The influence of turning angles on the success of non-oriented animal searches

Animal searches cover a full range of possibilities from highly deterministic to apparently completely random behaviors. However, even those stochastic components of animal movement can be adaptive, since not all random distributions lead to similar success in finding targets. Here we address the general problem of optimizing encounter rates in non-deterministic, non-oriented searches, both in homogeneous and patchy target landscapes. Specifically, we investigate how two different features related to turning angle distributions influence encounter success: (i) the shape (relative kurtosis) of the angular distribution and (ii) the correlations between successive relative orientations (directional memory). Such influence is analyzed in correlated random walk models using a proper choice of representative turning angle distributions of the recently proposed Jones and Pewsey class. We consider the cases of distributions with nearly the same shape but considerably distinct correlation lengths, and distributions with same correlation but with contrasting relative kurtosis. In homogeneous landscapes, we find that the correlation length has a large influence in the search efficiency. Moreover, similar search efficiencies can be reached by means of distinctly shaped turning angle distributions, provided that the resulting correlation length is the same. In contrast, in patchy landscapes the particular shape of the distribution also becomes relevant for the search efficiency, specially at high target densities. Excessively sharp distributions generate very inefficient searches in landscapes where local target density fluctuations are large. These results are of evolutionary interest. On the one hand, it is shown that equally successful directional memory can arise from contrasting turning behaviors, therefore increasing the likelihood of robust adaptive stochastic behavior. On the other hand, when target landscape is patchy, adequate tumbling may help to explore better local scale heterogeneities, being some details of the shape of the distribution also potentially adaptive.     

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.     

Exploring the relationship between funneled energy landscapes and two-state protein folding

It should take an astronomical time span for unfolded protein chains to find their native state based on an unguided conformational random search. The experimental observation that folding is fast can be rationalized by assuming that protein energy landscapes are sloped towards the native state minimum, such that rapid folding can proceed from virtually any point in conformational space. Folding transitions often exhibit two-state behavior, involving extensively disordered and highly structured conformers as the only two observable kinetic species. This study employs a simple Brownian dynamics model of "protein particles" moving in a spherically symmetrical potential. As expected, the presence of an overall slope towards the native state minimum is an effective means to speed up folding. However, the two-state nature of the transition is eradicated if a significant energetic bias extends too far into the non-native conformational space. The breakdown of two-state cooperativity under these conditions is caused by a continuous conformational drift of the unfolded proteins. Ideal two-state behavior can only be maintained on surfaces exhibiting large regions that are energetically flat, a result that is supported by other recent data in the literature (Kaya and Chan, Proteins: Struct Funct Genet 2003;52:510-523). Rapid two-state folding requires energy landscapes exhibiting the following features: (i) A large region in conformational space that is energetically flat, thus allowing for a significant degree of random sampling, such that unfolded proteins can retain a random coil structure; (ii) a trapping area that is strongly sloped towards the native state minimum.     

Evolutionary walks through a land plant morphospace

A model for mimicking land plant evolution is here expanded and re-evaluated. The model consists of (1) a morphospace containing on the order of 10⁹ phenotypic variants, (2) 15 different fitness landscapes, each defined on the basis of performing one or more of four tasks (i.e. maximizing light interception, mechanical stability and reproduction, and minimizing total surface area), and (3) an algorithm driving a search through fitness landscapes for more fit variants. The model is used to predict the effects of the number of simultaneously performed tasks ('complexity'), abrupt changes in environmental conditions (mimicked by random replacement of one fitness landscape with another), and developmental barriers (mimicked by barring searches from entering specific subdomains in the morphospace) on number and accessibility of variants occupying fitness maxima. The model predicts that (1) the number and accessibility of fitness peaks will increase (while the difference between the relative fitness of peaks and valleys will decrease) in proportion to functional complexity, (2) abrupt shifts in landscapes will increase rather than decrease phenotypic diversity, and (3) obstructed searches have an equal or higher probability of reaching fitness peaks than unfettered searches. These results follow axiomatically from the way hypothetical variants are spatially ordered in the morphospace, the manner in which relative fitness is defined, and the protocol used to locate fitness peaks. A critique of the model's predictions and desiderata for future research are provided.     

Prediction of protein coarse contact maps

Prediction of topological representations of proteins that are geometrically invariants can contribute towards the solution of fundamental open problems in structural genomics like folding. In this paper we focus on coarse grained protein contact maps, a representation that describes the spatial neighborhood relation between secondary structure elements such as helices, beta sheets, and random coils. Our methodology is based on searching the graph space. The search algorithm is guided by an adaptive evaluation function computed by a specialized noncausal recursive connectionist architecture. The neural network is trained using candidate graphs generated during examples of successful searches. Our results demonstrate the viability of the approach for predicting coarse contact maps.     

A systematic study of the vibrational free energies of polypeptides in folded and random states

Molecular vibrations, especially low frequency motions, may be used as an indication of the rigidity or the flatness of the protein folding energy landscape. We have studied the vibrational properties of native folded as well as random coil structures of more than 60 polypeptides. The picture we obtain allows us to perceive how and why the energy landscape progressively rigidifies while still allowing potential flexibility. Compared with random coil structures, both alpha-helices and beta-hairpins are vibrationally more flexible. The vibrational properties of loop structures are similar to those of the corresponding random coil structures. Inclusion of an alpha-helix tends to rigidify peptides and so-called building blocks of the structure, whereas the addition of a beta-structure has less effect. When small building blocks coalesce to form larger domains, the protein rigidifies. However, some folded native conformations are still found to be vibrationally more flexible than random coil structures, for example, beta(2)-microglobulin and the SH3 domain. Vibrational free energy contributes significantly to the thermodynamics of protein folding and affects the distribution of the conformational substates. We found a weak correlation between the vibrational folding energy and the protein size, consistent with both previous experimental estimates and theoretical partition of the heat capacity change in protein folding.     

Density guided importance sampling: application to a reduced model of protein folding

MOTIVATION: Monte Carlo methods are the most effective means of exploring the energy landscapes of protein folding. The rugged topography of folding energy landscapes causes sampling inefficiencies however, particularly at low, physiological temperatures. RESULTS: A hybrid Monte Carlo method, termed density guided importance sampling (DGIS), is presented that overcomes these sampling inefficiencies. The method is shown to be highly accurate and efficient in determining Boltzmann weighted structural metrics of a discrete off-lattice protein model. In comparison to the Metropolis Monte Carlo method, and the hybrid Monte Carlo methods, jump-walking, smart-walking and replica-exchange, the DGIS method is shown to be more efficient, requiring no parameter optimization. The method guides the simulation towards under-sampled regions of the energy spectrum and recognizes when equilibrium has been reached, avoiding arbitrary and excessively long simulation times. AVAILABILITY: Fortran code available from authors upon request. CONTACT: m.j.parker@leeds.ac.uk.     

Role of unfolded state heterogeneity and en-route ruggedness in protein folding kinetics

In order to improve our understanding of the physical bases of protein folding, there is a compelling need for better connections between experimental and computational approaches. This work addresses the role of unfolded state conformational heterogeneity and en-route intermediates, as an aid for planning and interpreting protein folding experiments. The expected kinetics were modeled for different types of energy landscapes, including multiple parallel folding routes, preferential paths dominated by one primary folding route, and distributed paths with a wide spectrum of microscopic folding rate constants. In the presence of one or more preferential routes, conformational exchange among unfolded state populations slows down the observed rates for native protein formation. We find this to be a general phenomenon, taking place even when unfolded conformations interconvert much faster than the "escape" rate constants to folding. Dramatic kinetic deceleration is expected in the presence of an increasing number of folding-incompetent unfolded conformations. This argues for the existence of parallel folding paths involving several folding-competent unfolded conformations, during the early stages of protein folding. Deviations from single-exponential behavior are observed for unfolded conformations exchanging at comparable rates or more slowly than folding events. Analysis of the effect of en-route (on-path) intermediate formation and landscape ruggedness on folding kinetics leads to the following unexpected conclusions: (1) intermediates, which often retard native state formation, may in some cases accelerate folding, and (2) rugged landscapes, usually associated with stretched exponentials, display single-exponential behavior in the presence of late high-friction paths.     

A critical assessment of the topomer search model of protein folding using a continuum explicit-chain model with extensive conformational sampling

Recently, a series of closely related theoretical constructs termed the "topomer search model" (TSM) has been proposed for the folding mechanism of small, single-domain proteins. A basic assumption of the proposed scenarios is that the rate-limiting step in folding is an essentially unbiased, diffusive search for a conformational state called the native topomer defined by an overall native-like topological pattern. Successes in correlating TSM-predicted folding rates with that of real proteins have been interpreted as experimental support for the model. To better delineate the physics entailed, key TSM concepts are examined here using extensive Langevin dynamics simulations of continuum C(alpha) chain models. The theoretical native topomers of four experimentally well-studied two-state proteins are characterized. Consistent with the TSM perspective, we found that the sizes of the native topomers increase with experimental folding rate. However, a careful determination of the corresponding probabilities that the native topomers are populated during a random search fails to reproduce the previously predicted folding rates. Instead, our results indicate that an unbiased TSM search for the native topomer amounts to a Levinthal-like process that would take an impossibly long average time to complete. Furthermore, intraprotein contacts in all four native topomers considered exhibit no apparent correlation with the experimental phi-values determined from the folding kinetics of these proteins. Thus, the present findings suggest that certain basic, generic yet essential energetic features in protein folding are not accounted for by TSM scenarios to date.     

Polymeric aspects of protein folding: a brief overview

Regardless of the differences in primary amino acid sequences, protein molecules in a number of conformational states behave as polymer homologues, allowing speculations as to the volume interactions being a driving force in formation of equilibrium structures. For instance, both native and molten globules exhibit key features of polymer globules, where the fluctuations of the molecular density are expected to be much less than the molecular density itself. Protein molecules in the compact denatured (pre-molten globule) states possess properties of squeezed coils. In fact, even high concentrations of strong denaturants (e.g., urea and GdmCl) more likely constitute bad solvents for protein chains. Thus, globular proteins are probably never random coils without positional correlations and biological polypeptide chains represent the macromolecular coils below a critical point even under harsh denaturing conditions. Several implications of these findings to protein folding are discussed.     

Folding landscapes of ankyrin repeat proteins: experiments meet theory

Nearly 6% of eukaryotic protein sequences contain ankyrin repeat (AR) domains, which consist of several repeats and often function in binding. AR proteins show highly cooperative folding despite a lack of long-range contacts. Both theory and experiment converge to explain that formation of the interface between elements is more favorable than formation of any individual repeat unit. IkappaBalpha and Notch both undergo partial folding upon binding perhaps influencing the binding free energy. The simple architecture, combined with identification of consensus residues that are important for stability, has enabled systematic perturbation of the energy landscape by single point mutations that affect stability or by addition of consensus repeats. The folding energy landscapes appear highly plastic, with small perturbations re-routing folding pathways.     

Using motion planning to map protein folding landscapes and analyze folding kinetics of known native structures.

We investigate a novel approach for studying the kinetics of protein folding. Our framework has evolved from robotics motion planning techniques called probabilistic roadmap methods (PRMs) that have been applied in many diverse fields with great success. In our previous work, we presented our PRM-based technique and obtained encouraging results studying protein folding pathways for several small proteins. In this paper, we describe how our motion planning framework can be used to study protein folding kinetics. In particular, we present a refined version of our PRM-based framework and describe how it can be used to produce potential energy landscapes, free energy landscapes, and many folding pathways all from a single roadmap which is computed in a few hours on a desktop PC. Results are presented for 14 proteins. Our ability to produce large sets of unrelated folding pathways may potentially provide crucial insight into some aspects of folding kinetics, such as proteins that exhibit both two-state and three-state kinetics that are not captured by other theoretical techniques.     

Reliable protein folding on complex energy landscapes: the free energy reaction path

A theoretical framework is developed to study the dynamics of protein folding. The key insight is that the search for the native protein conformation is influenced by the rate r at which external parameters, such as temperature, chemical denaturant, or pH, are adjusted to induce folding. A theory based on this insight predicts that 1), proteins with complex energy landscapes can fold reliably to their native state; 2), reliable folding can occur as an equilibrium or out-of-equilibrium process; and 3), reliable folding only occurs when the rate r is below a limiting value, which can be calculated from measurements of the free energy. We test these predictions against numerical simulations of model proteins with a single energy scale.     

Relaxation of backbone bond geometry improves protein energy landscape modeling

A key issue in macromolecular structure modeling is the granularity of the molecular representation. A fine‐grained representation can approximate the actual structure more accurately, but may require many more degrees of freedom than a coarse‐grained representation and hence make conformational search more challenging. We investigate this tradeoff between the accuracy and the size of protein conformational search space for two frequently used representations: one with fixed bond angles and lengths and one that has full flexibility. We performed large‐scale explorations of the energy landscapes of 82 protein domains under each model, and find that the introduction of bond angle flexibility significantly increases the average energy gap between native and non‐native structures. We also find that incorporating bonded geometry flexibility improves low resolution X‐ray crystallographic refinement. These results suggest that backbone bond angle relaxation makes an important contribution to native structure energetics, that current energy functions are sufficiently accurate to capture the energetic gain associated with subtle deformations from chain ideality, and more speculatively, that backbone geometry distortions occur late in protein folding to optimize packing in the native state.     

The highly cooperative folding of small naturally occurring proteins is likely the result of natural selection

To illuminate the evolutionary pressure acting on the folding free energy landscapes of naturally occurring proteins, we have systematically characterized the folding free energy landscape of Top7, a computationally designed protein lacking an evolutionary history. Stopped-flow kinetics, circular dichroism, and NMR experiments reveal that there are at least three distinct phases in the folding of Top7, that a nonnative conformation is stable at equilibrium, and that multiple fragments of Top7 are stable in isolation. These results indicate that the folding of Top7 is significantly less cooperative than the folding of similarly sized naturally occurring proteins, suggesting that the cooperative folding and smooth free energy landscapes observed for small naturally occurring proteins are not general properties of polypeptide chains that fold to unique stable structures but are instead a product of natural selection.     

Ab initio protein structure prediction

Steady progress has been made in the field of ab initio protein folding. A variety of methods now allow the prediction of low-resolution structures of small proteins or protein fragments up to approximately 100 amino acid residues in length. Such low-resolution structures may be sufficient for the functional annotation of protein sequences on a genome-wide scale. Although no consistently reliable algorithm is currently available, the essential challenges to developing a general theory or approach to protein structure prediction are better understood. The energy landscapes resulting from the structure prediction algorithms are only partially funneled to the native state of the protein. This review focuses on two areas of recent advances in ab initio structure prediction-improvements in the energy functions and strategies to search the caldera region of the energy landscapes.     

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.     

Cunning simplicity of protein folding landscapes

Funnel-like landscapes are widely used to visualize protein folding. It might seem that any funnel-like energy landscape helps to avoid the 'Levinthal paradox', i.e. to avoid sampling the impossibly large number of conformations for a folding protein. This cunning suggestion, reinforced by beautiful drawings of the energy funnels, stimulated some simple models of protein folding; one of them [D.J. Bicout and A. Szabo (2000) Protein Sci., 9, 452-465] is especially straightforward and instructive. A thorough analysis of this strict funnel model (which does not consider a nucleation of phase separation in the course of folding) shows that it cannot provide a simultaneous explanation for both major features observed for protein folding: (i) folding within non-astronomical time, and (ii) co-existence of the native and the unfolded states during the folding process. On the contrary, the nucleation mechanism of protein folding can account for both these major features simultaneously.     

Microsecond scale replica exchange molecular dynamic simulation of villin headpiece: an insight into the folding landscape

Reaching the experimental time scale of millisecond is a grand challenge for protein folding simulations. The development of advanced Molecular Dynamics techniques like Replica Exchange Molecular Dynamics (REMD) makes it possible to reach these experimental timescales. In this study, an attempt has been made to reach the multi microsecond simulation time scale by carrying out folding simulations on a three helix bundle protein, Villin, by combining REMD and Amber United Atom model. Twenty replicas having different temperatures ranging from 295 K to 390 K were simulated for 1.5 μs each. The lowest Root Mean Square Deviation (RMSD) structure of 2.5 ? was obtained with respect to native structure (PDB code 1VII), with all the helices formed. The folding population landscapes were built using segment-wise RMSD and Principal Components as reaction coordinates. These analyses suggest the two-stage folding for Villin. The combination of REMD and Amber United Atom model may be useful to understand the folding mechanism of various fast folding proteins.     

Protein folding at single-molecule resolution

The protein folding reaction carries great significance for cellular function and hence continues to be the research focus of a large interdisciplinary protein science community. Single-molecule methods are providing new and powerful tools for dissecting the mechanisms of this complex process by virtue of their ability to provide views of protein structure and dynamics without associated ensemble averaging. This review briefly introduces common FRET and force methods, and then explores several areas of protein folding where single-molecule experiments have yielded insights. These include exciting new information about folding landscapes, dynamics, intermediates, unfolded ensembles, intrinsically disordered proteins, assisted folding and biomechanical unfolding. Emerging and future work is expected to include advances in single-molecule techniques aimed at such investigations, and increasing work on more complex systems from both the physics and biology standpoints, including folding and dynamics of systems of interacting proteins and of proteins in cells and organisms. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.