Protein folding by motion planning

We investigate a novel approach for studying protein folding that has evolved from robotics motion planning techniques called probabilistic roadmap methods (PRMs). Our focus is to study issues related to the folding process, such as the formation of secondary and tertiary structures, assuming we know the native fold. A feature of our PRM-based framework is that the large sets of folding pathways in the roadmaps it produces, in just a few hours on a desktop PC, provide global information about the protein's energy landscape. This is an advantage over other simulation methods such as molecular dynamics or Monte Carlo methods which require more computation and produce only a single trajectory in each run. In our initial studies, we obtained encouraging results for several small proteins. In this paper, we investigate more sophisticated techniques for analyzing the folding pathways in our roadmaps. In addition to more formally revalidating our previous results, we present a case study showing that our technique captures known folding differences between the structurally similar proteins G and L.     

Using motion planning to study protein folding pathways.

We present a framework for studying protein folding pathways and potential landscapes which is based on techniques recently developed in the robotics motion planning community. Our focus in this work is to study the protein folding mechanism assuming we know the native fold. That is, instead of performing fold prediction, we aim to study issues related to the folding process, such as the formation of secondary and tertiary structure, and the dependence of the folding pathway on the initial denatured conformation. Our work uses probabilistic roadmap (PRM) motion planning techniques which have proven successful for problems involving high-dimensional configuration spaces. A strength of these methods is their efficiency in rapidly covering the planning space without becoming trapped in local minima. We have applied our PRM technique to several small proteins (~60 residues) and validated the pathways computed by comparing the secondary structure formation order on our paths to known hydrogen exchange experimental results. An advantage of the PRM framework over other simulation methods is that it enables one to easily and efficiently compute folding pathways from any denatured starting state to the (known) native fold. This aspect makes our approach ideal for studying global properties of the protein's potential landscape, most of which are difficult to simulate and study with other methods. For example, in the proteins we study, the folding pathways starting from different denatured states sometimes share common portions when they are close to the native fold, and moreover, the formation order of the secondary structure appears largely independent of the starting denatured conformation. Another feature of our technique is that the distribution of the sampled conformations is correlated with the formation of secondary structure and, in particular, appears to differentiate situations in which secondary structure clearly forms first and those in which the tertiary structure is obtained more directly. Overall, our results applying PRM techniques are very encouraging and indicate the promise of our approach for studying proteins for which experimental results are not available.     

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.     

Kinetics analysis methods for approximate folding landscapes

MOTIVATION: Protein motions play an essential role in many biochemical processes. Lab studies often quantify these motions in terms of their kinetics such as the speed at which a protein folds or the population of certain interesting states like the native state. Kinetic metrics give quantifiable measurements of the folding process that can be compared across a group of proteins such as a wild-type protein and its mutants. RESULTS: We present two new techniques, map-based master equation solution and map-based Monte Carlo simulation, to study protein kinetics through folding rates and population kinetics from approximate folding landscapes, models called maps. From these two new techniques, interesting metrics that describe the folding process, such as reaction coordinates, can also be studied. In this article we focus on two metrics, formation of helices and structure formation around tryptophan residues. These two metrics are often studied in the lab through circular dichroism (CD) spectra analysis and tryptophan fluorescence experiments, respectively. The approximated landscape models we use here are the maps of protein conformations and their associated transitions that we have presented and validated previously. In contrast to other methods such as the traditional master equation and Monte Carlo simulation, our techniques are both fast and can easily be computed for full-length detailed protein models. We validate our map-based kinetics techniques by comparing folding rates to known experimental results. We also look in depth at the population kinetics, helix formation and structure near tryptophan residues for a variety of proteins. AVAILABILITY: We invite the community to help us enrich our publicly available database of motions and kinetics analysis by submitting to our server: http://parasol.tamu.edu/foldingserver/.     

Repeat-protein folding: new insights into origins of cooperativity, stability, and topology

Although our understanding of globular protein folding continues to advance, the irregular tertiary structures and high cooperativity of globular proteins complicates energetic dissection. Recently, proteins with regular, repetitive tertiary structures have been identified that sidestep limitations imposed by globular protein architecture. Here we review recent studies of repeat-protein folding. These studies uniquely advance our understanding of both the energetics and kinetics of protein folding. Equilibrium studies provide detailed maps of local stabilities, access to energy landscapes, insights into cooperativity, determination of nearest-neighbor interaction parameters using statistical thermodynamics, relationships between consensus sequences and repeat-protein stability. Kinetic studies provide insight into the influence of short-range topology on folding rates, the degree to which folding proceeds by parallel (versus localized) pathways, and the factors that select among multiple potential pathways. The recent application of force spectroscopy to repeat-protein unfolding is providing a unique route to test and extend many of these findings.     

The building block folding model and the kinetics of protein folding.

Here we show that qualitatively, the building blocks folding model accounts for three-state versus the two-state protein folding. Additionally, it is consistent with the faster versus slower folding rates of the two-state proteins. Specifically, we illustrate that the building blocks size, their mode of associations in the native structure, the number of ways they can combinatorially assemble, their population times and the way they are split in the iterative, step-by-step structural dissection which yields the anatomy trees, explain a broad range of folding rates. We further show that proteins with similar general topologies may have different folding pathways, and hence different folding rates. On the other hand, the effect of mutations resembles that of changes in conditions, shifting the population times and hence the energy landscapes. Hence, together with the secondary structure type and the extent of local versus non-local interactions, a coherent, consistent rationale for folding kinetics can be outlined, in agreement with experimental results. Given the native structure of a protein, these guidelines enable a qualitative prediction of the folding kinetics. We further describe these in the context of the protein folding energy landscape. Quantitatively, in principle, the diffusion-collision model for the building block association can be used. However, the folding rates of the building blocks and traps in their formation and association, need to be considered.     

Simulating protein motions with rigidity analysis.

Protein motions, ranging from molecular flexibility to large-scale conformational change, play an essential role in many biochemical processes. Despite the explosion in our knowledge of structural and functional data, our understanding of protein movement is still very limited. In previous work, we developed and validated a motion planning based method for mapping protein folding pathways from unstructured conformations to the native state. In this paper, we propose a novel method based on rigidity theory to sample conformation space more effectively, and we describe extensions of our framework to automate the process and to map transitions between specified conformations. Our results show that these additions both improve the accuracy of our maps and enable us to study a broader range of motions for larger proteins. For example, we show that rigidity-based sampling results in maps that capture subtle folding differences between protein G and its mutants, NuG1 and NuG2, and we illustrate how our technique can be used to study large-scale conformational changes in calmodulin, a 148 residue signaling protein known to undergo conformational changes when binding to Ca(2+). Finally, we announce our web-based protein folding server which includes a publicly available archive of protein motions: (http://parasol.tamu.edu/foldingserver/).     

Untangling the complexity of membrane protein folding

Delineating the folding steps of helical-bundle membrane proteins has been a challenging task. Many questions remain unanswered, including the conformation and stability of the states populated during folding, the shape of the energy barriers between the states, and the role of lipids as a solvent in mediating the folding. Recently, theoretical frames have matured to a point that permits detailed dissection of the folding steps, and advances in experimental techniques at both single-molecule and ensemble levels enable selective modulation of specific steps for quantitative determination of the folding energy landscapes. We also discuss how lipid molecules would play an active role in shaping the folding energy landscape of membrane proteins, and how folding of multi-domain membrane proteins can be understood based on our current knowledge. We conclude this review by offering an outlook for emerging questions in the study of membrane protein folding.     

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.     

Matching simulation and experiment: a new simplified model for simulating protein folding.

Simulations of simplified protein folding models have provided much insight into solving the protein folding problem. We propose here a new off-lattice bead model, capable of simulating several different fold classes of small proteins. We present the sequence for an alpha/beta protein resembling the IgG-binding proteins L and G. The thermodynamics of the folding process for this model are characterized using the multiple multihistogram method combined with constant-temperature Langevin simulations. The folding is shown to be highly cooperative, with chain collapse nearly accompanying folding. Two parallel folding pathways are shown to exist on the folding free energy landscape. One pathway contains an intermediate--similar to experiments on protein G, and one pathway contains no intermediates-similar to experiments on protein L. The folding kinetics are characterized by tabulating mean-first passage times, and we show that the onset of glasslike kinetics occurs at much lower temperatures than the folding temperature. This model is expected to be useful in many future contexts: investigating questions of the role of local versus nonlocal interactions in various fold classes, addressing the effect of sequence mutations affecting secondary structure propensities, and providing a computationally feasible model for studying the role of solvation forces in protein folding.     

Stepwise formation of alpha-helices during cytochrome c folding

Two models have been proposed to describe the folding pathways of proteins. The framework model assumes the initial formation of the secondary structures whereas the hydrophobic collapse model supposes their formation after the collapse of backbone structures. To differentiate between these models for real proteins, we have developed a novel CD spectrometer that enables us to observe the submillisecond time frame of protein folding and have characterized the timing of secondary structure formation in the folding process of cytochrome c (cyt c). We found that approximately 20% of the native helical content was organized in the first phase of folding, which is completed within milliseconds. Furthermore, we suggest the presence of a second intermediate, which has alpha-helical content resembling that of the molten globule state. Our results indicate that many of the alpha-helices are organized after collapse in the folding mechanism of cyt c.     

Multiple parallel-pathway folding of proline-free Staphylococcal nuclease

When a protein exhibits complex kinetics of refolding, we often ascribe the complexity to slow isomerization events in the denatured protein, such as cis/trans isomerization of peptidyl prolyl bonds. Does the complex folding kinetics arise only from this well-known reason? Here, we have investigated the refolding of a proline-free variant of staphylococcal nuclease by stopped-flow, double-jump techniques, to examine the folding reactions without the slow prolyl isomerizations. As a result, the protein folds into the native state along at least two accessible parallel pathways, starting from a macroscopically single denatured-state ensemble. The presence of intermediates on the individual folding pathways has revealed the existence of multiple parallel pathways, and is characterized by multi-exponential folding kinetics with a lag phase. Therefore, a "single" amino acid sequence can fold along the multiple parallel pathways. This observation in staphylococcal nuclease suggests that the multiple folding may be more general than we have expected, because the multiple parallel-pathway folding cannot be excluded from proteins that show simpler kinetics.     

Probe-dependent and nonexponential relaxation kinetics: unreliable signatures of downhill protein folding

The theoretical suggestion that some proteins may encounter no significant free energy barriers during their folding raises an important question: What experimental signature does this downhill folding produce? Several authors have argued that nonexponential (and especially stretched exponential) or probe‐dependent kinetics represent useful experimental signatures of a downhill free energy surface. Here we examine more closely the connection between unusual kinetics and downhill energy surfaces. Simulation of diffusive relaxation dynamics on a variety of generically downhill, two‐dimensional free energy surfaces shows that these surfaces do not necessarily produce either probe‐dependent or significantly nonexponential kinetics. Conversely, we find that two‐dimensional surfaces with significant (>3k_B T) energy barriers can readily give rise to nonexponential and probe‐dependent kinetics. These results show that downhill folding does not constitute a necessary or sufficient condition for nonexponential and/or probe‐dependent folding kinetics. One cannot easily prove or disprove that a protein folds downhill simply by studying its relaxation kinetics. Proteins 2007. © 2007 Wiley‐Liss, Inc.     

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.     

Protein folding: then and now

Over the past three decades the protein folding field has undergone monumental changes. Originally a purely academic question, how a protein folds has now become vital in understanding diseases and our abilities to rationally manipulate cellular life by engineering protein folding pathways. We review and contrast past and recent developments in the protein folding field. Specifically, we discuss the progress in our understanding of protein folding thermodynamics and kinetics, the properties of evasive intermediates, and unfolded states. We also discuss how some abnormalities in protein folding lead to protein aggregation and human diseases.     

Folding versus aggregation: polypeptide conformations on competing pathways

Protein aggregation has now become recognised as an important and generic aspect of protein energy landscapes. Since the discovery that numerous human diseases are caused by protein aggregation, the biophysical characterisation of misfolded states and their aggregation mechanisms has received increased attention. Utilising experimental techniques and computational approaches established for the analysis of protein folding reactions has ensured rapid advances in the study of pathways leading to amyloid fibrils and amyloid-related aggregates. Here we describe recent experimental and theoretical advances in the elucidation of the conformational properties of dynamic, heterogeneous and/or insoluble protein ensembles populated on complex, multidimensional protein energy landscapes. We discuss current understanding of aggregation mechanisms in this context and describe how the synergy between biochemical, biophysical and cell-biological experiments are beginning to provide detailed insights into the partitioning of non-native species between protein folding and aggregation pathways.     

Protein energy landscape exploration with structure-based models

Exploring the multi-dimensional energy landscape of a large protein in detail is a computational challenge. Such investigations may include analysis of multiple folding pathways, rate constants for important conformational transitions, locating intermediate states populated during folding, estimating energetic and entropic barriers that separate populated basins, and visualising a high-dimensional surface. The complexity of the landscape can be simplified through coarse-grained structure-based models (SBMs). These widely used coarse-grained representations of proteins provide a minimalist approximation to the free energy landscape, which subsumes the folding behaviour of many single-domain proteins. Here we describe the combination of SBMs with discrete path sampling (DPS), and show how this approach can provide details of the landscape and folding pathways. Combining SBMs and DPS provides an efficient framework for sampling the protein free energy landscape and for calculating various kinetic and thermodynamic quantities.     

Kinetic pathways of beta-hairpin (un)folding in explicit solvent

We examine the dynamical (un)folding pathways of the C-terminal beta-hairpin of protein G-B1 at room temperature in explicit solvent, by employing transition path sampling algorithms. The path ensembles contain information on the folding kinetics, including solvent motion. We determine the transition state ensembles for the two main transitions: 1), the hydrophobic collapse; and 2), the backbone hydrogen bond formation. In both cases the transition state ensembles are characterized by a layer (1) or a strip (2) of water molecules in between the two hairpin strands, supporting the hypothesis of the solvent as lubricant in the folding process. The transition state ensembles do not correspond with saddle points in the equilibrium free-energy landscapes. The kinetic pathways are thus not completely determined by the free-energy landscape. This phenomenon can occur if the order parameters obey different timescales. Using the transition interface sampling technique, we calculate the rate constants for (un)folding and find them in reasonable agreement with experiments, thus supporting the validation of using all-atom force fields to study protein folding.     

The influence of intrinsic folding mechanism of an unfolded protein on the coupled folding-binding process during target recognition

Intrinsically disordered proteins (IDPs) are extensively involved in dynamic signaling processes which require a high association rate and a high dissociation rate for rapid binding/unbinding events and at the same time a sufficient high affinity for specific recognition. Although the coupled folding-binding processes of IDPs have been extensively studied, it is still impossible to predict whether an unfolded protein is suitable for molecular signaling via coupled folding-binding. In this work, we studied the interplay between intrinsic folding mechanisms and coupled folding-binding process for unfolded proteins through molecular dynamics simulations. We first studied the folding process of three representative IDPs with different folded structures, that is, c-Myb, AF9, and E3 rRNase. We found the folding free energy landscapes of IDPs are downhill or show low barriers. To further study the influence of intrinsic folding mechanism on the binding process, we modulated the folding mechanism of barnase via circular permutation and simulated the coupled folding-binding process between unfolded barnase permutant and folded barstar. Although folding of barnase was coupled to target binding, the binding kinetics was significantly affected by the intrinsic folding free energy barrier, where reducing the folding free energy barrier enhances binding rate up to two orders of magnitude. This accelerating effect is different from previous results which reflect the effect of structure flexibility on binding kinetics. Our results suggest that coupling the folding of an unfolded protein with no/low folding free energy barrier with its target binding may provide a way to achieve high specificity and rapid binding/unbinding kinetics simultaneously.