Theory for protein folding cooperativity: helix bundles

The intrinsically disordered protein glue of the myelin major dense line: Linking AlphaFold2 predictions to experimental data

Numerous human proteins are classified as intrinsically disordered proteins (IDPs). Due to their physicochemical properties, high-resolution structural information about IDPs is generally lacking. On the other hand, IDPs are known to adopt local ordered structures upon interactions with e.g. other proteins or lipid membrane surfaces. While recent developments in protein structure prediction have been revolutionary, their impact on IDP research at high resolution remains limited. We took a specific example of two myelin-specific IDPs, the myelin basic protein (MBP) and the cytoplasmic domain of myelin protein zero (P0ct). Both of these IDPs are crucial for normal nervous system development and function, and while they are disordered in solution, upon membrane binding, they partially fold into helices, being embedded into the lipid membrane. We carried out AlphaFold2 predictions of both proteins and analysed the models in light of experimental data related to protein structure and molecular interactions. We observe that the predicted models have helical segments that closely correspond to the membrane-binding sites on both proteins. We furthermore analyse the fits of the models to synchrotron-based X-ray scattering and circular dichroism data from the same IDPs. The models are likely to represent the membrane-bound state of both MBP and P0ct, rather than the conformation in solution. Artificial intelligence-based models of IDPs appear to provide information on the ligand-bound state of these proteins, instead of the conformers dominating free in solution. We further discuss the implications of the predictions for mammalian nervous system myelination and their relevance to understanding disease aspects of these IDPs.

Uncovering the Contributions of Charge Regulation to the Stability of Single Alpha Helices

The single alpha helix (SAH) is a recurring motif in biology. The consensus sequence has a di-block architecture that includes repeats of four consecutive glutamate residues followed by four consecutive lysine residues. Measurements show that the overall helicity of sequences with consensus E4 K4 repeats is insensitive to a wide range of pH values. Here, we use the recently introduced q-canonical ensemble, which allows us to decouple measurements of charge state and conformation, to explain the observed insensitivity of SAH helicity to pH. We couple the outputs from separate measurements of charge and conformation with atomistic simulations to derive residue-specific quantifications of preferences for being in an alpha helix and for the ionizable residues to be charged vs. uncharged. We find a clear preference for accommodating uncharged Glu residues within internal positions of SAH-forming sequences. The stabilities of alpha helical conformations increase with the number of E4 K4 repeats and so do the numbers of accessible charge states that are compatible with forming conformations of high helical content. There is conformational buffering whereby charge state heterogeneity buffers against large-scale conformational changes thus making the overall helicity insensitive to large changes in pH. Further, the results clearly argue against a single, rod-like alpha helical conformation being the only or even dominant conformation in the ensembles of so-called SAH sequences.

Conformational entropy limits the transition from nucleation to elongation in amyloid aggregation

The formation of β-sheet rich amyloid fibrils in Alzheimer's disease and other neurodegenerative disorders is limited by a slow nucleation event. To understand the initial formation of β-sheets from disordered peptides, we used all-atom simulations to parameterize a lattice model that treats each amino acid as a binary variable with β and non-β states. We show that translational and conformational entropy give the nascent β-sheet an anisotropic surface tension which can be used to describe the nucleus with two-dimensional Classical Nucleation Theory. Since translational entropy depends on concentration, the aspect ratio of the critical β-sheet changes with protein concentration. Our model explains the transition from the nucleation phase to elongation as the point where the β-sheet core becomes large enough to overcome the conformational entropy cost to straighten the terminal molecule. At this point the β-strands in the nucleus spontaneously elongate, which results in a larger binding surface to capture new molecules. These results suggest that nucleation is relatively insensitive to sequence differences in co-aggregation experiments because the nucleus only involves a small portion of the peptide.

Rules of Physical Mathematics Govern Intrinsically Disordered Proteins

In stark contrast to foldable proteins with a unique folded state, intrinsically disordered proteins and regions (IDPs) persist in perpetually disordered ensembles. Yet an IDP ensemble has conformational features-even when averaged-that are specific to its sequence. In fact, subtle changes in an IDP sequence can modulate its conformational features and its function. Recent advances in theoretical physics reveal a set of elegant mathematical expressions that describe the intricate relationships among IDP sequences, their ensemble conformations, and the regulation of their biological functions. These equations also describe the molecular properties of IDP sequences that predict similarities and dissimilarities in their functions and facilitate classification of sequences by function, an unmet challenge to traditional bioinformatics. These physical sequence-patterning metrics offer a promising new avenue for advancing synthetic biology at a time when multiple novel functional modes mediated by IDPs are emerging.

The sequence-ensemble relationship in fuzzy protein complexes

Intrinsically disordered proteins (IDPs) interact with globular proteins through a variety of mechanisms, resulting in the structurally heterogeneous ensembles known as fuzzy complexes. While there exists a reasonable comprehension on how IDP sequence determines the unbound IDP ensemble, little is known about what shapes the structural characteristics of IDPs bound to their targets. Using a statistical thermodynamic model, we show that the target-bound ensembles are determined by a simple code that combines the IDP sequence and the distribution of IDP-target interaction hotspots. These two parameters define the conformational space of target-bound IDPs and rationalize the observed structural heterogeneity of fuzzy complexes. The presented model successfully reproduces the dynamical signatures of target-bound IDPs from the NMR relaxation experiments as well as the changes of interaction affinity and the IDP helicity induced by mutations. The model explains how the target-bound IDP ensemble adapts to mutations in order to achieve an optimal balance between conformational freedom and interaction energy. Taken together, the presented sequence-ensemble relationship of fuzzy complexes explains the different manifestations of IDP disorder in folding-upon-binding processes.

Exposing the distinctive modular behavior of β-strands and α-helices in folded proteins

Although folded proteins are commonly depicted as simplistic combinations of β-strands and α-helices, the actual properties and functions of these secondary-structure elements in their native contexts are just partly understood. The principal reason is that the behavior of individual β- and α-elements is obscured by the global folding cooperativity. In this study, we have circumvented this problem by designing frustrated variants of the mixed α/β-protein S6, which allow the structural behavior of individual β-strands and α-helices to be targeted selectively by stopped-flow kinetics, X-ray crystallography, and solution-state NMR. Essentially, our approach is based on provoking intramolecular "domain swap." The results show that the α- and β-elements have quite different characteristics: The swaps of β-strands proceed via global unfolding, whereas the α-helices are free to swap locally in the native basin. Moreover, the α-helices tend to hybridize and to promote protein association by gliding over to neighboring molecules. This difference in structural behavior follows directly from hydrogen-bonding restrictions and suggests that the protein secondary structure defines not only tertiary geometry, but also maintains control in function and structural evolution. Finally, our alternative approach to protein folding and native-state dynamics presents a generally applicable strategy for in silico design of protein models that are computationally testable in the microsecond-millisecond regime.

The Anti-Aggregation Holdase Hsp33 Promotes the Formation of Folded Protein Structures

Holdase chaperones are known to be central to suppressing aggregation, but how they affect substrate conformations remains poorly understood. Here, we use optical tweezers to study how the holdase Hsp33 alters folding transitions within single maltose binding proteins and aggregation transitions between maltose binding protein substrates. Surprisingly, we find that Hsp33 not only suppresses aggregation but also guides the folding process. Two modes of action underlie these effects. First, Hsp33 binds unfolded chains, which suppresses aggregation between substrates and folding transitions within substrates. Second, Hsp33 binding promotes substrate states in which most of the chain is folded and modifies their structure, possibly by intercalating its intrinsically disordered regions. A statistical ensemble model shows how Hsp33 function results from the competition between these two contrasting effects. Our findings reveal an unexpectedly comprehensive functional repertoire for Hsp33 that may be more prevalent among holdases and dispels the notion of a strict chaperone hierarchy.

De novo protein design by citizen scientists

Online citizen science projects such as GalaxyZoo¹, Eyewire² and Phylo³ have been very successful for data collection, annotation, and processing, but for the most part have harnessed human pattern recognition skills rather than human creativity. An exception is the game EteRNA⁴, in which game players learn to build new RNA structures by exploring the discrete two-dimensional space of Watson-Crick base pairing possibilities. Building new proteins, however, is a more challenging task to present in a game, as both the representation and evaluation of a protein structure are intrinsically three-dimensional. We posed the challenge of de novo protein design in the online protein folding game Foldit⁵. Players were presented with a fully extended peptide chain and challenged to craft a folded protein structure with an amino acid sequence encoding that structure. After many iterations of player design, analysis of the top scoring solutions, and subsequent game improvement, Foldit players can now, starting from an extended polypeptide chain, generate a diversity of protein structures and sequences which encode them in silico. 146 Foldit player designs with sequences unrelated to naturally occurring proteins were encoded in synthetic genes; 56 were found to be expressed in E. coli with good solubility and to adopt stable monomeric folded structures in solution. The diversity of these structures is unprecedented in de novo protein design, representing 20 different folds—including a new fold not observed in natural proteins. High resolution structures were determined for four of the designs, and are nearly identical to the player models. This work makes explicit the considerable implicit knowledge contributing to success in de novo protein design, and shows that citizen scientists can discover creative new solutions to outstanding scientific challenges, such as the protein design problem.

Assembly Mechanism for Aggregation of Amyloid Fibrils

The assembly mechanism for aggregation of amyloid fibril is important and fundamental for any quantitative and physical descriptions because it needs to have a deep understanding of both molecular and statistical physics. A theoretical model with three states including coil, helix and sheet is presented to describe the amyloid formation. The corresponding general mathematical expression of N molecule systems are derived, including the partition function and thermodynamic quantities. We study the equilibrium properties of the system in the solution and find that three molecules have the extreme value of free energy. The denaturant effect on molecular assemble is also discussed. Furthermore, we apply the kinetic theories to take account of the nucleation and growth of the amyloid in the solution. It has been shown that our theoretical results can be compared with experimental results.

Secondary structures in synthetic polypeptides from N-carboxyanhydrides: design, modulation, association, and material applications

This article highlights the conformation-specific properties and functions of synthetic polypeptides derived from N-carboxyanhydrides.

Peptide strand length controls the energetics of self-assembly and morphology of β-sheet fibrils

Self-assembling peptides can be used as versatile, natural, and multifunctional building blocks to produce a variety of well-defined nanostructures, materials and devices for applications in medicine and nanotechnology. Here, we concentrate on the 1D self-assembly of de novo designed Px-2 peptide β-strands into anti-parallel β-sheet tapes and higher order aggregates. We study six members of the Px-2 family, ranging from 3 amino acids (aa) to 13 aa in length, using a range of complementary experimental techniques, computer simulation and theoretical statistical mechanics. The critical concentration for self-assembly (c*) is found to increase systematically with decreasing peptide length. The shortest peptide found to self-assemble into soluble β-tapes in water is a 5 amino acid residue peptide. These investigations help decipher the role of the peptide length in controlling self-assembly, aggregate morphology, and material properties. By extracting free energies from these data using a statistical mechanical analysis and combining the results with computer simulations at the atomistic level, we can extract the entropy of association for individual β-strands.

Folding Cooperativity of Synthetic Polypeptides with or without "Tertiary" Interactions

Model-based studies on helix-coil transition and folding cooperativity of synthetic polypeptides have contributed to the understanding of protein folding and stability and to the development of polypeptide-based functional materials. Polypeptide-containing macromolecules with complex architectures, however, remain a challenge in the model-based analysis. Herein, a modified Schellman-Zimm-Bragg model has been utilized to quantitatively analyze the folding cooperativity of polypeptide-containing macromolecules. While the helix-coil transition of homopolypeptides (e.g., poly(ε-benzyloxycarbonyl-l-lysine) (PZLL)) can be described by the classic model, the folding of grafted polypeptide chains in the comb macromolecules (e.g., polynorbornene-g-poly(ε-benzyloxycarbonyl-l-lysine) (PN-g-PZLL)) cannot be accurately predicted by the existing theories, due to the side-chain interactions between grafted polypeptides in the comb macromolecules. Incorporating nonlocal interaction explicability into the statistical mechanics treatment is found to be instructive to account for the possible "tertiary" interactions of polypeptides in the macromolecules with complex architectures.

Revisiting the Helical Cooperativity of Synthetic Polypeptides in Solution

Using synthetic polypeptides as a model system, the theories of helix-coil transition were developed into one of the most beautiful and fruitful subjects in macromolecular science. The classic models proposed by Schellman and Zimm-Bragg more than 50 years ago, differ in the assumption on whether the configuration of multiple helical sequences separated by random coil sections is allowed in a longer polypeptide chain. Zimm also calculated the critical chain lengths that facilitate such interrupted helices in different solvent conditions. The experimental validation of Zimm's prediction, however, was not carefully examined at that time. Herein, we synthesize a series of homopolypeptide samples with different lengths, to systematically examine their helix-coil transition and folding cooperativity in solution. We find that for longer chains, polypeptides do exist as interrupted helices with scattered coil sections even in helicogenic solvent conditions, as predicted in the Zimm-Bragg model. The critical chain lengths that facilitate such interrupted helices, however, are substantially smaller than Zimm's original estimation. The inaccuracy is in part due to an approximation that Zimm made in simplifying the calculation. But more importantly, we find there exist intramolecular interactions between different structural segments in the longer polypeptides, which are not considered in the classic helix-coil theories. As such, even the Zimm-Bragg model in its exact form cannot fully describe the transition behavior and folding cooperativity of longer polypeptides. The results suggest that long "all-helix" chains may be much less prevalent in solution than previously imagined, and a revised theory is required to accurately account for the helix-coil transition of the longer chains with potential "non-local" intramolecular interactions.

Theory of amyloid fibril nucleation from folded proteins

We present a theoretical model for the nucleation of amyloid fibrils. In our model we use helix-coil theory to describe the equilibrium between a soluble native state and an aggregation-prone unfolded state. We then extend the theory to include oligomers with β-sheet cores and calculate the free energy of these states using estimates for the energies of H-bonds, steric zipper interactions, and the conformational entropy cost of forming secondary structure. We find that states with fewer than ~10 β-strands are unstable relative to the dissociated state and three β-strands is the highest free energy state. We then use a modified version of Classical Nucleation Theory to compute the nucleation rate of fibrils from a supersaturated solution of monomers, dimers, and trimers. The nucleation rate has a non-monotonic dependence on denaturant concentration reflecting the competing effects of destabilizing the fibril and increasing the concentration of unfolded monomers. We estimate heterogeneous nucleation rates and discuss the application of our model to secondary nucleation.

Real-time monitoring of hydrophobic aggregation reveals a critical role of cooperativity in hydrophobic effect

The hydrophobic interaction drives nonpolar solutes to aggregate in aqueous solution, and hence plays a critical role in many fundamental processes in nature. An important property intrinsic to hydrophobic interaction is its cooperative nature, which is originated from the collective motions of water hydrogen bond networks surrounding hydrophobic solutes. This property is widely believed to enhance the formation of hydrophobic core in proteins. However, cooperativity in hydrophobic interactions has not been successfully characterized by experiments. Here, we quantify cooperativity in hydrophobic interactions by real-time monitoring the aggregation of hydrophobic solute (hexaphenylsilole, HPS) in a microfluidic mixer. We show that association of a HPS molecule to its aggregate in water occurs at sub-microsecond, and the free energy change is −5.8 to −13.6 kcal mol⁻¹. Most strikingly, we discover that cooperativity constitutes up to 40% of this free energy. Our results provide quantitative evidence for the critical role of cooperativity in hydrophobic interactions.

Hydrophobic interactions occur between nonpolar molecules in water and their experimental quantification can help the understanding of biological self-assembly. Here Jiang et al. examine the kinetics and thermodynamics of hydrophobic aggregation in a bulk environment and characterize its cooperativity.

Protein Folding Structures: Formation of Folding Structures Based on Probability Theory

To the best of our knowledge, this is the first study that shows that the X-ray structures of proteins can be dissected into their continuous folding structure units. Each folding structure unit was designed such that both the terminal di- or tri-peptide sequences shared common sequences with the two adjacent folding structure units. To encode the folding structure information of proteins into their amino acid sequences, we proposed 44 kinds of folding elements, which covered all of the amino acids in the protein chains, and defined all folding structure units. The folding element was defined to mean a minimum structural piece, which covered the frame of the main chain of each amino acid in a protein chain. A folding structure unit of a local sequence could be fully characterized by the sequential combination of individual folding elements assigned to each amino acid. The folding structure information showed amino acid preferences in various positions in folding structure units. Folding structure formation proceeded on the basis of probability theory. Strikingly, relative formation ability analysis clearly indicated that we can decode the types and the chain length of folding structure units from the amino acid sequence of a protein.

Tuning Oligovalent Biomacromolecular Interfaces Using Double-Layered α-Helical Coiled-Coil Nanoassemblies from Lariat-Type Building Blocks

The target affinity and selectivity of many biomacromolecules depend on the three-dimensional (3D) distribution of multiple ligands on their surfaces. Here, we devised a self-assembly strategy to control the target-tailored 3D distribution of multiple α-helical ligands on a coiled-coil core scaffold using novel lariat-type supramolecular building blocks. Depending on the coiled-coil composition and ligand grafting sites in the lariat building blocks, the structural and functional features of the self-assembled peptide nanostructures (SPNs) could be variably fine-tuned. Using oligovalent protein-RNA (Rev-RRE) interactions as a model system, we demonstrate that longer grafting reinforces the helicity of the peptide ligands, whereas shorter grafting strengthens the target binding affinity of the SPNs in both monovalent and oligovalent interactions. This supramolecular approach should be useful in developing precisely controllable multivalent ligands for biomacromolecular interactions.

Pseudo-one-dimensional nucleation in dilute polymer solutions

Pathogenic protein fibrils have been shown in vitro to have nucleation-dependent kinetics despite the fact that one-dimensional structures do not have the size-dependent surface energy responsible for the lag time in classical theory. We present a theory showing that the conformational entropy of the peptide chains creates a free-energy barrier that is analogous to the translational entropy barrier in higher dimensions. We find that the dynamics of polymer rearrangement make it very unlikely for nucleation to succeed along the lowest free-energy trajectory, meaning that most of the nucleation flux avoids the free-energy saddle point. We use these results to construct a three-dimensional model for amyloid nucleation that accounts for conformational entropy, backbone H bonds, and side-chain interactions to compute nucleation rates as a function of concentration.

A theoretical method to compute sequence dependent configurational properties in charged polymers and proteins

A general formalism to compute configurational properties of proteins and other heteropolymers with an arbitrary sequence of charges and non-uniform excluded volume interaction is presented. A variational approach is utilized to predict average distance between any two monomers in the chain. The presented analytical model, for the first time, explicitly incorporates the role of sequence charge distribution to determine relative sizes between two sequences that vary not only in total charge composition but also in charge decoration (even when charge composition is fixed). Furthermore, the formalism is general enough to allow variation in excluded volume interactions between two monomers. Model predictions are benchmarked against the all-atom Monte Carlo studies of Das and Pappu [Proc. Natl. Acad. Sci. U. S. A. 110, 13392 (2013)] for 30 different synthetic sequences of polyampholytes. These sequences possess an equal number of glutamic acid (E) and lysine (K) residues but differ in the patterning within the sequence. Without any fit parameter, the model captures the strong sequence dependence of the simulated values of the radius of gyration with a correlation coefficient of R(2) = 0.9. The model is then applied to real proteins to compare the unfolded state dimensions of 540 orthologous pairs of thermophilic and mesophilic proteins. The excluded volume parameters are assumed similar under denatured conditions, and only electrostatic effects encoded in the sequence are accounted for. With these assumptions, thermophilic proteins are found-with high statistical significance-to have more compact disordered ensemble compared to their mesophilic counterparts. The method presented here, due to its analytical nature, is capable of making such high throughput analysis of multiple proteins and will have broad applications in proteomic studies as well as in other heteropolymeric systems.

Topological and sequence information predict that foldons organize a partially overlapped and hierarchical structure

It has been suggested that proteins have substructures, called foldons, which can cooperatively fold into the native structure. However, several prior investigations define foldons in various ways, citing different foldon characteristics, thereby making the concept of a foldon ambiguous. In this study, we perform a Gō model simulation and analyze the characteristics of substructures that cooperatively fold into the native-like structure. Although some results do not agree well with the experimental evidence due to the simplicity of our coarse-grained model, our results strongly suggest that cooperatively folding units sometimes organize a partially overlapped and hierarchical structure. This view makes us easy to interpret some different proposal about the foldon as a difference of the hierarchical structure. On the basis of this finding, we present a new method to assign foldons and their hierarchy, using structural and sequence information. The results show that the foldons assigned by our method correspond to the intermediate structures identified by some experimental techniques. The new method makes it easy to predict whether a protein folds sequentially into the native structure or whether some foldons fold into the native structure in parallel.

Unraveling the forming mechanism of hierarchical helices via self-assembly of an achiral supramolecular polymer brush

We propose a detailed structural transition of the hierarchical helix or superhelix of PVP–PDP from straightforward experimental evidence.

General mechanism of two-state protein folding kinetics

We describe here a general model of the kinetic mechanism of protein folding. In the Foldon Funnel Model, proteins fold in units of secondary structures, which form sequentially along the folding pathway, stabilized by tertiary interactions. The model predicts that the free energy landscape has a volcano shape, rather than a simple funnel, that folding is two-state (single-exponential) when secondary structures are intrinsically unstable, and that each structure along the folding path is a transition state for the previous structure. It shows how sequential pathways are consistent with multiple stochastic routes on funnel landscapes, and it gives good agreement with the 9 order of magnitude dependence of folding rates on protein size for a set of 93 proteins, at the same time it is consistent with the near independence of folding equilibrium constant on size. This model gives estimates of folding rates of proteomes, leading to a median folding time in Escherichia coli of about 5 s.

Effects of desolvation barriers and sidechains on local–nonlocal coupling and chevron behaviors in coarse-grained models of protein folding

Coarse-grained protein chain models with desolvation barriers or sidechains lead to stronger local–nonlocal coupling and more linear chevron plots.

Difference of carboxybetaine and oligo(ethylene glycol) moieties in altering hydrophobic interactions: a molecular simulation study

Polycarboxybetaine and poly(ethylene glycol) materials resist nonspecific protein adsorption but differ in influencing biological functions such as enzymatic activity. To investigate this difference, we studied the influence of carboxybetaine and oligo(ethylene glycol) moieties on hydrophobic interactions using molecular simulations. We employed a model system composed of two non-polar plates and studied the potential of mean force of plate-plate association in carboxybetaine, (ethylene glycol)4, and (ethylene glycol)2 solutions using well-tempered metadynamics simulations. Water, trimethylamine N-oxide, and urea solutions were used as reference systems. We analyzed the variation of the potential of mean force in various solutions to study how carboxybetaine and oligo(ethylene glycol) moieties influence the hydrophobic interactions. To study the origin of their influence, we analyzed the normalized distributions of moieties and water molecules using molecular dynamics simulations. The simulation results showed that oligo(ethylene glycol) moieties repel water molecules away from the non-polar plates and weaken the hydrophobic interactions. Carboxybetaine moieties do not repel water molecules away from the plates and therefore do not influence the hydrophobic interactions.

Assessing the effect of dynamics on the closed-loop protein-folding hypothesis

The closed-loop (loop-n-lock) hypothesis of protein folding suggests that loops of about 25 residues, closed through interactions between the loop ends (locks), play an important role in protein structure. Coarse-grain elastic network simulations, and examination of loop lengths in a diverse set of proteins, each supports a bias towards loops of close to 25 residues in length between residues of high stability. Previous studies have established a correlation between total contact distance (TCD), a metric of sequence distances between contacting residues (cf. contact order), and the log-folding rate of a protein. In a set of 43 proteins, we identify an improved correlation (r² = 0.76), when the metric is restricted to residues contacting the locks, compared to the equivalent result when all residues are considered (r² = 0.65). This provides qualified support for the hypothesis, albeit with an increased emphasis upon the importance of a much larger set of residues surrounding the locks. Evidence of a similar-sized protein core/extended nucleus (with significant overlap) was obtained from TCD calculations in which residues were successively eliminated according to their hydrophobicity and connectivity, and from molecular dynamics simulations. Our results suggest that while folding is determined by a subset of residues that can be predicted by application of the closed-loop hypothesis, the original hypothesis is too simplistic; efficient protein folding is dependent on a considerably larger subset of residues than those involved in lock formation.

Stabilization and anomalous hydration of collagen fibril under heating

BACKGROUND

Type I collagen is the most common protein among higher vertebrates. It forms the basis of fibrous connective tissues (tendon, chord, skin, bones) and ensures mechanical stability and strength of these tissues. It is known, however, that separate triple-helical collagen macromolecules are unstable at physiological temperatures. We want to understand the mechanism of collagen stability at the intermolecular level. To this end, we study the collagen fibril, an intermediate level in the collagen hierarchy between triple-helical macromolecule and tendon.

METHODOLOGY/PRINCIPAL FINDING

When heating a native fibril sample, its Young's modulus decreases in temperature range 20-58°C due to partial denaturation of triple-helices, but it is approximately constant at 58-75°C, because of stabilization by inter-molecular interactions. The stabilization temperature range 58-75°C has two further important features: here the fibril absorbs water under heating and the internal friction displays a peak. We relate these experimental findings to restructuring of collagen triple-helices in fibril. A theoretical description of the experimental results is provided via a generalization of the standard Zimm-Bragg model for the helix-coil transition. It takes into account intermolecular interactions of collagen triple-helices in fibril and describes water adsorption via the Langmuir mechanism.

CONCLUSION/SIGNIFICANCE

We uncovered an inter-molecular mechanism that stabilizes the fibril made of unstable collagen macromolecules. This mechanism can be relevant for explaining stability of collagen.

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.

Comparing a simple theoretical model for protein folding with all-atom molecular dynamics simulations

Advances in computing have enabled microsecond all-atom molecular dynamics trajectories of protein folding that can be used to compare with and test critical assumptions of theoretical models. We show that recent simulations by the Shaw group (10, 11, 14, 15) are consistent with a key assumption of an Ising-like theoretical model that native structure grows in only a few regions of the amino acid sequence as folding progresses. The distribution of mechanisms predicted by simulating the master equation of this native-centric model for the benchmark villin subdomain, with only two adjustable thermodynamic parameters and one temperature-dependent kinetic parameter, is remarkably similar to the distribution in the molecular dynamics trajectories.

Statistical mechanical treatments of protein amyloid formation

Protein aggregation is an important field of investigation because it is closely related to the problem of neurodegenerative diseases, to the development of biomaterials, and to the growth of cellular structures such as cyto-skeleton. Self-aggregation of protein amyloids, for example, is a complicated process involving many species and levels of structures. This complexity, however, can be dealt with using statistical mechanical tools, such as free energies, partition functions, and transfer matrices. In this article, we review general strategies for studying protein aggregation using statistical mechanical approaches and show that canonical and grand canonical ensembles can be used in such approaches. The grand canonical approach is particularly convenient since competing pathways of assembly and dis-assembly can be considered simultaneously. Another advantage of using statistical mechanics is that numerically exact solutions can be obtained for all of the thermodynamic properties of fibrils, such as the amount of fibrils formed, as a function of initial protein concentration. Furthermore, statistical mechanics models can be used to fit experimental data when they are available for comparison.

Structure and mechanism of maximum stability of isolated alpha-helical protein domains at a critical length scale

The stability of alpha helices is important in protein folding, bioinspired materials design, and controls many biological properties under physiological and disease conditions. Here we show that a naturally favored alpha helix length of 9 to 17 amino acids exists at which the propensity towards the formation of this secondary structure is maximized. We use a combination of thermodynamical analysis, well-tempered metadynamics molecular simulation and statistical analyses of experimental alpha helix length distributions and find that the favored alpha helix length is caused by a competition between alpha helix folding, unfolding into a random coil and formation of higher-order tertiary structures. The theoretical result is suggested to be used to explain the statistical distribution of the length of alpha helices observed in natural protein structures. Our study provides mechanistic insight into fundamental controlling parameters in alpha helix structure formation and potentially other biopolymers or synthetic materials. The result advances our fundamental understanding of size effects in the stability of protein structures and may enable the design of de novo alpha-helical protein materials.

Hydrophobic enhancement of Dopa-mediated adhesion in a mussel foot protein

Dopa (3,4-dihydroxyphenylalanine) is recognized as a key chemical signature of mussel adhesion and has been adopted into diverse synthetic polymer systems. Dopa's notorious susceptibility to oxidation, however, poses significant challenges to the practical translation of mussel adhesion. Using a surface forces apparatus to investigate the adhesion of mussel foot protein 3 (Mfp3) "slow", a hydrophobic protein variant of the Mfp3 family in the plaque, we have discovered a subtle molecular strategy correlated with hydrophobicity that appears to compensate for Dopa instability. At pH 3, where Dopa is stable, Mfp3 slow, like Mfp3 "fast" adhesion to mica, is directly proportional to the mol % of Dopa present in the protein. At pH of 5.5 and 7.5, however, loss of adhesion in Mfp3 slow was less than half that occurring in Mfp3 fast, purportedly because Dopa in Mfp3 slow is less prone to oxidation. Indeed, cyclic voltammetry showed that the oxidation potential of Dopa in Mfp3 slow is significantly higher than in Mfp3 fast at pH of 7.5. A much greater difference between the two variants was revealed in the interaction energy of two symmetric Mfp3 slow films (E(ad) = -3 mJ/m(2)). This energy corresponds to the energy of protein cohesion which is notable for its reversibility and pH independence. Exploitation of aromatic hydrophobic sequences to protect Dopa against oxidation as well as to mediate hydrophobic and H-bonding interactions between proteins provides new insights for developing effective artificial underwater adhesives.

Coil-helix transition in poly(L-glutamic acid): Evidence for a 3-state non-cooperative process

A careful analysis of measurements of circular dichroism of poly(L-glutamic acid) (PGA) shows that the data can be very accurately described by introducing a third state for the PGA configuration, in addition to the helix and coil ones, and considering a simple equilibrium between these three states, without cooperativity. The third state is more conspicuous when high molecular weight polyethyleneglycol (PEG) is added. Excluded-volume effects shown by differences in the presence of short and long PEG chains suggest a direct interaction of PEG and PGA rather than an osmotic effect.

Structural propensities and entropy effects in peptide helix-coil transitions

The helix-coil transition in peptides is a critical structural transition leading to functioning proteins. Peptide chains have a large number of possible configurations that must be accounted for in statistical mechanical investigations. Using hydrogen bond and local helix propensity interaction terms, we develop a method for obtaining and incorporating the degeneracy factor that allows the exact calculation of the partition function for a peptide as a function of chain length. The partition function is used in calculations for engineered peptide chains of various lengths that allow comparison with a variety of different types of experimentally measured quantities, such as fraction of helicity as a function of both temperature and chain length, heat capacity, and denaturation studies. When experimental sensitivity in helicity measurements is properly accounted for in the calculations, the calculated curves fit well with the experimental curves. We determine values of interaction energies for comparison with known biochemical interactions, as well as quantify the difference in the number of configurations available to an amino acid in a random coil configuration compared to a helical configuration.

50 Years of Lifson-Roig Models: Application to Molecular Simulation Data

Simple helix-coil transition theories have been indispensable tools in the analysis of data reporting on the reversible folding of α-helical polypeptides. They provide a transferable means to not only characterize different systems but to also compare different techniques, viz., experimental probes monitoring helix-coil transitions in vitro or biomolecular force fields in silico. This article addresses several issues with the application of Lifson-Roig theory to helix-coil transition data. We use computer simulation to generate two sets of ensembles for the temperature-controlled, reversible folding of the 21-residue, alanine-rich FS peptide. Ensembles differ in the rigidity of backbone bond angles and are analyzed using two distinct descriptors of helicity. The analysis unmasks an underlying phase diagram that is surprisingly complex. The complexities give rise to fitted nucleation and propagation parameters that are difficult to interpret and that are inconsistent with the distribution of isolated residues in the α-helical basin. We show that enthalpies of helix formation are more robustly determined using van't Hoff analysis of simple measures of helicity rather than fitted propagation parameters. To overcome some of these issues, we design a simple variant of the Lifson-Roig model that recovers physical interpretability of the obtained parameters by allowing bundle formation to be described in simple fashion. The relevance of our results is discussed in relation to the applicability of Lifson-Roig models to both in silico and in vitro data.

Cooperativity, local-nonlocal coupling, and nonnative interactions: principles of protein folding from coarse-grained models

Coarse-grained, self-contained polymer models are powerful tools in the study of protein folding. They are also essential to assess predictions from less rigorous theoretical approaches that lack an explicit-chain representation. Here we review advances in coarse-grained modeling of cooperative protein folding, noting in particular that the Levinthal paradox was raised in response to the experimental discovery of two-state-like folding in the late 1960s, rather than to the problem of conformational search per se. Comparisons between theory and experiment indicate a prominent role of desolvation barriers in cooperative folding, which likely emerges generally from a coupling between local conformational preferences and nonlocal packing interactions. Many of these principles have been elucidated by native-centric models, wherein nonnative interactions may be treated perturbatively. We discuss these developments as well as recent applications of coarse-grained chain modeling to knotted proteins and to intrinsically disordered proteins.

Structural basis of folding cooperativity in model proteins: insights from a microcanonical perspective

Two-state cooperativity is an important characteristic in protein folding. It is defined by a depletion of states that lie energetically between folded and unfolded conformations. There are different ways to test for two-state cooperativity; however, most of these approaches probe indirect proxies of this depletion. Generalized-ensemble computer simulations allow us to unambiguously identify this transition by a microcanonical analysis on the basis of the density of states. Here, we present a detailed characterization of several helical peptides obtained by coarse-grained simulations. The level of resolution of the coarse-grained model allowed to study realistic structures ranging from small α-helices to a de novo three-helix bundle without biasing the force field toward the native state of the protein. By linking thermodynamic and structural features, we are able to show that whereas short α-helices exhibit two-state cooperativity, the type of transition changes for longer chain lengths because the chain forms multiple helix nucleation sites, stabilizing a significant population of intermediate states. The helix bundle exhibits signs of two-state cooperativity owing to favorable helix-helix interactions, as predicted from theoretical models. A detailed analysis of secondary and tertiary structure formation fits well into the framework of several folding mechanisms and confirms features that up to now have been observed only in lattice models.

Effect of hydrophilic polymer conjugation on heat-induced conformational changes in a protein

End-functional 2-methacryloyloxyethyl phosphorylcholine (MPC) co-polymers containing two different monomer units, 2-hydroxyethyl methacrylate (HEMA) and n-butyl methacrylate (BMA), with varying hydrophilicities were synthesized to investigate the effect of the conjugated hydrophilic polymer on the heat-induced conformational changes of a protein. MPC co-polymer-conjugated proteins containing the HEMA unit (PMH) could withstand thermal conformational changes better than those containing the more hydrophobic BMA unit (PMB). The changes in protein tertiary structures were estimated via the excitation of tryptophan. PMH-conjugated proteins could withstand heat-induced intensity changes better than the PMB-conjugated proteins. Thus, hydrophilic units in the conjugated polymer are probably essential in suppressing the heat-induced conformational changes of a protein. The changes in secondary and tertiary structures of poly(MPC)-(PMPC) and poly(HEMA) (PHEMA)-conjugated proteins were compared to validate the effect of MPC units on heat-induced conformational change. Although the thermally induced conformational changes in the secondary and tertiary structures of PHEMA-conjugated proteins were partially suppressed, the effect on PMPC-conjugated proteins was much greater, with significant conformational preservation. This is due to the specific hydration state of the hydrophilic PMPC chain, which reduces interaction between the protein molecules.

What drives amyloid molecules to assemble into oligomers and fibrils?

We develop a theory for three states of equilibrium of amyloid peptides: the monomer, oligomer, and fibril. We assume that the oligomeric state is a disordered micellelike collection of a few peptide chains held together loosely by hydrophobic interactions into a spherical hydrophobic core. We assume that fibrillar amyloid chains are aligned and further stabilized by steric zipper interactions-hydrogen bonding, steric packing, and specific hydrophobic side-chain contacts. The model makes a broad set of predictions that are consistent with experimental results: 1), Similar to surfactant micellization, amyloid oligomerization should increase with peptide concentration in solution. 2), The onset of fibrillization limits the concentration of oligomers in the solution. 3), The extent of Aβ fibrillization increases with peptide concentration. 4), The predicted average fibril length versus monomer concentration agrees with data on α-synuclein. 5), Full fibril length distributions agree with data on α-synuclein. 6), Denaturants should melt out fibrils. And finally, 7), added salt should stabilize fibrils by reducing repulsions between amyloid peptide chains. It is of interest that small changes in solvent conditions can tip the equilibrium balance between oligomer and fibril and cause large changes in rates through effects on the transition-state barrier. This model may provide useful insights into the physical processes underlying amyloid diseases.

Interplay between secondary and tertiary structure formation in protein folding cooperativity

Protein folding cooperativity is defined by the nature of the finite-size thermodynamic transition exhibited upon folding: two-state transitions show a free-energy barrier between the folded and unfolded ensembles, while downhill folding is barrierless. A microcanonical analysis, where the energy is the natural variable, has proved to be better suited than its canonical counterpart to unambiguously characterize the nature of the transition. Replica-exchange molecular dynamics simulations of a high-resolution coarse-grained model allow for the accurate evaluation of the density of states in order to extract precise thermodynamic information and measure its impact on structural features. The method has been applied to three helical peptides: a short helix shows sharp features of a two-state folder, while a longer helix and a three-helix bundle exhibit downhill and two-state transitions, respectively. Extending the results of lattice simulations and theoretical models, we have found that it is the interplay between secondary structure and the loss of non-native tertiary contacts that determines the nature of the transition.

The susceptibility of α-helical secondary structure to steric strain: Coarse-grained simulation of dendronized polypeptides

The propensity of a peptide chain for adopting helical secondary structure can be modulated not only through the solvation properties of its side chains but also through their size and shape. Here we examine a coarse-grained model for dendronized polypeptides that focuses on the susceptibility of α-helical structure to the steric strain exerted by hydrophilic pendant groups. Undecorated molecules exhibit a pronounced transition from random coil to helix upon cooling [J. P. Kemp and J. Z. Y. Chen, Biomacromolecules 2, 389 (2001)]. As gauged by specific heat and by order parameters characterizing helicity at several length scales, this transition is quite robust to the introduction of first- and second-generation dendron side chains. More highly branched side chains, however, reduce the entropy of compact states so severely that helical ordering is undetectable over the entire temperature range accessible to our importance sampling methods. Consistent with experimental observations for side chains comparable to those of our model in volume-excluding size and shape, we find the backbone of these third-generation molecules to assume a distended rodlike state that is both stiff and achiral.

Explicit Water Models Affect the Specific Solvation and Dynamics of Unfolded Peptides While the Conformational Behavior and Flexibility of Folded Peptides Remain Intact

Conventional molecular dynamics simulations on 50 ns to 1 μs time scales were used to study the effects of explicit solvent models on the conformational behavior and solvation of two oligopeptide solutes: α-helical EK-peptide (14 amino acids) and a β-hairpin chignolin (10 amino acids). The widely used AMBER force fields (ff99, ff99SB, and ff03) were combined with four of the most commonly used explicit solvent models (TIP3P, TIP4P, TIP5P, and SPC/E). Significant differences in the specific solvation of chignolin among the studied water models were identified. Chignolin was highly solvated in TIP5P, whereas reduced specific solvation was found in the TIP4P, SPC/E, and TIP3P models for kinetic, thermodynamic, and both kinetic and thermodynamic reasons, respectively. The differences in specific solvation did not influence the dynamics of structured parts of the folded peptide. However, substantial differences between TIP5P and the other models were observed in the dynamics of unfolded chignolin, stability of salt bridges, and specific solvation of the backbone carbonyls of EK-peptide. Thus, we conclude that the choice of water model may affect the dynamics of flexible parts of proteins that are solvent-exposed. On the other hand, all water models should perform similarly for well-structured folded protein regions. The merits of the TIP3P model include its high and overestimated mobility, which accelerates simulation processes and thus effectively increases sampling.

Cooperative formation of native-like tertiary contacts in the ensemble of unfolded states of a four-helix protein

In studies of the ensembles of unfolded structures of a four-helix bundle protein, we have detected the presence of potential precursors of native tertiary structures. These observations were based on the perturbation of NMR chemical shifts of the protein backbone atoms by single site mutations. Some mutations change the chemical shifts of residues remote from the site of mutation indicating the presence of an interaction between the mutated and the remote residues, suggesting that the formation of helix segments and helix-helix interactions is cooperative. We can begin to track down the folding mechanism of this protein using only experimental data by combining the information available for the rate limiting structure formation during the folding process with measurements of the site specific hydrogen bond formation in the burst phase, and with the existence prior to the folding reaction of tertiary structures in the ensemble of otherwise unfolded structures observed in the present study.