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Tsallis C. Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220293. [PMID: 37573876 DOI: 10.1098/rsta.2022.0293] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2022] [Accepted: 04/10/2023] [Indexed: 08/15/2023]
Abstract
The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional [Formula: see text]. The index [Formula: see text] and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.
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Affiliation(s)
- Constantino Tsallis
- Centro Brasileiro de Pesquisas Fisicas National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, 22290- Rio de Janeiro, Brazil
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, 87501 NM, USA
- Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
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Nagai T, Tsurumaki S, Urano R, Fujimoto K, Shinoda W, Okazaki S. Position-Dependent Diffusion Constant of Molecules in Heterogeneous Systems as Evaluated by the Local Mean Squared Displacement. J Chem Theory Comput 2020; 16:7239-7254. [DOI: 10.1021/acs.jctc.0c00448] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/10/2023]
Affiliation(s)
- Tetsuro Nagai
- Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
- Department of Materials Chemistry, Graduate School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
| | - Shuhei Tsurumaki
- Department of Materials Chemistry, Graduate School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
| | - Ryo Urano
- Department of Materials Chemistry, Graduate School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
| | - Kazushi Fujimoto
- Department of Materials Chemistry, Graduate School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
| | - Wataru Shinoda
- Department of Materials Chemistry, Graduate School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
| | - Susumu Okazaki
- Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
- Department of Materials Chemistry, Graduate School of Engineering, Nagoya University, Nagoya, Aichi 464-8603, Japan
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Narayan B, Herbert C, Yuan Y, Rodriguez BJ, Brooks BR, Buchete NV. Conformational analysis of replica exchange MD: Temperature-dependent Markov networks for FF amyloid peptides. J Chem Phys 2018; 149:072323. [PMID: 30134732 DOI: 10.1063/1.5027580] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
Abstract
Recent molecular modeling methods using Markovian descriptions of conformational states of biomolecular systems have led to powerful analysis frameworks that can accurately describe their complex dynamical behavior. In conjunction with enhanced sampling methods, such as replica exchange molecular dynamics (REMD), these frameworks allow the systematic and accurate extraction of transition probabilities between the corresponding states, in the case of Markov state models, and of statistically-optimized transition rates, in the case of the corresponding coarse master equations. However, applying automatically such methods to large molecular dynamics (MD) simulations, with explicit water molecules, remains limited both by the initial ability to identify good candidates for the underlying Markovian states and by the necessity to do so using good collective variables as reaction coordinates that allow the correct counting of inter-state transitions at various lag times. Here, we show that, in cases when representative molecular conformations can be identified for the corresponding Markovian states, and thus their corresponding collective evolution of atomic positions can be calculated along MD trajectories, one can use them to build a new type of simple collective variable, which can be particularly useful in both the correct state assignment and in the subsequent accurate counting of inter-state transition probabilities. In the case of the ubiquitously used root-mean-square deviation (RMSD) of atomic positions, we introduce the relative RMSD (RelRMSD) measure as a good reaction coordinate candidate. We apply this method to the analysis of REMD trajectories of amyloid-forming diphenylalanine (FF) peptides-a system with important nanotechnology and biomedical applications due to its self-assembling and piezoelectric properties-illustrating the use of RelRMSD in extracting its temperature-dependent intrinsic kinetics, without a priori assumptions on the functional form (e.g., Arrhenius or not) of the underlying conformational transition rates. The RelRMSD analysis enables as well a more objective assessment of the convergence of the REMD simulations. This type of collective variable may be generalized to other observables that could accurately capture conformational differences between the underlying Markov states (e.g., distance RMSD, the fraction of native contacts, etc.).
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Affiliation(s)
- Brajesh Narayan
- School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
| | - Colm Herbert
- School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
| | - Ye Yuan
- School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
| | - Brian J Rodriguez
- School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
| | - Bernard R Brooks
- Laboratory of Computational Biology, NHLBI, National Institutes of Health, Bethesda, Maryland 20892, USA
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Iwai R, Kasahara K, Takahashi T. Influence of various parameters in the replica-exchange molecular dynamics method: Number of replicas, replica-exchange frequency, and thermostat coupling time constant. Biophys Physicobiol 2018; 15:165-172. [PMID: 30250775 PMCID: PMC6145944 DOI: 10.2142/biophysico.15.0_165] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2018] [Accepted: 07/05/2018] [Indexed: 12/29/2022] Open
Abstract
The replica-exchange molecular dynamics (REMD) method has been used for conformational sampling of various biomolecular systems. To maximize sampling efficiency, some adjustable parameters must be optimized. Although it is agreed that shorter intervals between the replica-exchange attempts enhance traversals in the temperature space, details regarding the artifacts caused by these short intervals are controversial. In this study, we revisit this problem by performing REMD simulations on an alanine octapeptide in an implicit solvent. Fifty different sets of conditions, which are a combination of five replica-exchange periods, five different numbers of replicas, and two thermostat coupling time constants, were investigated. As a result, although short replica-exchange intervals enhanced the traversals in the temperature space, they led to artifacts in the ensemble average of the temperature, potential energy, and helix content. With extremely short replica-exchange intervals, i.e., attempted at every time step, the ensemble average of the temperature deviated from the thermostat temperature by ca. 7 K. Differences in the ensembles were observed even for larger replica-exchange intervals (between 100 and 1,000 steps). In addition, the shorter thermostat coupling time constant reduced the artifacts found when short replica-exchange intervals were used, implying that these artifacts are caused by insufficient thermal relaxation between the replica-exchange events. Our results will be useful to reduce the artifacts found in REMD simulations by adjusting some key parameters.
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Affiliation(s)
- Ryosuke Iwai
- Graduate School of Life Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
| | - Kota Kasahara
- College of Life Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
| | - Takuya Takahashi
- College of Life Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
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Leahy CT, Kells A, Hummer G, Buchete NV, Rosta E. Peptide dimerization-dissociation rates from replica exchange molecular dynamics. J Chem Phys 2018; 147:152725. [PMID: 29055328 DOI: 10.1063/1.5004774] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
We show how accurate rates of formation and dissociation of peptide dimers can be calculated using direct transition counting (DTC) from replica-exchange molecular dynamics (REMD) simulations. First, continuous trajectories corresponding to system replicas evolving at different temperatures are used to assign conformational states. Second, we analyze the entire REMD data to calculate the corresponding rates at each temperature directly from the number of transition counts. Finally, we compare the kinetics extracted directly, using the DTC method, with indirect estimations based on trajectory likelihood maximization using short-time propagators and on decay rates of state autocorrelation functions. For systems with relatively low-dimensional intrinsic conformational dynamics, the DTC method is simple to implement and leads to accurate temperature-dependent rates. We apply the DTC rate-extraction method to all-atom REMD simulations of dimerization of amyloid-forming NNQQ tetrapetides in explicit water. In an assessment of the REMD sampling efficiency with respect to standard MD, we find a gain of more than a factor of two at the lowest temperature.
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Affiliation(s)
- Cathal T Leahy
- School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
| | - Adam Kells
- Department of Chemistry, King's College London, London SE1 1DB, United Kingdom
| | - Gerhard Hummer
- Department of Theoretical Biophysics, Max Planck Institute of Biophysics, Max-von-Laue-Straße 3, 60438 Frankfurt am Main, Germany
| | | | - Edina Rosta
- Department of Chemistry, King's College London, London SE1 1DB, United Kingdom
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