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McDonald J, von Spakovsky MR, Reynolds WT. Predicting non-equilibrium folding behavior of polymer chains using the steepest-entropy-ascent quantum thermodynamic framework. J Chem Phys 2023; 158:104904. [PMID: 36922120 DOI: 10.1063/5.0137444] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023] Open
Abstract
The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework is used to explore the influence of heating and cooling on polymer chain folding kinetics. The framework predicts how a chain moves from an initial non-equilibrium state to stable equilibrium along a unique thermodynamic path. The thermodynamic state is expressed by occupation probabilities corresponding to the levels of a discrete energy landscape. The landscape is generated using the Replica Exchange Wang-Landau method applied to a polymer chain represented by a sequence of hydrophobic and polar monomers with a simple hydrophobic-polar amino acid model. The chain conformation evolves as energy shifts among the levels of the energy landscape according to the principle of steepest entropy ascent. This principle is implemented via the SEAQT equation of motion. The SEAQT framework has the benefit of providing insight into structural properties under non-equilibrium conditions. Chain conformations during heating and cooling change continuously without sharp transitions in morphology. The changes are more drastic along non-equilibrium paths than along quasi-equilibrium paths. The SEAQT-predicted kinetics are fitted to rates associated with the experimental intensity profiles of cytochrome c protein folding with Rouse dynamics.
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Affiliation(s)
- Jared McDonald
- Materials Science and Engineering Department, Virginia Tech, Blacksburg, Virginia 24061, USA
| | | | - William T Reynolds
- Materials Science and Engineering Department, Virginia Tech, Blacksburg, Virginia 24061, USA
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2
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Takahashi T, Chikenji G, Tokita K. Lattice protein design using Bayesian learning. Phys Rev E 2021; 104:014404. [PMID: 34412286 DOI: 10.1103/physreve.104.014404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2020] [Accepted: 06/11/2021] [Indexed: 01/01/2023]
Abstract
Protein design is the inverse approach of the three-dimensional (3D) structure prediction for elucidating the relationship between the 3D structures and amino acid sequences. In general, the computation of the protein design involves a double loop: A loop for amino acid sequence changes and a loop for an exhaustive conformational search for each amino acid sequence. Herein, we propose a novel statistical mechanical design method using Bayesian learning, which can design lattice proteins without the exhaustive conformational search. We consider a thermodynamic hypothesis of the evolution of proteins and apply it to the prior distribution of amino acid sequences. Furthermore, we take the water effect into account in view of the grand canonical picture. As a result, on applying the 2D lattice hydrophobic-polar (HP) model, our design method successfully finds an amino acid sequence for which the target conformation has a unique ground state. However, the performance was not as good for the 3D lattice HP models compared to the 2D models. The performance of the 3D model improves on using a 20-letter lattice proteins. Furthermore, we find a strong linearity between the chemical potential of water and the number of surface residues, thereby revealing the relationship between protein structure and the effect of water molecules. The advantage of our method is that it greatly reduces computation time, because it does not require long calculations for the partition function corresponding to an exhaustive conformational search. As our method uses a general form of Bayesian learning and statistical mechanics and is not limited to lattice proteins, the results presented here elucidate some heuristics used successfully in previous protein design methods.
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Affiliation(s)
- Tomoei Takahashi
- Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan
| | - George Chikenji
- Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
| | - Kei Tokita
- Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan
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3
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Shi G, Wüst T, Landau DP. Elucidating thermal behavior, native contacts, and folding funnels of simple lattice proteins using replica exchange Wang-Landau sampling. J Chem Phys 2018; 149:164913. [PMID: 30384708 DOI: 10.1063/1.5026256] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We studied the folding behavior of two coarse-grained, lattice models, the HP (hydrophobic-polar) model and the semi-flexible H0P model, whose 124 monomer long sequences were derived from the protein Ribonuclease A. Taking advantage of advanced parallel computing techniques, we applied replica exchange Wang-Landau sampling and calculated the density of states over the models entire energy ranges to high accuracy. We then determined both energetic and structural quantities in order to elucidate the folding behavior of each model completely. As a result of sufficiently long sequences and model complexity, yet computational accessibility, we were able to depict distinct free energy folding funnels for both models. In particular, we found that the HP model folds in a single-step process with a very highly degenerate native state and relatively flat low temperature folding funnel minimum. By contrast, the semi-flexible H0P model folds via a multi-step process and the native state is almost four orders of magnitude less degenerate than that for the HP model. In addition, for the H0P model, the bottom of the free energy folding funnel remains rough, even at low temperatures.
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Affiliation(s)
- Guangjie Shi
- Center for Simulational Physics, The University of Georgia, Athens, Georgia 30602-0002, USA
| | - Thomas Wüst
- Scientific IT Services, ETH Zurich, 8092 Zurich, Switzerland
| | - David P Landau
- Center for Simulational Physics, The University of Georgia, Athens, Georgia 30602-0002, USA
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4
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Farris ACK, Shi G, Wüst T, Landau DP. The role of chain-stiffness in lattice protein models: A replica-exchange Wang-Landau study. J Chem Phys 2018; 149:125101. [PMID: 30278675 DOI: 10.1063/1.5045482] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Using Monte Carlo simulations, we investigate simple, physically motivated extensions to the hydrophobic-polar lattice protein model for the small (46 amino acid) protein Crambin. We use two-dimensional replica-exchange Wang-Landau sampling to study the effects of a bond angle stiffness parameter on the folding and uncover a new step in the collapse process for particular values of this stiffness parameter. A physical interpretation of the folding is developed by analysis of changes in structural quantities, and the free energy landscape is explored. For these special values of stiffness, we find non-degenerate ground states, a property that is consistent with behavior of real proteins, and we use these unique ground states to elucidate the formation of native contacts during the folding process. Through this analysis, we conclude that chain-stiffness is particularly influential in the low energy, low temperature regime of the folding process once the lattice protein has partially collapsed.
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Affiliation(s)
- Alfred C K Farris
- Center for Simulational Physics, Department of Physics and Astronomy, The University of Georgia, Athens, Georgia 30602, USA
| | - Guangjie Shi
- Center for Simulational Physics, Department of Physics and Astronomy, The University of Georgia, Athens, Georgia 30602, USA
| | - Thomas Wüst
- Scientific IT Services, ETH Zürich, 8092 Zürich, Switzerland
| | - David P Landau
- Center for Simulational Physics, Department of Physics and Astronomy, The University of Georgia, Athens, Georgia 30602, USA
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5
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Wilson MS, Shi G, Wüst T, Li YW, Landau DP. Influence of substrate pattern on the adsorption of HP lattice proteins. MOLECULAR SIMULATION 2018. [DOI: 10.1080/08927022.2018.1471691] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
Affiliation(s)
- Matthew S. Wilson
- Centre for Simulational Physics, The University of Georgia, Athens, GA, USA
| | - Guangjie Shi
- Centre for Simulational Physics, The University of Georgia, Athens, GA, USA
| | - Thomas Wüst
- Scientific IT Services, ETH Zürich IT Services, Zürich, Switzerland
| | - Ying Wai Li
- National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, USA
| | - David P. Landau
- Centre for Simulational Physics, The University of Georgia, Athens, GA, USA
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6
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Shi G, Wüst T, Landau DP. Characterizing folding funnels with replica exchange Wang-Landau simulation of lattice proteins. Phys Rev E 2016; 94:050402. [PMID: 27967143 DOI: 10.1103/physreve.94.050402] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2016] [Indexed: 02/01/2023]
Abstract
We have studied the folding of ribonuclease A by mapping it onto coarse-grained lattice protein models. With replica exchange Wang-Landau sampling, we calculated the free energy vs end-to-end distance as a function of temperature. A mapping to the famous hydrophobic-polar (HP) model shows a relatively shallow folding funnel and flat free energy minimum, reflecting the high degeneracy of the ground state. In contrast, extending the HP model with an additional "neutral" monomer type (i.e., a mapping to the three-letter H0P model) has a well developed, rough free energy funnel with a low degeneracy ground state. In both cases, folding funnels are asymmetric with temperature dependent shape.
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Affiliation(s)
- Guangjie Shi
- Center for Simulational Physics, The University of Georgia, Athens, Georgia 30602, USA
| | - Thomas Wüst
- Scientific IT Services, ETH Zürich, 8092 Zürich, Switzerland
| | - David P Landau
- Center for Simulational Physics, The University of Georgia, Athens, Georgia 30602, USA
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Yang X, Lu ZY. Control globular structure formation of a copolymer chain through inverse design. J Chem Phys 2016; 144:224902. [PMID: 27306020 DOI: 10.1063/1.4953576] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023] Open
Abstract
A copolymer chain in dilute solution can exhibit various globular structures with characteristic morphologies, which makes it a potentially useful candidate for artificial materials design. However, the chain has a huge conformation space and may not naturally form the globular structure we desire. An ideal way to control globular structure formation should be inverse design, i.e., starting from the target structure and finding out what kind of polymers can effectively generate it. To accomplish this, we propose an inverse design procedure, which is combined with Wang-Landau Monte Carlo to fully and precisely explore the huge conformation space of the chain. Starting from a desired target structure, all the geometrically possible sequences are exactly enumerated. Interestingly, reasonable interaction strengths are obtained and found to be not specified for only one sequence. Instead, they can be combined with many other sequences and also achieve a relatively high yield for target structure, although these sequences may be rather different. These results confirm the possibility of controlling globular structure formation of a copolymer chain through inverse design and pave the way for targeted materials design.
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Affiliation(s)
- Xi Yang
- State Key Laboratory of Supramolecular Structure and Materials, Institute of Theoretical Chemistry, Jilin University, Changchun 130021, China
| | - Zhong-Yuan Lu
- State Key Laboratory of Supramolecular Structure and Materials, Institute of Theoretical Chemistry, Jilin University, Changchun 130021, China
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Belardinelli RE, Pereyra VD. Nonconvergence of the Wang-Landau algorithms with multiple random walkers. Phys Rev E 2016; 93:053306. [PMID: 27301004 DOI: 10.1103/physreve.93.053306] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2015] [Indexed: 12/11/2022]
Abstract
This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and 1/t algorithms. The classical algorithms are modified by the use of m-independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number π by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, t; then, the average over m walkers is performed. It is observed that the error goes as 1/sqrt[m]. However, if the number of walkers increases above a certain critical value m>m_{x}, the error reaches a constant value (i.e., it saturates). This occurs for both algorithms; however, it is shown that for a given system, the 1/t algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value m_{x}, since it does not reduce the error in the calculation. Therefore, the number of walkers does not guarantee convergence.
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Affiliation(s)
- R E Belardinelli
- Instituto de Física Aplicada (INFAP)-CONICET, San Luis, Argentina.,Departamento de Física, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina
| | - V D Pereyra
- Departamento de Física, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina
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Martins PHL, Bachmann M. Interlocking order parameter fluctuations in structural transitions between adsorbed polymer phases. Phys Chem Chem Phys 2016; 18:2143-51. [PMID: 26690091 DOI: 10.1039/c5cp05038c] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
By means of contact-density chain-growth simulations of a simple coarse-grained lattice model for a polymer grafted at a solid homogeneous substrate, we investigate the complementary behavior of the numbers of surface-monomer and monomer-monomer contacts under various solvent and thermal conditions. This pair of contact numbers represents an appropriate set of order parameters that enables the distinct discrimination of significantly different compact phases of polymer adsorption. Depending on the transition scenario, these order parameters can interlock in perfect cooperation. The analysis helps understand the transitions from compact filmlike adsorbed polymer conformations into layered morphologies and dissolved adsorbed structures, respectively, in more detail.
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Affiliation(s)
- Paulo H L Martins
- Instituto de Física, Universidade Federal de Mato Grosso, 78060-900 Cuiabá, MT, Brazil.
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10
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Janke W, Paul W. Thermodynamics and structure of macromolecules from flat-histogram Monte Carlo simulations. SOFT MATTER 2016; 12:642-657. [PMID: 26574738 DOI: 10.1039/c5sm01919b] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Over the last decade flat-histogram Monte Carlo simulations, especially multi-canonical and Wang-Landau simulations, have emerged as a strong tool to study the statistical mechanics of polymer chains. These investigations have focused on coarse-grained models of polymers on the lattice and in the continuum. Phase diagrams of chains in bulk as well as chains attached to surfaces were studied, for homopolymers as well as for protein-like models. Also, aggregation behavior in solution of these models has been investigated. We will present here the theoretical background for these simulations, explain the algorithms used and discuss their performance and give an overview over the systems studied with these methods in the literature, where we will limit ourselves to studies of coarse-grained model systems. Implementations of these algorithms on parallel computers will be also briefly described. In parallel to the development of these simulation methods, the power of a micro-canonical analysis of such simulations has been recognized, and we present the current state of the art in applying the micro-canonical analysis to phase transitions in nanoscopic polymer systems.
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Affiliation(s)
- Wolfhard Janke
- Institut für Theoretische Physik, Universität Leipzig, 04009 Leipzig, Germany.
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Hidaka T, Shimada A, Nakata Y, Kodama H, Kurihara H, Tokihiro T, Ihara S. Simple model of pH-induced protein denaturation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012709. [PMID: 26274205 DOI: 10.1103/physreve.92.012709] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2014] [Indexed: 06/04/2023]
Abstract
The pH-induced conformational changes of proteins are systematically studied in the framework of a hydrophobic-polar (HP) model, in which proteins are dramatically simplified as chains of hydrophobic (H) and polar (P) beads on a lattice. We express the electrostatic interaction, the principal driving force of pH-induced unfolding that is not included in the conventional HP model, as the repulsive energy term between P monomers. As a result of the exact enumeration of all of the 14- to 18-mers, it is found that lowest-energy states in many sequences change from single "native" conformations to multiple sets of "denatured" conformations with an increase in the electrostatic repulsion. The switching of the lowest-energy states occurs in quite a similar way to real proteins: it is almost always between two states, while in a small fraction of ≥16-mers it is between three states. We also calculate the structural fluctuations for all of the denatured states and find that the denatured states contain a broad range of incompletely unfolded conformations, similar to "molten globule" states referred to in acid or alkaline denatured real proteins. These results show that the proposed model provides a simple physical picture of pH-induced protein denaturation.
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Affiliation(s)
- T Hidaka
- Graduate School of Medicine and Faculty of Medicine, The University of Tokyo, Hongo, Tokyo 113-0033, Japan
| | - A Shimada
- Graduate School of Medicine and Faculty of Medicine, The University of Tokyo, Hongo, Tokyo 113-0033, Japan
| | - Y Nakata
- Institute for Biology and Mathematics of Dynamic Cellular Processes (iBMath), The University of Tokyo, Komaba, Tokyo 153-8904, Japan
- Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8902, Japan
| | - H Kodama
- Institute for Biology and Mathematics of Dynamic Cellular Processes (iBMath), The University of Tokyo, Komaba, Tokyo 153-8904, Japan
- Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8902, Japan
| | - H Kurihara
- Graduate School of Medicine and Faculty of Medicine, The University of Tokyo, Hongo, Tokyo 113-0033, Japan
- Institute for Biology and Mathematics of Dynamic Cellular Processes (iBMath), The University of Tokyo, Komaba, Tokyo 153-8904, Japan
- Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Chiyoda-ku, Tokyo 102-0076, Japan
| | - T Tokihiro
- Institute for Biology and Mathematics of Dynamic Cellular Processes (iBMath), The University of Tokyo, Komaba, Tokyo 153-8904, Japan
- Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8902, Japan
- Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Chiyoda-ku, Tokyo 102-0076, Japan
| | - S Ihara
- Institute for Biology and Mathematics of Dynamic Cellular Processes (iBMath), The University of Tokyo, Komaba, Tokyo 153-8904, Japan
- Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8902, Japan
- Research Center for Advanced Science and Technology, The University of Tokyo, Komaba, Tokyo 153-8904, Japan
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Wang W, Zhao P, Yang X, Lu ZY. Coil-to-globule transitions of homopolymers and multiblock copolymers. J Chem Phys 2014; 141:244907. [PMID: 25554180 DOI: 10.1063/1.4904888] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Wei Wang
- Institute of Theoretical Chemistry, State Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun 130023, China
| | - Peng Zhao
- Institute of Theoretical Chemistry, State Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun 130023, China
| | - Xi Yang
- Institute of Theoretical Chemistry, State Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun 130023, China
| | - Zhong-Yuan Lu
- Institute of Theoretical Chemistry, State Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun 130023, China
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