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Affiliation(s)
- Brooke E. Husic
- Department of Chemistry, Stanford University, Stanford, California 94305, United States
| | - Vijay S. Pande
- Department of Chemistry, Stanford University, Stanford, California 94305, United States
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2
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Abstract
The title, "Look to the past, Look to the present, and Look to the future," the motto of City College of New York, expresses how my family life and education led me to an academic career in physical chemistry and ultimately to a study of proteins. The economic depression of the 1930s left a lasting impression on my outlook and career aspirations. With fortunate experiences at several stages in my life, I was able to participate in the great adventure of the last half of the twentieth century: the revolution in biology that advanced the field of protein chemistry to so great an extent. The future is bright and limitless, with greater understanding of biology yet to come.
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Affiliation(s)
- Harold A Scheraga
- Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853-1301, USA.
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3
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Zheng W, Gallicchio E, Deng N, Andrec M, Levy RM. Kinetic network study of the diversity and temperature dependence of Trp-Cage folding pathways: combining transition path theory with stochastic simulations. J Phys Chem B 2011; 115:1512-23. [PMID: 21254767 PMCID: PMC3059588 DOI: 10.1021/jp1089596] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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.
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Affiliation(s)
- Weihua Zheng
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, the State University of New Jersey Piscataway, NJ 08854
| | - Emilio Gallicchio
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, the State University of New Jersey Piscataway, NJ 08854
| | - Nanjie Deng
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, the State University of New Jersey Piscataway, NJ 08854
| | - Michael Andrec
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, the State University of New Jersey Piscataway, NJ 08854
| | - Ronald M. Levy
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology, Rutgers, the State University of New Jersey Piscataway, NJ 08854
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4
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Klenin K, Strodel B, Wales DJ, Wenzel W. Modelling proteins: conformational sampling and reconstruction of folding kinetics. BIOCHIMICA ET BIOPHYSICA ACTA-PROTEINS AND PROTEOMICS 2010; 1814:977-1000. [PMID: 20851219 DOI: 10.1016/j.bbapap.2010.09.006] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2010] [Revised: 09/03/2010] [Accepted: 09/05/2010] [Indexed: 01/08/2023]
Abstract
In the last decades biomolecular simulation has made tremendous inroads to help elucidate biomolecular processes in-silico. Despite enormous advances in molecular dynamics techniques and the available computational power, many problems involve long time scales and large-scale molecular rearrangements that are still difficult to sample adequately. In this review we therefore summarise recent efforts to fundamentally improve this situation by decoupling the sampling of the energy landscape from the description of the kinetics of the process. Recent years have seen the emergence of many advanced sampling techniques, which permit efficient characterisation of the relevant family of molecular conformations by dispensing with the details of the short-term kinetics of the process. Because these methods generate thermodynamic information at best, they must be complemented by techniques to reconstruct the kinetics of the process using the ensemble of relevant conformations. Here we review recent advances for both types of methods and discuss their perspectives to permit efficient and accurate modelling of large-scale conformational changes in biomolecules. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.
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Affiliation(s)
- Konstantin Klenin
- Steinbuch Centre for Computing, Karlsruhe Institute of Technology, P.O. Box 3640, D-76021 Karlsruhe, Germany
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5
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Zheng W, Andrec M, Gallicchio E, Levy RM. Recovering kinetics from a simplified protein folding model using replica exchange simulations: a kinetic network and effective stochastic dynamics. J Phys Chem B 2009; 113:11702-9. [PMID: 19655770 PMCID: PMC2975981 DOI: 10.1021/jp900445t] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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.
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Affiliation(s)
- Weihua Zheng
- Department of Physics and Astronomy Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway NJ 08854, USA
| | - Michael Andrec
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology Rutgers, the State University of New Jersey, 610 Taylor Road, Piscataway NJ 08854, USA
| | - Emilio Gallicchio
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology Rutgers, the State University of New Jersey, 610 Taylor Road, Piscataway NJ 08854, USA
| | - Ronald M. Levy
- Department of Chemistry and Chemical Biology and BioMaPS Institute for Quantitative Biology Rutgers, the State University of New Jersey, 610 Taylor Road, Piscataway NJ 08854, USA
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6
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Andrec M, Felts AK, Gallicchio E, Levy RM. Protein folding pathways from replica exchange simulations and a kinetic network model. Proc Natl Acad Sci U S A 2005; 102:6801-6. [PMID: 15800044 PMCID: PMC1100763 DOI: 10.1073/pnas.0408970102] [Citation(s) in RCA: 125] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2004] [Indexed: 11/18/2022] Open
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.
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Affiliation(s)
- Michael Andrec
- Department of Chemistry and Chemical Biology and BIOMAPS Institute for Quantitative Biology, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA
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7
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Zhang W, Chen SJ. Analyzing the biopolymer folding rates and pathways using kinetic cluster method. J Chem Phys 2003; 119:8716-8729. [PMID: 19079645 DOI: 10.1063/1.1613255] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A kinetic cluster method enables us to analyze biopolymer folding kinetics with discrete rate-limiting steps by classifying biopolymer conformations into pre-equilibrated clusters. The overall folding kinetics is determined by the intercluster transitions. Due to the complex energy landscapes of biopolymers, the intercluster transitions have multiple pathways and can have kinetic intermediates (local free-energy minima) distributed on the intercluster pathways. We focus on the RNA secondary structure folding kinetics. The dominant folding pathways and the kinetic partitioning mechanism can be identified and quantified from the rate constants for different intercluster pathways. Moreover, the temperature dependence of the folding rate can be analyzed from the interplay between the stabilities of the on-pathway (nativelike) and off-pathway (misfolded) conformations and from the kinetic partitioning between different intercluster pathways. The predicted folding kinetics can be directly tested against experiments.
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Affiliation(s)
- Wenbing Zhang
- Department of Physics and Astronomy and Department of Biochemistry, University of Missouri, Columbia, Missouri 65211
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Zhang W, Chen SJ. Master equation approach to finding the rate-limiting steps in biopolymer folding. J Chem Phys 2003; 118:3413-3420. [PMID: 19079644 DOI: 10.1063/1.1538596] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
A master equation approach is developed to find the rate-limiting steps in biopolymer folding, where the folding kinetics is described as a linear combination of basic kinetic modes determined from the eigenvalues and eigenvectors of the rate matrix. Because the passage of a rate-limiting step is intrinsically related to the folding speed, it is possible to probe and to identify the rate-limiting steps through the folding from different unfolded initial conformations. In a master equation approach, slow and fast folding speeds are directly correlated to the large and small contributions of the (rate-limiting) slow kinetic modes. Because the contributions from the slow modes can be computed from the corresponding eigenvectors, the rate-limiting steps can be identified from the eigenvectors of the slow modes. Our rate-limiting searching method has been tested for a simplified hairpin folding kinetics model, and it may provide a general transition state searching method for biopolymer folding.
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Affiliation(s)
- Wenbing Zhang
- Department of Physics and Astronomy and Department of Biochemistry, University of Missouri, Columbia, Missouri 65211
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Ozkan SB, Dill KA, Bahar I. Computing the transition state populations in simple protein models. Biopolymers 2003; 68:35-46. [PMID: 12579578 DOI: 10.1002/bip.10280] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
We describe the master equation method for computing the kinetics of protein folding. We illustrate the method using a simple Go model. Presently most models of two-state fast-folding protein folding kinetics invoke the classical idea of a transition state to explain why there is a single exponential decay in time. However, if proteins fold via funnel-shaped energy landscapes, as predicted by many theoretical studies, then it raises the question of what is the transition state. Is it a specific structure, or a small ensemble of structures, as is expected from classical transition state theory? Or is it more like the denatured states of proteins, a very broad ensemble? The answer that is usually obtained depends on the assumptions made about the transition state. The present method is a rigorous way to find transition states, without assumptions or approximations, even for very nonclassical shapes of energy landscapes. We illustrate the method here, showing how the transition states in two-state protein folding can be very broad ensembles.
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Affiliation(s)
- S Banu Ozkan
- Center for Computational Biology, Department of Molecular Genetics, School of Medicine, University of Pittsburgh, PA 15213, USA
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Ghosh A, Elber R, Scheraga HA. An atomically detailed study of the folding pathways of protein A with the stochastic difference equation. Proc Natl Acad Sci U S A 2002; 99:10394-8. [PMID: 12140363 PMCID: PMC124925 DOI: 10.1073/pnas.142288099] [Citation(s) in RCA: 112] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
An algorithm is applied here to compute folding pathways of staphylococcal protein A, fragment B. Emphasis is on studies of the complete process, starting from an ensemble of fully denatured conformations and ending at the folded state. The stochastic difference equation algorithm is based on optimization of an action that makes it possible to use a large integration step. Motions with typical displacements that change rapidly on the size scale of the step are filtered out, providing numerically stable and approximate solutions. The present approach is unique in maintaining an atomically detailed picture while providing a systematic, controlled approximation to the classical equations of motion. Analysis of 130 trajectories suggests the following folding mechanism for protein A: At an early precollapse phase of the process, a few native hydrogen bonds form near the C terminus of the protein. The hydrogen bonds are formed mostly within the third helix. The next step is chain collapse that occurs in parallel to additional growth of secondary structure seeds. Therefore, the present study does not support a pure hydrophobic collapse, or substantial early formation of secondary structure. At the last step, native tertiary contacts are formed at the same time as the completion of the secondary structure elements. To a large extent, the process is parallel and not sequential. The early formation of the third helix of protein A, fragment B (in the calculation), is consistent with experimental data.
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Affiliation(s)
- Avijit Ghosh
- Department of Computer Science, Upson Hall 4130, Cornell University, Ithaca NY 14853-7501, USA
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Ozkan SB, Dill KA, Bahar I. Fast-folding protein kinetics, hidden intermediates, and the sequential stabilization model. Protein Sci 2002; 11:1958-70. [PMID: 12142450 PMCID: PMC2373683 DOI: 10.1110/ps.0207102] [Citation(s) in RCA: 65] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Abstract
Do two-state proteins fold by pathways or funnels? Native-state hydrogen exchange experiments show discrete nonnative structures in equilibrium with the native state. These could be called hidden intermediates (HI) because their populations are small at equilibrium, and they are not detected in kinetic experiments. HIs have been invoked as disproof of funnel models, because funnel pictures appear to indicate (1) no specific sequences of events in folding; (2) a continuum, rather than a discrete ladder, of structures; and (3) smooth landscapes. In the present study, we solve the exact dynamics of a simple model. We find, instead, that the present microscopic model is indeed consistent with HIs and transition states, but such states occur in parallel, rather than along the single pathway predicted by the sequential stabilization model. At the microscopic level, we observe a huge multiplicity of trajectories. But at the macroscopic level, we observe two pathways of specific sequences of events that are relatively traditional except that they are in parallel, so there is not a single reaction coordinate. Using singular value decomposition, we show an accurate representation of the shapes of the model energy landscapes. They are highly complex funnels.
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Affiliation(s)
- S Banu Ozkan
- Center for Computational Biology and Bioinformatics, and Department of Molecular Genetics and Biochemistry, School of Medicine, University of Pittsburgh, Pennsylvania 15213, USA
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