Unified perspective on proteins: a physics approach

A geometrical framework for thinking about proteins

We present a model, based on symmetry and geometry, for proteins. Using elementary ideas from mathematics and physics, we derive the geometries of discrete helices and sheets. We postulate a compatible solvent-mediated emergent pairwise attraction that assembles these building blocks, while respecting their individual symmetries. Instead of seeking to mimic the complexity of proteins, we look for a simple abstraction of reality that yet captures the essence of proteins. We employ analytic calculations and detailed Monte Carlo simulations to explore some consequences of our theory. The predictions of our approach are in accord with experimental data. Our framework provides a rationalization for understanding the common characteristics of proteins. Our results show that the free energy landscape of a globular protein is pre-sculpted at the backbone level, sequences and functionalities evolve in the fixed backdrop of the folds determined by geometry and symmetry, and that protein structures are unique in being simultaneously characterized by stability, diversity, and sensitivity.

Rigidity Dictates Spontaneous Helix Formation of Thermoresponsive Colloidal Chains in Poor Solvent

The formation of helical motifs typically requires specific directional interactions. Here, we demonstrate that isotropic interparticle attraction can drive self-assembly of colloidal chains into thermo-reversible helices, for chains with a critical level of backbone rigidity. We prepare thermoresponsive colloidal chains by cross-linking PNIPAM microgel-coated polystyrene colloids ("monomers"), aligned in an AC electric field. We control the chain rigidity by varying cross-linking time. Above the LCST of PNIPAM, there is an effective attraction between monomers so that the colloidal chains are in a bad solvent. On heating, the chains decrease in size. For the most rigid chains, the decrease is modest and is not accompanied by a change in shape. Much less rigid chains form relatively compact structures, resulting in a large increase in the local monomer density. Unusually, chains with intermediate rigidity spontaneously assemble into helical structures. The chain helicity increases with temperature and plateaus above the collapse transition temperature of the microgel particles. We simulate a minimal model that captures the spontaneous emergence of the helical conformations of the polymeric chain and provides insight into this shape transition. Our work suggests that a purely mechanical instability for semiflexible filaments can drive helix formation, without the need to invoke directional interactions.

Reframing the Protein Folding Problem: Entropy as Organizer

It has been a long-standing conviction that a protein's native fold is selected from a vast number of conformers by the optimal constellation of enthalpically favorable interactions. In marked contrast, this Perspective introduces a different mechanism, one that emphasizes conformational entropy as the principal organizer in protein folding while proposing that the conventional view is incomplete. This mechanism stems from the realization that hydrogen bond satisfaction is a thermodynamic necessity. In particular, a backbone hydrogen bond may add little to the stability of the native state, but a completely unsatisfied backbone hydrogen bond would be dramatically destabilizing, shifting the U(nfolded) ⇌ N(ative) equilibrium far to the left. If even a single backbone polar group is satisfied by solvent when unfolded but buried and unsatisfied when folded, that energy penalty alone, approximately +5 kcal/mol, would rival almost the entire free energy of protein stabilization, typically between -5 and -15 kcal/mol under physiological conditions. Consequently, upon folding, buried backbone polar groups must form hydrogen bonds, and they do so by assembling scaffolds of α-helices and/or strands of β-sheet, the only conformers in which, with rare exception, hydrogen bond donors and acceptors are exactly balanced. In addition, only a few thousand viable scaffold topologies are possible for a typical protein domain. This thermodynamic imperative winnows the folding population by culling conformers with unsatisfied hydrogen bonds, thereby reducing the entropy cost of folding. Importantly, conformational restrictions imposed by backbone···backbone hydrogen bonding in the scaffold are sequence-independent, enabling mutation─and thus evolution─without sacrificing the structure.

Building blocks of protein structures: Physics meets biology

The native state structures of globular proteins are stable and well packed indicating that self-interactions are favored over protein-solvent interactions under folding conditions. We use this as a guiding principle to derive the geometry of the building blocks of protein structures-α helices and strands assembled into β sheets-with no adjustable parameters, no amino acid sequence information, and no chemistry. There is an almost perfect fit between the dictates of mathematics and physics and the rules of quantum chemistry. Protein evolution is facilitated by sequence-independent platforms, which can elaborate sequence-dependent functional diversity. Our work highlights the vital role of discreteness in life and may have implications for the creation of artificial life and on the nature of life elsewhere in the cosmos.

Protein folding - seeing is deceiving

This Perspective is intended to raise questions about the conventional interpretation of protein folding. According to the conventional interpretation, developed over many decades, a protein population can visit a vast number of conformations under unfolding conditions, but a single dominant native population emerges under folding conditions. Accordingly, folding comes with a substantial loss of conformational entropy. How is this price paid? The conventional answer is that favorable interactions between and among the side chains can compensate for entropy loss, and moreover, these interactions are responsible for the structural particulars of the native conformation. Challenging this interpretation, the Perspective introduces a proposal that high energy (i.e., unfavorable) excluding interactions winnow the accessible population substantially under physical-chemical conditions that favor folding. Both steric clash and unsatisfied hydrogen bond donors and acceptors are classified as excluding interactions, so called because conformers with such disfavored interactions will be largely excluded from the thermodynamic population. Both excluding interactions and solvent factors that induce compactness are somewhat nonspecific, yet together they promote substantial chain organization. Moreover, proteins are built on a backbone scaffold consisting of α-helices and strands of β-sheet, where the number of hydrogen bond donors and acceptors is exactly balanced. These repetitive secondary structural elements are the only two conformers that can be both completely hydrogen-bond satisfied and extended indefinitely without encountering a steric clash. Consequently, the number of fundamental folds is limited to no more than ~10,000 for a protein domain. Once excluding interactions are taken into account, the issue of "frustration" is largely eliminated and the Levinthal paradox is resolved. Putting the "bottom line" at the top: it is likely that hydrogen-bond satisfaction represents a largely under-appreciated parameter in protein folding models.

Transient helix formation in charged semiflexible polymers without confinement effects

Switching on generic interactions e.g. the Coulomb potential or other long ranged spherically symmetric repulsive interactions between monomers of bead-spring model of a semi-flexible polymer induce instabilities in a semiflexible polymer chain to form transient helical structures. Our proposed mechanism could explain the spontaneous emergence of helical order in stiff (bio-) polymers as a chain gets charged from a neutral state. But since the obtained helical structures dissolve away with time, hydrogen bonding (or other additional mechanisms), would be required to form stabilized helical structures as observed in nature (such as in biological macro-molecules). The emergence of the helix is independent of the molecular details of the monomer constituent. The key factors which control the emergence of the helical structure is the persistence length and the charge density. We have avoided using torsional potentials to obtain the transient helical structures. Moreover, we can drive the semiflexible polymer to form helices in a recurring manner by periodically increasing and decreasing the effective charge of the monomers. If the two polymer ends are tethered to two surfaces separated by a distance equal to the contour length of the polymeric chain, which could be in the range 10 nm-μ, the life time of the helical structures formed is increased.

Sequence dependent aggregation of peptides and fibril formation

Deciphering the links between amino acid sequence and amyloid fibril formation is key for understanding protein misfolding diseases. Here we use Monte Carlo simulations to study the aggregation of short peptides in a coarse-grained model with hydrophobic-polar (HP) amino acid sequences and correlated side chain orientations for hydrophobic contacts. A significant heterogeneity is observed in the aggregate structures and in the thermodynamics of aggregation for systems of different HP sequences and different numbers of peptides. Fibril-like ordered aggregates are found for several sequences that contain the common HPH pattern, while other sequences may form helix bundles or disordered aggregates. A wide variation of the aggregation transition temperatures among sequences, even among those of the same hydrophobic fraction, indicates that not all sequences undergo aggregation at a presumable physiological temperature. The transition is found to be the most cooperative for sequences forming fibril-like structures. For a fibril-prone sequence, it is shown that fibril formation follows the nucleation and growth mechanism. Interestingly, a binary mixture of peptides of an aggregation-prone and a non-aggregation-prone sequence shows the association and conversion of the latter to the fibrillar structure. Our study highlights the role of a sequence in selecting fibril-like aggregates and also the impact of a structural template on fibril formation by peptides of unrelated sequences.

Signature of Pareto optimization in the Escherichia coli proteome

Proteins have coevolved with cellular environments to improve or preserve their functions, maintaining at the same time the degree of hydrophobicity necessary to fold correctly and enough solubility to perform their biological roles. Here, we study the Escherichia coli proteome using a Pareto front analysis in the solubility-hydrophobicity space. The results indicate the existence of a Pareto optimal front, a triangle whose vertices correspond to archetypal proteins specialized in distinct tasks, such as regulatory processes, membrane transport, outer-membrane pore formation, catalysis, and binding. The vertices are further enriched with proteins that occupy different subcellular compartments, namely, cytoplasmic, inner membrane, outer membrane, and outer membrane bounded periplasmic space. The combination of various enriching features offers an interpretation of how bacteria use the physico-chemical properties of proteins, both to drive them into their final destination in the cell and to have their tasks accomplished.

Sequence repeats and protein structure

Repeats are frequently found in known protein sequences. The level of sequence conservation in tandem repeats correlates with their propensities to be intrinsically disordered. We employ a coarse-grained model of a protein with a two-letter amino acid alphabet, hydrophobic (H) and polar (P), to examine the sequence-structure relationship in the realm of repeated sequences. A fraction of repeated sequences comprises a distinct class of bad folders, whose folding temperatures are much lower than those of random sequences. Imperfection in sequence repetition improves the folding properties of the bad folders while deteriorating those of the good folders. Our results may explain why nature has utilized repeated sequences for their versatility and especially to design functional proteins that are intrinsically unstructured at physiological temperatures.

Exploration of the relationship between topology and designability of conformations

Protein structures are evolutionarily more conserved than sequences, and sequences with very low sequence identity frequently share the same fold. This leads to the concept of protein designability. Some folds are more designable and lots of sequences can assume that fold. Elucidating the relationship between protein sequence and the three-dimensional (3D) structure that the sequence folds into is an important problem in computational structural biology. Lattice models have been utilized in numerous studies to model protein folds and predict the designability of certain folds. In this study, all possible compact conformations within a set of two-dimensional and 3D lattice spaces are explored. Complementary interaction graphs are then generated for each conformation and are described using a set of graph features. The full HP sequence space for each lattice model is generated and contact energies are calculated by threading each sequence onto all the possible conformations. Unique conformation giving minimum energy is identified for each sequence and the number of sequences folding to each conformation (designability) is obtained. Machine learning algorithms are used to predict the designability of each conformation. We find that the highly designable structures can be distinguished from other non-designable conformations based on certain graphical geometric features of the interactions. This finding confirms the fact that the topology of a conformation is an important determinant of the extent of its designability and suggests that the interactions themselves are important for determining the designability.

Molecular jamming--the cystine slipknot mechanical clamp in all-atom simulations

A recent survey of 17 134 proteins has identified a new class of proteins which are expected to yield stretching induced force peaks in the range of 1 nN. Such high force peaks should be due to forcing of a slip-loop through a cystine ring, i.e., by generating a cystine slipknot. The survey has been performed in a simple coarse grained model. Here, we perform all-atom steered molecular dynamics simulations on 15 cystine knot proteins and determine their resistance to stretching. In agreement with previous studies within a coarse grained structure based model, the level of resistance is found to be substantially higher than in proteins in which the mechanical clamp operates through shear. The large stretching forces arise through formation of the cystine slipknot mechanical clamp and the resulting steric jamming. We elucidate the workings of such a clamp in an atomic detail. We also study the behavior of five top strength proteins with the shear-based mechanostability in which no jamming is involved. We show that in the atomic model, the jamming state is relieved by moving one amino acid at a time and there is a choice in the selection of the amino acid that advances the first. In contrast, the coarse grained model also allows for a simultaneous passage of two amino acids.

Amino acid interaction preferences in proteins

Understanding the key factors that influence the interaction preferences of amino acids in the folding of proteins have remained a challenge. Here we present a knowledge-based approach for determining the effective interactions between amino acids based on amino acid type, their secondary structure, and the contact based environment that they find themselves in the native state structure as measured by their number of neighbors. We find that the optimal information is approximately encoded in a 60 x 60 matrix describing the 20 types of amino acids in three distinct secondary structures (helix, beta strand, and loop). We carry out a clustering scheme to understand the similarity between these interactions and to elucidate a nonredundant set. We demonstrate that the inferred energy parameters can be used for assessing the fit of a given sequence into a putative native state structure.

Mechanical strength of 17,134 model proteins and cysteine slipknots

A new theoretical survey of proteins' resistance to constant speed stretching is performed for a set of 17 134 proteins as described by a structure-based model. The proteins selected have no gaps in their structure determination and consist of no more than 250 amino acids. Our previous studies have dealt with 7510 proteins of no more than 150 amino acids. The proteins are ranked according to the strength of the resistance. Most of the predicted top-strength proteins have not yet been studied experimentally. Architectures and folds which are likely to yield large forces are identified. New types of potent force clamps are discovered. They involve disulphide bridges and, in particular, cysteine slipknots. An effective energy parameter of the model is estimated by comparing the theoretical data on characteristic forces to the corresponding experimental values combined with an extrapolation of the theoretical data to the experimental pulling speeds. These studies provide guidance for future experiments on single molecule manipulation and should lead to selection of proteins for applications. A new class of proteins, involving cystein slipknots, is identified as one that is expected to lead to the strongest force clamps known. This class is characterized through molecular dynamics simulations.

The advances in nanotechnology have allowed for manipulation of single biomolecules and determination of their elastic properties. Titin was among the first proteins studied in this way. Its unravelling by stretching requires a 204 pN force. The resistance to stretching comes mostly from a localized region known as a force clamp. In titin, the force clamp is simple as it is formed by two parallel β-strands that are sheared on pulling. Studies of a set of under a hundred proteins accomplished in the last decade have revealed a variety of the force clamps that lead to forces ranging from under 20 pN to about 500 pN. This set comprises only a tiny fraction of proteins known. Thus one needs guidance as to what proteins should be considered for specific mechanical properties. Such a guidance is provided here through simulations within simplified coarse-grained models on 17 134 proteins that are stretched at constant speed. We correlate their unravelling forces with two structure classification schemes. We identify proteins with large resistance to unravelling and characterize their force clamps. Quite a few top strength proteins owe their sturdiness to a new type of the force clamp: the cystein slipknot in which the force peak is due to dragging of a piece of the backbone through a closed ring formed by two other pieces of the backbone and two connecting disulphide bonds.

A condensation-ordering mechanism in nanoparticle-catalyzed peptide aggregation

Nanoparticles introduced in living cells are capable of strongly promoting the aggregation of peptides and proteins. We use here molecular dynamics simulations to characterise in detail the process by which nanoparticle surfaces catalyse the self-assembly of peptides into fibrillar structures. The simulation of a system of hundreds of peptides over the millisecond timescale enables us to show that the mechanism of aggregation involves a first phase in which small structurally disordered oligomers assemble onto the nanoparticle and a second phase in which they evolve into highly ordered as their size increases.

Protein misfolding and aggregation are associated with a wide variety of human disorders, which include Alzheimer's and Parkinson's diseases and late onset diabetes. It has been recently realised that the process of aggregation may be triggered by the presence of nanoparticles. We use here molecular dynamics simulations to characterise the molecular mechanism by which such nanoparticles are capable of enhancing the rate of formation of peptide aggregates. Our findings indicate that nanoparticle surfaces act as a catalyst that increases the local concentration of peptides, thus facilitating their subsequent assembly into stable fibrillar structures. The approach that we present, in addition to providing a description of the process of aggregation of peptides in the presence of nanoparticles, will enable the study of the mechanism of action of a variety of other potential aggregation-promoting agents present in living organisms, including lipid membranes and other cellular components.

Inter-helical interactions in membrane proteins: analysis based on the local backbone geometry and the side chain interactions

The availability of a significant number of the structures of helical membrane proteins has prompted us to investigate the mode of helix-helix packing. In the present study, we have considered a dataset of alpha-helical membrane proteins representing structures solved from all the known superfamilies. We have described the geometry of all the helical residues in terms of local coordinate axis at the backbone level. Significant inter-helical interactions have been considered as contacts by weighing the number of atom-atom contacts, including all the side-chain atoms. Such a definition of local axis and the contact criterion has allowed us to investigate the inter-helical interaction in a systematic and quantitative manner. We show that a single parameter (designated as alpha), which is derived from the parameters representing the mutual orientation of local axes, is able to accurately capture the details of helix-helix interaction. The analysis has been carried out by dividing the dataset into parallel, anti-parallel, and perpendicular orientation of helices. The study indicates that a specific range of alpha value is preferred for interactions among the anti-parallel helices. Such a preference is also seen among interacting residues of parallel helices, however to a lesser extent. No such preference is seen in the case of perpendicular helices, the contacts that arise mainly due to the interaction of surface helices with the end of the trans-membrane helices. The study supports the prevailing view that the anti-parallel helices are well packed. However, the interactions between helices of parallel orientation are non-trivial. The packing in alpha-helical membrane proteins, which is systematically and rigorously investigated in this study, may prove to be useful in modeling of helical membrane proteins.

Thermodynamics of tubelike flexible polymers

In this work, we discuss the general phase behavior of short tubelike flexible polymers. The geometric thickness constraint is implemented through the concept of the global radius of curvature. We use sophisticated Monte Carlo sampling methods to simulate small bead-stick polymer models with Lennard-Jones interaction among nonbonded monomers. We analyze energetic fluctuations and structural quantities to classify conformational pseudophases. We find that the tube thickness influences the thermodynamic behavior of simple tubelike polymers significantly, i.e., for a given temperature, the formation of secondary structures strongly depends on the tube thickness.

Self-templated nucleation in peptide and protein aggregation

Peptides and proteins exhibit a common tendency to assemble into highly ordered fibrillar aggregates, whose formation proceeds in a nucleation-dependent manner that is often preceded by the formation of oligomeric assemblies. This process has received much attention because disordered oligomeric aggregates have been associated with neurodegenerative disorders such as Alzheimer's and Parkinson's disease. Here we describe a self-templated nucleation mechanism that determines the transition between the initial condensation of polypeptide chains into disordered assemblies and their reordering into fibrillar structures. The results that we present show that at the molecular level this transition is due to the ability of polypeptide chains to reorder within oligomers into fibrillar assemblies whose surfaces act as templates that stabilize the disordered assemblies.

Folding pathways and rates for the three-stranded beta-sheet peptide Beta3s using discrete path sampling

The discrete path sampling method was used to investigate the folding of a three-stranded antiparallel beta-sheet peptide, Beta3s, described by an empirical potential and implicit solvent model. After application of a coarse-graining scheme that groups together sets of minima in local equilibrium, the calculated folding time was in reasonable agreement with other simulations and consistent with the experimental upper bound. The folding mechanism exhibited by the most significant discrete paths involves early formation of the C-terminal hairpin followed by docking of the N-terminal strand.

Emergence of secondary motifs in tubelike polymers in a solvent

We study the effects of two kinds of interactions in tubelike polymers and demonstrate that they result in the formation of secondary motifs. The first has an entropic origin and is a measure of the effective space available to the solvent. The second arises from solvophobic interactions of the solvent with the polymers and leads to an energy proportional to the contact surface between the tube and solvent particles. The solvent molecules are modeled as hard spheres and the two interactions are considered separately with the solvent density affecting their relative strength. In addition to analytical calculations, we present the results of numerical simulations in order to understand the role played by the finite length of short polymers and the discrete versus continuum descriptions of the system in determining the preferred conformation.

Kinetically driven helix formation during the homopolymer collapse process

Using Langevin simulations, we find that simple "generic" bead-and-spring homopolymer chains in a sufficiently bad solvent spontaneously develop helical order during the process of collapsing from an initially stretched conformation. The helix formation is initiated by the unstable modes of the straight chain, which drive the system towards a long-lived metastable transient state. The effect is most pronounced if hydrodynamic interactions are screened.

Denatured state is critical in determining the properties of model proteins designed on different folds

The thermodynamics of proteins designed on three common folds (SH3, chymotrypsin inhibitor 2 [CI2], and protein G) is studied with a simplified C(alpha) model and compared with the thermodynamics of proteins designed on random-generated folds. The model allows to design sequences to fold within a dRMSD ranging from 1.2 to 4.2 A from the crystallographic native conformation and to study properties that are hard to be measured experimentally. It is found that the denatured state of all of them is not random but is, to different extents, partially structured. The degree of structure is more abundant for SH3 and protein G, giving rise to a weaker stability but a more efficient folding kinetics than CI2 and, even more, than the random-generated folds. Consequently, the features of the unfolded state seem to be as important in the determination of the thermodynamic properties of these proteins as the features of the native state.

Alternative hydrogen bond implementations produce opposite effects on collapse cooperativity of lattice homopolypeptide models

We use complete enumeration of self-avoiding chains of up to N=26 monomers in two-dimensional lattices to investigate the effect of alternative implementations of backbone hydrogen bonds on the cooperativity of homopolypeptide collapse. Following a recent study on protein folding models, we use the square lattice with z=3 local conformations per monomer and lattice extensions containing diagonal steps which result in z=5 or z=7 and assume that only a subset of zh<z local conformations is compatible with hydrogen bond formation. As previously observed in heteropolymeric folding, a significant increase in cooperativity, as measured by kappa2 values, results from the coupling between hydrogen bonds and hydrophobic interactions, in such a way that hydrophobic contacts are favorable only when contacting monomers are involved in hydrogen bond formation. For some z/zh combinations the energy distribution is bimodal at the collapse transition temperature. The situation can be regarded as if all hydrophobic contacts actually decrease the energy by the same amount, 2h , with the addition of an energetic increase, epsilon2=h, as a penalty for each contacting monomer not satisfying the hydrogen bond condition. Cooperativity is little affected and might even decrease, however, when hydrogen bonds produce a decrease in energy by the same amount, epsilon1=h, for each bonding monomer. For the more general situation when the hydrogen bond effect is not equal, in modulus, to the hydrophobic interaction, i.e., epsilon2 not equalh or epsilon1 not equal h, we observe a pronounced increase in kappa2 for small epsilon2, with a maximum around epsilon2/h approximately 1.5, followed by a gradual decrease to a limiting value at large epsilon2. The opposite behavior is observed when epsilon1 is varied. The observed qualitative difference is shown to arise from opposite effects on the convexity of the total density of states of the system when subdensities corresponding to different numbers of hydrogen bonds are differently favored as opposed to the case when subdensities corresponding to different numbers of contacting monomers not forming hydrogen bonds are differently unfavored. Potential implications for the cooperativity of protein folding and protein unspecific collapse are discussed.

Physics of proteins

Globular proteins are a key component of the network of life. Over many decades much experimental data on proteins have been gathered, yet theoretical progress has been somewhat limited. We show that the results accumulated over the years inexorably lead to a unified framework for understanding proteins. The framework predicts the existence of a fixed menu of folds determined by geometry, clarifies the role of the amino acid sequence in selecting the native-state structure from this menu, and explains the propensity for amyloid formation. The experimental data and the new approach reveal an astonishing simplicity underlying the protein problem.

Ring polymer simulations with global radius of curvature

We simulate three-dimensional flexible off-lattice ring polymers of length L up to L=4000 for various values of the global radius of curvature Rgrc=0.25 , 0.48, and 1.0 and Rgrc=2.0 . We utilize two different ensembles: one with a delta -function constraint on the radius, and the other with a theta -function. For both cases the global radius of curvature provides a valid regularization of polymers with thickness D=2Rgrc . The Flory-type critical exponent nu SAW of self-avoiding rings at D=2 is found to be nu SAW=0.5869(5) from the radii of gyration chain length scaling, while other D values produce consistent results. For our current implementation, the numerical effort of chain thickness calculations is bounded by a number O(LlnL) per single update. We also study low-temperature configurations of spatially dense Lennard-Jones homopolymers on a ring and identify some conformational building blocks.

What determines the spectrum of protein native state structures?

We present a brief summary of the key factors underlying protein structure, as developed in the investigations of Pauling, Ramachandran, and Rose. We then outline a simplified physical model of proteins that focuses on geometry and symmetry. Although this model superficially appears unrelated to the detailed chemical descriptions commonly applied to proteins, we show that it captures the essential elements of the chemistry and provides a unified framework for understanding the common characteristics of folded proteins. We suggest that the spectrum of protein native state structures is determined by geometry and symmetry and the role of the sequence is to choose its native state structure from this predetermined menu.

A theoretical framework for CNS arousal

Rapid changes of state in central nervous systems (CNS), as required following stimuli that must arouse the CNS from a quiescent state in order to activate a behavioral response, constitute a particularly appropriate application of non-linear dynamics. Chaotic dynamics would provide tremendous amplification of neuronal activity needed for CNS arousal, sensitively dependent on the initial state of the CNS. This theoretical approach is attractive because it supposes dynamics that are deterministic and it links the elegant mathematics of chaos to the conception of a fundamental property of the CNS. However, a living system must be able to exit from chaotic dynamics in order to avoid widely divergent, biologically impossible outcomes. We hypothesize that, analogous to phase transitions in a liquid crystal, CNS arousal systems, having 'woken up the brain' to activate behavior, go through a phase transition and emerge under the control of orderly movement control systems.

A backbone-based theory of protein folding

Under physiological conditions, a protein undergoes a spontaneous disorder order transition called "folding." The protein polymer is highly flexible when unfolded but adopts its unique native, three-dimensional structure when folded. Current experimental knowledge comes primarily from thermodynamic measurements in solution or the structures of individual molecules, elucidated by either x-ray crystallography or NMR spectroscopy. From the former, we know the enthalpy, entropy, and free energy differences between the folded and unfolded forms of hundreds of proteins under a variety of solvent/cosolvent conditions. From the latter, we know the structures of approximately 35,000 proteins, which are built on scaffolds of hydrogen-bonded structural elements, alpha-helix and beta-sheet. Anfinsen showed that the amino acid sequence alone is sufficient to determine a protein's structure, but the molecular mechanism responsible for self-assembly remains an open question, probably the most fundamental open question in biochemistry. This perspective is a hybrid: partly review, partly proposal. First, we summarize key ideas regarding protein folding developed over the past half-century and culminating in the current mindset. In this view, the energetics of side-chain interactions dominate the folding process, driving the chain to self-organize under folding conditions. Next, having taken stock, we propose an alternative model that inverts the prevailing side-chain/backbone paradigm. Here, the energetics of backbone hydrogen bonds dominate the folding process, with preorganization in the unfolded state. Then, under folding conditions, the resultant fold is selected from a limited repertoire of structural possibilities, each corresponding to a distinct hydrogen-bonded arrangement of alpha-helices and/or strands of beta-sheet.

Characterization of the backbone geometry of protein native state structures

We present a simple method for analyzing the geometry of noncovalent residue-residue interactions stabilizing the protein structure, which takes into account the constraints on the local backbone geometry. We find that the principal geometrical constraints are amino acid aspecific and are associated with hydrogen bond formation in helices and sheets. In contrast, amino acid residues in nonhelical and nonextended conformations, which make noncovalent interactions stabilizing the protein tertiary structure, display greater flexibility. We apply the method to an analysis of the packing of helices in helical bundle proteins requiring an efficient packing of amino acid side-chains of the interacting helices.

Common attributes of native-state structures of proteins, disordered proteins, and amyloid

We show that a framework derived from the common character of globular proteins can be used to understand the design of protein sequences, the behavior of intrinsically unstructured proteins, and the formation of amyloid fibrils in a unified manner. Our studies provide compelling support for the idea that protein native-state structures, the structures adopted by intrinsically unstructured proteins on binding as well as those of amyloid aggregates, all reside in a physical state of matter in which the free energy landscape is sculpted not by the specific sequence of amino acids, but rather by considerations of geometry and symmetry. We elucidate the key role played by sequence design in selecting the structure of choice from the predetermined menu of putative native-state structures.

Geometry of proteins: hydrogen bonding, sterics, and marginally compact tubes

The functionality of proteins is governed by their structure in the native state. Protein structures are made up of emergent building blocks of helices and almost planar sheets. A simple coarse-grained geometrical model of a flexible tube barely subject to compaction provides a unified framework for understanding the common character of globular proteins. We argue that a recent critique of the tube idea is not well founded.

Three-bead rotating chain model shows universality in the stretching of proteins

We introduce a model of proteins in which all of the key atoms in the protein backbone are accounted for, thus extending the freely rotating chain model. We use average bond lengths and average angles from the Protein Data Bank as input parameters, leaving the number of residues as a single variable. The model is used to study the stretching of proteins in the entropic regime. The results of our Monte Carlo simulations are found to agree well with experimental data, suggesting that the force extension plot is universal and does not depend on the side chains or primary structure of the proteins.

Geometric and physical considerations for realistic protein models

Protein structure is generally conceptualized as the global arrangement or of smaller, local motifs of helices, sheets, and loops. These regular, recurring secondary structural elements have well understood and standardized definitions in terms of amino acid backbone geometry and the manner in which hydrogen bonding requirements are satisfied. Recently, "tube" models have been proposed to explain protein secondary structure in terms of the geometrically optimal packing of a featureless cylinder. However, atomically detailed simulations demonstrate that such packing considerations alone are insufficient for defining secondary structure; both excluded volume and hydrogen bonding must be explicitly modeled for helix formation. These results have fundamental implications for the construction and interpretation of realistic and meaningful biomacromolecular models.

Geometrical model for the native-state folds of proteins

We recently introduced a physical model [T.X. Hoang, A. Trovato, F. Seno, J.R. Banavar, A. Maritan, Geometry and symmetry pre-sculpt the free energy landscape of proteins. Proc. Natl. Acad. Sci. U. S. A. 101 (2004) 7960-7964, J.R. Banavar, T.X. Hoang, A. Maritan, F. Seno, A. Trovato, A unified perspective on proteins-a physics approach. Phys. Rev., E 70 (2004) 041905] for proteins which incorporates, in an approximate manner, several key features such as the inherent anisotropy of a chain molecule, the geometrical and energetic constraints placed by the hydrogen bonds and sterics, and the role played by hydrophobicity. Within this framework, marginally compact conformations resembling the native state folds of proteins emerge as broad competing minima in the free energy landscape even for a homopolymer. Here we show how the introduction of sequence heterogeneity using a simple scheme of just two types of amino acids, hydrophobic (H) and polar (P), and sequence design allows a selected putative native fold to become the free energy minimum at low temperature. The folding transition exhibits thermodynamic cooperativity, if one neglects the degeneracy between two different low energy conformations sharing the same fold topology.

Lattice tube model of proteins

We present a new lattice model for proteins that incorporates a tubelike anisotropy by introducing a preference for mutually parallel alignments in the conformations. The model is demonstrated to capture many aspects of real proteins.