of pathways. Stable intermediates were associated to kinking of the central helix close to a proline residue. A mutant stabilizing the central helix showed no intermediates in experiments, in agreement with simulation. Spectrin repeats may thus function as elastic elements, extendable to intermediate states at various lengths.

Energetics of glycerol conduction through aquaglyceroporin GlpF

Aquaglyceroporin GlpF selectively conducts water and linear polyalcohols, such as glycerol, across the inner membrane of Escherichia coli. We report steered molecular dynamics simulations of glycerol conduction through GlpF, in which external forces accelerate the transchannel conduction in a manner that preserves the intrinsic conduction mechanism. The simulations reveal channel-glycerol hydrogen bonding interactions and the stereoselectivity of the channel. Employing Jarzynski's identity between free energy and irreversible work, we reconstruct the potential of mean force along the conduction pathway through a time series analysis of molecular dynamics trajectories. This potential locates binding sites and barriers inside the channel; it also reveals a low energy periplasmic vestibule suited for efficient uptake of glycerol from the environment.

Steered molecular dynamics investigations of protein function

Molecular recognition and mechanical properties of proteins govern molecular processes in the cell that can cause disease and can be targeted for drug design. Single molecule measurement techniques have greatly advanced knowledge but cannot resolve enough detail to be interpreted in terms of protein structure. We seek to complement the observations through so-called Steered Molecular Dynamics (SMD) simulations that link directly to experiments and provide atomic-level descriptions of the underlying events. Such a research program has been initiated in our group and has involved, for example, studies of elastic properties of immunoglobulin and fibronectin domains as well as the binding of biotin and avidin. In this article we explain the SMD method and suggest how it can be applied to the function of three systems that are the focus of modern molecular biology research: force transduction by the muscle protein titin and extracellular matrix protein fibronectin, recognition of antibody-antigene pairs, and ion selective conductivity of the K+ channel.

Comparison of the early stages of forced unfolding for fibronectin type III modules

The structural changes accompanying stretch-induced early unfolding events were investigated for the four type III fibronectin (FN-III) modules, FN-III(7), FN-III(8), FN-III(9), and FN-III(10) by using steered molecular dynamics. Simulations revealed that two main energy barriers, I and II, have to be overcome to initiate unraveling of FN-III's tertiary structure. In crossing the first barrier, the two opposing beta-sheets of FN-III are rotated against each other such that the beta-strands of both beta-sheets align parallel to the force vector (aligned state). All further events in the unfolding pathway proceed from this intermediate state. A second energy barrier has to be overcome to break the first major cluster of hydrogen bonds between adjacent beta-strands. Simulations revealed that the height of barrier I varied significantly among the four modules studied, being largest for FN-III(7) and lowest for FN-III(10), whereas the height of barrier II showed little variation. Key residues affecting the mechanical stability of FN-III modules were identified. These results suggest that FN-III modules can be prestretched into an intermediate state with only minor changes to their tertiary structures. FN-III(10), for example, extends 12 A from the native "twisted" to the intermediate aligned state, and an additional 10 A from the aligned state to further unfolding where the first beta-strand is peeled away. The implications of the existence of intermediate states regarding the elasticity of fibrillar fibers and the stretch-induced exposure of cryptic sites are discussed.

Steered molecular dynamics and mechanical functions of proteins

Atomic force microscopy of single molecules, steered molecular dynamics and the theory of stochastic processes have established a new field that investigates mechanical functions of proteins, such as ligand-receptor binding/unbinding and elasticity of muscle proteins during stretching. The combination of these methods yields information on the energy landscape that controls mechanical function and on the force-bearing components of proteins, as well as on the underlying physical mechanisms.

Free energy reconstruction from nonequilibrium single-molecule pulling experiments

Laser tweezers and atomic force microscopes are increasingly used to probe the interactions and mechanical properties of individual molecules. Unfortunately, using such time-dependent perturbations to force rare molecular events also drives the system away from equilibrium. Nevertheless, we show how equilibrium free energy profiles can be extracted rigorously from repeated nonequilibrium force measurements on the basis of an extension of Jarzynski's remarkable identity between free energies and the irreversible work.

Measuring the forces that control protein interactions

Although the force fields and interaction energies that control protein behavior can be inferred indirectly from equilibrium and kinetic measurements, recent developments have made it possible to quantify directly (a) the ranges, magnitudes, and time dependence of the interaction energies and forces between biological materials; (b) the mechanical properties of isolated proteins; and (c) the strength of single receptor-ligand bonds. This review describes recent results obtained by using the atomic force microscope, optical tweezers, the surface force apparatus, and micropipette aspiration to quantify short-range protein-ligand interactions and the long-range, nonspecific forces that together control protein behavior. The examples presented illustrate the power of force measurements to quantify directly the force fields and energies that control protein behavior.

How do substrates enter and products exit the buried active site of cytochrome P450cam? 2. Steered molecular dynamics and adiabatic mapping of substrate pathways

Three possible channels by which substrates and products can exit from the buried active site of cytochrome P450cam have been identified by means of random expulsion molecular dynamics simulations. In the investigation described here, we computed estimates of the relative probabilities of ligand passage through the three channels using steered molecular dynamics and adiabatic mapping. For comparison, the same techniques are also applied to investigate substrate egress from cytochrome P450-BM3. The channel in cytochrome P450cam, for which there is the most supporting evidence from experiments (which we name pathway 2a), is computed to be the most probable ligand exit channel. It has the smallest computed unbinding work and force. For this channel, the ligand exits between the F/G loop and the B' helix. Two mechanistically distinct, but energetically similar routes through this channel were observed, showing that multiple pathways along one channel are possible. The probability of ligand exit via the next most probable channel (pathway 3), which is located between the I helix and the F and G helices, is estimated to be less than 1/10 of the probability of exit along pathway 2a. Low-frequency modes of the protein extracted from an essential dynamics analysis of a 1 ns duration molecular dynamics simulation of cytochrome P450cam with camphor bound, support the opening of pathway 2a on a longer timescale. On longer timescales, it is therefore expected that this pathway becomes more dominant than estimated from the present computations.

Model energy landscapes and the force-induced dissociation of ligand-receptor bonds

We discuss models for the force-induced dissociation of a ligand-receptor bond, occurring in the context of cell adhesion or single molecule unbinding force measurements. We consider a bond with a structured energy landscape which is modeled by a network of force dependent transition rates between intermediate states. The behavior of a model with only one intermediate state and a model describing a molecular zipper is studied. We calculate the bond lifetime as a function of an applied force and unbinding forces under an increasing applied load and determine the relationship between both quantities. The dissociation via an intermediate state can lead to distinct functional relations of the bond lifetime on force. One possibility is the occurrence of three force regimes where the lifetime of the bond is determined by different transitions within the energy landscape. This case can be related to recent experimental observations of the force-induced dissociation of single avidin-biotin bonds.

Thermal noise reduction of mechanical oscillators by actively controlled external dissipative forces

We show that the thermal fluctuations of very soft mechanical oscillators, such as the cantilever in an atomic force microscope (AFM), can be reduced without changing the stiffness of the spring or having to lower the environment temperature. We derive a theoretical relationship between the thermal fluctuations of an oscillator and an actively controlled external dissipative force. This relationship is verified by experiments with an AFM cantilever where the external active force is coupled through a magnetic field. With simple instrumentation, we have reduced the thermal noise amplitude of the cantilever by a factor of 3.4, achieving an apparent temperature of 25 K with the environment at 295 K. This active noise reduction approach can significantly improve the accuracy of static position or static force measurements in a number of practical applications.

Force probe measurements of antibody-antigen interactions

The surface force apparatus has been used to quantify directly the forces that govern the interactions between proteins and ligands. In this work, we describe the measured interactions between the antigen fluorescein and the Fab' fragment of the monoclonal 4-4-20 anti-fluorescyl IgG antibody. Here we first describe the use of the surface force apparatus to demonstrate directly the impact of the charge composition in the region of the antibody binding site on the antibody interactions. Several approaches are described for immobilizing antigens, antibodies, and proteins in general for direct force measurements. The measured force profiles presented are accompanied by an extensive discussion of protocols used to analyze the force-distance curves and to interpret them in terms of the antibody structure. In addition to long-range electrostatic forces, we also consider short-range forces that can affect the strength of adhesion between the Fab' and immobilized fluorescein. The latter investigations demonstrate the influence of interfacial properties on the recognition of surface-bound antigens.