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1.
PLoS Comput Biol ; 8(9): e1002675, 2012.
Article in English | MEDLINE | ID: mdl-22969420

ABSTRACT

HIV protease, an aspartyl protease crucial to the life cycle of HIV, is the target of many drug development programs. Though many protease inhibitors are on the market, protease eventually evades these drugs by mutating at a rapid pace and building drug resistance. The drug resistance mutations, called primary mutations, are often destabilizing to the enzyme and this loss of stability has to be compensated for. Using a coarse-grained biophysical energy model together with statistical inference methods, we observe that accessory mutations of charged residues increase protein stability, playing a key role in compensating for destabilizing primary drug resistance mutations. Increased stability is intimately related to correlations between electrostatic mutations - uncorrelated mutations would strongly destabilize the enzyme. Additionally, statistical modeling indicates that the network of correlated electrostatic mutations has a simple topology and has evolved to minimize frustrated interactions. The model's statistical coupling parameters reflect this lack of frustration and strongly distinguish like-charge electrostatic interactions from unlike-charge interactions for [Formula: see text] of the most significantly correlated double mutants. Finally, we demonstrate that our model has considerable predictive power and can be used to predict complex mutation patterns, that have not yet been observed due to finite sample size effects, and which are likely to exist within the larger patient population whose virus has not yet been sequenced.


Subject(s)
HIV Protease/chemistry , HIV Protease/genetics , Models, Chemical , Models, Genetic , Mutation/genetics , Amino Acid Sequence , Computer Simulation , Enzyme Stability/genetics , Models, Molecular , Molecular Sequence Data , Static Electricity , Statistics as Topic , Structure-Activity Relationship
2.
J Phys Chem B ; 115(6): 1512-23, 2011 Feb 17.
Article in English | MEDLINE | ID: mdl-21254767

ABSTRACT

We present a new approach to study a multitude of folding pathways and different folding mechanisms for the 20-residue mini-protein Trp-Cage using the combined power of replica exchange molecular dynamics (REMD) simulations for conformational sampling, transition path theory (TPT) for constructing folding pathways, and stochastic simulations for sampling the pathways in a high dimensional structure space. REMD simulations of Trp-Cage with 16 replicas at temperatures between 270 and 566 K are carried out with an all-atom force field (OPLSAA) and an implicit solvent model (AGBNP). The conformations sampled from all temperatures are collected. They form a discretized state space that can be used to model the folding process. The equilibrium population for each state at a target temperature can be calculated using the weighted-histogram-analysis method (WHAM). By connecting states with similar structures and creating edges satisfying detailed balance conditions, we construct a kinetic network that preserves the equilibrium population distribution of the state space. After defining the folded and unfolded macrostates, committor probabilities (P(fold)) are calculated by solving a set of linear equations for each node in the network and pathways are extracted together with their fluxes using the TPT algorithm. By clustering the pathways into folding "tubes", a more physically meaningful picture of the diversity of folding routes emerges. Stochastic simulations are carried out on the network, and a procedure is developed to project sampled trajectories onto the folding tubes. The fluxes through the folding tubes calculated from the stochastic trajectories are in good agreement with the corresponding values obtained from the TPT analysis. The temperature dependence of the ensemble of Trp-Cage folding pathways is investigated. Above the folding temperature, a large number of diverse folding pathways with comparable fluxes flood the energy landscape. At low temperature, however, the folding transition is dominated by only a few localized pathways.


Subject(s)
Peptides/chemistry , Algorithms , Amino Acid Sequence , Kinetics , Molecular Dynamics Simulation , Molecular Sequence Data , Protein Folding , Temperature , Thermodynamics
3.
BMC Bioinformatics ; 10 Suppl 8: S10, 2009 Aug 27.
Article in English | MEDLINE | ID: mdl-19758465

ABSTRACT

BACKGROUND: The reaction of HIV protease to inhibitor therapy is characterized by the emergence of complex mutational patterns which confer drug resistance. The response of HIV protease to drugs often involves both primary mutations that directly inhibit the action of the drug, and a host of accessory resistance mutations that may occur far from the active site but may contribute to restoring the fitness or stability of the enzyme. Here we develop a probabilistic approach based on connected information that allows us to study residue, pair level and higher-order correlations within the same framework. RESULTS: We apply our methodology to a database of approximately 13,000 sequences which have been annotated by the treatment history of the patients from which the samples were obtained. We show that including pair interactions is essential for agreement with the mutational data, since neglect of these interactions results in order-of-magnitude errors in the probabilities of the simultaneous occurence of many mutations. The magnitude of these pair correlations changes dramatically between sequences obtained from patients that were or were not exposed to drugs. Higher-order effects make a contribution of as much as 10% for residues taken three at a time, but increase to more than twice that for 10 to 15-residue groups. The sequence data is insufficient to determine the higher-order effects for larger groups. We find that higher-order interactions have a significant effect on the predicted frequencies of sequences with large numbers of mutations. While relatively rare, such sequences are more prevalent after multi-drug therapy. The relative importance of these higher-order interactions increases with the number of drugs the patient had been exposed to. CONCLUSION: Correlations are critical for the understanding of mutation patterns in HIV protease. Pair interactions have substantial qualitative effects, while higher-order interactions are individually smaller but may have a collective effect. Together they lead to correlations which could have an important impact on the dynamics of the evolution of cross-resistance, by allowing the virus to pass through otherwise unlikely mutational states. These findings also indicate that pairwise and possibly higher-order effects should be included in the models of protein evolution, instead of assuming that all residues mutate independently of one another.


Subject(s)
Computational Biology/methods , HIV Protease/genetics , HIV-1/enzymology , HIV-1/genetics , Mutation , Amino Acid Sequence , Databases, Protein , Drug Resistance, Viral , HIV Protease Inhibitors/pharmacology , Linear Models , Models, Molecular , Sequence Alignment , Thermodynamics
4.
J Phys Chem B ; 113(34): 11702-9, 2009 Aug 27.
Article in English | MEDLINE | ID: mdl-19655770

ABSTRACT

We present an approach to recover kinetics from a simplified protein folding model at different temperatures using the combined power of replica exchange (RE), a kinetic network, and effective stochastic dynamics. While RE simulations generate a large set of discrete states with the correct thermodynamics, kinetic information is lost due to the random exchange of temperatures. We show how we can recover the kinetics of a 2D continuous potential with an entropic barrier by using RE-generated discrete states as nodes of a kinetic network. By choosing the neighbors and the microscopic rates between the neighbors appropriately, the correct kinetics of the system can be recovered by running a kinetic simulation on the network. We fine-tune the parameters of the network by comparison with the effective drift velocities and diffusion coefficients of the system determined from short-time stochastic trajectories. One of the advantages of the kinetic network model is that the network can be built on a high-dimensional discretized state space, which can consist of multiple paths not consistent with a single reaction coordinate.


Subject(s)
Computer Simulation , Models, Chemical , Proteins/chemistry , Thermodynamics , Kinetics , Protein Folding
5.
J Phys Chem B ; 112(19): 6083-93, 2008 May 15.
Article in English | MEDLINE | ID: mdl-18251533

ABSTRACT

The efficiency of temperature replica exchange (RE) simulations hinge on their ability to enhance conformational sampling at physiological temperatures by taking advantage of more rapid conformational interconversions at higher temperatures. While temperature RE is a parallel simulation technique that is relatively straightforward to implement, kinetics in the RE ensemble is complicated, and there is much to learn about how best to employ RE simulations in computational biophysics. Protein folding rates often slow down above a certain temperature due to entropic bottlenecks. This "anti-Arrhenius" behavior represents a challenge for RE. However, it is far from straightforward to systematically explore the impact of this on RE by brute force molecular simulations, since RE simulations of protein folding are very difficult to converge. To understand some of the basic mechanisms that determine the efficiency of RE, it is useful to study simplified low dimensionality systems that share some of the key characteristics of molecular systems. Results are presented concerning the efficiency of temperature RE on a continuous two-dimensional potential that contains an entropic bottleneck. Optimal efficiency was obtained when the temperatures of the replicas did not exceed the temperature at which the harmonic mean of the folding and unfolding rates is maximized. This confirms a result we previously obtained using a discrete network model of RE. Comparison of the efficiencies obtained using the continuous and discrete models makes it possible to identify non-Markovian effects, which slow down equilibration of the RE ensemble on the more complex continuous potential. In particular, the rate of temperature diffusion and also the efficiency of RE is limited by the time scale of conformational rearrangements within free energy basins.


Subject(s)
Models, Biological , Protein Folding , Proteins/chemistry , Proteins/metabolism , Computer Simulation , Kinetics , Probability , Temperature , Thermodynamics
6.
Proc Natl Acad Sci U S A ; 104(39): 15340-5, 2007 Sep 25.
Article in English | MEDLINE | ID: mdl-17878309

ABSTRACT

Replica exchange (RE) is a generalized ensemble simulation method for accelerating the exploration of free-energy landscapes, which define many challenging problems in computational biophysics, including protein folding and binding. Although temperature RE (T-RE) is a parallel simulation technique whose implementation is relatively straightforward, kinetics and the approach to equilibrium in the T-RE ensemble are very complicated; there is much to learn about how to best employ T-RE to protein folding and binding problems. We have constructed a kinetic network model for RE studies of protein folding and used this reduced model to carry out "simulations of simulations" to analyze how the underlying temperature dependence of the conformational kinetics and the basic parameters of RE (e.g., the number of replicas, the RE rate, and the temperature spacing) all interact to affect the number of folding transitions observed. When protein folding follows anti-Arrhenius kinetics, we observe a speed limit for the number of folding transitions observed at the low temperature of interest, which depends on the maximum of the harmonic mean of the folding and unfolding transition rates at high temperature. The results shown here for the network RE model suggest ways to improve atomic-level RE simulations such as the use of "training" simulations to explore some aspects of the temperature dependence for folding of the atomic-level models before performing RE studies.


Subject(s)
Protein Folding , Biochemistry/methods , Computer Simulation , Kinetics , Markov Chains , Models, Chemical , Models, Molecular , Models, Statistical , Molecular Conformation , Protein Conformation , Protein Denaturation , Protein Structure, Secondary , Temperature , Thermodynamics
7.
Proteins ; 69(3): 449-65, 2007 Nov 15.
Article in English | MEDLINE | ID: mdl-17623851

ABSTRACT

The existence of a large number of proteins for which both nuclear magnetic resonance (NMR) and X-ray crystallographic coordinates have been deposited into the Protein Data Bank (PDB) makes the statistical comparison of the corresponding crystal and NMR structural models over a large data set possible, and facilitates the study of the effect of the crystal environment and other factors on structure. We present an approach for detecting statistically significant structural differences between crystal and NMR structural models which is based on structural superposition and the analysis of the distributions of atomic positions relative to a mean structure. We apply this to a set of 148 protein structure pairs (crystal vs NMR), and analyze the results in terms of methodological and physical sources of structural difference. For every one of the 148 structure pairs, the backbone root-mean-square distance (RMSD) over core atoms of the crystal structure to the mean NMR structure is larger than the average RMSD of the members of the NMR ensemble to the mean, with 76% of the structure pairs having an RMSD of the crystal structure to the mean more than a factor of two larger than the average RMSD of the NMR ensemble. On average, the backbone RMSD over core atoms of crystal structure to the mean NMR is approximately 1 A. If non-core atoms are included, this increases to 1.4 A due to the presence of variability in loops and similar regions of the protein. The observed structural differences are only weakly correlated with the age and quality of the structural model and differences in conditions under which the models were determined. We examine steric clashes when a putative crystalline lattice is constructed using a representative NMR structure, and find that repulsive crystal packing plays a minor role in the observed differences between crystal and NMR structures. The observed structural differences likely have a combination of physical and methodological causes. Stabilizing attractive interactions arising from intermolecular crystal contacts which shift the equilibrium of the crystal structure relative to the NMR structure is a likely physical source which can account for some of the observed differences. Methodological sources of apparent structural difference include insufficient sampling or other issues which could give rise to errors in the estimates of the precision and/or accuracy.


Subject(s)
Protein Conformation , Crystallography, X-Ray , Databases, Protein , Nuclear Magnetic Resonance, Biomolecular , Statistics as Topic
9.
Proc Natl Acad Sci U S A ; 102(19): 6801-6, 2005 May 10.
Article in English | MEDLINE | ID: mdl-15800044

ABSTRACT

We present an approach to the study of protein folding that uses the combined power of replica exchange simulations and a network model for the kinetics. We carry out replica exchange simulations to generate a large ( approximately 10(6)) set of states with an all-atom effective potential function and construct a kinetic model for folding, using an ansatz that allows kinetic transitions between states based on structural similarity. We use this network to perform random walks in the state space and examine the overall network structure. Results are presented for the C-terminal peptide from the B1 domain of protein G. The kinetics is two-state after small temperature perturbations. However, the coil-to-hairpin folding is dominated by pathways that visit metastable helical conformations. We propose possible mechanisms for the alpha-helix/beta-hairpin interconversion.


Subject(s)
Biophysics/methods , Computer Simulation , Hydrogen Bonding , Kinetics , Models, Chemical , Models, Molecular , Molecular Structure , Monte Carlo Method , Peptides/chemistry , Protein Conformation , Protein Folding , Protein Structure, Secondary , Protein Structure, Tertiary , Temperature , Thermodynamics , Time Factors
10.
J Phys Chem B ; 109(14): 6722-31, 2005 Apr 14.
Article in English | MEDLINE | ID: mdl-16851756

ABSTRACT

We analyzed the data from a replica exchange molecular dynamics simulation using the weighted histogram analysis method to combine data from all of the temperature replicas (T-WHAM) to obtain the room-temperature potential of mean force of the G-peptide (the C-terminal beta-hairpin of the B1 domain of protein G) in regions of conformational space not sampled at room temperature. We were able to determine the potential of mean force in the transition region between a minor alpha-helical population and the major beta-hairpin population and identify a possible transition path between them along which the peptide retains a significant amount of secondary structure. This observation provides new insights into a possible mechanism of formation of beta-sheet secondary structures in proteins. We developed a novel Bayesian statistical uncertainty estimation method for any quantity derived from WHAM and used it to validate the calculated potential of mean force. The feasibility of estimating regions of the potential of mean force with unfavorable free energy at room temperature by T-WHAM analysis of replica exchange simulations was further tested on a system that can be solved analytically and presented some of the same challenges found in more complex chemical systems.


Subject(s)
Chemistry, Physical/methods , Proteins/chemistry , Algorithms , Amino Acid Motifs , Bayes Theorem , Computer Simulation , Likelihood Functions , Molecular Conformation , Monte Carlo Method , Peptides/chemistry , Protein Conformation , Protein Structure, Secondary , Protein Structure, Tertiary , Temperature , Thermodynamics
11.
J Theor Biol ; 232(3): 427-41, 2005 Feb 07.
Article in English | MEDLINE | ID: mdl-15572066

ABSTRACT

One of the fundamental problems of cell biology is the understanding of complex regulatory networks. Such networks are ubiquitous in cells and knowledge of their properties is essential for the understanding of cellular behavior. In earlier work (Kholodenko et al. (PNAS 99: 12841), it was shown how the structure of biological networks can be quantified from experimental measurements of steady-state concentrations of key intermediates as a result of perturbations using a simple algorithm called "unravelling". Here, we study the effect of experimental uncertainty on the accuracy of the inferred structure (i.e. whether interactions are excitatory or inhibitory) of the networks determined using the unravelling algorithm. We show that the accuracy of the network structure depends not only on the noise level but on the strength of the interactions within the network. In particular, both very small and very large values of the connection strengths lead to large uncertainty in the inferred network. We describe a powerful geometric tool for the intuitive understanding of the effect of experimental error on the qualitative accuracy of the inferred network. In addition, we show that the use of additional data beyond that needed to minimally constrain the network not only improves the accuracy of the inferred network, but also may allow the detection of situations in which the initial assumptions of unravelling with respect to the network and the perturbations have been violated. Our ideas are illustrated using the mitogen-activated protein kinase (MAPK) signaling network as an example.


Subject(s)
Gene Expression Regulation/physiology , Models, Biological , Signal Transduction/physiology , Systems Biology/methods , Algorithms , Animals , Cell Physiological Phenomena
12.
Protein Eng ; 16(6): 407-14, 2003 Jun.
Article in English | MEDLINE | ID: mdl-12874373

ABSTRACT

Assembling short fragments from known structures has been a widely used approach to construct novel protein structures. To what extent there exist structurally similar fragments in the database of known structures for short fragments of a novel protein is a question that is fundamental to this approach. This work addresses that question for seven-, nine- and 15-residue fragments. For each fragment size, two databases, a query database and a template database of fragments from high-quality protein structures in SCOP20 and SCOP90, respectively, were constructed. For each fragment in the query database, the template database was scanned to find the lowest r.m.s.d. fragment among non-homologous structures. For seven-residue fragments, there is a 99% probability that there exists such a fragment within 0.7 A r.m.s.d. for each loop fragment. For nine-residue fragments there is a 96% probability of a fragment within 1 A r.m.s.d., while for 15-residue fragments there is a 91% probability of a fragment within 2 A r.m.s.d. These results, which update previous studies, show that there exists sufficient coverage to model even a novel fold using fragments from the Protein Data Bank, as the current database of known structures has increased enormously in the last few years. We have also explored the use of a grid search method for loop homology modeling and make some observations about the use of a grid search compared with a database search for the loop modeling problem.


Subject(s)
Databases, Factual , Peptide Fragments/chemistry , Protein Structure, Tertiary , Proteins/chemistry , Humans , Models, Molecular , Peptide Fragments/genetics
13.
J Phys Chem A ; 107(38): 7454-7464, 2003 Sep 03.
Article in English | MEDLINE | ID: mdl-19626138

ABSTRACT

The measurement of fluorescence from single protein molecules has become an important new tool in the study of dynamic processes, allowing for the direct visualization of the motions experienced by individual proteins and macromolecular complexes. The data from such single-molecule experiments are in the form of photon trajectories, consisting of arrival times and wavelength information on individual photons. The analysis of photon trajectories can be difficult, particularly if the motions are occurring at rates comparable to the photon arrival rate or in the presence of noise. In this paper, we introduce the use of hidden Markov models (HMMs) for the analysis of photon trajectory data that operate using the photon data directly, without the need for ensemble averaging of the data as implied by correlation function analysis. Using a simple kinetic model, we examine the relationship between the uncertainty in the estimates of the motional rate and the photon detection rate. Remarkably, we obtain relative uncertainties in the rate constants of as little as 3% even when the interconversion rate is equal to the photon detection rate, and the uncertainty increases to only 10% when the interconversion rate is 10 times the photon detection rate. This suggests that useful information can be obtained for much faster kinetic regimes than have typically been studied. We also examine the impact of background photons on the determination of the rate and demonstrate that the HMM-based approach is robust, displaying small uncertainties for background photon arrival rates approaching that of the signal. These results not only are relevant in establishing the theoretical limits on precision, but are also useful in the context of experimental design. Finally, to demonstrate how the methodology can be extended to more complex kinetic models and how it can allow one to make use of the full power of statistics for purposes of model evaluation and selection, we consider a four-state kinetic model for protein conformational transitions previously studied by Schenter et al. (J. Phys. Chem. A1999, 103, 10477). We show how an HMM can be used as an alternative to higher-order correlation function analysis for the detection of "conformational memory" and apparent non-Markovian dynamics arising from such temporally inhomogeneous kinetic schemes.

14.
J Biomol NMR ; 23(4): 263-70, 2002 Aug.
Article in English | MEDLINE | ID: mdl-12398347

ABSTRACT

The need for the structural characterization of proteins on a genomic scale has brought with it demands for new technology to speed the structure determination process. In NMR, one bottleneck is the sequential assignment of backbone resonances. In this paper, we explore the computational complexity of the sequential assignment problem using only 13C alpha chemical shift data and C alpha (i, i - 1) sequential connectivity information, all of which can potentially be obtained from a single three-dimensional NMR spectrum. Although it is generally believed that there is too much ambiguity in such data to provide sufficient information for sequential assignment, we show that a straightforward combinatorial search algorithm can be used to find correct and unambiguous sequential assignments in a reasonable amount of CPU time for small proteins (approximately 80 residues or smaller) when there is little missing data. The deleterious effect of missing or spurious peaks and the dependence on match tolerances is also explored. This simple algorithm could be used as part of a semi-automated, interactive assignment procedure, e.g., to test partial manually determined solutions for uniqueness and to extend these solutions.


Subject(s)
Algorithms , Nuclear Magnetic Resonance, Biomolecular/methods , Proteins/chemistry , Amino Acid Sequence , Carbon Isotopes , Time
15.
J Struct Funct Genomics ; 2(2): 103-11, 2002.
Article in English | MEDLINE | ID: mdl-12836667

ABSTRACT

Residual dipolar couplings provide significant structural information for proteins in the solution state, which makes them attractive for the rapid determination of protein structures. While dipolar couplings contain inherent structural ambiguities, these can be reduced via an overlap similarity measure that insists that protein fragments assigned to overlapping regions of the sequence must have self-consistent structures. This allows us to determine a backbone fold (including the correct Calpha-Cbeta bond orientations) using only residual dipolar coupling data from one ordering medium. The resulting backbone structures are of sufficient quality to allow for modeling of sidechain rotamer states using a rotamer prediction algorithm and a force field employing the Surface Generalized Born continuum solvation model. We demonstrate the applicability of the method using experimental data for ubiquitin. These results illustrate the synergies that are possible between protein structural database and molecular modeling methods and NMR spectroscopy, and we expect that the further development of these methods will lead to the extraction of high resolution structural information from minimal NMR data.


Subject(s)
Nuclear Magnetic Resonance, Biomolecular/methods , Protein Conformation , Computer Simulation , Crystallization , Databases, Protein , Models, Molecular , Monte Carlo Method , Peptide Fragments/chemistry , Protein Folding , Solubility , Ubiquitin/chemistry
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