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Tian X, Xu X, Chen Y, Chen J, Xu WS. Explicit analytical form for memory kernel in the generalized Langevin equation for end-to-end vector of Rouse chains. J Chem Phys 2022; 157:224901. [PMID: 36546812 DOI: 10.1063/5.0124925] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
The generalized Langevin equation (GLE) provides an attractive theoretical framework for investigating the dynamics of conformational fluctuations of polymeric systems. While the memory kernel is a central function in the GLE, explicit analytical forms for this function have been challenging to obtain, even for the simple models of polymer dynamics. Here, we achieve an explicit analytical expression for the memory kernel in the GLE for the end-to-end vector of Rouse chains in the overdamped limit. Our derivation takes advantage of the finding that the dynamics of the end-to-end vector of Rouse chains with both free ends are equivalent to those of Rouse chains with one free end and the other fixed. For the latter model, we first show that the equations of motion of the Rouse modes as well as their statistical properties can be obtained under the boundary conditions where the free end is held fixed temporarily. We then analytically solve the terms associated with intrachain interactions in the GLE. By formally comparing these terms with the GLE based on the Rouse modes, we obtain an explicit expression for the memory kernel, along with analytical forms for the potential field and the random colored noise force. Our analytical memory kernel is confirmed by numerical calculations in the Laplace space and is shown to yield asymptotic behaviors that are consistent with previous studies. Finally, we utilize our analytical result to simulate the cyclization dynamics of Rouse chains and discuss the scaling of the cyclization time with chain length.
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Affiliation(s)
- Xiaofei Tian
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
| | - Xiaolei Xu
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
| | - Ye Chen
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
| | - Jizhong Chen
- School of Chemical Engineering and Light Industry, Guangdong University of Technology, Guangzhou 510006, People's Republic of China
| | - Wen-Sheng Xu
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
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2
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Affiliation(s)
- Michael Lang
- Leibniz Institute of Polymer Research Dresden, Hohe Str. 6, 01069 Dresden, Germany
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3
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Importance of extreme value statistics in biophysical contexts. Phys Life Rev 2019; 28:94-95. [DOI: 10.1016/j.plrev.2019.03.001] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2019] [Accepted: 03/12/2019] [Indexed: 11/20/2022]
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Kappler J, Noé F, Netz RR. Cyclization and Relaxation Dynamics of Finite-Length Collapsed Self-Avoiding Polymers. PHYSICAL REVIEW LETTERS 2019; 122:067801. [PMID: 30822085 DOI: 10.1103/physrevlett.122.067801] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2018] [Revised: 11/23/2018] [Indexed: 06/09/2023]
Abstract
We study the cyclization and relaxation dynamics of ideal as well as interacting polymers as a function of chain length N. For the cyclization time τ_{cyc} of ideal chains we recover the known scaling τ_{cyc}∼N^{2} for different backbone models, for a self-avoiding slightly collapsed chain we obtain from Langevin simulations and scaling theory a modified scaling τ_{cyc}∼N^{5/3}. The cyclization and relaxation dynamics of a finite-length collapsed chain scale differently; this unexpected dynamic multiscale behavior is rationalized by the crossover between swollen and collapsed chain behavior.
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Affiliation(s)
- Julian Kappler
- Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
| | - Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany
| | - Roland R Netz
- Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
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Schuss Z, Basnayake K, Holcman D. Redundancy principle and the role of extreme statistics in molecular and cellular biology. Phys Life Rev 2019; 28:52-79. [PMID: 30691960 DOI: 10.1016/j.plrev.2019.01.001] [Citation(s) in RCA: 31] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2018] [Accepted: 01/03/2019] [Indexed: 12/17/2022]
Abstract
The paradigm of chemical activation rates in cellular biology has been shifted from the mean arrival time of a single particle to the mean of the first among many particles to arrive at a small activation site. The activation rate is set by extremely rare events, which have drastically different time scales from the mean times between activations, and depends on different structural parameters. This shift calls for reconsideration of physical processes used in deterministic and stochastic modeling of chemical reactions that are based on the traditional forward rate, especially for fast activation processes in living cells. Consequently, the biological activation time is not necessarily exponentially distributed. We review here the physical models, the mathematical analysis and the new paradigm of setting the scale to be the shortest time for activation that clarifies the role of population redundancy in selecting and accelerating transient cellular search processes. We provide examples in cellular transduction, gene activation, cell senescence activation or spermatozoa selection during fertilization, where the rate depends on numbers. We conclude that the statistics of the minimal time to activation set kinetic laws in biology, which can be very different from the ones associated to average times.
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Affiliation(s)
- Z Schuss
- Department of Applied Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel
| | - K Basnayake
- Computational Biology and Applied Mathematics, Ecole Normale Supérieure, Paris, France
| | - D Holcman
- Computational Biology and Applied Mathematics, Ecole Normale Supérieure, Paris, France; Churchill College, Univ. of Cambridge, CB30DS, UK.
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6
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Levernier N, Dolgushev M, Bénichou O, Blumen A, Guérin T, Voituriez R. Non-Markovian closure kinetics of flexible polymers with hydrodynamic interactions. J Chem Phys 2015; 143:204108. [PMID: 26627951 DOI: 10.1063/1.4935966] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Abstract
This paper presents a theoretical analysis of the closure kinetics of a polymer with hydrodynamic interactions. This analysis, which takes into account the non-Markovian dynamics of the end-to-end vector and relies on the preaveraging of the mobility tensor (Zimm dynamics), is shown to reproduce very accurately the results of numerical simulations of the complete nonlinear dynamics. It is found that Markovian treatments based on a Wilemski-Fixman approximation significantly overestimate cyclization times (up to a factor 2), showing the importance of memory effects in the dynamics. In addition, this analysis provides scaling laws of the mean first cyclization time (MFCT) with the polymer size N and capture radius b, which are identical in both Markovian and non-Markovian approaches. In particular, it is found that the scaling of the MFCT for large N is given by T ∼ N(3/2)ln(N/b(2)), which differs from the case of the Rouse dynamics where T ∼ N(2). The extension to the case of the reaction kinetics of a monomer of a Zimm polymer with an external target in a confined volume is also presented.
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Affiliation(s)
- N Levernier
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, Paris 75005, France
| | - M Dolgushev
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-St. 3, D-79104 Freiburg, Germany
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, Paris 75005, France
| | - A Blumen
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-St. 3, D-79104 Freiburg, Germany
| | - T Guérin
- Laboratoire Ondes et Matière d'Aquitaine, University of Bordeaux, Unité Mixte de Recherche 5798, CNRS, F-33400 Talence, France
| | - R Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, Paris 75005, France
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Dolgushev M, Guérin T, Blumen A, Bénichou O, Voituriez R. Contact Kinetics in Fractal Macromolecules. PHYSICAL REVIEW LETTERS 2015; 115:208301. [PMID: 26613478 DOI: 10.1103/physrevlett.115.208301] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2015] [Indexed: 06/05/2023]
Abstract
We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the mean first contact time for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the nonequilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the mean first contact time between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations.
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Affiliation(s)
- Maxim Dolgushev
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
| | - Thomas Guérin
- Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, 33400 Talence, France
| | - Alexander Blumen
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
| | - Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, 75005 Paris, France
| | - Raphaël Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, 75005 Paris, France
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Athanassoulis GA, Tsantili IC, Kapelonis ZG. Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time. Proc Math Phys Eng Sci 2015. [DOI: 10.1098/rspa.2015.0501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Uncertainty quantification for dynamical systems under non-white excitation is a difficult problem encountered across many scientific and engineering disciplines. Difficulties originate from the lack of Markovian character of system responses. The response–excitation (RE) theory, recently introduced by Sapsis & Athanassoulis (Sapsis & Athanassoulis 2008
Probabilistic Eng. Mech.
23, 289–306 (
doi:10.1016/j.probengmech.2007.12.028
)) and further studied by Venturi
et al.
(Venturi
et al.
2012
Proc. R. Soc. A
468, 759–783 (
doi:10.1098/rspa.2011.0186
)), is a new approach, based on a simple differential constraint which is exact but non-closed. The evolution equation obtained for the RE probability density function (pdf) has the form of a generalized Liouville equation, with the excitation time frozen in the time-derivative term. In this work, the missing information of the RE differential constraint is identified and a closure scheme is developed for the long-time, stationary, limit-state of scalar nonlinear random differential equations (RDEs) under coloured excitation. The closure scheme does not alter the RE evolution equation, but collects the missing information through the solution of local statistically linearized versions of the nonlinear RDE, and interposes it into the solution scheme. Numerical results are presented for two examples, and compared with Monte Carlo simulations.
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Affiliation(s)
- G. A. Athanassoulis
- School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece
- Research Center for High Performance Computing, ITMO University, St. Petersburg, Russian Federation
| | - I. C. Tsantili
- School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece
- Geostatistics Laboratory, School of Mineral Resources Engineering, Technical University of Crete, Chania, Greece
| | - Z. G. Kapelonis
- School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece
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Shin J, Cherstvy AG, Metzler R. Kinetics of polymer looping with macromolecular crowding: effects of volume fraction and crowder size. SOFT MATTER 2015; 11:472-88. [PMID: 25413029 DOI: 10.1039/c4sm02007c] [Citation(s) in RCA: 70] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
The looping of polymers such as DNA is a fundamental process in the molecular biology of living cells, whose interior is characterised by a high degree of molecular crowding. We here investigate in detail the looping dynamics of flexible polymer chains in the presence of different degrees of crowding. From the analysis of the looping-unlooping rates and the looping probabilities of the chain ends we show that the presence of small crowders typically slows down the chain dynamics but larger crowders may in fact facilitate the looping. We rationalise these non-trivial and often counterintuitive effects of the crowder size on the looping kinetics in terms of an effective solution viscosity and standard excluded volume. It is shown that for small crowders the effect of an increased viscosity dominates, while for big crowders we argue that confinement effects (caging) prevail. The tradeoff between both trends can thus result in the impediment or facilitation of polymer looping, depending on the crowder size. We also examine how the crowding volume fraction, chain length, and the attraction strength of the contact groups of the polymer chain affect the looping kinetics and hairpin formation dynamics. Our results are relevant for DNA looping in the absence and presence of protein mediation, DNA hairpin formation, RNA folding, and the folding of polypeptide chains under biologically relevant high-crowding conditions.
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Affiliation(s)
- Jaeoh Shin
- Institute for Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany.
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Guérin T, Dolgushev M, Bénichou O, Voituriez R, Blumen A. Cyclization kinetics of Gaussian semiflexible polymer chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052601. [PMID: 25493807 DOI: 10.1103/physreve.90.052601] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/26/2014] [Indexed: 06/04/2023]
Abstract
We consider the dynamics and the cyclization kinetics of Gaussian semiflexible chains, in which the interaction potential tends to align successive bonds. We provide asymptotic expressions for the cyclization time, for the eigenvalues and eigenfunctions, and for the mean square displacement at all time and length scales, with explicit dependence on the capture radius, on the positions of the reactive monomers in the chain, and on the finite number of beads. For the cyclization kinetics, we take into account non-Markovian effects by calculating the distribution of reactive conformations of the polymer, which are not taken into account in the classical Wilemski-Fixman theory. Comparison with numerical simulations confirms the accuracy of this non-Markovian theory.
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Affiliation(s)
- T Guérin
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, 75005 Paris, France and Laboratoire Ondes et Matière d'Aquitaine, University of Bordeaux, Unité Mixte de Recherche 5798, CNRS, F-33400 Talence, France
| | - M Dolgushev
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, 75005 Paris, France
| | - R Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, 75005 Paris, France
| | - A Blumen
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
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Dolgushev M, Guérin T, Blumen A, Bénichou O, Voituriez R. Gaussian semiflexible rings under angular and dihedral restrictions. J Chem Phys 2014; 141:014901. [PMID: 25005305 DOI: 10.1063/1.4885445] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
Semiflexible polymer rings whose bonds obey both angular and dihedral restrictions [M. Dolgushev and A. Blumen, J. Chem. Phys. 138, 204902 (2013)], are treated under exact closure constraints. This allows us to obtain semianalytic results for their dynamics, based on sets of Langevin equations. The dihedral restrictions clearly manifest themselves in the behavior of the mean-square monomer displacement. The determination of the equilibrium ring conformations shows that the dihedral constraints influence the ring curvature, leading to compact folded structures. The method for imposing such constraints in Gaussian systems is very general and it allows to account for heterogeneous (site-dependent) restrictions. We show it by considering rings in which one site differs from the others.
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Affiliation(s)
- Maxim Dolgushev
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Thomas Guérin
- Laboratoire de Physique Théorique de la Matière Condensée, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France
| | - Alexander Blumen
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
| | - Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France
| | - Raphaël Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France
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