Studies on protein folding, unfolding and fluctuations by computer simulation. III. Effect of short-range interactions.

The theoretical model of proteins on the two-dimensional square lattice, introduced previously, is extended to include the specific short-range interactions. Attractive long-range interactions with various specificities and non-specific repulsive long-range interactions in the form of self-avoidance of the polymer chain are also operative in the model. Dynamics of the model protein is studied by a Monte Carlo method. The short-range interactions are found to accelerate the folding and unfolding transitions. Non-specific part of the attractive long-range interactions have a competing effect of decelerating the transitions. When the short-range interactions are weighted beyond a certain extent over the attractive long-range interactions are weighted beyond a certain extent over the attractive long-range interactions, the all-or-none character of the folding and unfolding transitions is destroyed. How the destruction proceeds is quantitatively expressed in terms of the S-H curves. The limiting case of dominance of the specific short-range interactions over the attractive long-range interactions is studied in detail. The lattice polymer in this limit does not behave like a globular protein at all. This observation leads to a reexamination of the currently popular notion of the dominance of the short-range interactions. A new concept of consistency is proposed to replace it. Possible mechanisms of the acceleration of the transitions by the specific short-range interactions are discussed.     

Role of medium--and long-range interactions in discriminating globular and membrane proteins.

The analysis of inter-residue interactions in protein structures provides considerable insight to understand their folding and stability. We have previously analyzed the role of medium- and long-range interactions in the folding of globular proteins. In this work, we study the distinct role of such interactions in the three-dimensional structures of membrane proteins. We observed a higher number of long-range contacts in the termini of transmembrane helical (TMH) segments, implying their role in the stabilization of helix-helix interactions. The transmembrane strand (TMS) proteins are having appreciably higher long-range contacts than that in all-beta class of globular proteins, indicating closer packing of the strands in TMS proteins. The residues in membrane spanning segments of TMH proteins have 1.3 times higher medium-range contacts than long-range contacts whereas that of TMS proteins have 14 times higher long-range contacts than medium-range contacts. Residue-wise analysis indicates that in TMH proteins, the residues Cys, Glu, Gly, Pro, Gln, Ser and Tyr have higher long-range contacts than medium-range contacts in contrast with all-alpha class of globular proteins. The charged residue pairs have higher medium-range contacts in all-alpha proteins, whereas hydrophobic residue pairs are dominant in TMH proteins. The information on the preference of residue pairs to form medium-range contacts has been successfully used to discriminate the TMH proteins from all-alpha proteins. The statistical significance of the results obtained from the present study has been verified using randomized structures of TMH and TMS protein templates.     

Studies on protein folding,unfolding, and fluctuations by computer simulation. II. A. Three-dimensional lattice model of lysozyme

A three-dimensional lattice model of protein designed to assimilate lysozyme is introduced. An attractive interaction is assumed to work between preassigned specific pairs of units, when they occupy the nearest-nighbor lattice points. The behavior of this lattice lysozyme is studied by a Monte Carlo simulation method. Because of the specific interunit interactions,“native state” of the lattice lysozyme is stable at low temperatures. Conformational fluctuations in the native state are observed to occur at both termini and loop regions of the main chain existing on the surface. The process of unfolding and denatured states of this model are discussed. Complete refolding from a denatured state was not observed. However, by starting from partially folded structures, the native conformation could be attained. From these observation it is concluded that, in the process of folding of proteins as simplified in a lattice model, nulceation is a rate-limiting factor. The artificial character of this model and possible improvement are discussed.     

Studies on protein folding, unfolding and fluctuations by computer simulation. I. The effect of specific amino acid sequence represented by specific inter-unit interactions.

A lattice model of proteins is introduced. "A protein molecule" is a chain of nown-intersecting units of a given length on the two-dimensional square lattice. The copolymeric character of protein molecules is incorporated into the model in the form of specificities of inter-unit interactions. This model proved most effective for studying the statistical mechanical characteristics of protein folding, unfolding and fluctuations. The specificities of inter-unit interactions are shown to be the primary factors responsible for the all-or-none type transition from native to denatured states of globular proteins. The model has been studied by the Monte Carlo method of Metropolis et al., which is now shown applied to approximately simulating a kinetic process. In the strong limit of the specificity of the inter-unit interaction the native conformation was reached in this method by starting from an extended conformation. The possible generalization and application of this method for finding the native conformation of proteins form their amino sequence are discussed.     

A simple lattice model that exhibits a protein-like cooperative all-or-none folding transition

In a recent paper (D. Gront et al., Journal of Chemical Physics, Vol. 115, pp. 1569, 2001) we applied a simple combination of the Replica Exchange Monte Carlo and the Histogram methods in the computational studies of a simplified protein lattice model containing hydrophobic and polar units and sequence-dependent local stiffness. A well-defined, relatively complex Greek-key topology, ground (native) conformations was found; however, the cooperativity of the folding transition was very low. Here we describe a modified minimal model of the same Greek-key motif for which the folding transition is very cooperative and has all the features of the "all-or-none" transition typical of real globular proteins. It is demonstrated that the all-or-none transition arises from the interplay between local stiffness and properly defined tertiary interactions. The tertiary interactions are directional, mimicking the packing preferences seen in proteins. The model properties are compared with other minimal protein-like models, and we argue that the model presented here captures essential physics of protein folding (structurally well-defined protein-like native conformation and cooperative all-or-none folding transition).     

Why do proteins divide into domains? Insights from lattice model simulations

It is known that larger globular proteins are built from domains, relatively independent structural units. A domain size seems to be limited, and a single domain consists of from few tens to a couple of hundred amino acids. Based on Monte Carlo simulations of a reduced protein model restricted to the face centered simple cubic lattice, with a minimal set of short-range and long-range interactions, we have shown that some model sequences upon the folding transition spontaneously divide into separate domains. The observed domain sizes closely correspond to the sizes of real protein domains. Short chains with a proper sequence pattern of the hydrophobic and polar residues undergo a two-state folding transition to the structurally ordered globular state, while similar longer sequences follow a multistate transition. Homopolymeric (uniformly hydrophobic) chains and random heteropolymers undergo a continuous collapse transition into a single globule, and the globular state is much less ordered. Thus, the factors responsible for the multidomain structure of proteins are sufficiently long polypeptide chain and characteristic, protein-like, sequence patterns. These findings provide some hints for the analysis of real sequences aimed at prediction of the domain structure of large proteins.     

The effect of amino acid substitution on protein-folding and -unfolding transition studied by computer simulation

Protein-folding and -unfolding transitions were studied by the method of computer simulation. The protein was modeled as a two-dimensional lattice polymer. Various energy terms were assumed to be operative between units composing the polymer. But hydrophobic interactions were neglected explicitly. Both thermodynamic and kinetic quantities were obtained from the simulation, and from their temperature dependence in the transition zone characteristics of the conformational transition of proteins were discussed. Two amino acid substituted models, differing in the location of substitution, were studied and compared with the original in order to clarify the effect of substitution on conformational transition of proteins. The following conclusions were reached in this study: (1) The relaxation time of the slow mode, which reflects the overall folding and unfolding processes, shows a peak near the transition temperature, while that of the fast mode is almost independent of temperature. The peak of the slow mode occurs at a slightly lower temperature than the transition temperature. (2) The dependence of the logarithm of the rate constants on the inverse of temperature (Arrhenius plot) is linear. Therefore, the plot of the free energy of activation vs temperature is linear. (3) The values of kinetic parameters obtained suggest that in the activated state the intramolecular interactions are half broken, while the state is close to the native state on the entropy axis. (4) The amino acid substitution, which is modeled as having slightly unfavorable short-range interactions, causes the substituted ones to be slightly unstable. Moreover, it causes the folding transition to slow. From the analysis of the way slowing down is observed in the two substituted models, we conclude that a structure, designed to model a β-sheet, is formed before it gets assembled with other structures, which are designed to model α-helices. The process of assembly occurs nearly at the activated state of the folding and unfolding transition. (5) It is suggested from this study that the maximum of folding rate constant in the Arrhenius plot that has been observed experimentally in real proteins is likely due to hydrophobic interactions.     

Noninteracting local-structure model of folding and unfolding transition in globular proteins. II. Application to two-dimensional lattice proteins

The noninteracting local-structure model of the folding and unfolding transition in globular proteins, the formulation of which was given in the preceding paper, is applied to the analysis of the two-dimensional lattice model of proteins. The lattice model of proteins is a theoretical tool designed to study the statistical-mechanical aspect of the folding and unfolding transition. Its dynamics have been studied by a method of Monte Carlo simulation. The noninteracting local-structure model reproduces the equilibrium properties of the lattice model obtained previously by computer simulation remarkably well, when the specificity of the long-range interactions is strong. This observation indicates that the basic assumption of the noninteracting local-structure model is equivalent to the assumption of strong specificity of intramolecular interactions. It is argued that by assuming this strong specificity, we can emphasize the correct main paths of folding and unfolding transition. The way local structures grow and/or merge along the most probable path of folding in the lattice model is discussed by the noninteracting local-structure model.     

不同类型氨基酸网络参量与蛋白质折叠的关系

理论和实验研究表明,蛋白质天然拓扑结构对其折叠过程具有重要的影响．采用复杂网络的方法分析蛋白质天然结构的拓扑特征,并探索蛋白质结构特征与折叠速率之间的内在联系．分别构建了蛋白质氨基酸网络、疏水网、亲水网、亲水-疏水网以及相应的长程网络,研究了这些网络的匹配系数(assortativity coefficient)和聚集系数(clustering coefficient)的统计特性．结果表明,除了亲水-疏水网,上述各网络的匹配系数均为正值,并且氨基酸网和疏水网的匹配系数与折叠速率表现出明显的线性正相关,揭示了疏水残基间相互作用的协同性有助于蛋白质的快速折叠．同时,研究发现疏水网的聚集系数与折叠速率有明显的线性负相关关系,这表明疏水残基间三角结构(triangle construction)的形成不利于蛋白质快速折叠．还进一步构建了相应的长程网络,发现序列上间距较远的残基接触对的形成将使蛋白质折叠进程变慢．     

Influence of medium- and long-range interactions in different folding types of globular proteins

Recognition of protein fold from amino acid sequence is a challenging task. The structure and stability of proteins from different fold are mainly dictated by inter-residue interactions. In our earlier work, we have successfully used the medium- and long-range contacts for predicting the protein folding rates, discriminating globular and membrane proteins and for distinguishing protein structural classes. In this work, we analyze the role of inter-residue interactions in commonly occurring folds of globular proteins in order to understand their folding mechanisms. In the medium-range contacts, the globin fold and four-helical bundle proteins have more contacts than that of DNA-RNA fold although they all belong to all-alpha class. In long-range contacts, only the ribonuclease fold prefers 4-10 range and the other folding types prefer the range 21-30 in alpha/beta class proteins. Further, the preferred residues and residue pairs influenced by these different folds are discussed. The information about the preference of medium- and long-range contacts exhibited by the 20 amino acid residues can be effectively used to predict the folding type of each protein.     

Monte Carlo studies on equilibrium globular protein folding. II. Beta-barrel globular protein models

In the context of dynamic Monte Carlo simulations on a model protein confined to a tetrahedral lattice, the interplay of protein size and tertiary structure, and the requirements for an all-or-none transition to a unique native state, are investigated. Small model proteins having a primary sequence consisting of a central bend neutral region flanked by two tails having an alternating hydrophobic/hydrophilic pattern of residues are seen to undergo a continuous transition to a beta-hairpin collapsed state. On increasing the length of the tails, the beta-hairpin structural motif is found to be in equilibrium with a four-member beta-barrel. Further increase of the tail length results in the shift of the structural equilibrium to the four-member beta-barrel. The random coil to beta-barrel transition is of an all-or-none character, but while the central turn is always the desired native bend, the location of the turns involving the two external strands is variable. That is, beta-barrels having the external stands that are two residues out of register are also observed in the transition region. Introduction into the primary sequence of two additional regions that are at the very least neutral toward turn formation produces an all-or-none transition to the unique, native, four-member beta-barrel. Various factors that can augment the stability of the native conformation are explored. Overall, these folding simulations strongly indicate that the general rules of globular protein folding are rather robust--namely, one requires a general pattern of hydrophobic/hydrophilic residues that allow the protein to have a well-defined interior and exterior and the presence of regions in the amino acid sequence that at the very least are locally indifferent to turn formation. Since no site-specific interactions between hydrophobic and hydrophilic residues are required to produce a unique four-member beta-barrel, these simulations strongly suggest that site specificity is involved in structural fine-tuning.     

Folding pathway dependence on energetic frustration and interaction heterogeneity for a three-dimensional hydrophobic protein model

Monte Carlo simulations of a hydrophobic protein model of 40 monomers in the cubic lattice are used to explore the effect of energetic frustration and interaction heterogeneity on its folding pathway. The folding pathway is described by the dependence of relevant conformational averages on an appropriate reaction coordinate, pfold, defined as the probability for a given conformation to reach the native structure before unfolding. We compare the energetically frustrated and heterogeneous hydrophobic potential, according to which individual monomers have a higher or lower tendency to form contacts unspecifically depending on their hydrophobicities, to an unfrustrated homogeneous Go-type potential with uniformly attractive native interactions and neutral non-native interactions (called Go1 in this study), and to an unfrustrated heterogeneous potential with neutral non-native interactions and native interactions having the same energy as the hydrophobic potential (called Go2 in this study). Folding kinetics are slowed down dramatically when energetic frustration increases, as expected and previously observed in a two-dimensional model. Contrary to our previous results in two dimensions, however, it appears that the folding pathway and transition state ensemble can be significantly dependent on the energy function used to stabilize the native structure. The sequence of events along the reaction coordinate, or the order along this coordinate in which different regions of the native conformation become structured, turns out to be similar for the hydrophobic and Go2 potentials, but with analogous events tending to occur at lower pfold values in the first case. In particular, the transition state obtained from the ensemble around pfold = 0.5 is more structured for the hydrophobic potential. For Go1, not only the transition state ensemble but the order of events itself is modified, suggesting that interaction heterogeneity, in addition to energetic frustration, can have significant effects on the folding mechanism, most likely by modifying the probability of different contacts in the unfolded state, the starting point for the folding reaction. Although based on a simple model, these results provide interesting insight into how sequence-dependent switching between folding pathways might occur in real proteins.     

Hydrophobic folding units at protein-protein interfaces: implications to protein folding and to protein-protein association.

A hydrophobic folding unit cutting algorithm, originally developed for dissecting single-chain proteins, has been applied to a dataset of dissimilar two-chain protein-protein interfaces. Rather than consider each individual chain separately, the two-chain complex has been treated as a single chain. The two-chain parsing results presented in this work show hydrophobicity to be a critical attribute of two-state versus three-state protein-protein complexes. The hydrophobic folding units at the interfaces of two-state complexes suggest that the cooperative nature of the two-chain protein folding is the outcome of the hydrophobic effect, similar to its being the driving force in a single-chain folding. In analogy to the protein-folding process, the two-chain, two-state model complex may correspond to the formation of compact, hydrophobic nuclei. On the other hand, the three-state model complex involves binding of already folded monomers, similar to the association of the hydrophobic folding units within a single chain. The similarity between folding entities in protein cores and in two-state protein-protein interfaces, despite the absence of some chain connectivities in the latter, indicates that chain linkage does not necessarily affect the native conformation. This further substantiates the notion that tertiary, non-local interactions play a critical role in protein folding. These compact, hydrophobic, two-chain folding units, derived from structurally dissimilar protein-protein interfaces, provide a rich set of data useful in investigations of the role played by chain connectivity and by tertiary interactions in studies of binding and of folding. Since they are composed of non-contiguous pieces of protein backbones, they may also aid in defining folding nuclei.     

Amino acid replacements can selectively affect the interaction energy of autonomous folding units in the alpha subunit of tryptophan synthase.

Amino acid replacements were made at the interface between two autonomous folding units in the alpha subunit of tryptophan synthase from Salmonella typhimurium to test their mutual interaction energy. The results of equilibrium studies of the urea-induced unfolding reaction of the wild-type and mutant proteins in which phenylalanine 22 is replaced by leucine, isoleucine, and valine can be understood in terms of a selective decrease in the interaction energy between the two folding units; the intrinsic stability of each folding unit is not significantly altered. Kinetic studies of the rate-limiting step in unfolding show that the interaction energy appears in the transition state preceding the native conformation. Comparisons of the individual effects of these nonpolar side chains show that both hydrophobic and steric effects play important roles in the interaction energy between the folding units. The implication of these results is that the high cooperativity observed in the folding of many globular proteins may be reduced by appropriate amino acid replacements.     

On the relation between native geometry and conformational plasticity

In protein folding the term plasticity refers to the number of alternative folding pathways encountered in response to free energy perturbations such as those induced by mutation. Here we explore the relation between folding plasticity and a gross, generic feature of the native geometry, namely, the relative number of local and non-local native contacts. The results from our study, which is based on Monte Carlo simulations of simple lattice proteins, show that folding to a structure that is rich in local contacts is considerably more plastic than folding to a native geometry characterized by having a very large number of long-range contacts (i.e., contacts between amino acids that are separated by more than 12 units of backbone distance). The smaller folding plasticity of native geometries is probably a direct consequence of their higher folding cooperativity that renders the folding reaction more robust against single- and multiple-point mutations.     

Molecular basis of cooperativity in protein folding IV. Core: A general cooperative folding model

The cooperative nature of the protein folding process is independent of the characteristic fold and the specific secondary structure attributes of a globular protein. A general folding/unfolding model should, therefore, be based upon structural features that transcend the peculiarities of α-helices, β-sheets, and other structural motifs found in proteins. The studies presented in this paper suggest that a single structural characteristic common to all globular proteins is essential for cooperative folding. The formation of a partly folded state from the native state results in the exposure to solvent of two distinct regions: (1) the portions of the protein that are unfolded; and (2) the “complementary surfaces,” located in the regions of the protein that remain folded. The cooperative character of the folding/unfolding transition is determined largely by the energetics of exposing complementary surface regions to the solvent. By definition, complementary regions are present only in partly folded states; they are absent from the native and unfolded states. An unfavorable free energy lowers the probability of partly folded states and increases the cooperativity of the transition. In this paper we present a mathematical formulation of this behavior and develop a general cooperative folding/unfolding model, termed the “complementary region” (CORE) model. This model successfully reproduces the main properties of folding/unfolding transitions without limiting the number of partly folded states accessible to the protein, thereby permitting a systematic examination of the structural and solvent conditions under which intermediates become populated. It is shown that the CORE model predicts two-state folding/unfolding behavior, even though the two-state character is not assumed in the model. © 1993 Wiley-Liss, Inc.     

Monte Carlo studies on equilibrium globular protein folding. III. The four helix bundle

The nature of the equilibrium conformational transition from the denatured state to a four-member alpha-helical bundle was studied employing a dynamic Monte Carlo algorithm in which the model protein chain was confined to a tetrahedral lattice. The model chain was allowed to hunt over all phase space, the target native state was not assumed a priori, and no site-specific interactions were introduced. The exterior vs the interior part of the protein is distinguished by the pattern of hydrophilic and hydrophobic interactions encoded into the primary sequence. The importance of a statistical preference for forming bends, as a function of bend location in the primary sequence, and helical wheel type cooperative interactions were examined, and the necessary conditions for collapse of the chain to the unique native structure were investigated. It was found that an amphipathic pattern of hydrophobic/hydrophilic interactions along with a statistical preference of the central residues for bend formation are sufficient to obtain the four-helix bundle. The transition to the native state has an all-or-none character.     

A contribution to the theory of biological staining based on the principles for structural organization of biological macromolecules

Biological staining is to a large degree explainable based on the principles governing folding and aggregation of macromolecules in aqueous solution. Most macromolecules are polyions, which, except for heteropolysaccharides, have a large proportion of nonpolar or only slightly polar residues. Because they are amphiphilic, they react in water by a complex set of hydrophobic interactions involving charged residues, nonpolar residues and water molecules. The hydrophobic interactions lead to complex folding systems or micelle-like structures. Dyes are amphiphilic molecules with a tendency to form micelles, but with limitations due to geometric constraints and charge repulsion. Macromolecules and dyes react with each other in aqueous solution following the same principles as for the structural organization of macromolecules, as in protein folding for example. Dye binding requires near contact between nonpolar groups in both the dye and macromolecule, and this is accomplished by choosing a pH at which the dye and macromolecule have opposite net charges. Charge attraction is insufficient for binding in most cases, but it is directive because it determines which macromolecules a given dye ion is able to contact. These considerations apply to the staining of globular (cytoplasmic) proteins and to nucleic acid staining. The staining mechanism is by hydrophobic interactions. Above approximately pH 3.5, DNA may also bind dyes by hydrophobic intercalation between the bases of the double helix; at lower pH the double helix opens and dye binding is as for RNA and globular proteins. Heteroglycans (mucins) have virtually no nonpolar groups, so nonpolar interactions are restricted to the dye molecules. Metachromatic staining of heteroglycans is due to hydrophobic bonding or micelle formation between the monovalent planar dye molecules aided by charge neutralization by the negatively charged heteroglycans. Alternatively, as the charge attraction increases with the number of closely placed charges, acidic heteroglycans may be stained by a polycation such as alcian blue or colloidal iron. For elastic fiber and collagen staining, actual hydrophobic interactions are less important and hydrogen bonding and simple nonpolar interactions play a major role. These macromolecules may therefore be stained using a nonaqueous alcoholic solution.     

A contribution to the theory of biological staining based on the principles for structural organization of biological macromolecules

Biological staining is to a large degree explainable based on the principles governing folding and aggregation of macromolecules in aqueous solution. Most macromolecules are polyions, which, except for heteropolysaccharides, have a large proportion of nonpolar or only slightly polar residues. Because they are amphiphilic, they react in water by a complex set of hydrophobic interactions involving charged residues, nonpolar residues and water molecules. The hydrophobic interactions lead to complex folding systems or micelle-like structures. Dyes are amphiphilic molecules with a tendency to form micelles, but with limitations due to geometric constraints and charge repulsion. Macromolecules and dyes react with each other in aqueous solution following the same principles as for the structural organization of macromolecules, as in protein folding for example. Dye binding requires near contact between nonpolar groups in both the dye and macromolecule, and this is accomplished by choosing a pH at which the dye and macromolecule have opposite net charges. Charge attraction is insufficient for binding in most cases, but it is directive because it determines which macromolecules a given dye ion is able to contact. These considerations apply to the staining of globular (cytoplasmic) proteins and to nucleic acid staining. The staining mechanism is by hydrophobic interactions. Above approximately pH 3.5, DNA may also bind dyes by hydrophobic intercalation between the bases of the double helix; at lower pH the double helix opens and dye binding is as for RNA and globular proteins. Heteroglycans (mucins) have virtually no nonpolar groups, so nonpolar interactions are restricted to the dye molecules. Metachromatic staining of heteroglycans is due to hydrophobic bonding or micelle formation between the monovalent planar dye molecules aided by charge neutralization by the negatively charged heteroglycans. Alternatively, as the charge attraction increases with the number of closely placed charges, acidic heteroglycans may be stained by a polycation such as alcian blue or colloidal iron. For elastic fiber and collagen staining, actual hydrophobic interactions are less important and hydrogen bonding and simple nonpolar interactions play a major role. These macromolecules may therefore be stained using a nonaqueous alcoholic solution.     

The Gō model revisited: Native structure and the geometric coupling between local and long-range contacts

Monte Carlo simulations show that long-range interactions play a major role in determining the folding rates of 48-mer three-dimensional lattice polymers modeled by the Gō potential. For three target structures with different native geometries we found a sharp increase in the folding time when the relative contribution of the long-range interactions to the native state's energy is decreased from approximately 50% towards zero. However, the dispersion of the simulated folding times is strongly dependent on native geometry and Gō polymers folding to one of the target structures exhibits folding times spanning three orders of magnitude. We have also found that, depending on the target geometry, a strong geometric coupling may exist between local and long-range contacts, which means that, when this coupling exists, the formation of long-range contacts is forced by the previous formation of local contacts. The absence of a strong geometric coupling results in a kinetics that is more sensitive to the interaction energy parameters; in this case, the formation of local contacts is not capable of promoting the establishment of long-range ones when the latter are strongly penalized energetically and this results in longer folding times.