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Wicks TJ, Wattis JAD, Graham RS. Monte–Carlo simulation of crystallization in single‐chain square‐well homopolymers. POLYMER CRYSTALLIZATION 2021. [DOI: 10.1002/pcr2.10146] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Affiliation(s)
- Thomas J. Wicks
- School of Mathematical Sciences University of Nottingham Nottingham UK
| | | | - Richard S. Graham
- School of Mathematical Sciences University of Nottingham Nottingham UK
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Taylor MP, Prunty TM, O'Neil CM. All-or-none folding of a flexible polymer chain in cylindrical nanoconfinement. J Chem Phys 2020; 152:094901. [PMID: 33480730 DOI: 10.1063/1.5144818] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
Geometric confinement of a polymer chain results in a loss of conformational entropy. For a chain that can fold into a compact native state via a first-order-like transition, as is the case for many small proteins, confinement typically provides an entropic stabilization of the folded state, thereby shifting the location of the transition. This allows for the possibility of confinement (entropy) driven folding. Here, we investigate such confinement effects for a flexible square-well-sphere N-mer chain (monomer diameter σ) confined within a long cylindrical pore (diameter D) or a closed cylindrical box (height H = D). We carry out Wang-Landau simulations to construct the density of states, which provides access to the complete thermodynamics of the system. For a wide pore, an entropic stabilization of the folded state is observed. However, as the pore diameter approaches the size of the folded chain (D ∼ N1/3σ), we find a destabilization effect. For pore diameters smaller than the native ground-state, the chain folds into a different, higher energy, ground state ensemble and the T vs D phase diagram displays non-monotonic behavior as the system is forced into different ground states for different ranges of D. In this regime, isothermal reduction of the confinement dimension can induce folding, unfolding, or crystallite restructuring. For the cylindrical box, we find a monotonic stabilization effect with decreasing D. Scaling laws for the confinement free energy in the athermal limit are also investigated.
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Affiliation(s)
- Mark P Taylor
- Department of Physics, Hiram College, Hiram, Ohio 44234, USA
| | - Troy M Prunty
- Department of Physics, Hiram College, Hiram, Ohio 44234, USA
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Luettmer-Strathmann J. Configurational contribution to the Soret effect of a protein ligand system : An investigation with density-of-states simulations. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2019; 42:77. [PMID: 31222556 DOI: 10.1140/epje/i2019-11840-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Accepted: 05/13/2019] [Indexed: 06/09/2023]
Abstract
Many of the biological functions of proteins are closely associated with their ability to bind ligands and change conformations in response to changing conditions. Since binding state and conformation of a protein affect its response to a temperature gradient, they may be probed with thermophoresis. In recent years, thermophoretic techniques to investigate biomolecular interactions, quantify ligand binding, and probe conformational changes have become established. To develop a better understanding of the mechanisms underlying the thermophoretic behavior of proteins and ligands, we employ a simple, off-lattice model for a protein and ligand in explicit solvent. To investigate the partitioning of the particles in a temperature gradient, we perform Wang-Landau-type simulations in a divided simulation box and construct the density of states over a two-dimensional state space. This method gives us access to the entropy and energy of the divided system and allows us to estimate the configurational contribution to the Soret coefficient. In this work, we focus on dilute solutions of hydrophobic proteins and investigate the effect of ligand binding on their thermophoretic behavior. We find that our simple model captures important aspects of protein-ligand interactions and allows us to relate the binding energy to the change in Soret coefficient upon ligand binding.
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Affiliation(s)
- Jutta Luettmer-Strathmann
- Department of Physics and Department of Chemistry, The University of Akron, 44325-4001, Akron, OH, USA.
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Affiliation(s)
- Mark P. Taylor
- Department of Physics, Hiram College, Hiram, Ohio 44234, United States
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Taylor MP, Paul W, Binder K. On the polymer physics origins of protein folding thermodynamics. J Chem Phys 2017; 145:174903. [PMID: 27825238 DOI: 10.1063/1.4966645] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023] Open
Abstract
A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavior is also typical for flexible homopolymer chains with sufficiently short range interactions. Two-state behavior arises from the presence of a low entropy ground (folded) state separated from a set of high entropy disordered (unfolded) states by a free energy barrier. This homopolymer model exhibits a funneled free energy landscape that reveals a complex underlying dynamics involving competition between folding and non-folding pathways. Despite the presence of multiple pathways, this simple physics model gives the robust result of two-state thermodynamics for both the cases of folding from a basin of expanded coil states and from a basin of compact globule states.
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Affiliation(s)
- Mark P Taylor
- Department of Physics, Hiram College, Hiram, Ohio 44234, USA
| | - Wolfgang Paul
- Institut für Physik, Martin-Luther-Universität, D-06099 Halle (Saale), Germany
| | - Kurt Binder
- Institut für Physik, Johannes-Gutenberg-Universität, Staudinger Weg 7, D-55099 Mainz, Germany
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Molkenthin N, Timme M. Scaling Laws in Spatial Network Formation. PHYSICAL REVIEW LETTERS 2016; 117:168301. [PMID: 27792385 DOI: 10.1103/physrevlett.117.168301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2016] [Indexed: 06/06/2023]
Abstract
Geometric constraints impact the formation of a broad range of spatial networks, from amino acid chains folding to proteins structures to rearranging particle aggregates. How the network of interactions dynamically self-organizes in such systems is far from fully understood. Here, we analyze a class of spatial network formation processes by introducing a mapping from geometric to graph-theoretic constraints. Combining stochastic and mean field analyses yields an algebraic scaling law for the extent (graph diameter) of the resulting networks with system size, in contrast to logarithmic scaling known for networks without constraints. Intriguingly, the exponent falls between that of self-avoiding random walks and that of space filling arrangements, consistent with experimentally observed scaling of the radius of gyration of protein tertiary structures with their chain length.
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Affiliation(s)
- Nora Molkenthin
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
- Department of Physics, Technical University of Darmstadt, 64289 Darmstadt, Germany
- Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, 37077 Göttingen, Germany
| | - Marc Timme
- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
- Department of Physics, Technical University of Darmstadt, 64289 Darmstadt, Germany
- Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, 37077 Göttingen, Germany
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7
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Böker A, Paul W. Wang-Landau simulation of Gō model molecules. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2016; 39:5. [PMID: 26810395 DOI: 10.1140/epje/i2016-16005-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2015] [Accepted: 01/04/2016] [Indexed: 06/05/2023]
Abstract
Gō-like models are one of the oldest protein modeling concepts in computational physics and have proven their value over and over for forty years. The essence of a Gō model is to define a native contact matrix for a well-defined low-energy polymer configuration, e.g., the native state in the case of proteins or peptides. Many different potential shapes and many different cut-off distances in the definition of this native contact matrix have been proposed and applied. We investigate here the physical consequences of the choice for this cut-off distance in the Gō models derived for a square-well tangent sphere homopolymer chain. For this purpose we are performing flat-histogram Monte Carlo simulations of Wang-Landau type, obtaining the thermodynamic and structural properties of such models over the complete temperature range. Differences and similarities with Gō models for proteins and peptides are discussed.
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Affiliation(s)
- Arne Böker
- Institut für Physik, Martin-Luther-Universität, D-06099, Halle (Saale), Germany
| | - Wolfgang Paul
- Institut für Physik, Martin-Luther-Universität, D-06099, Halle (Saale), Germany.
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Chertovich A, Kos P. Crumpled globule formation during collapse of a long flexible and semiflexible polymer in poor solvent. J Chem Phys 2015; 141:134903. [PMID: 25296833 DOI: 10.1063/1.4896701] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023] Open
Abstract
By introducing explicit solvent particles and hydrodynamic interactions we demonstrate that crumpled globules are formed after the collapse of long polymer chains (N = 10(4)) in a poor solvent. During the collapse crumples of all sizes form sequentially, but small crumples are not stable and convert to blobs with Gaussian statistics. The observed effective mean squared distance R(2)(n) ∼ n(0.38) at n > Ne and contact probability index p(n) ∼ n(-0.5) at n ≫ Ne, which is not following either the model of a fractal globule, or the predictions for an equilibrium globule. Polymer chain stiffness pushes the system to form globular crystallite, and this freezes crumpled structure with R(2)(n) ∼ n(0.33) at n > Ne as a stable state. We note that there is some similarity to crumple globule formation and crystallization of polymer melt.
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Affiliation(s)
- A Chertovich
- Physics Department, Lomonosov Moscow State University, Moscow 119991, Russia
| | - P Kos
- Physics Department, Lomonosov Moscow State University, Moscow 119991, Russia
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Leitold C, Dellago C. Folding mechanism of a polymer chain with short-range attractions. J Chem Phys 2014; 141:134901. [DOI: 10.1063/1.4896560] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Christian Leitold
- Faculty of Physics, University of Vienna, Sensengasse 8, 1090 Vienna, Austria
| | - Christoph Dellago
- Faculty of Physics, University of Vienna, Sensengasse 8, 1090 Vienna, Austria
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Wiebe H, Weinberg N. Theoretical volume profiles as a tool for probing transition states: folding kinetics. J Chem Phys 2014; 140:124105. [PMID: 24697422 DOI: 10.1063/1.4868549] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Abstract
The mechanism by which conformational changes, particularly folding and unfolding, occur in proteins and other biopolymers has been widely discussed in the literature. Molecular dynamics (MD) simulations of protein folding present a formidable challenge since these conformational changes occur on a time scale much longer than what can be afforded at the current level of computational technology. Transition state (TS) theory offers a more economic description of kinetic properties of a reaction system by relating them to the properties of the TS, or for flexible systems, the TS ensemble (TSE). The application of TS theory to protein folding is limited by ambiguity in the definition of the TSE for this process. We propose to identify the TSE for conformational changes in flexible systems by comparison of its experimentally determined volumetric property, known as the volume of activation, to the structure-specific volume profile of the process calculated using MD. We illustrate this approach by its successful application to unfolding of a model chain system.
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Affiliation(s)
- H Wiebe
- Department of Chemistry, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
| | - N Weinberg
- Department of Chemistry, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
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Taylor MP, Paul W, Binder K. Applications of the Wang-Landau algorithm to phase transitions of a single polymer chain. POLYMER SCIENCE SERIES C 2013. [DOI: 10.1134/s1811238213060040] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Tiana G, Sutto L. Equilibrium properties of realistic random heteropolymers and their relevance for globular and naturally unfolded proteins. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:061910. [PMID: 22304119 DOI: 10.1103/physreve.84.061910] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2011] [Indexed: 05/31/2023]
Abstract
Random heteropolymers do not display the typical equilibrium properties of globular proteins, but are the starting point to understand the physics of proteins and, in particular, to describe their non-native states. So far, they have been studied with mean-field models in the thermodynamic limit, or with computer simulations of very small chains on lattice. After describing a self-adjusting parallel-tempering technique to sample efficiently the low-energy states of frustrated systems without the need of tuning the system-dependent parameters of the algorithm, we apply it to random heteropolymers moving in continuous space. We show that if the mean interaction between monomers is negative, the usual description through the random-energy model is nearly correct, provided that it is extended to account for noncompact conformations. If the mean interaction is positive, such a simple description breaks out and the system behaves in a way more similar to Ising spin glasses. The former case is a model for the denatured state of globular proteins, the latter of naturally unfolded proteins, whose equilibrium properties thus result as qualitatively different.
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Affiliation(s)
- G Tiana
- Department of Physics, Università degli Studi di Milano and Istituto Nazionale di Fisica Nucleare, via Celoria 16, I-20133 Milano, Italy
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Wang Z, He X. Phase transition of a single star polymer: A Wang-Landau sampling study. J Chem Phys 2011; 135:094902. [DOI: 10.1063/1.3629849] [Citation(s) in RCA: 12] [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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Abstract
The collapse of a homopolymer gaussian chain into a globule is represented as a transition between two states, viz., extended and collapsed. Appropriately, this model has been labeled as the all-or-none view of chain collapse. In the collapsed state, the single polymer partition function is expressed by a single Mayer diagram with the maximum number of f-bonds arising from nonbonded square well interactions. Our target is the dependence of the transition temperature on chain length and the interaction range of the square well, as indicated through the behavior of the radius of gyration and the constant volume heat capacity. Properties of the collapse transition are calculated exactly for chains with three to six backbone atoms and heuristically for long chains using arguments derived from the small chains and from conditions of integrability. Comparison with simulation studies is made.
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Affiliation(s)
- Glenn T Evans
- Department of Chemistry, Oregon State University, Corvallis, Oregon 97331, USA.
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