Modeling protein density of states: additive hydrophobic effects are insufficient for calorimetric two-state cooperativity

A well-established experimental criterion for two-state thermodynamic cooperativity in protein folding is that the van't Hoff enthalpy DeltaH(vH) around the transition midpoint is equal, or very nearly so, to the calorimetric enthalpy DeltaH(cal) of the entire transition. This condition is satisfied by many small proteins. We use simple lattice models to provide a statistical mechanical framework to elucidate how this calorimetric two-state picture may be reconciled with the hierarchical multistate scenario emerging from recent hydrogen exchange experiments. We investigate the feasibility of using inverse Laplace transforms to recover the underlying density of states (i.e., enthalpy distribution) from calorimetric data. We find that the constraint imposed by DeltaH(vH)/DeltaH(cal) approximately 1 on densities of states of proteins is often more stringent than other "two-state" criteria proposed in recent theoretical studies. In conjunction with reasonable assumptions, the calorimetric two-state condition implies a narrow distribution of denatured-state enthalpies relative to the overall enthalpy difference between the native and the denatured conformations. This requirement does not always correlate with simple definitions of "sharpness" of a transition and has important ramifications for theoretical modeling. We find that protein models that assume capillarity cooperativity can exhibit overall calorimetric two-state-like behaviors. However, common heteropolymer models based on additive hydrophobic-like interactions, including highly specific two-dimensional Gō models, fail to produce proteinlike DeltaH(vH)/DeltaH(cal) approximately 1. A simple model is constructed to illustrate a proposed scenario in which physically plausible local and nonlocal cooperative terms, which mimic helical cooperativity and environment-dependent hydrogen bonding strength, can lead to thermodynamic behaviors closer to experiment. Our results suggest that proteinlike thermodynamic cooperativity may require a cooperative interplay between local and nonlocal interactions. The prospect of using calorimetric data to constrain Z-scores of knowledge-based potentials is discussed.     

Polymer principles and protein folding.

This paper surveys the emerging role of statistical mechanics and polymer theory in protein folding. In the polymer perspective, the folding code is more a solvation code than a code of local phipsi propensities. The polymer perspective resolves two classic puzzles: (1) the Blind Watchmaker's Paradox that biological proteins could not have originated from random sequences, and (2) Levinthal's Paradox that the folded state of a protein cannot be found by random search. Both paradoxes are traditionally framed in terms of random unguided searches through vast spaces, and vastness is equated with impossibility. But both processes are partly guided. The searches are more akin to balls rolling down funnels than balls rolling aimlessly on flat surfaces. In both cases, the vastness of the search is largely irrelevant to the search time and success. These ideas are captured by energy and fitness landscapes. Energy landscapes give a language for bridging between microscopics and macroscopics, for relating folding kinetics to equilibrium fluctuations, and for developing new and faster computational search strategies.     

The calorimetric criterion for a two-state process revisited.

The "calorimetric criterion" is one of the important experimental approaches for determining whether protein folding is an "all-or-none" two-state transition (i.e., whether intermediates are present at equilibrium). The calorimetric criterion states that the equivalence of the "measured" calorimetric enthalpy change and the effective two-state van't Hoff enthalpy change demonstrates that there is a two-state transition. This paper addresses the essential question of whether the calorimetric criterion is a necessary and sufficient condition for a two-state process and shows that it is necessary but not sufficient by means of specific examples. Analysis of simple models indicates that the heat capacity curve, regardless of whether it originates from a two-state process or not, can always be decomposed in such a way that the calorimetric criterion is satisfied. Exact results for a three-state model and a homopolymer tetramer demonstrate that the deviation from the calorimetric criterion is not simply related to the population of intermediate states. Analysis of a three-helix bundle protein model, which has a two-state folding from a random coil to ordered (molten) globule, shows that the calorimetric criterion may not be satisfied if the standard linear interpolation of baselines (weighted or unweighted) is employed. A specific example also suggests that the more recently introduced deconvolution method is not necessarily better than the simple calorimetric criterion for distinguishing a two-state transition from a three-state transition. Although the calorimetric criterion is not a sufficient condition for a two-state process, it is likely to continue to be of practical utility, particularly when its results are shown to be consistent with those from other experimental methods.     

Pathways in two-state protein folding

Thermodynamic measurements of proteins indicate that the folding to the native state takes place either through stable intermediates or through a two-state process without intermediates. The rather short folding times of proteins indicate that folding is guided through some sequence of contact bindings. We discuss the possibility of reconciling a two-state folding event with a sequential folding process in a schematic model of protein folding. We propose a new dynamical transition temperature that is lower than the temperature at which proteins in equilibrium unfold. This is in qualitative agreement with observations of in vivo protein folding activity quantified by chaperone concentration in Escherichia coli. Finally, we discuss our framework in connection with the unfolding of proteins at low temperatures.     

Cooperativity in two-state protein folding kinetics

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.     

The peculiar nature of the guanidine hydrochloride-induced two-state denaturation of staphylococcal nuclease: a calorimetric study.

This work determines the ratio of DeltaH(vH) /DeltaH(cal) for staphylococcal nuclease (SN) denaturation in guanidine hydrochloride (GdnHCl) to test whether GdnHCl-induced denaturation is two-state. Heats of mixing of SN as a function of [GdnHCl] were determined at pH 7.0 and 25 degrees C. The resulting plot of DeltaH(mix) vs [GdnHCl] exhibits a sigmoid shaped curve with linear pre- and post-denaturational base lines. Extending the pre- and post-denaturational lines to zero [GdnHCl] gives a calorimetric DeltaH (DeltaH(cal)) of 24.1 +/- 1.0 kcal/mol, for SN denaturation in the limit of zero GdnHCl concentration. Guanidine hydrochloride-induced denaturation Gibbs energy changes in the limit of zero denaturant concentration (DeltaG degrees (N)(-)(D)) at pH 7. 0 were determined for SN from fluorescence measurements at fixed temperatures over the range from 15 to 35 degrees C. Analysis of the resulting temperature-dependent DeltaG degrees (N)(-)(D) data defines a van't Hoff denaturation enthalpy change (DeltaH(vH)) of 26. 4 +/- 2.8 kcal/mol. The model-dependent van't Hoff DeltaH(vH) divided by the model-independent DeltaH(cal) gives a ratio of 1.1 +/- 0.1 for DeltaH(vH)/DeltaH(cal), a result that rules out the presence of thermodynamically important intermediate states in the GdnHCl-induced denaturation of SN. The likelihood that GdnHCl-induced SN denaturation involves a special type of two-state denaturation, known as a variable two-state process, is discussed in terms of the thermodynamic implications of the process.     

Hydrogen-bond cooperativity in protein secondary structure

Hydrogen bonding in the α-helix and β-sheet has been studied by ab initio molecular orbital calculations carried out on complexes of formamide. Hydrogen-bond geometries were taken from x-ray crystallography of polypeptides. Positive cooperativity is found in all cases. The limiting value for infinite chains is obtained by use of a double-reciprocal plot and indicates an increase in the effective bond strength of 25% over that of a single isolated bond. Parallel calculations based on a classical electrostatic model yield qualitatively similar trends but underestimate the cooperativity by half. Charge redistribution accompanying cooperativity is characterized by a new type of charge-density difference plot, the cooperativity map. The magnitude and distance over which cooperativity acts suggest several significant biological consequences. Thus the average of α-helices and the number of β-sheet strands found in protein may be influenced by cooperativity. Cooperativity in the interpeptide hydrogen bond may also be partly responsible for the rapid formation of secondary structure in renaturing proteins and help stabilize secondary structure relative to the random-coil conformation.     

The meaning of interaction parameters in two-state protein complexes

The theory of interaction parameters has thus far been based on the free-energy relationships in the formation of ternary complexes formed between a pair of ligands and a protein molecule. The concept has been formulted in terms of a thermodynamic square comprised of the free protein, the two binary complexes, and the ternary complex. However, an increasing number of proteins have been found to exist as equilibrium mixtures of two macrostates. The equilibrium constants for such two-state transitions vary quite considerably between the various binary and ternary complexes of a given protein. We show here that the interpretations of interaction parameters in such two-state systems, requiring the use of a thermodynamic cube, are much more complex than those based on the classic thermodynamic square commonly employed. We demonstrate the use of enthalpies of interaction and heat capacities of interaction to analyze the source of observed free enerigies of interaction in such systems. Specifically, we find that measured negative interaction parameters may arise simply from the inability of a system to achieve all of the positive component effects anticipated by the conventional formulation.     

Designing cooperativity into the designed protein Top7

The topology of the designed protein Top7 is not found in natural proteins. Top7 shows signatures of non‐cooperative folding in both experimental studies and computer simulations. In particular, molecular dynamics of coarse‐grained structure‐based models of Top7 show a well‐populated C‐terminal folding‐intermediate. Since most similarly sized globular proteins are cooperative folders, the non‐natural topology of Top7 has been suggested as a reason for its non‐cooperative folding. Here, we computationally examine the folding of Top7 with the intent of making it cooperative. We find that its folding cooperativity can be increased in two ways: (a) Optimization of packing interactions in the N‐terminal half of the protein enables further folding of the C‐terminal intermediate. (b) Reduction in the packing density of the C‐terminal region destabilizes the intermediate. In practice, these strategies are implemented in our Top7 model through modifications to the contact‐map. These modifications do not alter the topology of Top7 but result in cooperative folding. Amino‐acid mutations that mimic these modifications also lead to a significant increase in folding cooperativity. Finally, we devise a method to randomize the sizes of amino‐acids within the same topology, and confirm that the structure of Top7 makes its folding sensitive to packing interactions. In contrast, the ribosomal protein S6, which has secondary structure similar to Top7, has folding which is much less sensitive to packing perturbations. Thus, it should be possible to make a sequence fold cooperatively to the structure of Top7, but to do so its side‐chain packing needs to be carefully designed. Proteins 2014; 82:364–374. © 2013 Wiley Periodicals, Inc.     

Diffusional barrier crossing in a two-state protein folding reaction.

There has been some debate as to whether protein folding involves diffusive chain motions and thus depends on solvent viscosity. The interpretation of folding kinetics in viscous solvents has remained difficult and controversial, in that viscogenic agents affect folding rates not only by increasing solvent viscosity but also by increasing protein stability. By carefully choosing experimental conditions, we can now eliminate the effect on stability and show that the folding dynamics of the cold shock protein CspB are viscosity dependent. Thus Kramers' theory of reaction rates rather than transition state theory should be used to describe this folding reaction.     

Design principles of protein switches

Protein switches perform essential roles in many biological processes and are exciting targets for de novo protein design, which aims to produce proteins of arbitrary shape and functionality. However, the biophysical requirements for switch function — multiple conformational states, fine-tuned energetics, and stimuli-responsiveness — pose a formidable challenge for design by computation (or intuition). A variety of methods have been developed toward tackling this challenge, usually taking inspiration from the wealth of sequence and structural information available for naturally occurring protein switches. More recently, modular switches have been designed computationally, and new methods have emerged for sampling unexplored structure space, providing promising new avenues toward the generation of purpose-built switches and de novo signaling systems for cellular engineering.     

Calculation of the free energy and cooperativity of protein folding

Calculation of the free energy of protein folding and delineation of its pre-organization are of foremost importance for understanding, predicting and designing biological macromolecules. Here, we introduce an energy smoothing variant of parallel tempering replica exchange Monte Carlo (REMS) that allows for efficient configurational sampling of flexible solutes under the conditions of molecular hydration. Its usage to calculate the thermal stability of a model globular protein, Trp cage TC5b, achieves excellent agreement with experimental measurements. We find that the stability of TC5b is attained through the coupled formation of local and non-local interactions. Remarkably, many of these structures persist at high temperature, concomitant with the origin of native-like configurations and mesostates in an otherwise macroscopically disordered unfolded state. Graph manifold learning reveals that the conversion of these mesostates to the native state is structurally heterogeneous, and that the cooperativity of their formation is encoded largely by the unfolded state ensemble. In all, these studies establish the extent of thermodynamic and structural pre-organization of folding of this model globular protein, and achieve the calculation of macromolecular stability ab initio, as required for ab initio structure prediction, genome annotation, and drug design.     

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.     

Physical principles of protein structure and protein folding

Physical principles determining the protein structure and protein folding are reviewed: (i) the molecular theory of protein secondary structure and the method of its prediction based on this theory; (ii) the existence of a limited set of thermodynamically favourable folding patterns of α- and β-regions in a compact globule which does not depend on the details of the amino acid sequence; (iii) the moderns approaches to the prediction of the folding patterns of α- and β-regions in concrete proteins; (iv) experimental approaches to the mechanism of protein folding. The review reflects theoretical and experimental works of the author and his collaborators as well as those of other groups.