1
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Yuan Y, Mao X, Pan X, Zhang R, Su W. Kinetic Ensemble of Tau Protein through the Markov State Model and Deep Learning Analysis. J Chem Theory Comput 2024; 20:2947-2958. [PMID: 38501645 DOI: 10.1021/acs.jctc.3c01211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/20/2024]
Abstract
The ordered assembly of Tau protein into filaments characterizes Alzheimer's and other neurodegenerative diseases, and thus, stabilization of Tau protein is a promising avenue for tauopathies therapy. To dissect the underlying aggregation mechanisms on Tau, we employ a set of molecular simulations and the Markov state model to determine the kinetics of ensemble of K18. K18 is the microtubule-binding domain of Tau protein and plays a vital role in the microtubule assembly, recycling processes, and amyloid fibril formation. Here, we efficiently explore the conformation of K18 with about 150 μs lifetimes in silico. Our results observe that all four repeat regions (R1-R4) are very dynamic, featuring frequent conformational conversion and lacking stable conformations, and the R2 region is more flexible than the R1, R3, and R4 regions. Additionally, it is worth noting that residues 300-310 in R2-R3 and residues 319-336 in R3 tend to form sheet structures, indicating that K18 has a broader functional role than individual repeat monomers. Finally, the simulations combined with Markov state models and deep learning reveal 5 key conformational states along the transition pathway and provide the information on the microsecond time scale interstate transition rates. Overall, this study offers significant insights into the molecular mechanism of Tau pathological aggregation and develops novel strategies for both securing tauopathies and advancing drug discovery.
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Affiliation(s)
- Yongna Yuan
- School of Information Science & Engineering, Lanzhou University, South Tianshui Road, Lanzhou 730000, Gansu, China
| | - Xuqi Mao
- School of Information Science & Engineering, Lanzhou University, South Tianshui Road, Lanzhou 730000, Gansu, China
| | - Xiaohang Pan
- School of Information Science & Engineering, Lanzhou University, South Tianshui Road, Lanzhou 730000, Gansu, China
| | - Ruisheng Zhang
- School of Information Science & Engineering, Lanzhou University, South Tianshui Road, Lanzhou 730000, Gansu, China
| | - Wei Su
- School of Information Science & Engineering, Lanzhou University, South Tianshui Road, Lanzhou 730000, Gansu, China
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2
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Gupta C, Sarkar D, Tieleman DP, Singharoy A. The ugly, bad, and good stories of large-scale biomolecular simulations. Curr Opin Struct Biol 2022; 73:102338. [PMID: 35245737 DOI: 10.1016/j.sbi.2022.102338] [Citation(s) in RCA: 27] [Impact Index Per Article: 13.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2021] [Revised: 11/29/2021] [Accepted: 01/24/2022] [Indexed: 12/20/2022]
Abstract
Molecular modeling of large biomolecular assemblies exemplifies a disruptive area holding both promises and contentions. Propelled by peta and exascale computing, several simulation methodologies have now matured into user-friendly tools that are successfully employed for modeling viruses, membranous nano-constructs, and key pieces of the genetic machinery. We present three unifying biophysical themes that emanate from some of the most recent multi-million atom simulation endeavors. Despite connecting molecular changes with phenotypic outcomes, the quality measures of these simulations remain questionable. We discuss the existing and upcoming strategies for constructing representative ensembles of large systems, how new computing technologies will boost this area, and make a point that integrative modeling guided by experimental data is the future of biomolecular computations.
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Affiliation(s)
- Chitrak Gupta
- School of Molecular Sciences, Center for Applied Structural Discovery, Arizona State University at Tempe, Tempe, AZ, 85282, USA; Biodesign Institute, Tempe, AZ, 85281, USA. https://twitter.com/ChitrakGupta2
| | - Daipayan Sarkar
- School of Molecular Sciences, Center for Applied Structural Discovery, Arizona State University at Tempe, Tempe, AZ, 85282, USA; MSU-DOE Plant Research Laboratory, Michigan State University, East Lansing, MI, 48824-1319, USA. https://twitter.com/17Dsarkar
| | - D Peter Tieleman
- Centre for Molecular Simulation and Department of Biological Sciences, University of Calgary, Calgary, AB T2N 1N4, Canada.
| | - Abhishek Singharoy
- School of Molecular Sciences, Center for Applied Structural Discovery, Arizona State University at Tempe, Tempe, AZ, 85282, USA; Biodesign Institute, Tempe, AZ, 85281, USA.
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3
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Suárez E, Wiewiora RP, Wehmeyer C, Noé F, Chodera JD, Zuckerman DM. What Markov State Models Can and Cannot Do: Correlation versus Path-Based Observables in Protein-Folding Models. J Chem Theory Comput 2021; 17:3119-3133. [PMID: 33904312 PMCID: PMC8127341 DOI: 10.1021/acs.jctc.0c01154] [Citation(s) in RCA: 25] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Markov state models (MSMs) have been widely applied to study the kinetics and pathways of protein conformational dynamics based on statistical analysis of molecular dynamics (MD) simulations. These MSMs coarse-grain both configuration space and time in ways that limit what kinds of observables they can reproduce with high fidelity over different spatial and temporal resolutions. Despite their popularity, there is still limited understanding of which biophysical observables can be computed from these MSMs in a robust and unbiased manner, and which suffer from the space-time coarse-graining intrinsic in the MSM model. Most theoretical arguments and practical validity tests for MSMs rely on long-time equilibrium kinetics, such as the slowest relaxation time scales and experimentally observable time-correlation functions. Here, we perform an extensive assessment of the ability of well-validated protein folding MSMs to accurately reproduce path-based observable such as mean first-passage times (MFPTs) and transition path mechanisms compared to a direct trajectory analysis. We also assess a recently proposed class of history-augmented MSMs (haMSMs) that exploit additional information not accounted for in standard MSMs. We conclude with some practical guidance on the use of MSMs to study various problems in conformational dynamics of biomolecules. In brief, MSMs can accurately reproduce correlation functions slower than the lag time, but path-based observables can only be reliably reproduced if the lifetimes of states exceed the lag time, which is a much stricter requirement. Even in the presence of short-lived states, we find that haMSMs reproduce path-based observables more reliably.
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Affiliation(s)
- Ernesto Suárez
- Advanced Biomedical Computational Science, Frederick National Laboratory for Cancer Research, Frederick, MD 21702
| | - Rafal P. Wiewiora
- Computational and Systems Biology Program, Sloan Kettering Institute, Memorial Sloan Kettering Cancer Center, New York, NY 10065
| | | | | | - John D. Chodera
- Computational and Systems Biology Program, Sloan Kettering Institute, Memorial Sloan Kettering Cancer Center, New York, NY 10065
| | - Daniel M. Zuckerman
- Department of Biomedical Engineering, Oregon Health and Science University, Portland, OR 97239
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4
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Löhr T, Kohlhoff K, Heller GT, Camilloni C, Vendruscolo M. A kinetic ensemble of the Alzheimer's Aβ peptide. NATURE COMPUTATIONAL SCIENCE 2021; 1:71-78. [PMID: 38217162 DOI: 10.1038/s43588-020-00003-w] [Citation(s) in RCA: 30] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2020] [Accepted: 11/24/2020] [Indexed: 01/15/2024]
Abstract
The conformational and thermodynamic properties of disordered proteins are commonly described in terms of structural ensembles and free energy landscapes. To provide information on the transition rates between the different states populated by these proteins, it would be desirable to generalize this description to kinetic ensembles. Approaches based on the theory of stochastic processes can be particularly suitable for this purpose. Here, we develop a Markov state model and apply it to determine a kinetic ensemble of Aβ42, a disordered peptide associated with Alzheimer's disease. Through the Google Compute Engine, we generated 315-µs all-atom molecular dynamics trajectories. Using a probabilistic-based definition of conformational states in a neural network approach, we found that Aβ42 is characterized by inter-state transitions on the microsecond timescale, exhibiting only fully unfolded or short-lived, partially folded states. Our results illustrate how kinetic ensembles provide effective information about the structure, thermodynamics and kinetics of disordered proteins.
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Affiliation(s)
- Thomas Löhr
- Department of Chemistry, University of Cambridge, Cambridge, UK
| | | | | | - Carlo Camilloni
- Dipartimento di Bioscienze, Università degli Studi di Milano, Milano, Italy
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5
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Zhang P, Wang D, Yang W, Marszalek PE. Piecewise All-Atom SMD Simulations Reveal Key Secondary Structures in Luciferase Unfolding Pathway. Biophys J 2020; 119:2251-2261. [PMID: 33130123 DOI: 10.1016/j.bpj.2020.10.023] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2020] [Revised: 09/24/2020] [Accepted: 10/14/2020] [Indexed: 01/08/2023] Open
Abstract
Although the folding of single-domain proteins is well characterized theoretically and experimentally, the folding of large multidomain proteins is less well known. Firefly luciferase, a 550 residue three-domain protein, has been commonly used as a substrate to study chaperone reactions and as a model system for the study of folding of long polypeptide chains, including related phenomena such as cotranslational folding. Despite being characterized by various experimental techniques, the atomic-level contributions of various secondary structures of luciferase to its fold's mechanical stability remain unknown. Here, we developed a piecewise approach for all-atom steered molecular dynamics simulations to examine specific secondary structures that resist mechanical unfolding while minimizing the amount of computational resources required by the large water box of standard all-atom steered molecular dynamics simulations. We validated the robustness of this approach with a small NI3C protein and used our approach to elucidate the specific secondary structures that provide the largest contributions to luciferase mechanostability. In doing so, we show that piecewise all-atom steered molecular dynamics simulations can provide novel atomic resolution details regarding mechanostability and can serve as a platform for novel mutagenesis studies as well as a point for comparison with high-resolution force spectroscopy experiments.
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Affiliation(s)
- Pan Zhang
- Department of Chemistry, Duke University, Durham, North Carolina
| | - David Wang
- Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina
| | - Weitao Yang
- Department of Chemistry, Duke University, Durham, North Carolina.
| | - Piotr E Marszalek
- Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina.
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6
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Affiliation(s)
- Lavi S. Bigman
- Department of Structural BiologyWeizmann Institute of Science Rehovot 76100 Israel
| | - Yaakov Levy
- Department of Structural BiologyWeizmann Institute of Science Rehovot 76100 Israel
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7
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Sidky H, Chen W, Ferguson AL. High-Resolution Markov State Models for the Dynamics of Trp-Cage Miniprotein Constructed Over Slow Folding Modes Identified by State-Free Reversible VAMPnets. J Phys Chem B 2019; 123:7999-8009. [DOI: 10.1021/acs.jpcb.9b05578] [Citation(s) in RCA: 23] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023]
Affiliation(s)
- Hythem Sidky
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States
| | - Wei Chen
- Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, United States
| | - Andrew L. Ferguson
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States
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8
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Fritch B, Kosolapov A, Hudson P, Nissley DA, Woodcock HL, Deutsch C, O'Brien EP. Origins of the Mechanochemical Coupling of Peptide Bond Formation to Protein Synthesis. J Am Chem Soc 2018; 140:5077-5087. [PMID: 29577725 DOI: 10.1021/jacs.7b11044] [Citation(s) in RCA: 26] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Abstract
Mechanical forces acting on the ribosome can alter the speed of protein synthesis, indicating that mechanochemistry can contribute to translation control of gene expression. The naturally occurring sources of these mechanical forces, the mechanism by which they are transmitted 10 nm to the ribosome's catalytic core, and how they influence peptide bond formation rates are largely unknown. Here, we identify a new source of mechanical force acting on the ribosome by using in situ experimental measurements of changes in nascent-chain extension in the exit tunnel in conjunction with all-atom and coarse-grained computer simulations. We demonstrate that when the number of residues composing a nascent chain increases, its unstructured segments outside the ribosome exit tunnel generate piconewtons of force that are fully transmitted to the ribosome's P-site. The route of force transmission is shown to be through the nascent polypetide's backbone, not through the wall of the ribosome's exit tunnel. Utilizing quantum mechanical calculations we find that a consequence of such a pulling force is to decrease the transition state free energy barrier to peptide bond formation, indicating that the elongation of a nascent chain can accelerate translation. Since nascent protein segments can start out as largely unfolded structural ensembles, these results suggest a pulling force is present during protein synthesis that can modulate translation speed. The mechanism of force transmission we have identified and its consequences for peptide bond formation should be relevant regardless of the source of the pulling force.
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Affiliation(s)
- Benjamin Fritch
- Department of Chemistry , Pennsylvania State University , University Park , Pennsylvania 16802 , United States
| | - Andrey Kosolapov
- Department of Physiology , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
| | - Phillip Hudson
- Department of Chemistry , University of South Florida , Tampa , Florida 33620 , United States.,Laboratory of Computational Biology , National Institutes of Health , Bethesda , Maryland 20892 , United States
| | - Daniel A Nissley
- Department of Chemistry , Pennsylvania State University , University Park , Pennsylvania 16802 , United States
| | - H Lee Woodcock
- Department of Chemistry , University of South Florida , Tampa , Florida 33620 , United States
| | - Carol Deutsch
- Department of Physiology , University of Pennsylvania , Philadelphia , Pennsylvania 19104 , United States
| | - Edward P O'Brien
- Department of Chemistry , Pennsylvania State University , University Park , Pennsylvania 16802 , United States.,Bioinformatics and Genomics Graduate Program, The Huck Institutes of the Life Sciences , Pennsylvania State University , University Park , Pennsylvania 16802 , United States
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9
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Clote P, Bayegan AH. RNA folding kinetics using Monte Carlo and Gillespie algorithms. J Math Biol 2017; 76:1195-1227. [PMID: 28780735 DOI: 10.1007/s00285-017-1169-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2016] [Revised: 07/09/2017] [Indexed: 11/26/2022]
Abstract
RNA secondary structure folding kinetics is known to be important for the biological function of certain processes, such as the hok/sok system in E. coli. Although linear algebra provides an exact computational solution of secondary structure folding kinetics with respect to the Turner energy model for tiny ([Formula: see text]20 nt) RNA sequences, the folding kinetics for larger sequences can only be approximated by binning structures into macrostates in a coarse-grained model, or by repeatedly simulating secondary structure folding with either the Monte Carlo algorithm or the Gillespie algorithm. Here we investigate the relation between the Monte Carlo algorithm and the Gillespie algorithm. We prove that asymptotically, the expected time for a K-step trajectory of the Monte Carlo algorithm is equal to [Formula: see text] times that of the Gillespie algorithm, where [Formula: see text] denotes the Boltzmann expected network degree. If the network is regular (i.e. every node has the same degree), then the mean first passage time (MFPT) computed by the Monte Carlo algorithm is equal to MFPT computed by the Gillespie algorithm multiplied by [Formula: see text]; however, this is not true for non-regular networks. In particular, RNA secondary structure folding kinetics, as computed by the Monte Carlo algorithm, is not equal to the folding kinetics, as computed by the Gillespie algorithm, although the mean first passage times are roughly correlated. Simulation software for RNA secondary structure folding according to the Monte Carlo and Gillespie algorithms is publicly available, as is our software to compute the expected degree of the network of secondary structures of a given RNA sequence-see http://bioinformatics.bc.edu/clote/RNAexpNumNbors .
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Affiliation(s)
- Peter Clote
- Department of Biology, Boston College, Chestnut Hill, MA, 02467, USA.
| | - Amir H Bayegan
- Department of Biology, Boston College, Chestnut Hill, MA, 02467, USA
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10
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Influence of Glu/Arg, Asp/Arg, and Glu/Lys Salt Bridges on α-Helical Stability and Folding Kinetics. Biophys J 2017; 110:2328-2341. [PMID: 27276251 DOI: 10.1016/j.bpj.2016.04.015] [Citation(s) in RCA: 52] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2015] [Revised: 03/17/2016] [Accepted: 04/08/2016] [Indexed: 01/24/2023] Open
Abstract
Using a combination of ultraviolet circular dichroism, temperature-jump transient-infrared spectroscopy, and molecular dynamics simulations, we investigate the effect of salt bridges between different types of charged amino-acid residue pairs on α-helix folding. We determine the stability and the folding and unfolding rates of 12 alanine-based α-helical peptides, each of which has a nearly identical composition containing three pairs of positively and negatively charged residues (either Glu(-)/Arg(+), Asp(-)/Arg(+), or Glu(-)/Lys(+)). Within each set of peptides, the distance and order of the oppositely charged residues in the peptide sequence differ, such that they have different capabilities of forming salt bridges. Our results indicate that stabilizing salt bridges (in which the interacting residues are spaced and ordered such that they favor helix formation) speed up α-helix formation by up to 50% and slow down the unfolding of the α-helix, whereas salt bridges with an unfavorable geometry have the opposite effect. Comparing the peptides with different types of charge pairs, we observe that salt bridges between side chains of Glu(-) and Arg(+) are most favorable for the speed of folding, probably because of the larger conformational space of the salt-bridging Glu(-)/Arg(+) rotamer pairs compared to Asp(-)/Arg(+) and Glu(-)/Lys(+). We speculate that the observed impact of salt bridges on the folding kinetics might explain why some proteins contain salt bridges that do not stabilize the final, folded conformation.
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11
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Theoretical and computational validation of the Kuhn barrier friction mechanism in unfolded proteins. Sci Rep 2017; 7:269. [PMID: 28325911 PMCID: PMC5428071 DOI: 10.1038/s41598-017-00287-5] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2016] [Accepted: 02/20/2017] [Indexed: 11/08/2022] Open
Abstract
A long time ago, Kuhn predicted that long polymers should approach a limit where their global motion is controlled by solvent friction alone, with ruggedness of their energy landscapes having no consequences for their dynamics. In contrast, internal friction effects are important for polymers of modest length. Internal friction in proteins, in particular, affects how fast they fold or find their binding targets and, as such, has attracted much recent attention. Here we explore the molecular origins of internal friction in unfolded proteins using atomistic simulations, coarse-grained models and analytic theory. We show that the characteristic internal friction timescale is directly proportional to the timescale of hindered dihedral rotations within the polypeptide chain, with a proportionality coefficient b that is independent of the chain length. Such chain length independence of b provides experimentally testable evidence that internal friction arises from concerted, crankshaft-like dihedral rearrangements. In accord with phenomenological models of internal friction, we find the global reconfiguration timescale of a polypeptide to be the sum of solvent friction and internal friction timescales. At the same time, the time evolution of inter-monomer distances within polypeptides deviates both from the predictions of those models and from a simple, one-dimensional diffusion model.
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12
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Zhang BW, Dai W, Gallicchio E, He P, Xia J, Tan Z, Levy RM. Simulating Replica Exchange: Markov State Models, Proposal Schemes, and the Infinite Swapping Limit. J Phys Chem B 2016; 120:8289-301. [PMID: 27079355 DOI: 10.1021/acs.jpcb.6b02015] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Abstract
Replica exchange molecular dynamics is a multicanonical simulation technique commonly used to enhance the sampling of solvated biomolecules on rugged free energy landscapes. While replica exchange is relatively easy to implement, there are many unanswered questions about how to use this technique most efficiently, especially because it is frequently the case in practice that replica exchange simulations are not fully converged. A replica exchange cycle consists of a series of molecular dynamics steps of a set of replicas moving under different Hamiltonians or at different thermodynamic states followed by one or more replica exchange attempts to swap replicas among the different states. How the replica exchange cycle is constructed affects how rapidly the system equilibrates. We have constructed a Markov state model of replica exchange (MSMRE) using long molecular dynamics simulations of a host-guest binding system as an example, in order to study how different implementations of the replica exchange cycle can affect the sampling efficiency. We analyze how the number of replica exchange attempts per cycle, the number of MD steps per cycle, and the interaction between the two parameters affects the largest implied time scale of the MSMRE simulation. The infinite swapping limit is an important concept in replica exchange. We show how to estimate the infinite swapping limit from the diagonal elements of the exchange transition matrix constructed from MSMRE "simulations of simulations" as well as from relatively short runs of the actual replica exchange simulations.
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Affiliation(s)
- Bin W Zhang
- Center for Biophysics and Computational Biology, Department of Chemistry and Institute for Computational Molecular Science, Temple University , Philadelphia, Pennsylvania 19122, United States
| | - Wei Dai
- Department of Physics and Astronomy, Rutgers, the State University of New Jersey , Piscataway, New Jersey 08854, United States
| | - Emilio Gallicchio
- Department of Chemistry, Brooklyn College of the City University of New York , Brooklyn, New York 11210, United States
| | - Peng He
- Center for Biophysics and Computational Biology, Department of Chemistry and Institute for Computational Molecular Science, Temple University , Philadelphia, Pennsylvania 19122, United States
| | - Junchao Xia
- Center for Biophysics and Computational Biology, Department of Chemistry and Institute for Computational Molecular Science, Temple University , Philadelphia, Pennsylvania 19122, United States
| | - Zhiqiang Tan
- Department of Statistics, Rutgers, the State University of New Jersey , Piscataway, New Jersey 08854, United States
| | - Ronald M Levy
- Center for Biophysics and Computational Biology, Department of Chemistry and Institute for Computational Molecular Science, Temple University , Philadelphia, Pennsylvania 19122, United States
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13
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Levy RM. Recollection. Protein Sci 2015; 25:9-11. [PMID: 26575588 DOI: 10.1002/pro.2844] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2015] [Accepted: 11/12/2015] [Indexed: 11/05/2022]
Affiliation(s)
- Ronald M Levy
- Center for Biophysics and Computational Biology, Department of chemistry, and Institute for Computational Molecular Science, Temple University, Philadelphia, Pennsylvania, 19122-1801
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14
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Dai W, Sengupta AM, Levy RM. First Passage Times, Lifetimes, and Relaxation Times of Unfolded Proteins. PHYSICAL REVIEW LETTERS 2015; 115:048101. [PMID: 26252709 PMCID: PMC4531052 DOI: 10.1103/physrevlett.115.048101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2015] [Indexed: 06/04/2023]
Abstract
The dynamics of proteins in the unfolded state can be quantified in computer simulations by calculating a spectrum of relaxation times which describes the time scales over which the population fluctuations decay to equilibrium. If the unfolded state space is discretized, we can evaluate the relaxation time of each state. We derive a simple relation that shows the mean first passage time to any state is equal to the relaxation time of that state divided by the equilibrium population. This explains why mean first passage times from state to state within the unfolded ensemble can be very long but the energy landscape can still be smooth (minimally frustrated). In fact, when the folding kinetics is two-state, all of the unfolded state relaxation times within the unfolded free energy basin are faster than the folding time. This result supports the well-established funnel energy landscape picture and resolves an apparent contradiction between this model and the recently proposed kinetic hub model of protein folding. We validate these concepts by analyzing a Markov state model of the kinetics in the unfolded state and folding of the miniprotein NTL9 (where NTL9 is the N-terminal domain of the ribosomal protein L9), constructed from a 2.9 ms simulation provided by D. E. Shaw Research.
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Affiliation(s)
- Wei Dai
- Department of Physics and Astronomy, Rutgers the State University of New Jersey, Piscataway, New Jersey 08854, USA
| | - Anirvan M Sengupta
- Department of Physics and Astronomy, Rutgers the State University of New Jersey, Piscataway, New Jersey 08854, USA
| | - Ronald M Levy
- Center for Biophysics and Computational Biology and Institute for Computational Molecular Science, Temple University, Philadelphia, Pennsylvania 19122-1801, USA and Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122-1801, USA
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15
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Nam GM, Makarov DE. Extracting intrinsic dynamic parameters of biomolecular folding from single-molecule force spectroscopy experiments. Protein Sci 2015; 25:123-34. [PMID: 26088347 DOI: 10.1002/pro.2727] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2015] [Revised: 06/17/2015] [Accepted: 06/17/2015] [Indexed: 11/12/2022]
Abstract
Single-molecule studies in which a mechanical force is transmitted to the molecule of interest and the molecular extension or position is monitored as a function of time are versatile tools for probing the dynamics of protein folding, stepping of molecular motors, and other biomolecular processes involving activated barrier crossing. One complication in interpreting such studies, however, is the fact that the typical size of a force probe (e.g., a dielectric bead in optical tweezers or the atomic force microscope tip/cantilever assembly) is much larger than the molecule itself, and so the observed molecular motion is affected by the hydrodynamic drag on the probe. This presents the experimenter with a nontrivial task of deconvolving the intrinsic molecular parameters, such as the intrinsic free energy barrier and the effective diffusion coefficient exhibited while crossing the barrier from the experimental signal. Here we focus on the dynamical aspect of this task and show how the intrinsic diffusion coefficient along the molecular reaction coordinate can be inferred from single-molecule measurements of the rates of biomolecular folding and unfolding. We show that the feasibility of accomplishing this task is strongly dependent on the relationship between the intrinsic molecular elasticity and that of the linker connecting the molecule to the force probe and identify the optimal range of instrumental parameters allowing determination of instrument-free molecular dynamics.
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Affiliation(s)
- Gi-Moon Nam
- Department of Chemistry, University of Texas at Austin, Austin, Texas, 78712
| | - Dmitrii E Makarov
- Department of Chemistry, University of Texas at Austin, Austin, Texas, 78712.,Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, 78712
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16
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English CA, García AE. Charged Termini on the Trp-Cage Roughen the Folding Energy Landscape. J Phys Chem B 2015; 119:7874-81. [DOI: 10.1021/acs.jpcb.5b02040] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Charles A. English
- Department of Physics and Astronomy and The Center for
Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, New York 12180, United States
| | - Angel E. García
- Department of Physics and Astronomy and The Center for
Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, New York 12180, United States
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17
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Chekmarev SF. Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario. PLoS One 2015; 10:e0121640. [PMID: 25848943 PMCID: PMC4388825 DOI: 10.1371/journal.pone.0121640] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2014] [Accepted: 02/11/2015] [Indexed: 11/19/2022] Open
Abstract
The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson’s equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.
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Affiliation(s)
- Sergei F. Chekmarev
- Institute of Thermophysics, 630090 Novosibirsk, Russia and Department of Physics, Novosibirsk State University, 630090 Novosibirsk, Russia
- * E-mail:
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Kalgin IV, Chekmarev SF. Folding of a β-sheet miniprotein: probability fluxes, streamlines, and the potential for the driving force. J Phys Chem B 2015; 119:1380-7. [PMID: 25544646 DOI: 10.1021/jp5112795] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
In this work we continue the study of the first-passage folding of an antiparallel β-sheet miniprotein (beta3s) that was initiated in the previous work [Kalgin et al. J. Phys. Chem. B, 2014, 118, 4287]. We consider a larger ensemble of folding trajectories, which allows us to gain a closer insight into the folding dynamics. In particular, we calculate the potential for the driving force of folding in a reduced space of collective variables. The potential has two components. One component (Φ) is responsible for the source and sink of the folding flow, which are formed, respectively, in the regions of the unfolded and native states of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the sides of the reaction channel and provides the canalization of the folding flow between the source and sink. We show that both components obey Poisson's equations with the corresponding source/sink terms. The resulting components have a very simple form: the Φ-surface consists of two well-defined peaks of different signs, which correspond, respectively, to the source and sink of the folding flow, and the Ψ-surface consists of two ridges of different signs that connect the source and sink of the flow.
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Affiliation(s)
- Igor V Kalgin
- Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia
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Lane TJ, Schwantes CR, Beauchamp KA, Pande VS. Probing the origins of two-state folding. J Chem Phys 2014; 139:145104. [PMID: 24116650 DOI: 10.1063/1.4823502] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/26/2023] Open
Abstract
Many protein systems fold in a two-state manner. Random models, however, rarely display two-state kinetics and thus such behavior should not be accepted as a default. While theories for the prevalence of two-state kinetics have been presented, none sufficiently explain the breadth of experimental observations. A model, making minimal assumptions, is introduced that suggests two-state behavior is likely for any system with an overwhelmingly populated native state. We show two-state folding is a natural consequence of such two-state thermodynamics, and is strengthened by increasing the population of the native state. Further, the model exhibits hub-like behavior, with slow interconversions between unfolded states. Despite this, the unfolded state equilibrates quickly relative to the folding time. This apparent paradox is readily understood through this model. Finally, our results compare favorable with measurements of folding rates as a function of chain length and Keq, providing new insight into these relations.
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Affiliation(s)
- Thomas J Lane
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
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Kalgin IV, Chekmarev SF, Karplus M. First passage analysis of the folding of a β-sheet miniprotein: is it more realistic than the standard equilibrium approach? J Phys Chem B 2014; 118:4287-99. [PMID: 24669953 PMCID: PMC4002127 DOI: 10.1021/jp412729r] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
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Simulations of first-passage folding
of the antiparallel β-sheet
miniprotein beta3s, which has been intensively studied under equilibrium
conditions by A. Caflisch and co-workers, show that the kinetics and
dynamics are significantly different from those for equilibrium folding.
Because the folding of a protein in a living system generally corresponds
to the former (i.e., the folded protein is stable and unfolding is
a rare event), the difference is of interest. In contrast to equilibrium
folding, the Ch-curl conformations become very rare because they contain
unfavorable parallel β-strand arrangements, which are difficult
to form dynamically due to the distant N- and C-terminal strands.
At the same time, the formation of helical conformations becomes much
easier (particularly in the early stage of folding) due to short-range
contacts. The hydrodynamic descriptions of the folding reaction have
also revealed that while the equilibrium flow field presented a collection
of local vortices with closed ”streamlines”, the first-passage
folding is characterized by a pronounced overall flow from the unfolded
states to the native state. The flows through the locally stable structures
Cs-or and Ns-or, which are conformationally close to the native state,
are negligible due to detailed balance established between these structures
and the native state. Although there are significant differences in
the general picture of the folding process from the equilibrium and
first-passage folding simulations, some aspects of the two are in
agreement. The rate of transitions between the clusters of characteristic
protein conformations in both cases decreases approximately exponentially
with the distance between the clusters in the hydrogen bond distance
space of collective variables, and the folding time distribution in
the first-passage segments of the equilibrium trajectory is in good
agreement with that for the first-passage folding simulations.
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Affiliation(s)
- Igor V Kalgin
- Department of Physics, Novosibirsk State University , 630090 Novosibirsk, Russia
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Meuzelaar H, Tros M, Huerta-Viga A, van Dijk CN, Vreede J, Woutersen S. Solvent-Exposed Salt Bridges Influence the Kinetics of α-Helix Folding and Unfolding. J Phys Chem Lett 2014; 5:900-904. [PMID: 24634715 PMCID: PMC3948500 DOI: 10.1021/jz500029a] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2014] [Accepted: 02/14/2014] [Indexed: 06/03/2023]
Abstract
Salt bridges are known to play an essential role in the thermodynamic stability of the folded conformation of many proteins, but their influence on the kinetics of folding remains largely unknown. Here, we investigate the effect of Glu-Arg salt bridges on the kinetics of α-helix folding using temperature-jump transient-infrared spectroscopy and steady-state UV circular dichroism. We find that geometrically optimized salt bridges (Glu- and Arg+ are spaced four peptide units apart, and the Glu/Arg order is such that the side-chain rotameric preferences favor salt-bridge formation) significantly speed up folding and slow down unfolding, whereas salt bridges with unfavorable geometry slow down folding and slightly speed up unfolding. Our observations suggest a possible explanation for the surprising fact that many biologically active proteins contain salt bridges that do not stabilize the native conformation: these salt bridges might have a kinetic rather than a thermodynamic function.
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