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Ooka K, Liu R, Arai M. The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics. Molecules 2022; 27:molecules27144460. [PMID: 35889332 PMCID: PMC9319528 DOI: 10.3390/molecules27144460] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2022] [Revised: 07/07/2022] [Accepted: 07/08/2022] [Indexed: 11/16/2022] Open
Abstract
Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.
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Affiliation(s)
- Koji Ooka
- Department of Physics, Graduate School of Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan;
- Komaba Organization for Educational Excellence, College of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
| | - Runjing Liu
- Department of Life Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan;
| | - Munehito Arai
- Department of Physics, Graduate School of Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan;
- Department of Life Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan;
- Correspondence:
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2
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Ancona M, Bentivoglio A, Caraglio M, Gonnella G, Pelizzola A. Emergence of effective temperatures in an out-of-equilibrium model of biopolymer folding. Phys Rev E 2021; 103:062415. [PMID: 34271706 DOI: 10.1103/physreve.103.062415] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2020] [Accepted: 05/26/2021] [Indexed: 11/07/2022]
Abstract
We investigate the possibility of extending the notion of temperature in a stochastic model for the RNA or protein folding driven out of equilibrium. We simulate the dynamics of a small RNA hairpin subject to an external pulling force, which is time-dependent. First, we consider a fluctuation-dissipation relation (FDR) whereby we verify that various effective temperatures can be obtained for different observables, only when the slowest intrinsic relaxation timescale of the system regulates the dynamics of the system. Then, we introduce a different nonequilibrium temperature, which is defined from the rate of heat exchanged with a weakly interacting thermal bath. Notably, this "kinetic" temperature can be defined for any frequency of the external switching force. We also discuss and compare the behavior of these two emerging parameters, by discriminating the time-delayed nature of the FDR temperature from the instantaneous character of the kinetic temperature. The validity of our numerics are corroborated by a simple four-state Markov model which describes the long-time behavior of the RNA molecule.
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Affiliation(s)
- Marco Ancona
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
| | - Alessandro Bentivoglio
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
| | - Michele Caraglio
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria
| | - Giuseppe Gonnella
- Dipartimento di Fisica, Università degli Studi di Bari and INFN, Sezione di Bari, 70126 Bari, Italy
| | - Alessandro Pelizzola
- Dipartimento Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.,INFN, Sezione di Torino, via Pietro Giuria 1, 10125 Torino, Italy
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3
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Hutton RD, Wilkinson J, Faccin M, Sivertsson EM, Pelizzola A, Lowe AR, Bruscolini P, Itzhaki LS. Mapping the Topography of a Protein Energy Landscape. J Am Chem Soc 2015; 137:14610-25. [PMID: 26561984 DOI: 10.1021/jacs.5b07370] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Abstract
Protein energy landscapes are highly complex, yet the vast majority of states within them tend to be invisible to experimentalists. Here, using site-directed mutagenesis and exploiting the simplicity of tandem-repeat protein structures, we delineate a network of these states and the routes between them. We show that our target, gankyrin, a 226-residue 7-ankyrin-repeat protein, can access two alternative (un)folding pathways. We resolve intermediates as well as transition states, constituting a comprehensive series of snapshots that map early and late stages of the two pathways and show both to be polarized such that the repeat array progressively unravels from one end of the molecule or the other. Strikingly, we find that the protein folds via one pathway but unfolds via a different one. The origins of this behavior can be rationalized using the numerical results of a simple statistical mechanics model that allows us to visualize the equilibrium behavior as well as single-molecule folding/unfolding trajectories, thereby filling in the gaps that are not accessible to direct experimental observation. Our study highlights the complexity of repeat-protein folding arising from their symmetrical structures; at the same time, however, this structural simplicity enables us to dissect the complexity and thereby map the precise topography of the energy landscape in full breadth and remarkable detail. That we can recapitulate the key features of the folding mechanism by computational analysis of the native structure alone will help toward the ultimate goal of designed amino-acid sequences with made-to-measure folding mechanisms-the Holy Grail of protein folding.
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Affiliation(s)
- Richard D Hutton
- Hutchison/MRC Research Centre , Hills Road, Cambridge CB2 0XZ, U.K
| | - James Wilkinson
- Hutchison/MRC Research Centre , Hills Road, Cambridge CB2 0XZ, U.K
| | - Mauro Faccin
- ICTEAM, Université Catholique de Lovain , Euler Building 4, Avenue Lemaître, B-1348 Louvain-la-Neuve, Belgium
| | - Elin M Sivertsson
- Department of Pharmacology, University of Cambridge , Tennis Court Road, Cambridge CB2 1PD, U.K
| | - Alessandro Pelizzola
- Dipartimento di Scienza Applicata e Tecnologia, CNISM, and Center for Computational Studies, Politecnico di Torino , Corso Duca degli Abruzzi 24, I-10129 Torino, Italy.,INFN, Sezione di Torino , via Pietro Giuria 1, I-10125 Torino, Italy.,Human Genetics Foundation (HuGeF) , Via Nizza 52, I-10126 Torino, Italy
| | - Alan R Lowe
- Institute of Structural and Molecular Biology and London Centre for Nanotechnology, University College London and Birkbeck College , London WC1E 7HX, U.K
| | - Pierpaolo Bruscolini
- Departamento de Física Teórica and Instituto de Biocomputacíon y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza , c/Mariano Esquillor s/n, 50018 Zaragoza, Spain
| | - Laura S Itzhaki
- Department of Pharmacology, University of Cambridge , Tennis Court Road, Cambridge CB2 1PD, U.K
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Aioanei D, Brucale M, Tessari I, Bubacco L, Samorì B. Worm-like Ising model for protein mechanical unfolding under the effect of osmolytes. Biophys J 2012; 102:342-50. [PMID: 22339871 PMCID: PMC3260691 DOI: 10.1016/j.bpj.2011.12.007] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2011] [Revised: 10/12/2011] [Accepted: 12/06/2011] [Indexed: 10/14/2022] Open
Abstract
We show via single-molecule mechanical unfolding experiments that the osmolyte glycerol stabilizes the native state of the human cardiac I27 titin module against unfolding without shifting its unfolding transition state on the mechanical reaction coordinate. Taken together with similar findings on the immunoglobulin-binding domain of streptococcal protein G (GB1), these experimental results suggest that osmolytes act on proteins through a common mechanism that does not entail a shift of their unfolding transition state. We investigate the above common mechanism via an Ising-like model for protein mechanical unfolding that adds worm-like-chain behavior to a recent generalization of the Wako-Saitô-Muñoz-Eaton model with support for group-transfer free energies. The thermodynamics of the model are exactly solvable, while protein kinetics under mechanical tension can be simulated via Monte Carlo algorithms. Notably, our force-clamp and velocity-clamp simulations exhibit no shift in the position of the unfolding transition state of GB1 and I27 under the effect of various osmolytes. The excellent agreement between experiment and simulation strongly suggests that osmolytes do not assume a structural role at the mechanical unfolding transition state of proteins, acting instead by adjusting the solvent quality for the protein chain analyte.
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Affiliation(s)
- Daniel Aioanei
- Department of Biochemistry G. Moruzzi, University of Bologna, Bologna, Italy.
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5
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Caraglio M, Pelizzola A. Effects of confinement on thermal stability and folding kinetics in a simple Ising-like model. Phys Biol 2012; 9:016006. [PMID: 22313579 DOI: 10.1088/1478-3975/9/1/016006] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
In a cellular environment, confinement and macromolecular crowding play an important role in thermal stability and folding kinetics of a protein. We have resorted to a generalized version of the Wako-Saitô-Muñoz-Eaton model for protein folding to study the behavior of six different protein structures confined between two walls. Changing the distance 2R between the walls, we found, in accordance with previous studies, two confinement regimes: starting from large R and decreasing R, confinement first enhances the stability of the folded state as long as this is compact and until a given value of R; then a further decrease of R leads to a decrease of folding temperature and folding rate. We found that in the low confinement regime, both unfolding temperatures and logarithm of folding rates scale as R(-γ) where γ values lie in between 1.42 and 2.35.
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Affiliation(s)
- M Caraglio
- Dipartimento di Fisica, CNISM and Center for Computational Studies, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy.
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6
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Caraglio M, Imparato A, Pelizzola A. Direction-dependent mechanical unfolding and green fluorescent protein as a force sensor. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021918. [PMID: 21929030 DOI: 10.1103/physreve.84.021918] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2011] [Revised: 07/05/2011] [Indexed: 05/31/2023]
Abstract
An Ising-like model of proteins is used to investigate the mechanical unfolding of the green fluorescent protein along different directions. When the protein is pulled from its ends, we recover the major and minor unfolding pathways observed in experiments. Upon varying the pulling direction, we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the direction dependence of the unfolding force at equilibrium, we propose a force sensor whose luminescence depends on the applied force.
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Affiliation(s)
- M Caraglio
- Dipartimento di Fisica and CNISM, Politecnico di Torino, c. Duca degli Abruzzi 24, Torino, Italy.
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7
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Faccin M, Bruscolini P, Pelizzola A. Analysis of the equilibrium and kinetics of the ankyrin repeat protein myotrophin. J Chem Phys 2011; 134:075102. [PMID: 21341874 DOI: 10.1063/1.3535562] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We apply the Wako-Saito-Muñoz-Eaton model to the study of myotrophin, a small ankyrin repeat protein, whose folding equilibrium and kinetics have been recently characterized experimentally. The model, which is a native-centric with binary variables, provides a finer microscopic detail than the Ising model that has been recently applied to some different repeat proteins, while being still amenable for an exact solution. In partial agreement with the experiments, our results reveal a weakly three-state equilibrium and a two-state-like kinetics of the wild-type protein despite the presence of a nontrivial free-energy profile. These features appear to be related to a careful "design" of the free-energy landscape, so that mutations can alter this picture, stabilizing some intermediates and changing the position of the rate-limiting step. Also, the experimental findings of two alternative pathways, an N-terminal and a C-terminal one, are qualitatively confirmed, even if the variations in the rates upon the experimental mutations cannot be quantitatively reproduced. Interestingly, the folding and unfolding pathways appear to be different, even if closely related: a property that is not generally considered in the phenomenological interpretation of the experimental data.
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Affiliation(s)
- Mauro Faccin
- Departamento de Física Teórica & Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, Zaragoza, Spain
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8
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Aioanei D, Tessari I, Bubacco L, Samorì B, Brucale M. Observing the osmophobic effect in action at the single molecule level. Proteins 2011; 79:2214-23. [DOI: 10.1002/prot.23045] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2010] [Revised: 03/04/2011] [Accepted: 03/13/2011] [Indexed: 11/08/2022]
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Caraglio M, Imparato A, Pelizzola A. Pathways of mechanical unfolding of FnIII10: Low force intermediates. J Chem Phys 2010; 133:065101. [DOI: 10.1063/1.3464476] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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10
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Morgado WAM, Soares-Pinto DO. Exact nonequilibrium work generating function for a small classical system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:021112. [PMID: 20866780 DOI: 10.1103/physreve.82.021112] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2009] [Revised: 07/25/2010] [Indexed: 05/29/2023]
Abstract
We obtain the exact nonequilibrium work generating function (NEWGF) for a small system consisting of a massive Brownian particle connected to internal and external springs. The external work is provided to the system for a finite-time interval. The Jarzynski equality, obtained in this case directly from the NEWGF, is shown to be valid for the present model, in an exact way regardless of the rate of external work.
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Affiliation(s)
- W A M Morgado
- Departamento de Física, Pontifícia Universidade Católica and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, RJ, Brazil.
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11
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Luccioli S, Imparato A, Mitternacht S, Irbäck A, Torcini A. Unfolding times for proteins in a force clamp. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:010902. [PMID: 20365316 DOI: 10.1103/physreve.81.010902] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2009] [Revised: 11/09/2009] [Indexed: 05/29/2023]
Abstract
The escape process from the native valley for proteins subjected to a constant stretching force is examined using a model for a beta barrel. For a wide range of forces, the unfolding dynamics can be treated as one-dimensional diffusion, parametrized in terms of the end-to-end distance. In particular, the escape times can be evaluated as first passage times for a Brownian particle moving on the protein free-energy landscape, using the Smoluchowski equation. At strong forces, the unfolding process can be viewed as a diffusive drift away from the native state, while at weak forces thermal activation is the relevant mechanism. An escape-time analysis within this approach reveals a crossover from an exponential to an inverse Gaussian escape-time distribution upon passing from weak to strong forces. Moreover, a single expression valid at weak and strong forces can be devised both for the average unfolding time as well as for the corresponding variance. The analysis offers a possible explanation of recent experimental findings for the proteins ddFLN4 and ubiquitin.
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Affiliation(s)
- Stefano Luccioli
- Istituto dei Sistemi Complessi, CNR, via Madonna del Piano, Sesto Fiorentino, Italy
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12
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Zamparo M, Pelizzola A. Nearly symmetrical proteins: folding pathways and transition states. J Chem Phys 2009; 131:035101. [PMID: 19624233 DOI: 10.1063/1.3170984] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The folding pathways of the B domain of protein A have been the subject of many experimental and computational studies. Based on a statistical mechanical model, it has been suggested that the native state symmetry leads to multiple pathways, highly dependent on temperature and denaturant concentration. Experiments, however, have not confirmed this scenario. By considering four nearly symmetrical proteins, one of them being the above molecule, here we show that, if contact energies are properly taken into account, a different picture emerges from kinetic simulations of the above-mentioned model. This is characterized by a dominant folding pathway, which is consistent with the most recent experimental results. Given the simplicity of the model, we also report on a direct sampling of the transition state.
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Affiliation(s)
- Marco Zamparo
- Dipartimento di Fisica, CNISM Unità di Torino and INFN, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
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Imparato A, Pelizzola A, Zamparo M. Equilibrium properties and force-driven unfolding pathways of RNA molecules. PHYSICAL REVIEW LETTERS 2009; 103:188102. [PMID: 19905835 DOI: 10.1103/physrevlett.103.188102] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2009] [Indexed: 05/28/2023]
Abstract
The mechanical unfolding of a simple RNA hairpin and of a 236-base portion of the Tetrahymena thermophila ribozyme is studied by means of an Ising-like model. Phase diagrams and free energy landscapes are computed exactly and suggest a simple two-state behavior for the hairpin and the presence of intermediate states for the ribozyme. Nonequilibrium simulations give the possible unfolding pathways for the ribozyme, and the dominant pathway corresponds to the experimentally observed one.
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Affiliation(s)
- A Imparato
- Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, Building 1520, DK-8000 Aarhus C, Denmark.
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15
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Changing the mechanical unfolding pathway of FnIII10 by tuning the pulling strength. Biophys J 2009; 96:429-41. [PMID: 19167294 DOI: 10.1016/j.bpj.2008.09.043] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2008] [Accepted: 09/30/2008] [Indexed: 11/21/2022] Open
Abstract
We investigate the mechanical unfolding of the tenth type III domain from fibronectin (FnIII(10)) both at constant force and at constant pulling velocity, by all-atom Monte Carlo simulations. We observe both apparent two-state unfolding and several unfolding pathways involving one of three major, mutually exclusive intermediate states. All three major intermediates lack two of seven native beta-strands, and share a quite similar extension. The unfolding behavior is found to depend strongly on the pulling conditions. In particular, we observe large variations in the relative frequencies of occurrence for the intermediates. At low constant force or low constant velocity, all three major intermediates occur with a significant frequency. At high constant force or high constant velocity, one of them, with the N- and C-terminal beta-strands detached, dominates over the other two. Using the extended Jarzynski equality, we also estimate the equilibrium free-energy landscape, calculated as a function of chain extension. The application of a constant pulling force leads to a free-energy profile with three major local minima. Two of these correspond to the native and fully unfolded states, respectively, whereas the third one can be associated with the major unfolding intermediates.
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Imparato A, Pelizzola A. Mechanical unfolding and refolding pathways of ubiquitin. PHYSICAL REVIEW LETTERS 2008; 100:158104. [PMID: 18518158 DOI: 10.1103/physrevlett.100.158104] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2007] [Indexed: 05/26/2023]
Abstract
Mechanical unfolding and refolding of ubiquitin are studied by Monte Carlo simulations of a Gō model with binary variables. The exponential dependence of the time constants on the force is verified, and folding and unfolding lengths are computed, with good agreement with experimental results. Furthermore, the model exhibits intermediate kinetic states, as observed in experiments. Unfolding and refolding pathways and intermediate states, obtained by tracing single secondary structure elements, are consistent with simulations of previous all-atom models and with the experimentally observed step sizes.
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Affiliation(s)
- A Imparato
- Dipartimento di Fisica and CNISM, INFN Sezione di Torino, Politecnico di Torino, c. Duca degli Abruzzi 24, Turin, Italy
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