The nucleation-collapse mechanism in protein folding: evidence for the non-uniqueness of the folding nucleus

Impact of Hydrodynamic Interactions on the Kinetic Pathway of Protein Folding

Protein folding is a fundamental process critical to cellular function and human health, but it remains a grand challenge in biophysics. Hydrodynamic interaction (HI) plays a vital role in the self-organization of soft and biological materials, yet its role in protein folding is not fully understood despite folding occurring in a fluid environment. Here, we use the fluid particle dynamics method to investigate many-body hydrodynamic couplings between amino acid residues and fluid motion in the folding kinetics of a coarse-grained four-α-helices bundle protein. Our results reveal that HI helps select fast folding pathways to the native state without being kinetically trapped, significantly speeding up the folding kinetics compared to its absence. First, the directional flow along the protein backbone expedites protein collapse. Then, the incompressibility-induced squeezing flow effects retard the accumulation of non-native hydrophobic contacts, thus preventing the protein from being trapped in local energy minima during the conformational search of the native structure. We also find that the significance of HI in folding kinetics depends on temperature, with a pronounced effect under biologically relevant conditions. Our findings suggest that HI, particularly the short-range squeezing effect, may be crucial in avoiding protein misfolding.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4-5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

Determining the atomic charge of calcium ion requires the information of its coordination geometry in an EF-hand motif

It is challenging to parameterize the force field for calcium ions (Ca²⁺) in calcium-binding proteins because of their unique coordination chemistry that involves the surrounding atoms required for stability. In this work, we observed a wide variation in Ca²⁺ binding loop conformations of the Ca²⁺-binding protein calmodulin, which adopts the most populated ternary structures determined from the molecular dynamics simulations, followed by ab initio quantum mechanical (QM) calculations on all 12 amino acids in the loop that coordinate Ca²⁺ in aqueous solution. Ca²⁺ charges were derived by fitting to the electrostatic potential in the context of a classical or polarizable force field (PFF). We discovered that the atomic radius of Ca²⁺ in conventional force fields is too large for the QM calculation to capture the variation in the coordination geometry of Ca²⁺ in its ionic form, leading to unphysical charges. Specifically, we found that the fitted atomic charges of Ca²⁺ in the context of PFF depend on the coordinating geometry of electronegative atoms from the amino acids in the loop. Although nearby water molecules do not influence the atomic charge of Ca²⁺, they are crucial for compensating for the coordination of Ca²⁺ due to the conformational flexibility in the EF-hand loop. Our method advances the development of force fields for metal ions and protein binding sites in dynamic environments.

Prediction of the initial folding sites and the entire folding processes for Ig-like beta-sandwich proteins

Describing the whole story of protein folding is currently the main enigmatic problem in molecular bioinformatics study. Protein folding mechanisms have been intensively investigated with experimental as well as simulation techniques. Since a protein folds into its specific 3D structure from a unique amino acid sequence, it is interesting to extract as much information as possible from the amino acid sequence of a protein. Analyses based on inter-residue average distance statistics and a coarse-grained Gō-model simulation were conducted on Ig and FN3 domains of a titin protein to decode the folding mechanisms from their sequence data and native structure data, respectively. The central region of all domains was predicted to be an initial folding unit, that is, stable in an early state of folding. This common feature coincides well with the experimental results and underscores the significance of the β-sandwich proteins' common structure, namely, the key strands for folding and the Greek-key motif, which is located in the central region. We confirmed that our sequence-based techniques were able to predict the initial folding event just next to the denatured state and that a 3D-based Gō-model simulation can be used to investigate the whole process of protein folding.

Prediction of folding mechanisms for Ig-like beta sandwich proteins based on inter-residue average distance statistics methods

To understand the folding mechanism of a protein is one of the goals in bioinformatics study. Nowadays, it is enigmatic and difficult to extract folding information from amino acid sequence using standard bioinformatics techniques or even experimental protocols which can be time consuming. To overcome these problems, we aim to extract the initial folding unit for titin protein (Ig and fnIII domains) by means of inter-residue average distance statistics, Average Distance Map (ADM) and contact frequency analysis (F-value). TI I27 and TNfn3 domains are used to represent the Ig-domain and fnIII-domain, respectively. Beta-strands 2, 3, 5, and 6 are significant for the initial folding processes of TI I27. The central strands of TNfn3 were predicted as a primary folding segment. Known 3D structure and unknown 3D structure domains were investigated by structure or non-structure based multiple sequence alignment, respectively, to learn the conserved hydrophobic residues and predicted compact region relevant to evolution. Our results show good correspondence to experimental data, phi-value and protection factor from H-D exchange experiments. The significance of conserved hydrophobic residues near F-value peaks for structural stability using hydrophobic packing is confirmed. Our prediction methods once again could extract a folding mechanism only knowing the amino acid sequence.

Molecular dynamics simulations suggest changes in electrostatic interactions as a potential mechanism through which serine phosphorylation inhibits DNA polymerase β activity

DNA polymerase β is a 39 kDa enzyme that is a major component of Base Excision Repair in human cells. The enzyme comprises two major domains, a 31 kDa domain responsible for the polymerase activity and an 8 kDa domain, which bind ssDNA and has a deoxyribose phosphate (dRP) lyase activity. DNA polymerase β was shown to be phosphorylated in vitro with protein kinase C (PKC) at serines 44 and 55 (S44 and S55), resulting in loss of its polymerase enzymic activity, but not its ability to bind ssDNA. In this study, we investigate the potential phosphorylation-induced structural changes for DNA polymerase β using molecular dynamics simulations. The simulations show drastic conformational changes of the polymerase structure as a result of S44 phosphorylation. Phosphorylation-induced conformational changes transform the closed (active) enzyme structure into an open one. Further analysis of the results points to a key hydrogen bond and newly formed salt bridges as potential drivers of these structural fluctuations. The changes observed with S55/44 and S55 phosphorylation were less dramatic and the integrity of the H-bond was not compromised. Thus the phosphorylation of S44 is the major contributor to structural fluctuations that lead to loss of enzymatic activity.

Molecular dynamics simulations suggest changes in electrostatic interactions as a potential mechanism through which serine phosphorylation inhibits DNA Polymerase β's activity

DNA polymerase β is a 39kDa enzyme that is a major component of Base Excision Repair in human cells. The enzyme comprises two major domains, a 31kDa domain responsible for the polymerase activity and an 8kDa domain, which bind ssDNA and has a deoxyribose phosphate (dRP) lyase activity. DNA polymerase β was shown to be phosphorylated in vitro with protein kinase C (PKC) at serines 44 and 55 (S44 and S55), resulting in loss of its polymerase enzymic activity, but not its ability to bind ssDNA. In this study, we investigate the potential phosphorylation-induced structural changes for DNA polymerase β using molecular dynamics. The simulations show drastic conformational changes of the polymerase structure as a result of S44 phosphorylation. Phosphorylation-induced conformational changes transform the closed (active) enzyme structure into an open one. Further analysis of the results points to a key hydrogen bond and newly formed salt bridges as potential drivers of these structural fluctuations. The changes observed with S44/55 and S55 phosphorylation were less dramatic than S44 and the integrity of the H-bond was not compromised. Thus the phosphorylation of S44 is likely the major contributor to structural fluctuations that lead to loss of enzymatic activity.

Opposing Intermolecular Tuning of Ca2+ Affinity for Calmodulin by Neurogranin and CaMKII Peptides

We investigated the impact of bound calmodulin (CaM)-target compound structure on the affinity of calcium (Ca²⁺) by integrating coarse-grained models and all-atomistic simulations with nonequilibrium physics. We focused on binding between CaM and two specific targets, Ca²⁺/CaM-dependent protein kinase II (CaMKII) and neurogranin (Ng), as they both regulate CaM-dependent Ca²⁺ signaling pathways in neurons. It was shown experimentally that Ca²⁺/CaM (holoCaM) binds to the CaMKII peptide with overwhelmingly higher affinity than Ca²⁺-free CaM (apoCaM); the binding of CaMKII peptide to CaM in return increases the Ca²⁺ affinity for CaM. However, this reciprocal relation was not observed in the Ng peptide (Ng_13–49), which binds to apoCaM or holoCaM with binding affinities of the same order of magnitude. Unlike the holoCaM-CaMKII peptide, whose structure can be determined by crystallography, the structural description of the apoCaM-Ng_13–49 is unknown due to low binding affinity, therefore we computationally generated an ensemble of apoCaM-Ng_13–49 structures by matching the changes in the chemical shifts of CaM upon Ng_13–49 binding from nuclear magnetic resonance experiments. Next, we computed the changes in Ca²⁺ affinity for CaM with and without binding targets in atomistic models using Jarzynski’s equality. We discovered the molecular underpinnings of lowered affinity of Ca²⁺ for CaM in the presence of Ng_13–49 by showing that the N-terminal acidic region of Ng peptide pries open the β-sheet structure between the Ca²⁺ binding loops particularly at C-domain of CaM, enabling Ca²⁺ release. In contrast, CaMKII peptide increases Ca²⁺ affinity for the C-domain of CaM by stabilizing the two Ca²⁺ binding loops. We speculate that the distinctive structural difference in the bound complexes of apoCaM-Ng_13–49 and holoCaM-CaMKII delineates the importance of CaM’s progressive mechanism of target binding on its Ca²⁺ binding affinities.

Contribution to the prediction of the fold code: application to immunoglobulin and flavodoxin cases

Background

Folding nucleus of globular proteins formation starts by the mutual interaction of a group of hydrophobic amino acids whose close contacts allow subsequent formation and stability of the 3D structure. These early steps can be predicted by simulation of the folding process through a Monte Carlo (MC) coarse grain model in a discrete space. We previously defined MIRs (Most Interacting Residues), as the set of residues presenting a large number of non-covalent neighbour interactions during such simulation. MIRs are good candidates to define the minimal number of residues giving rise to a given fold instead of another one, although their proportion is rather high, typically [15-20]% of the sequences. Having in mind experiments with two sequences of very high levels of sequence identity (up to 90%) but different folds, we combined the MIR method, which takes sequence as single input, with the “fuzzy oil drop” (FOD) model that requires a 3D structure, in order to estimate the residues coding for the fold. FOD assumes that a globular protein follows an idealised 3D Gaussian distribution of hydrophobicity density, with the maximum in the centre and minima at the surface of the “drop”. If the actual local density of hydrophobicity around a given amino acid is as high as the ideal one, then this amino acid is assigned to the core of the globular protein, and it is assumed to follow the FOD model. Therefore one obtains a distribution of the amino acids of a protein according to their agreement or rejection with the FOD model.

Results

We compared and combined MIR and FOD methods to define the minimal nucleus, or keystone, of two populated folds: immunoglobulin-like (Ig) and flavodoxins (Flav). The combination of these two approaches defines some positions both predicted as a MIR and assigned as accordant with the FOD model. It is shown here that for these two folds, the intersection of the predicted sets of residues significantly differs from random selection. It reduces the number of selected residues by each individual method and allows a reasonable agreement with experimentally determined key residues coding for the particular fold. In addition, the intersection of the two methods significantly increases the specificity of the prediction, providing a robust set of residues that constitute the folding nucleus.

Denaturant-dependent folding of GFP

We use molecular simulations using a coarse-grained model to map the folding landscape of Green Fluorescent Protein (GFP), which is extensively used as a marker in cell biology and biotechnology. Thermal and Guanidinium chloride (GdmCl) induced unfolding of a variant of GFP, without the chromophore, occurs in an apparent two-state manner. The calculated midpoint of the equilibrium folding in GdmCl, taken into account using the Molecular Transfer Model (MTM), is in excellent agreement with the experiments. The melting temperatures decrease linearly as the concentrations of GdmCl and urea are increased. The structural features of rarely populated equilibrium intermediates, visible only in free energy profiles projected along a few order parameters, are remarkably similar to those identified in a number of ensemble experiments in GFP with the chromophore. The excellent agreement between simulations and experiments show that the equilibrium intermediates are stabilized by the chromophore. Folding kinetics, upon temperature quench, show that GFP first collapses and populates an ensemble of compact structures. Despite the seeming simplicity of the equilibrium folding, flux to the native state flows through multiple channels and can be described by the kinetic partitioning mechanism. Detailed analysis of the folding trajectories show that both equilibrium and several kinetic intermediates, including misfolded structures, are sampled during folding. Interestingly, the intermediates characterized in the simulations coincide with those identified in single molecule pulling experiments. Our predictions, amenable to experimental tests, show that MTM is a practical way to simulate the effect of denaturants on the folding of large proteins.

A physics-based approach of coarse-graining the cytoplasm of Escherichia coli (CGCYTO)

We have investigated protein stability in an environment of Escherichia coli cytoplasm using coarse-grained computer simulations. To coarse-grain a small slide of E. coli's cytoplasm consisting of over 16 million atoms, we have developed a self-assembled clustering algorithm (CGCYTO). CGCYTO uses the shape parameter and asphericity as well as a parameter λ (ranging from 0 to 1) that measures the covolume of a test protein and a macromolecule against the covolume of a test protein and a sphere of equal volume as that of a macromolecule for the criteria of coarse-graining a cytoplasmic model. A cutoff λ(c) = 0.8 was chosen based on the size of a test protein and computational resources and it determined the resolution of a coarse-grained cytoplasm. We compared the results from a polydisperse cytoplasmic model (PD model) produced by CGCYTO with two other coarse-grained hard-sphere cytoplasmic models: 1), F70 model, macromolecules in the cytoplasm were modeled by homogeneous hard spheres with a radius of 55 Å, the size of Ficoll70 and 2), HS model, each macromolecule in the cytoplasm was modeled by a hard sphere of equal volume. It was found that the folding temperature T(f) of a test protein (apoazurin) in a PD model is ~5° greater than that in a F70 model. In addition, the deviation of T(f) in a PD model is twice as much as that in a HS model when an apoazurin is randomly placed at different voids formed by particle fluctuations in PD models.

Comparison of chemical and thermal protein denaturation by combination of computational and experimental approaches. II

Chemical and thermal denaturation methods have been widely used to investigate folding processes of proteins in vitro. However, a molecular understanding of the relationship between these two perturbation methods is lacking. Here, we combined computational and experimental approaches to investigate denaturing effects on three structurally different proteins. We derived a linear relationship between thermal denaturation at temperature T(b) and chemical denaturation at another temperature T(u) using the stability change of a protein (ΔG). For this, we related the dependence of ΔG on temperature, in the Gibbs-Helmholtz equation, to that of ΔG on urea concentration in the linear extrapolation method, assuming that there is a temperature pair from the urea (T(u)) and the aqueous (T(b)) ensembles that produces the same protein structures. We tested this relationship on apoazurin, cytochrome c, and apoflavodoxin using coarse-grained molecular simulations. We found a linear correlation between the temperature for a particular structural ensemble in the absence of urea, T(b), and the temperature of the same structural ensemble at a specific urea concentration, T(u). The in silico results agreed with in vitro far-UV circular dichroism data on apoazurin and cytochrome c. We conclude that chemical and thermal unfolding processes correlate in terms of thermodynamics and structural ensembles at most conditions; however, deviations were found at high concentrations of denaturant.

Collapse kinetics and chevron plots from simulations of denaturant-dependent folding of globular proteins

Quantitative description of how proteins fold under experimental conditions remains a challenging problem. Experiments often use urea and guanidinium chloride to study folding whereas the natural variable in simulations is temperature. To bridge the gap, we use the molecular transfer model that combines measured denaturant-dependent transfer free energies for the peptide group and amino acid residues, and a coarse-grained C(α)-side chain model for polypeptide chains to simulate the folding of src SH(3) domain. Stability of the native state decreases linearly as [C] (the concentration of guanidinium chloride) increases with the slope, m, that is in excellent agreement with experiments. Remarkably, the calculated folding rate at [C] = 0 is only 16-fold larger than the measured value. Most importantly ln k(obs) (k(obs) is the sum of folding and unfolding rates) as a function of [C] has the characteristic V (chevron) shape. In every folding trajectory, the times for reaching the native state, interactions stabilizing all the substructures, and global collapse coincide. The value of (m(f) is the slope of the folding arm of the chevron plot) is identical to the fraction of buried solvent accessible surface area in the structures of the transition state ensemble. In the dominant transition state, which does not vary significantly at low [C], the core of the protein and certain loops are structured. Besides solving the long-standing problem of computing the chevron plot, our work lays the foundation for incorporating denaturant effects in a physically transparent manner either in all-atom or coarse-grained simulations.

Multiscale investigation of chemical interference in proteins

We developed a multiscale approach (MultiSCAAL) that integrates the potential of mean force obtained from all-atomistic molecular dynamics simulations with a knowledge-based energy function for coarse-grained molecular simulations in better exploring the energy landscape of a small protein under chemical interference such as chemical denaturation. An excessive amount of water molecules in all-atomistic molecular dynamics simulations often negatively impacts the sampling efficiency of some advanced sampling techniques such as the replica exchange method and it makes the investigation of chemical interferences on protein dynamics difficult. Thus, there is a need to develop an effective strategy that focuses on sampling structural changes in protein conformations rather than solvent molecule fluctuations. In this work, we address this issue by devising a multiscale simulation scheme (MultiSCAAL) that bridges the gap between all-atomistic molecular dynamics simulation and coarse-grained molecular simulation. The two key features of this scheme are the Boltzmann inversion and a protein atomistic reconstruction method we previously developed (SCAAL). Using MultiSCAAL, we were able to enhance the sampling efficiency of proteins solvated by explicit water molecules. Our method has been tested on the folding energy landscape of a small protein Trp-cage with explicit solvent under 8M urea using both the all-atomistic replica exchange molecular dynamics and MultiSCAAL. We compared computational analyses on ensemble conformations of Trp-cage with its available experimental NOE distances. The analysis demonstrated that conformations explored by MultiSCAAL better agree with the ones probed in the experiments because it can effectively capture the changes in side-chain orientations that can flip out of the hydrophobic pocket in the presence of urea and water molecules. In this regard, MultiSCAAL is a promising and effective sampling scheme for investigating chemical interference which presents a great challenge when modeling protein interactions in vivo.

Refolding dynamics of stretched biopolymers upon force quench

Single-molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench by using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, tau(F), depends asymmetrically on deltaf(S) = f (S) - f (m) and deltaf (Q) = f (m) - f (Q) where f (S) (f (Q)) is the stretch (quench) force and f (m) is the transition midforce of the RNA hairpin. In accord with experiments, the relaxation kinetics of the molecular extension, R(t), occurs in three stages: A rapid initial decrease in the extension is followed by a plateau and finally, an abrupt reduction in R(t) occurs as the native state is approached. The duration of the plateau increases as lambda = tau (Q)/tau (F) decreases (where tau (Q) is the time in which the force is reduced from f (S) to f (Q)). Variations in the mechanisms of force-quench relaxation as lambda is altered are reflected in the experimentally measurable time-dependent entropy, which is computed directly from the folding trajectories. An analytical solution of the de Gennes model under tension reproduces the multistage stage kinetics in R(t). The prediction that the initial stages of collapse should also be a generic feature of polymers is validated by simulation of the kinetics of toroid (globule) formation in semiflexible (flexible) homopolymers in poor solvents upon quenching the force from a fully stretched state. Our findings give a unified explanation for multiple disparate experimental observations of protein folding.

Residue-specific analysis of frustration in the folding landscape of repeat beta/alpha protein apoflavodoxin

Flavodoxin adopts the common repeat beta/alpha topology and folds in a complex kinetic reaction with intermediates. To better understand this reaction, we analyzed a set of Desulfovibrio desulfuricans apoflavodoxin variants with point mutations in most secondary structure elements by in vitro and in silico methods. By equilibrium unfolding experiments, we first revealed how different secondary structure elements contribute to overall protein resistance to heat and urea. Next, using stopped-flow mixing coupled with far-UV circular dichroism, we probed how individual residues affect the amount of structure formed in the experimentally detected burst-phase intermediate. Together with in silico folding route analysis of the same point-mutated variants and computation of growth in nucleation size during early folding, computer simulations suggested the presence of two competing folding nuclei at opposite sides of the central beta-strand 3 (i.e., at beta-strands 1 and 4), which cause early topological frustration (i.e., misfolding) in the folding landscape. Particularly, the extent of heterogeneity in folding nuclei growth correlates with the in vitro burst-phase circular dichroism amplitude. In addition, phi-value analysis (in vitro and in silico) of the overall folding barrier to apoflavodoxin's native state revealed that native-like interactions in most of the beta-strands must form in transition state. Our study reveals that an imbalanced competition between the two sides of apoflavodoxin's central beta-sheet directs initial misfolding, while proper alignment on both sides of beta-strand 3 is necessary for productive folding.

Macromolecular crowding modulates folding mechanism of alpha/beta protein apoflavodoxin

Protein dynamics in cells may be different from those in dilute solutions in vitro, because the environment in cells is highly concentrated with other macromolecules. This volume exclusion because of macromolecular crowding is predicted to affect both equilibrium and kinetic processes involving protein conformational changes. To quantify macromolecular crowding effects on protein folding mechanisms, we investigated the folding energy landscape of an alpha/beta protein, apoflavodoxin, in the presence of inert macromolecular crowding agents, using in silico and in vitro approaches. By means of coarse-grained molecular simulations and topology-based potential interactions, we probed the effects of increased volume fractions of crowding agents (phi(c)) as well as of crowding agent geometry (sphere or spherocylinder) at high phi(c). Parallel kinetic folding experiments with purified Desulfovibro desulfuricans apoflavodoxin in vitro were performed in the presence of Ficoll (sphere) and Dextran (spherocylinder) synthetic crowding agents. In conclusion, we identified the in silico crowding conditions that best enhance protein stability, and discovered that upon manipulation of the crowding conditions, folding routes experiencing topological frustrations can be either enhanced or relieved. Our test-tube experiments confirmed that apoflavodoxin's time-resolved folding path is modulated by crowding agent geometry. Macromolecular crowding effects may be a tool for the manipulation of protein-folding and function in living cells.

Probing the mechanisms of fibril formation using lattice models

Using exhaustive Monte Carlo simulations we study the kinetics and mechanism of fibril formation using lattice models as a function of temperature (T) and the number of chains (M). While these models are, at best, caricatures of peptides, we show that a number of generic features thought to govern fibril assembly are captured by the toy model. The monomer, which contains eight beads made from three letters (hydrophobic, polar, and charged), adopts a compact conformation in the native state. In both the single-layered protofilament (seen for M10) structures, the monomers are arranged in an antiparallel fashion with the "strandlike" conformation that is perpendicular to the fibril axis. Partial unfolding of the folded monomer that populates an aggregation prone conformation (N(*)) is required for ordered assembly. The contacts in the N(*) conformation, which is one of the four structures in the first "excited" state of the monomer, are also present in the native conformation. The time scale for fibril formation is a minimum in the T-range when the conformation N(*) is substantially populated. The kinetics of fibril assembly occurs in three distinct stages. In each stage there is a cascade of events that transforms the monomers and oligomers to ordered structures. In the first "burst" stage, highly mobile oligomers of varying sizes form. The conversion to the N(*) conformation occurs within the oligomers during the second stage in which a vast number of interchain contacts are established. As time progresses, a dominant cluster emerges that contains a majority of the chains. In the final stage, the aggregation of N(*) particles serve as a template onto which smaller oligomers or monomers can dock and undergo conversion to fibril structures. The overall time for growth in the latter stages is well described by the Lifshitz-Slyazov growth kinetics for crystallization from supersaturated solutions. The detailed analysis shows that elements of the three popular models, namely, nucleation and growth, templated assembly, and nucleated conformational conversion are present at various stages of fibril assembly.

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.

Manipulating biopolymer dynamics by anisotropic nanoconfinement

How the geometry of nanosized confinement affects dynamics of biomaterials is interesting yet poorly understood. An elucidation of structural details upon nanosized confinement may benefit manufacturing pharmaceuticals in biomaterial sciences and medicine. The behavior of biopolymers in nanosized confinement is investigated using coarse-grained models and molecular simulations. Particularly, we address the effects of shapes of a confinement on protein-folding dynamics by measuring folding rates and dissecting structural properties of the transition states in nanosized spheres and ellipsoids. We find that when the form of a confinement resembles the geometrical properties of the transition states, the rates of folding kinetics are most enhanced. This knowledge of shape selectivity in identifying optimal conditions for reactions will have a broad impact in nanotechnology and pharmaceutical sciences.

A framework for describing topological frustration in models of protein folding

In a natively folded protein of moderate or larger size, the protein backbone may weave through itself in complex ways, raising questions about what sequence of events might have to occur in order for the protein to reach its native configuration from the unfolded state. A mathematical framework is presented here for describing the notion of a topological folding barrier, which occurs when a protein chain must pass through a hole or opening, formed by other regions of the protein structure. Different folding pathways encounter different numbers of such barriers and therefore different degrees of frustration. A dynamic programming algorithm finds the optimal theoretical folding path and minimal degree of frustration for a protein based on its natively folded configuration. Calculations over a database of protein structures provide insights into questions such as whether the path of minimal frustration might tend to favor folding from one or from many sites of folding nucleation, or whether proteins favor folding around the N terminus, thereby providing support for the hypothesis that proteins fold co-translationally. The computational methods are applied to a multi-disulfide bonded protein, with computational findings that are consistent with the experimentally observed folding pathway. Attention is drawn to certain complex protein folds for which the computational method suggests there may be a preferred site of nucleation or where folding is likely to proceed through a relatively well-defined pathway or intermediate. The computational analyses lead to testable models for protein folding.

Are residues in a protein folding nucleus evolutionarily conserved?

Protein is the working molecule of the cell, and evolution is the hallmark of life. It is important to understand how protein folding and evolution influence each other. Several studies correlating experimental measurement of residue participation in folding nucleus and sequence conservation have reached different conclusions. These studies are based on assessment of sequence conservation at folding nucleus sites using entropy or relative entropy measurement derived from multiple sequence alignment. Here we report analysis of conservation of folding nucleus using an evolutionary model alternative to entropy-based approaches. We employ a continuous time Markov model of codon substitution to distinguish mutation fixed by evolution and mutation fixed by chance. This model takes into account bias in codon frequency, bias-favoring transition over transversion, as well as explicit phylogenetic information. We measure selection pressure using the ratio omega of synonymous versus non-synonymous substitution at individual residue site. The omega-values are estimated using the PAML method, a maximum-likelihood estimator. Our results show that there is little correlation between the extent of kinetic participation in protein folding nucleus as measured by experimental phi-value and selection pressure as measured by omega-value. In addition, two randomization tests failed to show that folding nucleus residues are significantly more conserved than the whole protein, or the median omega value of all residues in the protein. These results suggest that at the level of codon substitution, there is no indication that folding nucleus residues are significantly more conserved than other residues. We further reconstruct candidate ancestral residues of the folding nucleus and suggest possible test tube mutation studies for testing folding behavior of ancient folding nucleus.

Protein folding

The problem of protein folding is that how proteins acquire their native unique three-dimensional structure in the physiological milieu. To solve the problem, the following key questions should be answered: do proteins fold co- or post-translationally, i.e. during or after biosynthesis, what is the mechanism of protein folding, and what is the explanation for fast folding of proteins? The two first questions are discussed in the current review. The general lines are to show that the opinion, that proteins fold after they are synthesized is hardly substantiated and suitable for solving the problem of protein folding and why proteins should fold cotranslationally. A possible tentative model for the mechanism of protein folding is also suggested. To this end, a thorough analysis is made of the biosynthesis, delivery to the folding compartments, and the rates of the biosynthesis, translocation and folding of proteins. A cursory attention is assigned to the role of GroEL/ES-like chaperonins in protein folding.

Master equation approach to finding the rate-limiting steps in biopolymer folding

A master equation approach is developed to find the rate-limiting steps in biopolymer folding, where the folding kinetics is described as a linear combination of basic kinetic modes determined from the eigenvalues and eigenvectors of the rate matrix. Because the passage of a rate-limiting step is intrinsically related to the folding speed, it is possible to probe and to identify the rate-limiting steps through the folding from different unfolded initial conformations. In a master equation approach, slow and fast folding speeds are directly correlated to the large and small contributions of the (rate-limiting) slow kinetic modes. Because the contributions from the slow modes can be computed from the corresponding eigenvectors, the rate-limiting steps can be identified from the eigenvectors of the slow modes. Our rate-limiting searching method has been tested for a simplified hairpin folding kinetics model, and it may provide a general transition state searching method for biopolymer folding.

Pathway heterogeneity in protein folding

We generate ab initio folding pathways in two single-domain proteins, hyperthermophile variant of protein G domain (1gb4) and ubiquitin (1ubi), both presumed to be two-state folders. Both proteins are endowed with the same topology but, as shown in this work, rely to a different extent on large-scale context to find their native folds. First, we demonstrate a generic feature of two-state folders: A downsizing of structural fluctuations is achieved only when the protein reaches a stationary plateau maximizing the number of highly protected hydrogen bonds. This enables us to identify the folding nucleus and show that folding does not become expeditious until a topology is generated that is able to protect intramolecular hydrogen bonds from water attack. Pathway heterogeneity is shown to be dependent on the extent to which the protein relies on large-scale context to fold, rather than on contact order: Proteins that can only stabilize native secondary structure by packing it against scaffolding hydrophobic moieties are meant to have a heterogeneous transition-state ensemble if they are to become successful folders (otherwise, successful folding would be too fortuitous an event.) We estimate mutational Phi values as ensemble averages and deconvolute individual-route contributions to the averaged two-state kinetic picture. Our results find experimental corroboration in the well-studied chymotrypsin inhibitor (CI2), while leading to verifiable predictions for the other two study cases.

Time-resolved backbone desolvation and mutational hot spots in folding proteins

A method is presented to identify hot mutational spots and predict the extent of surface burial at the transition state relative to the native fold in two-state folding proteins. The method is based on ab initio simulations of folding histories in which transitions between coarsely defined conformations and pairwise interactions are dependent on the solvent environments created by the chain. The highly conserved mammalian ubiquitin is adopted as a study case to make predictions. The evolution in time of the chain topology suggests a nucleation process with a critical point signaled by a sudden quenching of structural fluctuations. The occurrence of this nucleus is shown to be concurrent with a sudden escalation in the number of three-body correlations whereby hydrophobic units approach residue pairs engaged in amide-carbonyl hydrogen bonding. These correlations determine a pattern designed to structure the surrounding solvent, protecting intramolecular hydrogen bonds from water attack. Such correlations are shown to be required to stabilize the nucleus, with kinetic consequences for the folding process. Those nuclear residues that adopt the dual role of protecting and being protected while engaged in hydrogen bonds are predicted to be the hottest mutational spots. Some such residues are shown not to retain the same protecting role in the native fold. This kinetic treatment of folding nucleation is independently validated vis-a-vis a Phi-value analysis on chymotrypsin inhibitor 2, a protein for which extensive mutational data exists.

How do we probe ubiquitin's pathway heterogeneity?

We identify folding pathways for ubiquitin and assess its extent of transition state (TS) heterogeneity using a kinetically controlled ab initio algorithm that generates a coarse-grained description of torsional dynamics. The algorithm computes the time evolution of backbone-motion constraints, finds optimized conformations within such constraints, evaluates local solvent environments, and rescales accordingly the energetic contributions to determine the transition to the next set of torsional constraints. Native and nonnative structural features are found in the TS ensemble determined from a pool of 72 successful runs whose final folds are within 4-5 A RMSD from native. Certain nonnative features at the TS are shown to be necessary to create a large-scale context that overrides local propensities. Such misfolds undergo a subsequent rearrangement on the downhill side of the energy profile. The effects of tunable bi-histidine metal-binding sites, point mutations, negative Phi-value mutations, and denaturant on kinetics are predicted.

Native and non-native interactions along protein folding and unfolding pathways

Gō-type models, which include only native contact interactions in the energy function, are being used increasingly to describe the protein folding reaction. To investigate the validity of such models, we determine the role of native and non-native interactions along folding and unfolding pathways. For this purpose, we use a molecular mechanics energy function with an implicit solvation model (an effective energy function or potential of mean force) that can be expressed in a pairwise decomposable form. We find that for the native state and a wide range of other configurations, the contact energy is an accurate description, in part due to the cancellation of non-zero contributions from more distant residues. However, significant errors in the energy are introduced for non-native structures if the energy is calculated from the native contacts alone. Non-native contacts tend to make a significant contribution, particularly for molten globules and collapsed states along the unfolding pathways. The implication of these results for the use of Gō-type models in studies of protein folding are discussed.

Stiffness of the distal loop restricts the structural heterogeneity of the transition state ensemble in SH3 domains

Protein engineering experiments and Phi(F)-value analysis of SH3 domains reveal that their transition state ensemble (TSE) is conformationally restricted, i.e. the fluctuations in the transition state (TS) structures are small. In the TS of src SH3 and alpha-spectrin SH3 the distal loop and the associated hairpin are fully structured, while the rest of the protein is relatively disordered. If native structure predominantly determines the folding mechanism, the findings for SH3 folds raise the question: What are the features of the native topology that determine the nature of the TSE? We propose that the presence of stiff loops in the native state that connect local structural elements (such as the distal hairpin in SH3 domains) conformationally restricts TSE. We validate this hypothesis using the simulations of a "control" system (16 residue beta-hairpin forming C-terminal fragment of the GBl protein) and its variants. In these fragments the role of bending rigidity in determining the nature of the TSE can be directly examined without complications arising from interactions with the rest of the protein. The TSE structures in the beta-hairpins are determined computationally using cluster analysis and limited Phi(F)-value analysis. Both techniques prove that the conformational heterogeneity decreases as the bending rigidity of the loop increases. To extend this finding to SH3 domains a measure of bending rigidity based on loop curvature, which utilizes native structures in the Protein Data Bank (PDB), is introduced. Using this measure we show that, with few exceptions, the ordering of stiffness of the distal, n-src, and RT loops in the 29 PDB structures of SH3 domains is conserved. Combining the simulation results for beta-hairpins and the analysis of PDB structures for SH3 domains, we propose that the stiff distal loop restricts the conformational fluctuations in the TSE. We also predict that constraining the distal loop to be preformed in the denatured ensemble should not alter the nature of TSE. On the other hand, if the amino and carboxy terminals are cross-linked to form a circular polypeptide chain, the pathways and TSs are altered. These contrasting scenarios are illustrated using simulations of cross-linked WT beta-hairpin fragments. Computations of bending rigidities for immunoglobulin-like domain proteins reveal no clear separation in the stiffness of their loops. In the beta-sandwich proteins, which have large fractions of non-local native contacts, the nature of the TSE cannot be apparently determined using purely local structural characteristics. Nevertheless, the measure of loop stiffness still provides qualitative predictions of the ordered regions in the TSE of Ig27 and TenFn3.

Pathway diversity and concertedness in protein folding: an ab-initio approach

Making use of an ab-initio folding simulator, we generate in vitro pathways leading to the native fold in moderate size single- domain proteins. The assessment of pathway diversity is not biased by any a priori information on the native fold. We focus on two study cases, hyperthermophile variant of protein G domain (1gb4) and ubiquitin (1ubi), with the same topology but different context dependence in their native folds. We demonstrate that a quenching of structural fluctuations is achieved once the proteins find a stationary plateau maximizing the number of highly protected hydrogen bonds. This enables us to identify the folding nucleus and show that folding does not become expeditious until a concerted event takes place generating a topology able to prevent water attack on a maximal number of hydrogen bonds. This result is consistent with the standard nucleation mechanism postulated for two-state folders. Pathway diversity is correlated with the extent of conflict between local structural propensity and large-scale context, rather than with contact order: In highly context-dependent proteins, the success of folding cannot rely on a single fortuitous event in which local propensity is overruled by large-scale effects. We predict mutational Pi values on individual pathways, compute ensemble averages and predict extent of surface burial and percentage of hydrogen bonding on each component of the transition state ensemble, thus deconvoluting individual folding-route contributions to the averaged two-state kinetic picture. Our predicted kinetic isotopic effects find experimental support and lead to further probes. Finally, the molecular redesign potentiality of the method, aimed at increasing folding expediency, is explored.

Structural transitions in neutral and charged proteins in vacuo

In vacuo proteins provide a simple laboratory to explore the roles of sequence, temperature, charge state, and initial configuration in protein folding. Moreover, by the very absence of solvent, the study of anhydrous proteins in vacuo will also help us to understand specific environmental effects. From the experimental viewpoint, these systems are now beginning to be characterized at low resolution. Molecular dynamics (MD) simulations, in combination with tools for protein shape analysis, can complement experiments and provide further insights on the folding-unfolding transitions of these proteins. We review some aspects of this issue by using the results from a detailed MD study of hen egg-white lysozyme. For lysozyme ions, unfolding can be triggered by Coulombic repulsion. In neutral lysozyme, unfolding can be induced by centrifugal forces and also by weakening the monomer-monomer interaction. In both cases, the resulting unfolded transients can be used as initial configurations for relaxation dynamics. All trajectories are analyzed in terms of global molecular shape features of the backbone, including its anisometry and chain entanglement complexity. This strategy allows us to quantify separately the degree of polymer collapse and the evolution of large-scale folding features. Using these last two notions, we discuss some basic questions regarding the nature of the accessible paths associated with unfolding from, and refolding into, compact conformers.

Multiple protein folding nuclei and the transition state ensemble in two-state proteins

Using exhaustive simulations of lattice models with side-chains, we show that optimized two-state folders reach the native state by a nucleation-collapse mechanism with multiple folding nuclei (MFN). For both the full model and the Go version, there are certain contacts that on an average participate in the critical nuclei with higher probability than the others. The high- (> or = 0.5) probability contacts are largely determined by the structure of the native state. Comparison of the results for the full sequence and the Go model shows that non-native interactions compromise the degree of cooperativity and stability of the native state. From an extremely detailed analysis of the folding kinetics, we find that non-native interactions are present in the folding nuclei. The folding times decrease if the non-native interactions in the folding nuclei are made neutral or repulsive. Using cluster analysis and making no prior assumption about reaction coordinate, we show that both full and Go models have three distinct transition states that give a structural description for the MFN. In the transition states, on an average, about two-thirds of the sequence is structured, whereas the rest is disordered, reminiscent of the polarized transition state in the SH3 domain. Our studies show that Go models cannot describe the transition state characteristics of two-state folders at the molecular level. As a byproduct of our investigations, we establish that our method of computing the transition state ensemble is numerically equivalent to the technique based on the stochastic separatrix, which also does not require a priori knowledge of the folding reaction coordinate.

Characterisation of the transition states for protein folding: towards a new level of mechanistic detail in protein engineering analysis

The field of protein folding now offers considerable excitement. Comparative studies of the transition-state structures for a series of protein families with analogous structures have helped to uncover the overall rules for protein folding. In addition, new protein engineering experiments that continuously follow the growth of the folding nucleus have started to fill in the missing details.

Mechanisms and kinetics of beta-hairpin formation

Thermodynamics and kinetics of off-lattice models with side chains for the beta-hairpin fragment of immunoglobulin-binding protein and its variants are reported. For all properties (except refolding time tau(F)) there are no qualitative differences between the full model and the Go version. The validity of the models is established by comparison of the calculated native structure with the Protein Data Bank coordinates and by reproducing the experimental results for the degree of cooperativity and tau(F). For the full model tau(F) approximately 2 micros at the folding temperature (experimental value is 6 micros); the Go model folds 50 times faster. Upon refolding, structural changes take place over three time scales. On the collapse time scale compact structures with intact hydrophobic cluster form. Subsequently, hydrogen bonds form, predominantly originating from the turn by a kinetic zipping mechanism. The assembly of the hairpin is complete when most of the interstrand contacts (the rate-limiting step) is formed. The dominant transition state structure (located by using cluster analysis) is compact and structured. We predict that when hydrophobic cluster is moved to the loop tau(F) marginally increases, whereas moving the hydrophobic cluster closer to the termini results in significant decrease in tau(F) relative to wild type. The mechanism of hairpin formation is predicted to depend on turn stiffness.

Unexpected scenario of glass transition in polymer globules: an exactly enumerable model

We introduce a lattice model of glass transition in polymer globules. This model exhibits ergodicity breaking in which the disjoint regions of phase space do not arise uniformly, but as small chambers whose number increases exponentially with polymer density. Chamber sizes obey power law distribution, making phase space similar to a fractal foam. This clearly demonstrates the importance of the phase space geometry and topology in describing any glass-forming system, such as semicompact polymers during protein folding.

Redesigning the hydrophobic core of a model beta-sheet protein: destabilizing traps through a threading approach

An off-lattice 46-bead model of a small all-beta protein has been recently criticized for possessing too many traps and long-lived intermediates compared with the folding energy landscape predicted for real proteins and models using the principle of minimal frustration. Using a novel sequence design approach based on threading for finding beneficial mutations for destabilizing traps, we proposed three new sequences for folding in the beta-sheet model. Simulated annealing on these sequences found the global minimum more reliably, indicative of a smoother energy landscape, and simulated thermodynamic variables found evidence for a more cooperative collapse transition, lowering of the collapse temperature, and higher folding temperatures. Folding and unfolding kinetics were acquired by calculating first-passage times, and the new sequences were found to fold significantly faster than the original sequence, with a concomitant lowering of the glass temperature, although none of the sequences have highly stable native structures. The new sequences found here are more representative of real proteins and are good folders in the T(f) > T(g) sense, and they should prove useful in future studies of the details of transition states and the nature of folding intermediates in the context of simplified folding models. These results show that our sequence design approach using threading can improve models possessing glasslike folding dynamics.

Towards understanding the mechanisms of molecular recognition by computer simulations of ligand-protein interactions

The thermodynamic and kinetic aspects of molecular recognition for the methotrexate (MTX)-dihydrofolate reductase (DHFR) ligand-protein system are investigated by the binding energy landscape approach. The impact of 'hot' and 'cold' errors in ligand mutations on the thermodynamic stability of the native MTX-DHFR complex is analyzed, and relationships between the molecular recognition mechanism and the degree of ligand optimization are discussed. The nature and relative stability of intermediates and thermodynamic phases on the ligand-protein association pathway are studied, providing new insights into connections between protein folding and molecular recognition mechanisms, and cooperativity of ligand-protein binding. The results of kinetic docking simulations are rationalized based on the thermodynamic properties determined from equilibrium simulations and the shape of the underlying binding energy landscape. We show how evolutionary ligand selection for a receptor active site can produce well-optimized ligand-protein systems such as MTX-DHFR complex with the thermodynamically stable native structure and a direct transition mechanism of binding from unbound conformations to the unique native structure.

Why reversing the sequence of the alpha domain of human metallothionein-2 does not change its metal-binding and folding characteristics

A novel peptide, the backward reading sequence of human metallothionein-2 alpha domain, was synthesized and its chemical and spectroscopic properties analyzed. This folded retro-alpha domain was able to bind Cd(II) in identical stoichiometries with the chemically synthesized alpha domain of metallothionein-2. Nearly identical to the alpha domain, Cd-binding retro-alpha domain showed a characteristic ultraviolet absorption spectrum with a shoulder at 245-250 nm (due to cadmium-thiolate charge transfer), and the absorption shoulder was abolished by acidification [suggesting mercaptide bonding between Cd(II) and the cysteine residues]. Similar metal-binding capabilities between alpha domain and retro-alpha domain were observed also by pH titration and in the reaction with the sulfhydryl reagent 5,5'-dithiobis(2-nitrobenzoic acid). A two-state cooperativity of the metal-cluster formation was observed spectroscopically in the titration of the retro-alpha domain, indicating that the retro-protein is foldable. In contrast to other proteins, our results indicate that the reversion of the amino acid sequence for the alpha domain does not change its foldability and metal-binding capacity, suggesting that the order of its sequence is not critical to the formation of a critical metal-tetrathiolate nucleus. However, CD spectra of the Cd-binding alpha domain and retro-alpha domain showed that the reversal direction of the domain sequence backbone significantly affects the formation of structure even when it is foldable.

Is protein unfolding the reverse of protein folding? A lattice simulation analysis

Simulations and experiments that monitor protein unfolding under denaturing conditions are commonly employed to study the mechanism by which a protein folds to its native state in a physiological environment. Due to the differences in conditions and the complexity of the reaction, unfolding is not necessarily the reverse of folding. To assess the relevance of temperature initiated unfolding studies to the folding problem, we compare the folding and unfolding of a 125-residue protein model by Monte Carlo dynamics at two temperatures; the lower one corresponds to the range used in T -jump experiments and the higher one to the range used in unfolding simulations of all-atom models. The trajectories that lead from the native state to the denatured state at these elevated temperatures are less diverse than those observed in the folding simulations. At the lower temperature, the system unfolds through a mandatory intermediate that corresponds to a local free energy minimum. At the higher temperature, no such intermediate is observed, but a similar pathway is followed. The structures contributing to the unfolding pathways resemble most closely those that make up the "fast track" of folding. The transition state for unfolding at the lower temperature (above Tm) is determined and is found to be more structured than the transition state for folding below the melting temperature. This shift towards the native state is consistent with the Hammond postulate. The implications for unfolding simulations of higher resolution models and for unfolding experiments of proteins are discussed.

Mapping the interactions present in the transition state for unfolding/folding of FKBP12

The structure of the transition state for folding/unfolding of the immunophilin FKBP12 has been characterised using a combination of protein engineering techniques, unfolding kinetics, and molecular dynamics simulations. A total of 34 mutations were made at sites throughout the protein to probe the extent of secondary and tertiary structure in the transition state. The transition state for folding is compact compared with the unfolded state, with an approximately 30 % increase in the native solvent-accessible surface area. All of the interactions are substantially weaker in the transition state, as probed by both experiment and molecular dynamics simulations. In contrast to some other proteins of this size, no element of structure is fully formed in the transition state; instead, the transition state is similar to that found for smaller, single-domain proteins, such as chymotrypsin inhibitor 2 and the SH3 domain from alpha-spectrin. For FKBP12, the central three strands of the beta-sheet, beta-strand 2, beta-strand 4 and beta-strand 5, comprise the most structured region of the transition state. In particular Val101, which is one of the most highly buried residues and located in the middle of the central beta-strand, makes approximately 60 % of its native interactions. The outer beta-strands and the ends of the central beta-strands are formed to a lesser degree. The short alpha-helix is largely unstructured in the transition state, as are the loops. The data are consistent with a nucleation-condensation model of folding, the nucleus of which is formed by side-chains within beta-strands 2, 4 and 5, and the C terminus of the alpha-helix. The precise residues involved in the nucleus differ in the two simulated transition state ensembles, but the interacting regions of the protein are conserved. These residues are distant in the primary sequence, demonstrating the importance of tertiary interactions in the transition state. The two independently derived transition state ensembles are structurally similar, which is consistent with a Bronsted analysis confirming that the transition state is an ensemble of states close in structure.