Spatial propagation of protein polymerization.

We consider the spatial dependence of filamentous protein self-assembly. Through studying the cases where the spreading of aggregated material is dominated either by diffusion or by growth, we derive analytical results for the spatial evolution of filamentous protein aggregation, which we validate against Monte Carlo simulations. Moreover, we compare the predictions of our theory with experimental measurements of two systems for which we identify the propagation as either growth or diffusion controlled. Our results connect the macroscopic observables that characterize the spatial propagation of protein self-assembly with the underlying microscopic processes and provide physical limits on spatial propagation and prionlike behavior associated with protein aggregation.

Influence of the fluctuations of the alignment tensor on the analysis of the structure and dynamics of proteins using residual dipolar couplings.

It has been suggested that the fluctuations of the alignment tensor can affect the results of procedures for characterizing the structure and the dynamics of proteins using residual dipolar couplings. We show here that the very significant fluctuations of the steric alignment tensor caused by the dynamics of proteins can be safely ignored when they do not correlate with those of the bond vectors. A detailed analysis of these correlations in the protein ubiquitin reveals that their effects are negligible for the analysis of backbone motions within secondary structure elements, but also that they may be significant in turns, loops and side chains, especially for bond vectors that have small residual dipolar couplings. Our results suggest that methods that explicitly consider the motions of the alignment tensor will be needed to study the large-scale structural fluctuations that take place on the millisecond timescale, which are often important for the biological function of proteins, from residual dipolar coupling measurements.

Form of growing strings.

Patterns and forms adopted by nature are often the results of simple dynamical paradigms. Here we show that a growing self-interacting string attached to a tracking origin, modeled to resemble nascent polypeptides in vivo, develops helical structures which are more pronounced at the growing end. We also show that the dynamic growth ensemble shares several features of an equilibrium ensemble in which the growing end of the polymer is under an effective stretching force. A statistical analysis of native states of proteins shows that the signature of this nonequilibrium phenomenon has been fixed by evolution at the C terminus, the growing end of a nascent protein. These findings suggest how evolution may have built on the properties of a generic nonequilibrium growth process in favoring helical structures in nascent chains.

Small-world view of the amino acids that play a key role in protein folding.

We use geometrical considerations to provide a different perspective on the fact that a few selected amino acids, the so-called "key residues," act as nucleation centers for protein folding. By constructing graphs corresponding to protein structures we show that they have the "small-world" feature of having a limited set of vertices with large connectivity. These vertices correspond to the key residues that play the role of "hubs" in the network of interactions that stabilize the structure of the transition state.

Statistical properties of contact vectors.

We study the statistical properties of contact vectors, a construct to characterize a protein's structure. The contact vector of an N-residue protein is a list of N integers n(i), representing the number of residues in contact with residue i. We study analytically (at mean-field level) and numerically the amount of structural information contained in a contact vector. Analytical calculations reveal that a large variance in the contact numbers reduces the degeneracy of the mapping between contact vectors and structures. Exact enumeration for lengths up to N=16 on the three-dimensional cubic lattice indicates that the growth rate of number of contact vectors as a function of N is only 3% less than that for contact maps. In particular, for compact structures we present numerical evidence that, practically, each contact vector corresponds to only a handful of structures. We discuss how this information can be used for better structure prediction.

How to guarantee optimal stability for most representative structures in the Protein Data Bank.

We proposed recently an optimization method to derive energy parameters for simplified models of protein folding. The method is based on the maximization of the thermodynamic average of the overlap between protein native structures and a Boltzmann ensemble of alternative structures. Such a condition enforces protein models whose ground states are most similar to the corresponding native states. We present here an extensive testing of the method for a simple residue-residue contact energy function and for alternative structures generated by threading. The optimized energy function guarantees high stability and a well-correlated energy landscape to most representative structures in the PDB database. Failures in the recognition of the native structure can be attributed to the neglect of interactions between different chains in oligomeric proteins or with cofactors. When these are taken into account, only very few X-ray structures are not recognized. Most of them are short inhibitors or fragments and one is a structure that presents serious inconsistencies. Finally, we discuss the reasons that make NMR structures more difficult to recognizeCopyright 2001 Wiley-Liss, Inc.

Three key residues form a critical contact network in a protein folding transition state.

Determining how a protein folds is a central problem in structural biology. The rate of folding of many proteins is determined by the transition state, so that a knowledge of its structure is essential for understanding the protein folding reaction. Here we use mutation measurements--which determine the role of individual residues in stabilizing the transition state--as restraints in a Monte Carlo sampling procedure to determine the ensemble of structures that make up the transition state. We apply this approach to the experimental data for the 98-residue protein acylphosphatase, and obtain a transition-state ensemble with the native-state topology and an average root-mean-square deviation of 6 A from the native structure. Although about 20 residues with small positional fluctuations form the structural core of this transition state, the native-like contact network of only three of these residues is sufficient to determine the overall fold of the protein. This result reveals how a nucleation mechanism involving a small number of key residues can lead to folding of a polypeptide chain to its unique native-state structure.

Assessment of the quality of energy functions for protein folding by using a criterion derived with the help of the noisy go model.

We propose a simple criterion based on the Z-scoreto assess the quality of energy functions for protein folding: one should obtain Z>-10 for the equilibrium ensembleat about native conditions. We derive this criterionby studying a Go model with random errors added to the native interactions. The dependence of the Z-score on the thermodynamic parameters,including the noise, can be precisely obtained in this case,as the ground state of the model is known exactly.We apply this criterion to rapidly rule out two otherwise promisingpairwise energy approximations.The advantage of adopting the present criterionis that it is not necessary to know the ground state of an energy function to assess its quality. It is sufficient to compute the Z-scorefrom a single equilibrium simulation at around the folding temperature.

Comparison of two optimization methods to derive energy parameters for protein folding: perceptron and Z score.

Two methods were proposed recently to derive energy parameters from known native protein conformations and corresponding sets of decoys. One is based on finding, by means of a perceptron learning scheme, energy parameters such that the native conformations have lower energies than the decoys. The second method maximizes the difference between the native energy and the average energy of the decoys, measured in terms of the width of the decoys' energy distribution (Z-score). Whereas the perceptron method is sensitive mainly to "outlier" (i.e., extremal) decoys, the Z-score optimization is governed by the high density regions in decoy-space. We compare the two methods by deriving contact energies for two very different sets of decoys: the first obtained for model lattice proteins and the second by threading. We find that the potentials derived by the two methods are of similar quality and fairly closely related. This finding indicates that standard, naturally occurring sets of decoys are distributed in a way that yields robust energy parameters (that are quite insensitive to the particular method used to derive them). The main practical implication of this finding is that it is not necessary to fine-tune the potential search method to the particular set of decoys used.

Toward an energy function for the contact map representation of proteins.

We analyzed several energy functions for predicting the native state of proteins from an energy minimization procedure. We derived the parameters of a given energy function by imposing the basic requirement that the energy of the native conformation of a protein is lower than that of any conformation chosen from a set of decoys. Our work is motivated by a recent result which proved that the simple pairwise contact approximation of the energy is insufficient to satisfy simultaneously such a basic requirement for all the proteins in a database. Here, we investigate the reasons of such negative results and show how to improve the predictive power of methods based on energy minimization. We generated decoys by gapless threading, and we derive energy parameters by perceptron learning. We first considered hydrophobic contributions to the energy, defined in several ways, and showed that the additional hydrophobic terms enlarge slightly the number of proteins that can be stabilized together. Next, we performed various modifications of the pairwise energy term. We introduced (1) a distinction between inter-residue contacts on the surface and in the core of a protein and (2) a simple distance-dependent pairwise interaction in which a two-tier definition of contact replaces the original (single-tier) one. Our results suggest that a detailed treatment of the pairwise potential is likely to be more relevant than the consideration of other forces.

A statistical mechanical method to optimize energy functions for protein folding.

We present a method for deriving energy functions for protein folding by maximizing the thermodynamic average of the overlap with the native state. The method has been tested by using the pairwise contact approximation of the energy function and generating alternative structures by threading sequences over a database of 1, 169 structures. With the derived energy function, most native structures: (i) have minimal energy and (ii) are thermodynamically rather stable, and (iii) the corresponding energy landscapes are smooth. Precisely, 92% of the 1,013 x-ray structures are stabilized. Most failures can be attributed to the neglect of interactions between chains forming polychain proteins and of interactions with cofactors. When these are considered, only nine cases remain unexplained. In contrast, 38% of NMR structures are not assigned properly.

Protein folding using contact maps.

We discuss the problem of representations of protein structure and give the definition of contact maps. We present a method to obtain a three-dimensional polypeptide conformation from a contact map. We also explain how to deal with the case of nonphysical contact maps. We describe a stochastic method to perform dynamics in contact map space. We explain how the motion is restricted to physical regions of the space. First, we introduce the exact free energy of a contact map and discuss two simple approximations to it. Second, we present a method to derive energy parameters based on perception learning. We prove in an extensive number of situations that the pairwise contact approximation both when alone and when supplemented with a hydrophobic term is unsuitable for stabilizing proteins' native states.

Can a pairwise contact potential stabilize native protein folds against decoys obtained by threading?

We present a method to derive contact energy parameters from large sets of proteins. The basic requirement on which our method is based is that for each protein in the database the native contact map has lower energy than all its decoy conformations that are obtained by threading. Only when this condition is satisfied one can use the proposed energy function for fold identification. Such a set of parameters can be found (by perceptron learning) if Mp, the number of proteins in the database, is not too large. Other aspects that influence the existence of such a solution are the exact definition of contact and the value of the critical distance Rc, below which two residues are considered to be in contact. Another important novel feature of our approach is its ability to determine whether an energy function of some suitable proposed form can or cannot be parameterized in a way that satisfies our basic requirement. As a demonstration of this, we determine the region in the (Rc, Mp) plane in which the problem is solvable, i.e., we can find a set of contact parameters that stabilize simultaneously all the native conformations. We show that for large enough databases the contact approximation to the energy cannot stabilize all the native folds even against the decoys obtained by gapless threading.

Neutral evolution of model proteins: diffusion in sequence space and overdispersion.

We stimulate the evolution of model protein sequences subject to mutations. A mutation is considered neutral if it conserves (1) the structure of the ground state, (2) its thermodynamic stability and (3) its kinetic accessibility. All other mutations are considered lethal and are rejected. We adopt a lattice model, amenable to a reliable solution of the protein folding problem. We prove the existence of extended neutral networks in sequence space-sequences can evolve until their similarity with the starting point is almost the same as for random sequences. Furthermore, we find that the rate of neutral mutations has a broad distribution in sequence space. Due to this fact, the substitution process is overdispersed (the ratio between variance and mean is larger than 1). This result is in contrast with the simplest model of neutral evolution, which assumes a Poisson process for substitutions, and in qualitative agreement with the biological data.

Folding Lennard-Jones proteins by a contact potential.

We studied the possibility to approximate a Lennard-Jones interaction by a pairwise contact potential. First we used a Lennard-Jones potential to design off-lattice, protein-like heteropolymer sequences, whose lowest energy (native) conformations were then identified by molecular dynamics. Then we turned to investigate whether one can find a pairwise contact potential, whose ground states are the contact maps associated with these native conformations. We show that such a requirement cannot be satisfied exactly, i.e., no such contact parameters exist. Nevertheless, we found that one can find contact energy parameters for which an energy minimization procedure, acting in the space of contact maps, yields maps whose corresponding structures are close to the native ones. Finally, we show that when these structures are used as the initial point of a molecular dynamics energy minimization process, the correct native folds are recovered with high probability.

Efficient dynamics in the space of contact maps.

BACKGROUND: Two problems are of major importance in protein fold prediction: how to generate plausible conformations, and how to choose an energy function to identify the native state. Contact maps are a simple representation of protein structure and offer a promising framework to address these two issues. RESULTS: In this work we develop Monte Carlo dynamics in contact map space. The procedure is divided into four steps: non-local dynamics, in which large-scale "cluster" moves are performed (clusters are in approximate correspondence with secondary structure elements); local dynamics, in which secondary structure location is optimized; reconstruction, in which the physicality of the contact map is restored; and refinement, which consists of a further Monte Carlo energy minimization in real space. We demonstrate that such a dynamical procedure is effective in producing uncorrelated low-energy states. CONCLUSIONS: The procedure introduced in this paper very effectively generates a representative ensemble of conformations. We are able to show that existing sets of pairwise contact energy parameters are not suitable to single out the native state within this ensemble. The remaining outstanding issue in protein folding is to find an energy function that can discriminate the native state from decoys.

Recovery of protein structure from contact maps.

BACKGROUND: Prediction of a protein's structure from its amino acid sequence is a key issue in molecular biology. While dynamics, performed in the space of two-dimensional contact maps, eases the necessary conformational search, it may also lead to maps that do not correspond to any real three-dimensional structure. To remedy this, an efficient procedure is needed to reconstruct three-dimensional conformations from their contact maps. RESULTS: We present an efficient algorithm to recover the three-dimensional structure of a protein from its contact map representation. We show that when a physically realizable map is used as target, our method generates a structure whose contact map is essentially similar to the target. furthermore, the reconstructed and original structures are similar up to the resolution of the contact map representation. Next, we use nonphysical target maps, obtained by corrupting a physical one; in this case, our method essentially recovers the underlying physical map and structure. Hence, our algorithm will help to fold proteins, using dynamics in the space of contact maps. Finally, we investigate the manner in which the quality of the recovered structure degrades when the number of contacts is reduced. CONCLUSIONS: The procedure is capable of assigning quickly and reliably a three-dimensional structure to a given contact map. It is well suited for use in parallel with dynamics in contact map space to project a contact map onto its closest physically allowed structural counterpart.