Direct measurement of protein energy landscape roughness

Transition Path Dynamics of Non-Markovian Systems across a Rough Potential Barrier

Transition paths refer to rare events in physics, chemistry, and biology where the molecules cross barriers separating stable molecular conformations. The conventional analysis of the transition path times employs a diffusive and memoryless transition over a smooth potential barrier. However, it is widely acknowledged that the free energy profile between two minima in biomolecular processes is inherently not smooth. In this article, we discuss a theoretical model with a parabolic rough potential barrier and obtain analytical results of the transition path distribution and mean transition path times by incorporating absorbing boundary conditions across the boundaries under the driving of Gaussian white noise. Further, the influence of anomalous dynamics in rough potential driven by a power-law memory kernel is analyzed by deriving a time-dependent scaled diffusion coefficient that coarse-grains the effects of roughness, and the system's dynamics is reduced to a scaled diffusion on a smooth potential. Our theoretical results are tested and validated against numerical simulations. The findings of our study show the influence of the boundary conditions, barrier height, barrier roughness, and memory effect on the transition path time distributions in a rough potential, and the validity of the scaling diffusion coefficient has been discussed.

Fluctuating landscapes and heavy tails in animal behavior

Animal behavior is shaped by a myriad of mechanisms acting on a wide range of scales, which hampers quantitative reasoning and the identification of general principles. Here, we combine data analysis and theory to investigate the relationship between behavioral plasticity and heavy-tailed statistics often observed in animal behavior. Specifically, we first leverage high-resolution recordings of C. elegans locomotion to show that stochastic transitions among long-lived behaviors exhibit heavy-tailed first passage time distributions and correlation functions. Such heavy tails can be explained by slow adaptation of behavior over time. This particular result motivates our second step of introducing a general model where we separate fast dynamics on a quasi-stationary multi-well potential, from non-ergodic, slowly varying modes. We then show that heavy tails generically emerge in such a model, and we provide a theoretical derivation of the resulting functional form, which can become a power law with exponents that depend on the strength of the fluctuations. Finally, we provide direct support for the generality of our findings by testing them in a C. elegans mutant where adaptation is suppressed and heavy tails thus disappear, and recordings of larval zebrafish swimming behavior where heavy tails are again prevalent.

Fluctuating landscapes and heavy tails in animal behavior

Animal behavior is shaped by a myriad of mechanisms acting on a wide range of scales. This immense variability hampers quantitative reasoning and renders the identification of universal principles elusive. Through data analysis and theory, we here show that slow non-ergodic drives generally give rise to heavy-tailed statistics in behaving animals. We leverage high-resolution recordings of C. elegans locomotion to extract a self-consistent reduced order model for an inferred reaction coordinate, bridging from sub-second chaotic dynamics to long-lived stochastic transitions among metastable states. The slow mode dynamics exhibits heavy-tailed first passage time distributions and correlation functions, and we show that such heavy tails can be explained by dynamics on a time-dependent potential landscape. Inspired by these results, we introduce a generic model in which we separate faster mixing modes that evolve on a quasi-stationary potential, from slower non-ergodic modes that drive the potential landscape, and reflect slowly varying internal states. We show that, even for simple potential landscapes, heavy tails emerge when barrier heights fluctuate slowly and strongly enough. In particular, the distribution of first passage times and the correlation function can asymptote to a power law, with related exponents that depend on the strength and nature of the fluctuations. We support our theoretical findings through direct numerical simulations.

Roughness induced current reversal in fractional hydrodynamic memory

The existence of a corrugated surface is of great importance and ubiquity in biological systems, exhibiting diverse dynamic behaviors. However, it has remained unclear whether such rough surface leads to the current reversal in fractional hydrodynamic memory. We investigate the transport of a particle within a rough potential under external forces in a subdiffusive media with fractional hydrodynamic memory. The results demonstrate that roughness induces current reversal and a transition from no transport to transport. These phenomena are analyzed through the subdiffusion, Peclet number, useful work, input power, and thermodynamic efficiency. The analysis reveals that transport results from energy conversion, wherein time-dependent periodic force is partially converted into mechanical energy to drive transport against load, and partially dissipated through environmental absorption. In addition, the findings indicate that the size and shape of ratchet tune the occurrence and disappearance of the current reversal, and control the number of times of the current reversal occurring. Furthermore, we find that temperature, friction, and load tune transport, resonant-like activity, and enhanced stability of the system, as evidenced by thermodynamic efficiency. These findings may have implications for understanding dynamics in biological systems and may be relevant for applications involving molecular devices for particle separation at the mesoscopic scale.

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable micro-states which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each micro-state. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is ∼3k_BT. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

Roughness in the periodic potential induces absolute negative mobility in a driven Brownian ratchet

Absolute negative mobility, where particles move opposite to the direction as governed by the external load, is an anomalous transport property of a Brownian ratchet and has technological implications in mass separation and bioanalytical applications. We numerically investigated here the effect of roughness in symmetric periodic potential on the negative mobility of a driven inertial Brownian ratchet in the presence of an external load. We show that the microscopic spatial heterogeneity of the potential can generate negative mobility which would not otherwise be possible under smooth potential in the concerned parameter space. We determined the optimal condition in terms of parameter space for such anomalous behavior. Our calculations indicate that the shift of balance towards the negative velocity phase in the temporal oscillations of velocity and weakly chaotic dynamics are responsible factors for roughness-induced negative mobility. These calculations highlight a constructive role of roughness in the anomalous transport properties of Brownian ratchet.

Roughness in the periodic potential enhances transport in a driven inertial ratchet

We study the effects of roughness in the asymmetric periodic potential on the transport and diffusion of an inertial Brownian particle driven by a time-periodic force in a Gaussian environment. We find that moderate roughness leads to the loss of transient anomalous diffusion, and it helps to establish normal diffusion in the weak noise limit. We uncover a contrasting effect of roughness on the transport of particles in the weak and moderate to large noise limit. In the weak noise limit, small amplitude roughness results in the increase of directed transport, whereas in the moderate to large noise limit, roughness hinders transport. The deterministic dynamics of the system reveals that the purely periodic system under smooth potential transits into a chaotic system due to the moderate roughness in the potential. Therefore our calculations demonstrate the constructive role of roughness in the transport of particles in the inertial regime.

Quantifying residue-specific conformational dynamics of a highly reactive 29-mer peptide

Understanding structural transitions within macromolecules remains an important challenge in biochemistry, with important implications for drug development and medicine. Insight into molecular behavior often requires residue-specific dynamics measurement at micromolar concentrations. We studied MP01-Gen4, a library peptide selected to rapidly undergo bioconjugation, by using electron paramagnetic resonance (EPR) to measure conformational dynamics. We mapped the dynamics of MP01-Gen4 with residue-specificity and identified the regions involved in a structural transformation related to the conjugation reaction. Upon reaction, the conformational dynamics of residues near the termini slow significantly more than central residues, indicating that the reaction induces a structural transition far from the reaction site. Arrhenius analysis demonstrates a nearly threefold decrease in the activation energy of conformational diffusion upon reaction (8.0 kBT to 3.4 kBT), which occurs across the entire peptide, independently of residue position. This novel approach to EPR spectral analysis provides insight into the positional extent of disorder and the nature of the energy landscape of a highly reactive, intrinsically disordered library peptide before and after conjugation.

First-passage-time distribution in a moving parabolic potential with spatial roughness

In this paper, we investigate the first-passage-time distribution (FPTD) within a time-dependent parabolic potential in the presence of roughness with two methods: the Kramers theory and a nonsingular integral equation. By spatially averaging, the rough potential is equivalent to the combination of an effective smooth potential and an effective diffusion coefficient. Based on the Kramers theory, we first obtain Kramers approximations (KAs) of FPTD for both smooth and rough potentials. As expected, KA is valid only for high barriers and small external forces, and generally applicable for high barriers in rough potentials. To overcome the shortcoming of KA, a probability asymptotic approximation (PAA) based on an integral equation is proposed, which uses the transient probability density function (PDF) of the natural boundary conditions instead of the absorbing boundary conditions. We find that PAA fits very well even for large external forces. This enables us to analytically solve the FPTD for large external forces and low barriers as a strong extension to KA. In addition, we show that in the presence of a rough potential, the PAA of FPTD is in good agreement with numerical simulations for low barrier potentials. The PAA makes it possible to investigate the first-passage problem with ultrafast varying potentials and short exiting time. Thus, KA and PAA are complementary in determining the FPTD both for various barriers and external forces. Finally, the mean first-passage time (MFPT) is studied, which illustrates that the PAA of MFPT is effective almost in the whole range of external forces, while the KA of MFPT is valid only for small external forces.

Diffusion crossing over a barrier in a random rough metastable potential

We carry out a detailed study of escape dynamics of a particle driven by a white noise over a metastable potential corrugated by spatial disorder in the form of zero-mean random correlated potential. The approach of double-averaging over test particles and statistic ensemble is proposed to calculate the escape rate in a finite-size random rough metastable potential, moreover, the interference mechanism of test particles multi-passing over the saddle point is considered. Through analyzing the dependence of the steady escape rate on various modelled potentials and parameters, we demonstrate that the obstruction induced by roughness leads to a decrease in the steady escape rate with the increase of rough intensity. We also add the random correlated potential into the vicinity of the saddle-point of metastable potentials of three kinds and show an enhancement phenomenon of escape rate similar to the previous study of a surmounting fluctuating sharp barrier.

Roughness-enhanced transport in a tilted ratchet driven by Lévy noise

The enhanced transport of particles by roughness in a tilted rough ratchet potential subject to a Lévy noise is investigated in this paper. Due to the roughness, the transport process exhibits quite different properties compared to the smooth case. We find that the roughness on the potential wall functions like a ladder to provide the convenience for particles to climb up but hinder them to slide down. The mean first passage time from one well to its right adjacent well and the mean velocity are, respectively, calculated versus the roughness, the external force, and the Lévy stability index. Our results show that the roughness is able to induce an enhancement on the mean velocity of particles and accelerate the barrier crossing process. The general conditions require a small external force and a small Lévy stability index. We find that with increasing external forces, the enhancement areas of roughness and Lévy stability index both shrink. However, for the Lévy stability index within the enhancement area, its increase will enlarge the enhancement area of roughness. On the contrary, under the same conditions we observe that for a Gaussian noise the roughness always reduces the corresponding mean velocity which is very different from the case of Lévy noise.

Transports in a rough ratchet induced by Lévy noises

We study the transport of a particle subjected to a Lévy noise in a rough ratchet potential which is constructed by superimposing a fast oscillating trigonometric function on a common ratchet background. Due to the superposition of roughness, the transport process exhibits significantly different properties under the excitation of Lévy noises compared to smooth cases. The influence of the roughness on the directional motion is explored by calculating the mean velocities with respect to the Lévy stable index α and the spatial asymmetry parameter q of the ratchet. Variations in the splitting probability have been analyzed to illustrate how roughness affects the transport. In addition, we have examined the influences of roughness on the mean first passage time to know when it accelerates or slows down the first passage process. We find that the roughness can lead to a fast reduction of the absolute value of the mean velocity for small α, however the influence is small for large α. We have illustrated that the ladder-like roughness on the potential wall increases the possibility for particles to cross the gentle side of the ratchet, which results in an increase of the splitting probability to right for the right-skewed ratchet potential. Although the roughness increases the corresponding probability, it does not accelerate the mean first passage process to the right adjacent well. Our results show that the influences of roughness on the mean first passage time are sensitive to the combination of q and α. Hence, the proper q and α can speed up the passage process, otherwise it will slow down it.

The thermodynamics of protein aggregation reactions may underpin the enhanced metabolic efficiency associated with heterosis, some balancing selection, and the evolution of ploidy levels

Identifying the physical basis of heterosis (or "hybrid vigor") has remained elusive despite over a hundred years of research on the subject. The three main theories of heterosis are dominance theory, overdominance theory, and epistasis theory. Kacser and Burns (1981) identified the molecular basis of dominance, which has greatly enhanced our understanding of its importance to heterosis. This paper aims to explain how overdominance, and some features of epistasis, can similarly emerge from the molecular dynamics of proteins. Possessing multiple alleles at a gene locus results in the synthesis of different allozymes at reduced concentrations. This in turn reduces the rate at which each allozyme forms soluble oligomers, which are toxic and must be degraded, because allozymes co-aggregate at low efficiencies. The model developed in this paper can explain how heterozygosity impacts the metabolic efficiency of an organism. It can also explain why the viabilities of some inbred lines seem to decline rapidly at high inbreeding coefficients (F > 0.5), which may provide a physical basis for truncation selection for heterozygosity. Finally, the model has implications for the ploidy level of organisms. It can explain why polyploids are frequently found in environments where severe physical stresses promote the formation of soluble oligomers. The model can also explain why complex organisms, which need to synthesize aggregation-prone proteins that contain intrinsically unstructured regions (IURs) and multiple domains because they facilitate complex protein interaction networks (PINs), tend to be diploid while haploidy tends to be restricted to relatively simple organisms.

Lévy-noise-induced transport in a rough triple-well potential

Rough energy landscape and noisy environment are two common features in many subjects, such as protein folding. Due to the wide findings of bursting or spiking phenomenon in biology science, small diffusions mixing large jumps are adopted to model the noisy environment that can be properly described by Lévy noise. We combine the Lévy noise with the rough energy landscape, modeled by a potential function superimposed by a fast oscillating function, and study the transport of a particle in a rough triple-well potential excited by Lévy noise, rather than only small perturbations. The probabilities of a particle staying in the middle well are considered under different amplitudes of roughness to find out how roughness affects the steady-state probability density function. Variations in the mean first passage time from the middle well to the right well have been investigated with respect to Lévy parameters and amplitudes of the roughness. In addition, we have examined the influences of roughness on the splitting probabilities of the first escape from the middle well. We uncover that the roughness can enhance significantly the first escape of a particle from the middle well, especially for different skewness parameters, but weak differences are found for stability index and noise intensity on the probabilities a particle staying in the middle well and splitting probability to the right.

Structural Mechanisms of Voltage Sensing in G Protein-Coupled Receptors

G-protein-coupled receptors (GPCRs) form the largest superfamily of membrane proteins and one-third of all drug targets in humans. A number of recent studies have reported evidence for substantial voltage regulation of GPCRs. However, the structural basis of GPCR voltage sensing has remained enigmatic. Here, we present atomistic simulations on the δ-opioid and M2 muscarinic receptors, which suggest a structural and mechanistic explanation for the observed voltage-induced functional effects. The simulations reveal that the position of an internal Na(+) ion, recently detected to bind to a highly conserved aqueous pocket in receptor crystal structures, strongly responds to voltage changes. The movements give rise to gating charges in excellent agreement with previous experimental recordings. Furthermore, free energy calculations show that these rearrangements of Na(+) can be induced by physiological membrane voltages. Due to its role in receptor function and signal bias, the repositioning of Na(+) has important general implications for signal transduction in GPCRs.

Transition path time distribution and the transition path free energy barrier

Free energy profile, showing why the transition path barrier is lower than the free energy of activation.

Energy landscape investigation by wavelet transform analysis of atomic force spectroscopy data in a biorecognition experiment

Force fluctuations recorded in an atomic force spectroscopy experiment, during the approach of a tip functionalized with biotin towards a substrate charged with avidin, have been analyzed by a wavelet transform. The observation of strong transient changes only when a specific biorecognition process between the partners takes place suggests a drastic modulation of the force fluctuations when biomolecules recognize each other. Such an analysis allows to investigate the peculiar features of a biorecognition process. These results are discussed in connection with the possible role of energy minima explored by biomolecules during the biorecognition process.

Colloidal transport and diffusion over a tilted periodic potential: dynamics of individual particles

A tilted two-layer colloidal system is constructed for the study of force-assisted barrier-crossing dynamics over a periodic potential. The periodic potential is provided by the bottom layer colloidal spheres forming a fixed crystalline pattern on a glass substrate. The corrugated surface of the bottom colloidal crystal provides a gravitational potential field for the top layer diffusing particles. By tilting the sample at an angle θ with respect to the vertical (gravity) direction, a tangential component of the gravitational force F is applied to the diffusing particles. The measured mean drift velocity v(F, Eb) and diffusion coefficient D(F, Eb) of the particles as a function of F and energy barrier height Eb agree well with the exact results of the one-dimensional drift velocity (R. L. Stratonovich, Radiotekh. Elektron, 1958, 3, 497) and diffusion coefficient (P. Reimann, et al., Phys. Rev. Lett., 2001, 87, 010602 and P. Reimann, et al., Phys. Rev. E, 2002, 65, 031104). Based on these exact results, we show analytically and verify experimentally that there exists a scaling region, in which v(F, Eb) and D(F, Eb) both scale as ν'(F)exp[-E(F)/kBT], where the Arrhenius pre-factor ν'(F) and effective barrier height E(F) are both modified by F. The experiment demonstrates the applications of this model system in evaluating different scaling forms of ν'(F) and E(F) and their accuracy, in order to extract useful information about the external potential, such as the intrinsic barrier height Eb.

The roughness of the protein energy landscape results in anomalous diffusion of the polypeptide backbone

Recombination of photolysed protein disulfide bonds confirms subdiffusional backbone motion and measures the roughness of the protein's energy landscape.

Reconstructing folding energy landscapes by single-molecule force spectroscopy

Folding may be described conceptually in terms of trajectories over a landscape of free energies corresponding to different molecular configurations. In practice, energy landscapes can be difficult to measure. Single-molecule force spectroscopy (SMFS), whereby structural changes are monitored in molecules subjected to controlled forces, has emerged as a powerful tool for probing energy landscapes. We summarize methods for reconstructing landscapes from force spectroscopy measurements under both equilibrium and nonequilibrium conditions. Other complementary, but technically less demanding, methods provide a model-dependent characterization of key features of the landscape. Once reconstructed, energy landscapes can be used to study critical folding parameters, such as the characteristic transition times required for structural changes and the effective diffusion coefficient setting the timescale for motions over the landscape. We also discuss issues that complicate measurement and interpretation, including the possibility of multiple states or pathways and the effects of projecting multiple dimensions onto a single coordinate.

Evidence of disorder in biological molecules from single molecule pulling experiments

Heterogeneity in biological molecules, resulting in molecule-to-molecule variations in their dynamics and function, is an emerging theme. To elucidate the consequences of heterogeneous behavior at the single molecule level, we propose an exactly solvable model in which the unfolding rate due to mechanical force depends parametrically on an auxiliary variable representing an entropy barrier arising from fluctuations in internal dynamics. When the rate of fluctuations--a measure of dynamical disorder--is comparable to or smaller than the rate of force-induced unbinding, we show that there are two experimentally observable consequences: nonexponential survival probability at constant force, and a heavy-tailed rupture force distribution at constant loading rate. By fitting our analytical expressions to data from single molecule pulling experiments on proteins and DNA, we quantify the extent of disorder. We show that only by analyzing data over a wide range of forces and loading rates can the role of disorder due to internal dynamics be quantitatively assessed.

Ensemble-based characterization of unbound and bound states on protein energy landscape

Physicochemical description of numerous cell processes is fundamentally based on the energy landscapes of protein molecules involved. Although the whole energy landscape is difficult to reconstruct, increased attention to particular targets has provided enough structures for mapping functionally important subspaces associated with the unbound and bound protein structures. The subspace mapping produces a discrete representation of the landscape, further called energy spectrum. We compiled and characterized ensembles of bound and unbound conformations of six small proteins and explored their spectra in implicit solvent. First, the analysis of the unbound-to-bound changes points to conformational selection as the binding mechanism for four proteins. Second, results show that bound and unbound spectra often significantly overlap. Moreover, the larger the overlap the smaller the root mean square deviation (RMSD) between the bound and unbound conformational ensembles. Third, the center of the unbound spectrum has a higher energy than the center of the corresponding bound spectrum of the dimeric and multimeric states for most of the proteins. This suggests that the unbound states often have larger entropy than the bound states. Fourth, the exhaustively long minimization, making small intrarotamer adjustments (all-atom RMSD ≤ 0.7 Å), dramatically reduces the distance between the centers of the bound and unbound spectra as well as the spectra extent. It condenses unbound and bound energy levels into a thin layer at the bottom of the energy landscape with the energy spacing that varies between 0.8-4.6 and 3.5-10.5 kcal/mol for the unbound and bound states correspondingly. Finally, the analysis of protein energy fluctuations showed that protein vibrations itself can excite the interstate transitions, including the unbound-to-bound ones.

Atomic force microscopy and spectroscopy to probe single membrane proteins in lipid bilayers

The atomic force microscope (AFM) has opened vast avenues hitherto inaccessible to the biological scientist. The high temporal (millisecond) and spatial (nanometer) resolutions of the AFM are suited for studying many biological processes in their native conditions. The AFM cantilever stylus is aptly termed as a "lab on a tip" owing to its versatility as an imaging tool as well as a handle to manipulate single bonds and proteins. Recent examples assert that the AFM can be used to study the mechanical properties and monitor processes of single proteins and single cells, thus affording insight into important mechanistic details. This chapter specifically focuses on practical and analytical protocols of single-molecule AFM methodologies related to high-resolution imaging and single-molecule force spectroscopy of membrane proteins. Both these techniques are operator oriented, and require specialized working knowledge of the instrument, theoretical, and practical skills.

Unwinding and rewinding the nucleosome inner turn: force dependence of the kinetic rate constants

A simple model for the force-dependent unwinding and rewinding rates of the nucleosome inner turn is constructed and quantitatively compared to the results of recent measurements [A. H. Mack et al., J. Mol. Biol. 423, 687 (2012)]. First, a coarse-grained model for the histone-DNA free-energy landscape that incorporates both an elastic free-energy barrier and specific histone-DNA bonds is developed. Next, a theoretical expression for the rate of transitions across a piecewise linear free-energy landscape with multiple minima and maxima is presented. Then, the model free-energy landscape, approximated as a piecewise linear function, and the theoretical expression for the transition rates are combined to construct a model for the force-dependent unwinding and rewinding rates of the nucleosome inner turn. Least-mean-squares fitting of the model rates to the rates observed in recent experiments rates demonstrates that this model is able to well describe the force-dependent unwinding and rewinding rates of the nucleosome inner turn, observed in the recent experiments, except at the highest forces studied, where an additional ad hoc term is required to describe the data, which may be interpreted as an indication of an alternate high-force nucleosome disassembly pathway, that bypasses simple unwinding. The good agreement between the measurements and the model at lower forces demonstrates that both specific histone-DNA contacts and an elastic free-energy barrier play essential roles for nucleosome winding and unwinding, and quantifies their relative contributions.

Effects of pH on proteins: predictions for ensemble and single-molecule pulling experiments

Protein conformations change among distinct thermodynamic states as solution conditions (temperature, denaturants, pH) are altered or when they are subjected to mechanical forces. A quantitative description of the changes in the relative stabilities of the various thermodynamic states is needed to interpret and predict experimental outcomes. We provide a framework based on the Molecular Transfer Model (MTM) to account for pH effects on the properties of globular proteins. The MTM utilizes the partition function of a protein calculated from molecular simulations at one set of solution conditions to predict protein properties at another set of solution conditions. To take pH effects into account, we utilized experimentally measured pK(a) values in the native and unfolded states to calculate the free energy of transferring a protein from a reference pH to the pH of interest. We validate our approach by demonstrating that the native-state stability as a function of pH is accurately predicted for chymotrypsin inhibitor 2 (CI2) and protein G. We use the MTM to predict the response of CI2 and protein G subjected to a constant force (f) and varying pH. The phase diagrams of CI2 and protein G as a function of f and pH are dramatically different and reflect the underlying pH-dependent stability changes in the absence of force. The calculated equilibrium free energy profiles as functions of the end-to-end distance of the two proteins show that, at various pH values, CI2 unfolds via an intermediate when subjected to f. The locations of the two transition states move toward the more unstable state as f is changed, which is in accord with the Hammond-Leffler postulate. In sharp contrast, force-induced unfolding of protein G occurs in a single step. Remarkably, the location of the transition state with respect to the folded state is independent of f, which suggests that protein G is mechanically brittle. The MTM provides a natural framework for predicting the outcomes of ensemble and single-molecule experiments for a wide range of solution conditions.

Temperature dependence of internal friction in enzyme reactions

Our aim was to elucidate the physical background of internal friction of enzyme reactions by investigating the temperature dependence of internal viscosity. By rapid transient kinetic methods, we directly measured the rate constant of trypsin 4 activation, which is an interdomain conformational rearrangement, as a function of temperature and solvent viscosity. We found that the apparent internal viscosity shows an Arrhenius-like temperature dependence, which can be characterized by the activation energy of internal friction. Glycine and alanine mutations were introduced at a single position of the hinge of the interdomain region to evaluate how the flexibility of the hinge affects internal friction. We found that the apparent activation energies of the conformational change and the internal friction are interconvertible parameters depending on the protein flexibility. The more flexible a protein was, the greater proportion of the total activation energy of the reaction was observed as the apparent activation energy of internal friction. Based on the coupling of the internal and external movements of the protein during its conformational change, we constructed a model that quantitatively relates activation energy, internal friction, and protein flexibility.

Compaction and tensile forces determine the accuracy of folding landscape parameters from single molecule pulling experiments

We establish a framework for assessing whether the transition state location of a biopolymer, which can be inferred from single molecule pulling experiments, corresponds to the ensemble of structures that have equal probability of reaching either the folded or unfolded states (P(fold)=0.5). Using results for the forced unfolding of a RNA hairpin, an exactly soluble model, and an analytic theory, we show that P(fold) is solely determined by s, an experimentally measurable molecular tensegrity parameter, which is a ratio of the tensile force and a compaction force that stabilizes the folded state. Applications to folding landscapes of DNA hairpins and a leucine zipper with two barriers provide a structural interpretation of single molecule experimental data. Our theory can be used to assess whether molecular extension is a good reaction coordinate using measured free energy profiles.

Optimization of Monte Carlo trial moves for protein simulations

Closed rigid-body rotations of residue segments under bond-angle restraints are simple and effective Monte Carlo moves for searching the conformational space of proteins. The efficiency of these moves is examined here as a function of the number of moving residues and the magnitude of their displacement. It is found that the efficiency of folding and equilibrium simulations can be significantly improved by tailoring the distribution of the number of moving residues to the simulation temperature. In general, simulations exploring compact conformations are more efficient when the average number of moving residues is smaller. It is also demonstrated that the moves do not require additional restrictions on the magnitude of the rotation displacements and perform much better than other rotation moves that do not restrict the bond angles a priori. As an example, these results are applied to the replica exchange method. By assigning distributions that generate a smaller number of moving residues to lower temperature replicas, the simulation times are decreased as long as the higher temperature replicas are effective.

Theoretical perspectives on protein folding

Understanding how monomeric proteins fold under in vitro conditions is crucial to describing their functions in the cellular context. Significant advances in theory and experiments have resulted in a conceptual framework for describing the folding mechanisms of globular proteins. The sizes of proteins in the denatured and folded states, cooperativity of the folding transition, dispersions in the melting temperatures at the residue level, and timescales of folding are, to a large extent, determined by N, the number of residues. The intricate details of folding as a function of denaturant concentration can be predicted by using a novel coarse-grained molecular transfer model. By watching one molecule fold at a time, using single-molecule methods, investigators have established the validity of the theoretically anticipated heterogeneity in the folding routes and the N-dependent timescales for the three stages in the approach to the native state. Despite the successes of theory, of which only a few examples are documented here, we conclude that much remains to be done to solve the protein folding problem in the broadest sense.

The implications of gene heterozygosity for protein folding and protein turnover

The offspring of closely related parents often suffer from inbreeding depression, sometimes resulting in a slower growth rate for inbred offspring relative to non-inbred offspring. Previous research has shown that some of the slower growth rate of inbred organisms can be attributed to the inbred organisms' increased levels of protein turnover. This paper attempts to show that the higher levels of protein turnover among inbred organisms can be attributed to accumulations of misfolded and aggregated proteins that require degradation by the inbred organisms' protein quality control systems. The accumulation of misfolded and aggregated proteins within inbred organisms are the result of more negative free energies of folding for proteins encoded at homozygous gene loci and higher concentrations of potentially aggregating non-native protein species within the cell. The theory presented here makes several quantitative predictions that suggest a connection between protein misfolding/aggregation and polyploidy that can be tested by future research.

The effect of different force applications on the protein-protein complex Barnase-Barstar

Steered molecular dynamics simulations are a tool to examine the energy landscape of protein-protein complexes by applying external forces. Here, we analyze the influence of the velocity and geometry of the probing forces on a protein complex using this tool. With steered molecular dynamics, we probe the stability of the protein-protein complex Barnase-Barstar. The individual proteins are mechanically labile. The Barnase-Barstar binding site is more stable than the folds of the individual proteins. By using different force protocols, we observe a variety of responses of the system to the applied tension.

Periodic forces trigger a complex mechanical response in ubiquitin

Mechanical forces govern physiological processes in all living organisms. Many cellular forces, for example, those generated in cyclic conformational changes of biological machines, have repetitive components. In apparent contrast, little is known about how dynamic protein structures respond to periodic mechanical information. Ubiquitin is a small protein found in all eukaryotes. We developed molecular dynamics simulations to unfold single and multimeric ubiquitins with periodic forces. By using a coarse-grained representation, we were able to model forces with periods about 2 orders of magnitude longer than the protein's relaxation time. We found that even a moderate periodic force weakened the protein and shifted its unfolding pathways in a frequency- and amplitude-dependent manner. A complex dynamic response with secondary structure refolding and an increasing importance of local interactions was revealed. Importantly, repetitive forces with broadly distributed frequencies elicited very similar molecular responses compared to fixed-frequency forces. When testing the influence of pulling geometry on ubiquitin's mechanical stability, it was found that the linkage involved in the mechanical degradation of cellular proteins renders the protein remarkably insensitive to periodic forces. We also devised a complementary kinetic energy landscape model that traces these observations and explains periodic-force, single-molecule measurements. In turn, this analytical model is capable of predicting dynamic protein responses. These results provide new insights into ubiquitin mechanics and a potential mechanical role during protein degradation, as well as first frameworks for dynamic protein stability and the modeling of repetitive mechanical processes.

Mutations in gp120 contribute to the resistance of human immunodeficiency virus type 1 to membrane-anchored C-peptide maC46

Binding of the human immunodeficiency virus (HIV) envelope glycoprotein (Env) to the cellular CD4 receptor and a chemokine coreceptor initiates a series of conformational changes in the Env subunits gp120 and gp41. Eventually, the trimeric gp41 folds into a six-helix bundle, thereby inducing fusion of the viral and cellular membranes. C peptides derived from the C-terminal heptad repeat (CHR) of gp41 are efficient entry inhibitors as they block the six-helix bundle formation. Previously, we developed a membrane-anchored C peptide (maC46) expressed from a retroviral vector that also shows high activity against virus strains resistant to enfuvirtide (T-20), an antiviral C peptide approved for clinical use. Here, we present a systematic analysis of mutations in Env that confer resistance of HIV type 1 (HIV-1) to maC46. We selected an HIV-1 BaL strain with 10-fold reduced sensitivity to maC46 (BaL_C46) by passaging virus for nearly 200 days in the presence of gradually increasing concentrations of maC46. In comparison to wild-type BaL, BaL_C46 had five mutations at highly conserved positions in Env, three in gp120, one in the N-terminal heptad-repeat (NHR), and one in the CHR of gp41. No mutations were found in the NHR domain around the GIV motif that are known to cause resistance to enfuvirtide. Instead, maC46 resistance was found to depend on complementary mutations in the NHR and CHR that considerably favor binding of the mutated NHR to the mutated CHR over binding to maC46. In addition, resistance was highly dependent on mutations in gp120 that accelerated entry. Taken together, resistance to maC46 did not develop readily and required multiple cooperating mutations at conserved positions of the viral envelope glycoproteins gp120 and gp41.

The ruggedness of protein-protein energy landscape and the cutoff for 1/r(n) potentials

MOTIVATION

Computational studies of the energetics of protein association are important for revealing the underlying fundamental principles and for designing better tools to model protein complexes. The interaction cutoff contribution to the ruggedness of protein-protein energy landscape is studied in terms of relative energy fluctuations for 1/r(n) potentials based on a simplistic model of a protein complex. This artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase.

RESULTS

The critical values of the cutoff were calculated for each of 11 popular power-type potentials with n=0/9, 12 and for two thresholds of 5% and 10%. The artificial ruggedness decreases to tolerable thresholds for cutoffs larger than the critical ones. The results showed that for both thresholds the critical cutoff is a non-monotonic function of the potential power n. The functions reach the maximum at n=3/4 and then decrease with the increase of the potential power. The difference between two cutoffs for 5% and 10% artificial ruggedness becomes negligible for potentials decreasing faster than 1/r(12). The analytical results obtained for the simple model of protein complexes agree with the analysis of artificial ruggedness in a dataset of 62 protein-protein complexes, with different parameterizations of soft Lennard-Jones potential and two types of protein representations: all-atom and coarse-grained. The results suggest that cutoffs larger than the critical ones can be recommended for protein-protein potentials.

Characterizing antibody specificity to different protein morphologies by AFM

Protein misfolding and aggregation can lead to several neurodegenerative diseases including Alzheimer's Disease (AD), Parkinson's Disease (PD) and Huntington's Disease (HD). While the respective proteins involved in each disease differ in their pathological effects and amino acid sequences, the aggregated forms all share a common cross beta-sheet conformation. Substantial controversy exists over the roles of the different aggregate morphologies in disease onset and progression, and analytical tools such as morphology specific antibodies are needed to distinguish between the different protein morphologies in situ. Here we utilize atomic force microscopy (AFM) to characterize the binding of three single chain antibody fragments (scFvs) to different morphologies of alpha-synuclein (alphaS). From the topographic images generated using the AFM, we were able to show that one scFv bound all morphologies of alphaS, a second bound only oligomeric alphaS, and a third bound only fibrillar alphaS by comparing the height distribution of the different alphaS morphologies with and without addition of the different scFvs. These results demonstrate the versatility of the AFM-based technique as an easy tool to characterize specific antigen-antibody binding and the potential applications of scFvs as promising immunodiagnostics for protein misfolding diseases.

Internal friction of single polypeptide chains at high stretch

Experiments that measure the viscoelasticity of single molecules from the Brownian fluctuations of an atomic force microscope (AFM) have provided a new window onto their internal dynamics in an underlying conformational landscape. Here we develop and apply these methods to examine the internal friction of unfolded polypeptide chains at high stretch. The results reveal a power law dependence of internal friction with tension (exponent 1.3 +/- 0.5) and a relaxation time approximately independent of force. To explain these results we develop a frictional worm-like chain (FWLC) model based on the Rayleigh dissipation function of a stiff chain with dynamical resistance to local bending. We analyse the dissipation rate integrated over the chain length by its Fourier components to calculate an effective tension-dependent friction constant for the end-to-end vector of the chain. The result is an internal friction that increases as a power law with tension with an exponent 3/2, consistent with experiment. Extracting the intrinsic bending friction constant of the chain it is found to be approximately 7 orders of magnitude greater than expected from solvent friction alone; a possible explanation we offer is that the underlying energy landscape for bending amino acids and/or peptide bond is rough, consistent with recent results on both proteins and polysaccharides.

Protein energy landscape roughness

The ‘new view’ of proteins sees protein reactions as parallel processes occurring along funnelled energy landscapes. These landscapes are generally not smooth, but are superimposed by hills and valleys of different heights and widths leading to roughness on the energy surface. In the present paper, we describe the origins of protein energy landscape roughness, measurements of its scale and its implications.

The effect of temperature on mechanical resistance of the native and intermediate states of I27

We investigated the effect of temperature on the mechanical unfolding of I27 from human cardiac titin, employing a custom-built temperature control device for single-molecule atomic force microscopy measurement. A sawtooth pattern was observed in the force curves where each force peak reports on the unfolding of an I27 domain. In early unfolding events, we observed a "hump-like" deviation due to the detachment of beta-strand A from each I27 domain. The force at which the humps appear was approximately 130 pN and showed no temperature dependence, at least in the temperature range of 2 degrees C-30 degrees C. The hump structure was successfully analyzed with a two-state worm-like chain model, and the Gibbs free energy difference of the detachment reaction was estimated to be 11.6 +/- 0.58 kcal/mol and found to be temperature independent. By contrast, upon lowering the temperature, the mean unfolding force from the partly unfolded intermediate state was found to markedly increase and the unfolding force distribution to broaden significantly, suggesting that the distance (x(u)) between the folded and transition states in the energy landscape along the pulling direction is decreased. These results suggest that the local structure of beta-strand A are stabilized by topologically simple local hydrogen-bond network and that the temperature does not affect the detachment reaction thermodynamically and kinetically, whereas the interaction between the beta-strands A' and G, which is a critical region for its mechanical stability, is strongly dependent on the temperature.

Observations on rate theory for rugged energy landscapes

The potential energy profile for many complex reactions of proteins, such as folding or allosteric conformational change, involves many different scales of molecular motion along the reaction coordinate. Although it is natural to model the dynamics of motion along such rugged energy landscapes as diffusional (the Smoluchowski equation; SE), problems arise because the frictional forces generated by the molecular surround are typically not strong enough to justify the use of the SE. Here, we discuss the fundamental theory behind the SE and note that it may be justified through a master equation when reduced to its continuum limit. However, the SE cannot be used for rough energy landscapes, where the continuum limit is ill defined. Instead, we suggest that one should use a mean first passage time expression derived from a master equation, and show how this approach can be used to glean information about the underlying dynamics of barrier crossing. We note that the potential profile in the SE is that of the microbarriers between conformational substates, and that there is a temperature-dependent, effective friction associated with the long residence time in the microwells that populate the rough landscape. The number of recrossings of the overall barrier is temperature-dependent, governed by the microbarriers and not by the effective friction. We derive an explicit expression for the mean number of recrossings and its temperature dependence. Finally, we note that the mean first passage time can be used as a departure point for measuring the roughness of the landscape.

Force-dependent hopping rates of RNA hairpins can be estimated from accurate measurement of the folding landscapes

The sequence-dependent folding landscapes of nucleic acid hairpins reflect much of the complexity of biomolecular folding. Folding trajectories, generated by using single-molecule force-clamp experiments by attaching semiflexible polymers to the ends of hairpins, have been used to infer their folding landscapes. Using simulations and theory, we study the effect of the dynamics of the attached handles on the handle-free RNA free-energy profile F(o)(eq)(z(m)), where z(m) is the molecular extension of the hairpin. Accurate measurements of F(o)(eq)(z(m)) requires stiff polymers with small L/l(p), where L is the contour length of the handle, and l(p) is the persistence length. Paradoxically, reliable estimates of the hopping rates can only be made by using flexible handles. Nevertheless, we show that the equilibrium free-energy profile F(o)(eq)(z(m)) at an external tension f(m), the force (f) at which the folded and unfolded states are equally populated, in conjunction with Kramers' theory, can provide accurate estimates of the force-dependent hopping rates in the absence of handles at arbitrary values of f. Our theoretical framework shows that z(m) is a good reaction coordinate for nucleic acid hairpins under tension.

From valleys to ridges: exploring the dynamic energy landscape of single membrane proteins

Membrane proteins are involved in essential biological processes such as energy conversion, signal transduction, solute transport and secretion. All biological processes, also those involving membrane proteins, are steered by molecular interactions. Molecular interactions guide the folding and stability of membrane proteins, determine their assembly, switch their functional states or mediate signal transduction. The sequential steps of molecular interactions driving these processes can be described by dynamic energy landscapes. The conceptual energy landscape allows to follow the complex reaction pathways of membrane proteins while its modifications describe why and how pathways are changed. Single-molecule force spectroscopy (SMFS) detects, quantifies and locates interactions within and between membrane proteins. SMFS helps to determine how these interactions change with temperature, point mutations, oligomerization and the functional states of membrane proteins. Applied in different modes, SMFS explores the co-existence and population of reaction pathways in the energy landscape of the protein and thus reveals detailed insights into local mechanisms, determining its structural and functional relationships. Here we review how SMFS extracts the defining parameters of an energy landscape such as the barrier position, reaction kinetics and roughness with high precision.