Entropy and barrier-controlled fluctuations determine conformational viscoelasticity of single biomolecules

Biological macromolecules have complex and nontrivial energy landscapes, endowing them with a unique conformational adaptability and diversity in function. Hence, understanding the processes of elasticity and dissipation at the nanoscale is important to molecular biology and emerging fields such as nanotechnology. Here we analyze single molecule fluctuations in an atomic force microscope, using a generic model of biopolymer viscoelasticity that includes local "internal" conformational dissipation. Comparing two biopolymers, dextran and cellulose (polysaccharides with and without local bistable transitions), demonstrates that signatures of simple conformational change are minima in both the elastic and internal friction constants around a characteristic force. A novel analysis of dynamics on a bistable energy landscape provides a simple explanation: an elasticity driven by the entropy, and friction by a barrier-controlled hopping time of populations between states, which is surprisingly distinct to the well-known relaxation time. This nonequilibrium microscopic analysis thus provides a means of quantifying new dynamical features of the energy landscape of the glucopyranose ring, revealing an unexpected underlying roughness and information on the shape of the barrier of the chair-boat transition in dextran. The results presented herein provide a basis toward probing the viscoelasticity of macromolecular conformational transitions on more complex energy landscapes, such as during protein folding.     

Analysis of the conformational energy landscape of human snRNA with a metric based on tree representation of RNA structures

It is an outstanding problem to clarify how the RNA sequence is related to its structure and biological functions. We developed a simplified definition of a metric for tree representation of RNA secondary structures and analyzed the conformational energy landscapes of human spliceosomal snRNAs. We discuss the structural properties of the biological sequence by calculating the conformational energy landscapes based on the structural distance between each of the pairs in the set of suboptimal structures. The new index value is introduced for estimating the shapes of distribution patterns in conformational energy landscapes. We apply our method to the five human snRNAs and show that U1 snRNA has a multi-valley profile of the landscape, whereas the landscapes of the other four snRNAs have one steep valley. This result reflects different biological functions of these snRNAs in the pre-mRNA splicing process. The results of analyzing tRNAs and rRNAs show that the conformational energy landscapes of these sequences have multi-valley profiles.     

The Topology of the Energy Landscapes of Macromolecules in the Torsion Angle Space and the Principle of the Minimum Energy Dissipation Rate in Conformational Relaxation

This article discusses some basic problems of structural biology and molecular dynamics simulation methods that need to be taken into account when considering, the protein folding problem, and prediction of 3D-structures for biopolymers. A multidimensional Fourier series expansions were formulated for the energy landscapes of the systems with conformational mobility, These energy landscape representations are correct from the viewpoint of the topology of the macromolecule configuration spaces. The problem of the single global minimum on the energy landscape for proteins is discussed and is formulated in tems of phase rules for the component of Fourier expansions. This rule is formally similar to the problem of diffraction on a multidimensional cubic lattice. The calibration of biopolymer force fields and their correspondence to topologically correct energy landscapes are discussed. Equations of motion were obtained in a matrix form for the relaxation of a representative point position on a multidimensional potential energy surface. The solutions of the equations for conformational relaxation were shown to obey the principle of the minimum energy dissipation rate at a given relaxation rate of potential energy (or folding rate).     

Impact of enzyme motion on activity

Experimental and theoretical data imply that enzyme motion plays an important role in enzymatic reactions. Enzyme motion can influence both the activation free energy barrier and the degree of barrier recrossing. A hybrid theoretical approach has been developed for the investigation of the relation between enzyme motion and activity. This approach includes both electronic and nuclear quantum effects. It distinguishes between thermally averaged promoting motions that influence the activation free energy barrier and dynamical motions that influence the barrier recrossings. Applications to hydride transfer in liver alcohol dehydrogenase and dihydrofolate reductase resulted in the identification and characterization of important enzyme motions. These applications have also led to the proposal of a network of coupled promoting motions in enzymatic reactions. These concepts have important implications for protein engineering and drug design.     

Using motion planning to study RNA folding kinetics.

We propose a novel, motion planning based approach to approximately map the energy landscape of an RNA molecule. A key feature of our method is that it provides a sparse map that captures the main features of the energy landscape which can be analyzed to compute folding kinetics. Our method is based on probabilistic roadmap motion planners that we have previously successfully applied to protein folding. In this paper, we provide evidence that this approach is also well suited to RNA. We compute population kinetics and transition rates on our roadmaps using the master equation for a few moderately sized RNA and show that our results compare favorably with results of other existing methods.     

Scope and utility of hydrogen exchange as a tool for mapping landscapes

The ability to determine conformational parameters of protein-folding landscapes is critical for understanding the link between conformation, function, and disease. Monitoring hydrogen exchange (HX) of labile protons at equilibrium enables direct extraction of thermodynamic or kinetic landscape parameters in two limiting extremes. Here, we establish a quantitative framework for relating HX behavior to landscape. We use this framework to demonstrate that the range of predicted global HX behavior for the majority of a set of characterized two-state proteins under near-native conditions does not readily span between both extremes. For most, stability may be quantitatively determined under physiological conditions, with semiquantitative boundaries on kinetics additionally determined using modest experimental perturbations to shift HX behavior. The framework and relationships derived in the simple context of two-state global folding highlight the importance of understanding HX across the entire continuum of behavior, in order to apply HX to map landscapes.     

Heterogeneous folding of the trpzip hairpin: full atom simulation and experiment

The beta-hairpin trpzip2 can be tuned continuously from a two-state folder to folding on a rough energy landscape without a dominant refolding barrier. At high denaturant concentration, this extremely stable peptide exhibits a single apparent "two-state" transition temperature when monitored by different spectroscopic techniques. However, under optimal folding conditions the hairpin undergoes an unusual folding process with three clusters of melting transitions ranging from 15 degrees C to 160 degrees C, as monitored by 12 different experimental and computational observables. We explain this behavior in terms of a rough free energy landscape of the unfolded peptide caused by multiple tryptophan interactions and alternative backbone conformations. The landscape is mapped out by potentials of mean force derived from replica-exchange molecular dynamics simulations. Implications for deducing cooperativity from denaturant titrations, for the origin of folding cooperativity, and for the folding of thermophilic proteins are pointed out. trpzip is an excellent small tunable model system for the glass-like folding transitions predicted by landscape theory.     

The energetics of T4 lysozyme reveal a hierarchy of conformations.

We have used native state exchange to examine the energy landscape of the well-characterized protein T4 lysozyme. Although the protein exhibits two-state behavior by traditional probes, the energy landscape determined here is much more complex. The average stability of the C-terminal subdomain is significantly higher than that for the N-terminus suggesting at least two regions of unfolding. At a more detailed level, there appears to be a broad continuum of stabilities throughout each region. The overall subdomain hierarchy of energies does not mirror data on the folding pathway for this protein, challenging the relationship between energy landscapes and folding trajectories.     

Replication and mutation on neutral networks

Folding of RNA sequences into secondary structures is viewed as a map that assigns a uniquely defined base pairing pattern to every sequence. The mapping is non-invertible since many sequences fold into the same minimum free energy (secondary) structure or shape. The pre-images of this map, called neutral networks, are uniquely associated with the shapes and vice versa. Random graph theory is used to construct networks in sequence space which are suitable models for neutral networks. The theory of molecular quasispecies has been applied to replication and mutation on single-peak fitness landscapes. This concept is extended by considering evolution on degenerate multi-peak landscapes which originate from neutral networks by assuming that one particular shape is fitter than all the others. On such a single-shape landscape the superior fitness value is assigned to all sequences belonging to the master shape. All other shapes are lumped together and their fitness values are averaged in a way that is reminiscent of mean field theory. Replication and mutation on neutral networks are modeled by phenomenological rate equations as well as by a stochastic birth-and-death model. In analogy to the error threshold in sequence space the phenotypic error threshold separates two scenarios: (i) a stationary (fittest) master shape surrounded by closely related shapes and (ii) populations drifting through shape space by a diffusion-like process. The error classes of the quasispecies model are replaced by distance classes between the master shape and the other structures. Analytical results are derived for single-shape landscapes, in particular, simple expressions are obtained for the mean fraction of master shapes in a population and for phenotypic error thresholds. The analytical results are complemented by data obtained from computer simulation of the underlying stochastic processes. The predictions of the phenomenological approach on the single-shape landscape are very well reproduced by replication and mutation kinetics of tRNA(phe). Simulation of the stochastic process at a resolution of individual distance classes yields data which are in excellent agreement with the results derived from the birth-and-death model.     

Simple physical models connect theory and experiment in protein folding kinetics

Our understanding of the principles underlying the protein-folding problem can be tested by developing and characterizing simple models that make predictions which can be compared to experimental data. Here we extend our earlier model of folding free energy landscapes, in which each residue is considered to be either folded as in the native state or completely disordered, by investigating the role of additional factors representing hydrogen bonding and backbone torsion strain, and by using a hybrid between the master equation approach and the simple transition state theory to evaluate kinetics near the free energy barrier in greater detail. Model calculations of folding phi-values are compared to experimental data for 19 proteins, and for more than half of these, experimental data are reproduced with correlation coefficients between r=0.41 and 0.88; calculations of transition state free energy barriers correlate with rates measured for 37 single domain proteins (r=0.69). The model provides insight into the contribution of alternative-folding pathways, the validity of quasi-equilibrium treatments of the folding landscape, and the magnitude of the Arrhenius prefactor for protein folding. Finally, we discuss the limitations of simple native-state-based models, and as a more general test of such models, provide predictions of folding rates and mechanisms for a comprehensive set of over 400 small protein domains of known structure.     

From mutations to mechanisms and dysfunction via computation and mining of protein energy landscapes

Background

The protein energy landscape underscores the inherent nature of proteins as dynamic molecules interconverting between structures with varying energies. Reconstructing a protein’s energy landscape holds the key to characterizing a protein’s equilibrium conformational dynamics and its relationship to function. Many pathogenic mutations in protein sequences alter the equilibrium dynamics that regulates molecular interactions and thus protein function. In principle, reconstructing energy landscapes of a protein’s healthy and diseased variants is a central step to understanding how mutations impact dynamics, biological mechanisms, and function.

Results

Recent computational advances are yielding detailed, sample-based representations of protein energy landscapes. In this paper, we propose and describe two novel methods that leverage computed, sample-based representations of landscapes to reconstruct them and extract from them informative local structures that reveal the underlying organization of an energy landscape. Such structures constitute landscape features that, as we demonstrate here, can be utilized to detect alterations of landscapes upon mutation.

Conclusions

The proposed methods detect altered protein energy landscape features in response to sequence mutations. By doing so, the methods allow formulating hypotheses on the impact of mutations on specific biological activities of a protein. This work demonstrates that the availability of energy landscapes of healthy and diseased variants of a protein opens up new avenues to harness the quantitative information embedded in landscapes to summarize mechanisms via which mutations alter protein dynamics to percolate to dysfunction.

     

Conformational transition states of a beta-hairpin peptide between the ordered and disordered conformations in explicit water

The conformational transition states of a beta-hairpin peptide in explicit water were identified from the free energy landscapes obtained from the multicanonical ensemble, using an enhanced conformational sampling calculation. The beta-hairpin conformations were significant at 300 K in the landscape, and the typical nuclear Overhauser effect signals were reproduced, consistent with the previously reported experiment. In contrast, the disordered conformations were predominant at higher temperatures. Among the stable conformations at 300 K, there were several free energy barriers, which were not visible in the landscapes formed with the conventional parameters. We identified the transition states around the saddle points along the putative folding and unfolding paths between the beta-hairpin and the disordered conformations in the landscape. The characteristic features of these transition states are the predominant hydrophobic contacts and the several hydrogen bonds among the side-chains, as well as some of the backbone hydrogen bonds. The unfolding simulations at high temperatures, 400 K and 500 K, and their principal component analyses also provided estimates for the transition state conformations, which agreed well with those at 400 K and 500 K deduced from the current free energy landscapes at 400 K and 500 K, respectively. However, the transition states at high temperatures were much more widely distributed on the landscape than those at 300 K, and their conformations were different.     

Energy budget changes caused by varying solar angles,cloud scenarios,and air temperatures in contrasting landscapes

The combination of two physically-based models of solar radiationcloud attenuation and a steady state surface energy budget made possible the systematic examination of the causal effects of varying solar altitudes, selected cloud types and amounts, and air temperatures on the resultant energy budget components extant in two contrasting (grassy and barren) landscapes. The grassy landscape reacted more conservatively to the forcing by changing solar angles, cloud, and air temperature regimes. The results are considered as universal and they are assumed to encompass many possible extremes encountered in the continuum formed by real world landscapes.     

Predictability of evolutionary trajectories in fitness landscapes

Experimental studies on enzyme evolution show that only a small fraction of all possible mutation trajectories are accessible to evolution. However, these experiments deal with individual enzymes and explore a tiny part of the fitness landscape. We report an exhaustive analysis of fitness landscapes constructed with an off-lattice model of protein folding where fitness is equated with robustness to misfolding. This model mimics the essential features of the interactions between amino acids, is consistent with the key paradigms of protein folding and reproduces the universal distribution of evolutionary rates among orthologous proteins. We introduce mean path divergence as a quantitative measure of the degree to which the starting and ending points determine the path of evolution in fitness landscapes. Global measures of landscape roughness are good predictors of path divergence in all studied landscapes: the mean path divergence is greater in smooth landscapes than in rough ones. The model-derived and experimental landscapes are significantly smoother than random landscapes and resemble additive landscapes perturbed with moderate amounts of noise; thus, these landscapes are substantially robust to mutation. The model landscapes show a deficit of suboptimal peaks even compared with noisy additive landscapes with similar overall roughness. We suggest that smoothness and the substantial deficit of peaks in the fitness landscapes of protein evolution are fundamental consequences of the physics of protein folding.     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

Disparate degrees of hypervariable loop flexibility control T-cell receptor cross-reactivity, specificity, and binding mechanism

αβ T-cell receptors (TCRs) recognize multiple antigenic peptides bound and presented by major histocompatibility complex molecules. TCR cross-reactivity has been attributed in part to the flexibility of TCR complementarity-determining region (CDR) loops, yet there have been limited direct studies of loop dynamics to determine the extent of its role. Here we studied the flexibility of the binding loops of the αβ TCR A6 using crystallographic, spectroscopic, and computational methods. A significant role for flexibility in binding and cross-reactivity was indicated only for the CDR3α and CDR3β hypervariable loops. Examination of the energy landscapes of these two loops indicated that CDR3β possesses a broad, smooth energy landscape, leading to rapid sampling in the free TCR of a range of conformations compatible with different ligands. The landscape for CDR3α is more rugged, resulting in more limited conformational sampling that leads to specificity for a reduced set of peptides as well as the major histocompatibility complex protein. In addition to informing on the mechanisms of cross-reactivity and specificity, the energy landscapes of the two loops indicate a complex mechanism for TCR binding, incorporating elements of both conformational selection and induced fit in a manner that blends features of popular models for TCR recognition.     

Stochastic ensembles, conformationally adaptive teamwork, and enzymatic detoxification

It has been appreciated for a long time that enzymes exist as conformational ensembles throughout multiple stages of the reactions they catalyze, but there is renewed interest in the functional implications. The energy landscape that results from conformationlly diverse poteins is a complex surface with an energetic topography in multiple dimensions, even at the transition state(s) leading to product formation, and this represents a new paradigm. At the same time there has been renewed interest in conformational ensembles, a new paradigm concerning enzyme function has emerged, wherein catalytic promiscuity has clear biological advantages in some cases. "Useful", or biologically functional, promiscuity or the related behavior of "multifunctionality" can be found in the immune system, enzymatic detoxification, signal transduction, and the evolution of new function from an existing pool of folded protein scaffolds. Experimental evidence supports the widely held assumption that conformational heterogeneity promotes functional promiscuity. The common link between these coevolving paradigms is the inherent structural plasticity and conformational dynamics of proteins that, on one hand, lead to complex but evolutionarily selected energy landscapes and, on the other hand, promote functional promiscuity. Here we consider a logical extension of the overlap between these two nascent paradigms: functionally promiscuous and multifunctional enzymes such as detoxification enzymes are expected to have an ensemble landscape with more states accessible on multiple time scales than substrate specific enzymes. Two attributes of detoxification enzymes become important in the context of conformational ensembles: these enzymes metabolize multiple substrates, often in substrate mixtures, and they can form multiple products from a single substrate. These properties, combined with complex conformational landscapes, lead to the possibility of interesting time-dependent, or emergent, properties. Here we demonstrate these properties with kinetic simulations of nonequilibrium steady state (NESS) behavior resulting from energy landscapes expected for detoxification enzymes. Analogous scenarios with other promiscuous enzymes may be worthy of consideration.