Bell-Evans Model Research Articles

Bell-Evans model and steered molecular dynamics in uncovering the dissociation kinetics of ligands targeting G-protein-coupled receptors

Recently, academic and industrial scientific communities involved in kinetics-based drug development have become immensely interested in predicting the drug target residence time. Screening drug candidates in terms of their computationally predicted residence times, which is a measure of drug efficacy in vivo, and simultaneously assessing computational binding affinities are becoming inevitable. Non-equilibrium molecular simulation approaches are proven to be useful in this purpose. Here, we have implemented an optimized approach of combining the data derived from steered molecular dynamics simulations and the Bell-Evans model to predict the absolute residence times of the antagonist ZMA241385 and agonist NECA that target the A2A adenosine receptor of the G-protein-coupled receptor (GPCR) protein family. We have predicted the absolute ligand residence times on the timescale of seconds. However, our predictions were many folds shorter than those determined experimentally. Additionally, we calculated the thermodynamics of ligand binding in terms of ligand binding energies and the per-residue contribution of the receptor. Subsequently, binding pocket hotspot residues that would be important for further computational mutagenesis studies were identified. In the experiment, similar sets of residues were found to be in significant contact with both ligands under study. Our results build a strong foundation for further improvement of our approach by rationalizing the kinetics of ligand unbinding with the thermodynamics of ligand binding.

Nanomechanical Study of Enzyme: Coenzyme Complexes: Bipartite Sites in Plastidic Ferredoxin-NADP+ Reductase for the Interaction with NADP.

Plastidic ferredoxin-NADP+ reductase (FNR) transfers two electrons from two ferredoxin or flavodoxin molecules to NADP+, generating NADPH. The forces holding the Anabaena FNR:NADP+ complex were analyzed by dynamic force spectroscopy, using WT FNR and three C-terminal Y303 variants, Y303S, Y303F, and Y303W. FNR was covalently immobilized on mica and NADP+ attached to AFM tips. Force–distance curves were collected for different loading rates and specific unbinding forces were analyzed under the Bell–Evans model to obtain the mechanostability parameters associated with the dissociation processes. The WT FNR:NADP+ complex presented a higher mechanical stability than that reported for the complexes with protein partners, corroborating the stronger affinity of FNR for NADP+. The Y303 mutation induced changes in the FNR:NADP+ interaction mechanical stability. NADP+ dissociated from WT and Y303W in a single event related to the release of the adenine moiety of the coenzyme. However, two events described the Y303S:NADP+ dissociation that was also a more durable complex due to the strong binding of the nicotinamide moiety of NADP+ to the catalytic site. Finally, Y303F shows intermediate behavior. Therefore, Y303, reported as crucial for achieving catalytically competent active site geometry, also regulates the concerted dissociation of the bipartite nucleotide moieties of the coenzyme.

Force Dependence of Proteins' Transition State Position and the Bell-Evans Model.

Single-molecule force spectroscopy has opened a new field of research in molecular biophysics and biochemistry. Pulling experiments on individual proteins permit us to monitor conformational transitions with high temporal resolution and measure their free energy landscape. The force–extension curves of single proteins often present large hysteresis, with unfolding forces that are higher than refolding ones. Therefore, the high energy of the transition state (TS) in these molecules precludes kinetic rates measurements in equilibrium hopping experiments. In irreversible pulling experiments, force-dependent kinetic rates measurements show a systematic discrepancy between the sum of the folding and unfolding TS distances derived by the kinetic Bell–Evans model and the full molecular extension predicted by elastic models. Here, we show that this discrepancy originates from the force-induced movement of TS. Specifically, we investigate the highly kinetically stable protein barnase, using pulling experiments and the Bell–Evans model to characterize the position of its kinetic barrier. Experimental results show that while the TS stays at a roughly constant distance relative to the native state, it shifts with force relative to the unfolded state. Interestingly, a conversion of the protein extension into amino acid units shows that the TS position follows the Leffler–Hammond postulate: the higher the force, the lower the number of unzipped amino acids relative to the native state. The results are compared with the quasi-reversible unfolding–folding of a short DNA hairpin.

Molecular Models of Human Elastin and Elastin Biomaterials

The elastin polymer, assembled from its molecular precursor tropoelastin, is the dominant component of elastic fibers, which confer elasticity and structural integrity to skin, lung, connective and vascular tissue. Historically, elastin's dynamic nature has precluded traditional approaches such as X-ray crystallography to understand its detailed features. Here, I describe recent work using atomistic and coarse-grained models for predicting elastin's molecular structure, mechanical properties and mechanisms, as well as dynamics. We use the models to probe the function of key molecular regions, investigate disease etiology and explore implications for hierarchical assembly. From the materials perspective, elastin-based materials display tunable thermal sensitivity, presenting opportunities to mimic and control these responsive features for biomedical applications. We characterize the temperature response spectrum of elastin-like peptides to design synthetic polymers with tunable switching, resolving effects of peptide chemistry, chain length, and solvent environment. We also study a chimera silk-elastin-like protein polymer that combines silk's strength with elastin's extensibility and responsive features to identify temperature transition effects on molecular-scale mechanics. We analyze the associated free-energy landscape with the Bell-Evans model to interpret temperature-induced phase transitions. Such a feedback loop between simulation and experiment for predictive biomaterial design may enable new applications in drug delivery and tissue engineering.

Forced-rupture of cell-adhesion complexes reveals abrupt switch between two brittle states.

Cell adhesion complexes (CACs), which are activated by ligand binding, play key roles in many cellular functions ranging from cell cycle regulation to mediation of cell extracellular matrix adhesion. Inspired by single molecule pulling experiments using atomic force spectroscopy on leukocyte function-associated antigen-1 (LFA-1), expressed in T-cells, bound to intercellular adhesion molecules (ICAM), we performed constant loading rate (rf) and constant force (F) simulations using the self-organized polymer model to describe the mechanism of ligand rupture from CACs. The simulations reproduce the major experimental finding on the kinetics of the rupture process, namely, the dependence of the most probable rupture forces (f*s) on ln rf (rf is the loading rate) exhibits two distinct linear regimes. The first, at low rf, has a shallow slope, whereas the slope at high rf is much larger, especially for a LFA-1/ICAM-1 complex with the transition between the two occurring over a narrow rf range. Locations of the two transition states (TSs) extracted from the simulations show an abrupt change from a high value at low rf or constant force, F, to a low value at high rf or F. This unusual behavior in which the CACs switch from one brittle (TS position is a constant over a range of forces) state to another brittle state is not found in forced-rupture in other protein complexes. We explain this novel behavior by constructing the free energy profiles, F(Λ)s, as a function of a collective reaction coordinate (Λ), involving many key charged residues and a critical metal ion (Mg2+). The TS positions in F(Λ), which quantitatively agree with the parameters extracted using the Bell-Evans model, change abruptly at a critical force, demonstrating that it, rather than the molecular extension, is a good reaction coordinate. Our combined analyses using simulations performed in both the pulling modes (constant rf and F) reveal a new mechanism for the two loading regimes observed in the rupture kinetics in CACs.

Force feedback effects on single molecule hopping and pulling experiments.

Single-molecule experiments with optical tweezers have become an important tool to study the properties and mechanisms of biological systems, such as cells and nucleic acids. In particular, force unzipping experiments have been used to extract the thermodynamics and kinetics of folding and unfolding reactions. In hopping experiments, a molecule executes transitions between the unfolded and folded states at a preset value of the force [constant force mode (CFM) under force feedback] or trap position [passive mode (PM) without feedback] and the force-dependent kinetic rates extracted from the lifetime of each state (CFM) and the rupture force distributions (PM) using the Bell-Evans model. However, hopping experiments in the CFM are known to overestimate molecular distances and folding free energies for fast transitions compared to the response time of the feedback. In contrast, kinetic rate measurements from pulling experiments have been mostly done in the PM while the CFM is seldom implemented in pulling protocols. Here, we carry out hopping and pulling experiments in a short DNA hairpin in the PM and CFM at three different temperatures (6 °C, 25 °C, and 45 °C) exhibiting largely varying kinetic rates. As expected, we find that equilibrium hopping experiments in the CFM and PM perform well at 6 °C (where kinetics are slow), whereas the CFM overestimates molecular parameters at 45 °C (where kinetics are fast). In contrast, nonequilibrium pulling experiments perform well in both modes at all temperatures. This demonstrates that the same kind of feedback algorithm in the CFM leads to more reliable determination of the folding reaction parameters in irreversible pulling experiments.

Monodisperse measurement of the biotin-streptavidin interaction strength in a well-defined pulling geometry.

The widely used interaction of the homotetramer streptavidin with the small molecule biotin has been intensively studied by force spectroscopy and has become a model system for receptor ligand interaction. However, streptavidin’s tetravalency results in diverse force propagation pathways through the different binding interfaces. This multiplicity gives rise to polydisperse force spectroscopy data. Here, we present an engineered monovalent streptavidin tetramer with a single cysteine in its functional subunit that allows for site-specific immobilization of the molecule, orthogonal to biotin binding. Functionality of streptavidin and its binding properties for biotin remain unaffected. We thus created a stable and reliable molecular anchor with a unique high-affinity binding site for biotinylated molecules or nanoparticles, which we expect to be useful for many single-molecule applications. To characterize the mechanical properties of the bond between biotin and our monovalent streptavidin, we performed force spectroscopy experiments using an atomic force microscope. We were able to conduct measurements at the single-molecule level with 1:1-stoichiometry and a well-defined geometry, in which force exclusively propagates through a single subunit of the streptavidin tetramer. For different force loading rates, we obtained narrow force distributions of the bond rupture forces ranging from 200 pN at 1,500 pN/s to 230 pN at 110,000 pN/s. The data are in very good agreement with the standard Bell-Evans model with a single potential barrier at Δx0 = 0.38 nm and a zero-force off-rate koff,0 in the 10−6 s-1 range.

Modeling and Experiment Reveal Structure and Nanomechanics across the Inverse Temperature Transition in B. mori Silk-Elastin-like Protein Polymers.

Silk and elastin are exemplary protein materials that exhibit exceptional material properties. Silk is uniquely strong, surpassing engineering materials such as Kevlar and steel, while elastin has exquisite flexibility and can reversibly fold into a more structured form at high temperatures when many other proteins would unfold and denature. This phenomenon in elastin is termed the inverse temperature transition. It is a reversible, controllable process that motivates applications in drug delivery, shape change materials, and biomimetic devices. Silk-elastinlike protein polymers (SELPs), which combine repeating B. mori silk and elastin blocks, have been introduced as biologically inspired materials that combine the distinctive properties of the component parts to achieve strong and extensible, tunable biomaterials. Here, we considered a single SELP sequence to examine temperature transition effects at the molecular scale. SELP molecular models were created using Replica Exchange Molecular Dynamics, an accelerated sampling method, and confirmed in experiment by comparing secondary structure distributions. A molecular collapse of the SELP molecule was observed with increased temperature in both molecular simulation and experiment. Temperature-specific differences were observed in the mechanical properties and the unfolding pathways of the polypeptide. Using the Bell-Evans model, we analyzed the free energy landscape associated with molecular unfolding at temperatures below and above the transition temperature range (Tt) of the polypeptide. We found that at physiological pulling rates, the energy barrier to unfold SELPs was counterintuitively higher above Tt. Our findings offer a foundational perspective on the molecular scale mechanisms of temperature-induced phase transition in SELPs, and suggest a novel approach to combine simulation and experiment to study materials for multifunctional biomimetic applications.

Force spectroscopy of multivalent binding of riboflavin-conjugated dendrimers to riboflavin binding protein.

Putative riboflavin receptors are considered as biomarkers due to their overexpression in breast and prostate cancers. Hence, these receptors can be potentially exploited for use in targeted drug delivery systems where dendrimer nanoparticles with multivalent ligand attachments can lead to greater specificity in cellular interactions. In this study, the single molecule force spectroscopy technique was used to assess the physical strength of multivalent interactions by employing a riboflavin (RF)-conjugated generation 5 PAMAM dendrimer G5(RF)n nanoparticle. By varying the average RF ligand valency (n = 0, 3, 5), the rupture force was measured between G5(RF)n and the riboflavin binding protein (RFBP). The rupture force increased when the valency of RF increased. We observed at the higher valency (n = 5) three binding events that increased in rupture force with increasing loading rate. Assuming a single energy barrier, the Bell-Evans model was used to determine the kinetic off-rate and barrier width for all binding interactions. The analysis of our results appears to indicate that multivalent interactions are resulting in changes to rupture force and kinetic off-rates.

Transition model for ricin-aptamer interactions with multiple pathways and energy barriers.

We develop a transition model to interpret single-molecule ricin-aptamer interactions with multiple unbinding pathways and energy barriers measured by atomic force microscopy dynamic force spectroscopy. Molecular simulations establish the relationship between binding conformations and the corresponding unbinding pathways. Each unbinding pathway follows a Bell-Evans multiple-barrier model. Markov-type transition matrices are developed to analyze the redistribution of unbinding events among the pathways under different loading rates. Our study provides detailed information about complex behaviors in ricin-aptamer unbinding events.

Force spectroscopy of polymer desorption: theory and molecular dynamics simulations

Forced detachment of a single polymer chain, strongly adsorbed on a solid substrate, is investigated by two complementary methods: a coarse-grained analytical dynamical model, based on the Onsager stochastic equation, and Molecular Dynamics (MD) simulations with a Langevin thermostat. The suggested approach makes it possible to go beyond the limitations of the conventional Bell-Evans model. We observe a series of characteristic force spikes when the pulling force is measured against the cantilever displacement during detachment at constant velocity vc (displacement control mode) and find that the average magnitude of this force increases as vc increases. The probability distributions of the pulling force and the end-monomer distance from the surface at the moment of the final detachment are investigated for different adsorption energies ε and pulling velocities vc. Our extensive MD simulations validate and support the main theoretical findings. Moreover, the simulations reveal a novel behavior: for a strong-friction and massive cantilever the force spike pattern is smeared out at large vc. As a challenging task for experimental bio-polymer sequencing in future we suggest the fabrication of a stiff, super-light, nanometer-sized AFM probe.

Comparison of three competing dynamic force spectroscopy models to study binding forces of amyloid-β (1–42)

We performed single molecule dynamic force spectroscopy experiments to study the dimerization of two amyloid-β (1-42) peptides and compared three different theoretical models used to fit experimental data: Bell-Evans, Dudko-Hummer-Szabo, and Friddle-De Yoreo. Using these models we extracted values of the dissociation rate at zero force, k0, and height and the width of the energy barrier, ΔG and xβ. We show the importance of including the effect of the linker molecule. All three models corrected for the linker effect give comparable results for xβ and show more discrepancy for k0 and ΔG values, ΔG parameter correlates well between Dudko-Hummer-Szabo and Friddle-De Yoreo models but differs for the Bell-Evans model.

Dependence of the most probable and average bond rupture force on the force loading rate: First order correction to the Bell–Evans model

The Bell–Evans model which predicts the linear dependence of the most probable intermolecular bond rupture force on the logarithm of stretching force loading rate is usually used to discuss the dynamic force spectroscopy experiment data. This model is consistent with the Kramers' theory of the bond dissociation rate only if one presupposes an independence of the pre-exponential factors in the Kramers relation on the acting force and a linear decrease of the dissociation barrier height on this same force, and for this to be true rather special shape of the interaction landscape is required. Here, we present a first order correction to this model (first terms of corresponding Taylor expansions are taken into account), discuss its implication for the interpretation of dynamic force spectroscopy experiment data and compare our model with the Monte Carlo simulation of a specially designed single molecule dynamic force spectroscopy experiment. In addition to the most probable bond rupture force, an average rupture force values are also calculated. All approximations made and the range of applicability of the obtained results are carefully described and compared with those for some other models in the field.

Dependence of the Most Probable and Average Bond Rupture Forces on the Force Loading Rate in Single Molecule Dynamic Force Spectroscopy Experiment: First Order Correction to the Bell - Evans Model

The Bell-Evans model which predicts the linear dependence of the most probable bond rupture force on the logarithm of the force loading rate is usually used to discuss the dynamic force spectroscopy experiment data. This model is consistent with the Kramers' theory of the bond dissociation rate only if one presupposes an independence of the pre-exponential factors in the Kramers relation on the acting force F and a linear decrease of the dissociation barrier height on this same force, and for this to be true a rather special shape of the interaction landscape is required.

Force Measurement Enabling Precise Analysis by Dynamic Force Spectroscopy

Dynamic force spectroscopy (DFS) makes it possible to investigate specific interactions between two molecules such as ligand-receptor pairs at the single-molecule level. In the DFS method based on the Bell-Evans model, the unbinding force applied to a molecular bond is increased at a constant rate, and the force required to rupture the molecular bond is measured. By analyzing the relationship between the modal rupture force and the logarithm of the loading rate, microscopic potential barrier landscapes and the lifetimes of bonds can be obtained. However, the results obtained, for example, in the case of streptavidin/biotin complexes, have differed among previous studies and some results have been inconsistent with theoretical predictions. In this study, using an atomic force microscopy technique that enables the precise analysis of molecular interactions on the basis of DFS, we investigated the effect of the sampling rate on DFS analysis. The shape of rupture force histograms, for example, was significantly deformed at a sampling rate of 1 kHz in comparison with that of histograms obtained at 100 kHz, indicating the fundamental importance of ensuring suitable experimental conditions for further advances in the DFS method.

Single-Molecule Force Spectroscopy of Multimodular Proteins: A New Method to Extract Kinetic Unfolding Parameters

When multimodular proteins composed of a number of identical modules are unfolded in the course of a single-molecule force-spectroscopy experiment, both the probability distribution of the unfolding force and the most probable unfolding force depend on the number of not-yet-unfolded modules in the protein. We show that this unfolding can be described in the framework of a generalized Bell-Evans model, based on certain minimal assumptions concerning the force loading conditions. In particular, our modified equations for the unfolding force predict, for a multimodular protein containing N not-yet-unfolded modules, a linear dependence of the most probable unfolding force on ln(N). This conclusion has been confirmed in experiments with GB1 proteins, which verify in addition further consequences of the proposed model, including a modified probability distribution of the unfolding force. Based on these results, we propose a new method for analyzing single-molecule force-spectroscopy data for multimodular proteins, which enables one to extract kinetic unfolding parameters from experimental data with no additional assumptions about the unfolding process.

Single-Molecule Force Spectroscopy of Cartilage Aggrecan Self-Adhesion

We investigated self-adhesion between highly negatively charged aggrecan macromolecules extracted from bovine cartilage extracellular matrix by performing atomic force microscopy (AFM) imaging and single-molecule force spectroscopy (SMFS) in saline solutions. By controlling the density of aggrecan molecules on both the gold substrate and the gold-coated tip surface at submonolayer densities, we were able to detect and quantify the Ca 2+-dependent homodimeric interaction between individual aggrecan molecules at the single-molecule level. We found a typical nonlinear sawtooth profile in the AFM force-versus-distance curves with a molecular persistence length of l p = 0.31 ± 0.04 nm. This is attributed to the stepwise dissociation of individual glycosaminoglycan (GAG) side chains in aggrecans, which is very similar to the known force fingerprints of other cell adhesion proteoglycan systems. After studying the GAG-GAG dissociation in a dynamic, loading-rate-dependent manner (dynamic SMFS) and analyzing the data according to the stochastic Bell-Evans model for a thermally activated decay of a metastable state under an external force, we estimated for the single glycan interaction a mean lifetime of τ = 7.9 ± 4.9 s and a reaction bond length of x β = 0.31 ± 0.08 nm. Whereas the x β -value compares well with values from other cell adhesion carbohydrate recognition motifs in evolutionary distant marine sponge proteoglycans, the rather short GAG interaction lifetime reflects high intermolecular dynamics within aggrecan complexes, which may be relevant for the viscoelastic properties of cartilage tissue.

Single-molecule force spectroscopy of supramolecular heterodimeric capsules

The specific interaction of a supramolecular binding motif was quantitatively evaluated by dynamic single-molecule force spectroscopy (SMFS) using an atomic force microscope (AFM). The supramolecular capsule forms by two different cavitands stitched together by four hydrogen bonds between carboxylic acid and pyridyl groups. The tetra(carboxyl)cavitand is monofunctionalized at the lower rim with a flexible poly(ethylene glycol) linker and attached to the AFM sensor tip. Single-molecule association experiments are accomplished using a diluted self-assembled monolayer (SAM) of the tetra(pyridyl)cavitand on a gold substrate. The measured single-molecule dissociation forces of the heterodimeric capsule represent the mechanical stability of the supramolecular system and allow a quantitative evaluation of the interaction according to the Bell-Evans model yielding dissociation rate constant k(off) = (0.14 +/- 0.14) s(-1), reaction length x(beta) = (0.56 +/- 0.076) nm and an estimated value of DeltaG(0) = -27 kJ mol(-1).

Force-Driven Separation of Short Double-Stranded DNA

Short double-stranded DNA is used in a variety of nanotechnological applications, and for many of them, it is important to know for which forces and which force loading rates the DNA duplex remains stable. In this work, we develop a theoretical model that describes the force-dependent dissociation rate for DNA duplexes tens of basepairs long under tension along their axes (“shear geometry”). Explicitly, we set up a three-state equilibrium model and apply the canonical transition state theory to calculate the kinetic rates for strand unpairing and the rupture-force distribution as a function of the separation velocity of the end-to-end distance. Theory is in excellent agreement with actual single-molecule force spectroscopy results and even allows for the prediction of the rupture-force distribution for a given DNA duplex sequence and separation velocity. We further show that for describing double-stranded DNA separation kinetics, our model is a significant refinement of the conventionally used Bell-Evans model.

A Single Molecule Study of Enzyme Inhibitor Interactions

Recently developed microscopic models by Dudko et al. were used to estimate the apparent kinetic and thermodynamic parameters in a single molecule force spectroscopy study of the carbonic anhydrase enzyme and a sulfonamide inhibitor. The most probable rupture force for the enzyme-inhibitor interaction demonstrates a nonlinear dependency on the log-loading rate. Estimates for the kinetic and thermodynamic parameters were obtained by fitting the nonlinear dependency to linear cubic potential and cusp potential models and compared to the Bell-Evans model. The reliability of the estimated parameters was verified by modeling the experimental rupture force distributions by the theoretically predicted distributions at rupture. We also report that an increase in the inhibitor tether length has a significant effect on the apparent kinetic and thermodynamic parameters while extending the length of the linkers which attach the enzyme to the surface has a minimal effect.