Lattice model for rapidly folding protein-like heteropolymers.

Protein folding is a relatively fast process considering the astronomical number of conformations in which a protein could find itself. Within the framework of a lattice model, we show that one can design rapidly folding sequences by assigning the strongest attractive couplings to the contacts present in a target native state. Our protein design can be extended to situations with both attractive and repulsive contacts. Frustration is minimized by ensuring that all the native contacts are again strongly attractive. Strikingly, this ensures the inevitability of folding and accelerates the folding process by an order of magnitude. The evolutionary implications of our findings are discussed.

Exploring the origins of topological frustration: Design of a minimally frustrated model of fragment B of protein A

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This Gō-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht φ values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate φ, whereas highly frustrated systems display a bimodal distribution peaked at low and high φ values. The distribution of φ values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at φ = 0.3 with tails extending to low and high φ values, indicating the presence of a small amount of topological frustration. The contacts with high φ values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of φ values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

Stochastic processes strongly influence HIV-1 evolution during suboptimal protease-inhibitor therapy

It has long been assumed that HIV-1 evolution is best described by deterministic evolutionary models because of the large population size. Recently, however, it was suggested that the effective population size (Ne) may be rather small, thereby allowing chance to influence evolution, a situation best described by a stochastic evolutionary model. To gain experimental evidence supporting one of the evolutionary models, we investigated whether the development of resistance to the protease inhibitor ritonavir affected the evolution of the env gene. Sequential serum samples from five patients treated with ritonavir were used for analysis of the protease gene and the V3 domain of the env gene. Multiple reverse transcription–PCR products were cloned, sequenced, and used to construct phylogenetic trees and to calculate the genetic variation and Ne. Genotypic resistance to ritonavir developed in all five patients, but each patient displayed a unique combination of mutations, indicating a stochastic element in the development of ritonavir resistance. Furthermore, development of resistance induced clear bottleneck effects in the env gene. The mean intrasample genetic variation, which ranged from 1.2% to 5.7% before treatment, decreased significantly (P < 0.025) during treatment. In agreement with these findings, Ne was estimated to be very small (500–15,000) compared with the total HIV-1 RNA copy number. This study combines three independent observations, strong population bottlenecking, small Ne, and selection of different combinations of protease-resistance mutations, all of which indicate that HIV-1 evolution is best described by a stochastic evolutionary model.

Folding funnels and frustration in off-lattice minimalist protein landscapes

A full quantitative understanding of the protein folding problem is now becoming possible with the help of the energy landscape theory and the protein folding funnel concept. Good folding sequences have a landscape that resembles a rough funnel where the energy bias towards the native state is larger than its ruggedness. Such a landscape leads not only to fast folding and stable native conformations but, more importantly, to sequences that are robust to variations in the protein environment and to sequence mutations. In this paper, an off-lattice model of sequences that fold into a β-barrel native structure is used to describe a framework that can quantitatively distinguish good and bad folders. The two sequences analyzed have the same native structure, but one of them is minimally frustrated whereas the other one exhibits a high degree of frustration.

Folding protein models with a simple hydrophobic energy function: The fundamental importance of monomer inside/outside segregation

The present study explores a “hydrophobic” energy function for folding simulations of the protein lattice model. The contribution of each monomer to conformational energy is the product of its “hydrophobicity” and the number of contacts it makes, i.e., E(h⃗, c⃗) = −Σi=1N cihi = −(h⃗.c⃗) is the negative scalar product between two vectors in N-dimensional cartesian space: h⃗ = (h1, … , hN), which represents monomer hydrophobicities and is sequence-dependent; and c⃗ = (c1, … , cN), which represents the number of contacts made by each monomer and is conformation-dependent. A simple theoretical analysis shows that restrictions are imposed concomitantly on both sequences and native structures if the stability criterion for protein-like behavior is to be satisfied. Given a conformation with vector c⃗, the best sequence is a vector h⃗ on the direction upon which the projection of c⃗ − c̄⃗ is maximal, where c̄⃗ is the diagonal vector with components equal to c̄, the average number of contacts per monomer in the unfolded state. Best native conformations are suggested to be not maximally compact, as assumed in many studies, but the ones with largest variance of contacts among its monomers, i.e., with monomers tending to occupy completely buried or completely exposed positions. This inside/outside segregation is reflected on an apolar/polar distribution on the corresponding sequence. Monte Carlo simulations in two dimensions corroborate this general scheme. Sequences targeted to conformations with large contact variances folded cooperatively with thermodynamics of a two-state transition. Sequences targeted to maximally compact conformations, which have lower contact variance, were either found to have degenerate ground state or to fold with much lower cooperativity.

Folding and aggregation of designed proteins

Protein aggregation is studied by following the simultaneous folding of two designed identical 20-letter amino acid chains within the framework of a lattice model and using Monte Carlo simulations. It is found that protein aggregation is determined by elementary structures (partially folded intermediates) controlled by local contacts among some of the most strongly interacting amino acids and formed at an early stage in the folding process.

An iterative method for extracting energy-like quantities from protein structures.

We present a method (ENERGI) for extracting energy-like quantities from a data base of protein structures. In this paper, we use the method to generate pairwise additive amino acid "energy" scores. These scores are obtained by iteration until they correctly discriminate a set of known protein folds from decoy conformations. The method succeeds in lattice model tests and in the gapless threading problem as defined by Maiorov and Crippen [Maiorov, V. N. & Crippen, G. M. (1992) J. Mol. Biol. 227, 876-888]. A more challenging test of threading a larger set of test proteins derived from the representative set of Hobohm and Sander [Hobohm, U. & Sander, C. (1994) Protein Sci. 3, 522-524] is used as a "workbench" for exploring how the ENERGI scores depend on their parameter sets.

Mass vaccination to control chickenpox: the influence of zoster.

The impact of transmission events from patients with shingles (zoster) on the epidemiology of varicella is examined before and after the introduction of mass immunization by using a stochastic mathematical model of transmission dynamics. Reactivation of the virus is shown to damp stochastic fluctuations and move the dynamics toward simple annual oscillations. The force of infection due to zoster cases is estimated by comparison of simulated and observed incidence time series. The presence of infectious zoster cases reduces the tendency for mass immunization to increase varicella incidence at older ages when disease severity is typically greater.

How optimization of potential functions affects protein folding.

The relationship between the optimization of the potential function and the foldability of theoretical protein models is studied based on investigations of a 27-mer cubic-lattice protein model and a more realistic lattice model for the protein crambin. In both the simple and the more complicated systems, optimization of the energy parameters achieves significant improvements in the statistical-mechanical characteristics of the systems and leads to foldable protein models in simulation experiments. The foldability of the protein models is characterized by their statistical-mechanical properties--e.g., by the density of states and by Monte Carlo folding simulations of the models. With optimized energy parameters, a high level of consistency exists among different interactions in the native structures of the protein models, as revealed by a correlation function between the optimized energy parameters and the native structure of the model proteins. The results of this work are relevant to the design of a general potential function for folding proteins by theoretical simulations.

Cooperative enhancement of specificity in a lattice of T cell receptors

Two of the most important models to account for the specificity and sensitivity of the T cell receptor (TCR) are the kinetic proofreading and serial ligation models. However, although kinetic proofreading provides a means for individual TCRs to measure accurately the length of time they are engaged and signal appropriately, the stochastic nature of ligand dissociation means the kinetic proofreading model implies that at high concentrations the response of the cell will be relatively nonspecific. Recent ligand experiments have revealed the phenomenon of both negative and positive crosstalk among neighboring TCRs. By using a Monte Carlo simulation of a lattice of TCRs, we integrate receptor crosstalk with the kinetic proofreading and serial ligation models and discover that receptor cooperativity can enhance T cell specificity significantly at a very modest cost to the sensitivity of the response.

A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems

We propose a general procedure for solving incomplete data estimation problems. The procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model, the nonlinear structural measurement error model, and the random effects model. The procedure is based on the general principle of stochastic approximation and the Markov chain Monte-Carlo method. Applying the theory on adaptive algorithms, we derive conditions under which the proposed procedure converges. Simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model.

Models for stochastic climate prediction

There has been a recent burst of activity in the atmosphere/ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction. Here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory. This analysis demonstrates the potential need for more sophisticated models beyond stable linear Langevin equations. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. The strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta-plane with topography and a mean flow. In this example, the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom.

Folding thermodynamics of a model three-helix-bundle protein

The calculated folding thermodynamics of a simple off-lattice three-helix-bundle protein model under equilibrium conditions shows the experimentally observed protein transitions: a collapse transition, a disordered-to-ordered globule transition, a globule to native-state transition, and the transition from the active native state to a frozen inactive state. The cooperativity and physical origin of the various transitions are explored with a single “optimization” parameter and characterized with the Lindemann criterion for liquid versus solid-state dynamics. Below the folding temperature, the model has a simple free energy surface with a single basin near the native state; the surface is similar to that calculated from a simulation of the same three-helix-bundle protein with an all-atom representation [Boczko, E. M. & Brooks III, C. L. (1995) Science 269, 393–396].

Adsorption from a one-dimensional lattice gas and the Brunauer–Emmett–Teller equation

An exact treatment of adsorption from a one-dimensional lattice gas is used to eliminate and correct a well-known inconsistency in the Brunauer–Emmett–Teller (B.E.T.) equation—namely, Gibbs excess adsorption is not taken into account and the Gibbs integral diverges at the transition point. However, neither model should be considered realistic for experimental adsorption systems.