Varying the Resolution of the Rouse Model on Temporal and Spatial Scales: Application to Multiscale Modeling of DNA Dynamics
Abstract
A multiresolution bead-spring model for polymer dynamics is developed as a generalization of the Rouse model. A polymer chain is described using beads of variable sizes connected by springs with variable spring constants. A numerical scheme which can use different timesteps to advance the positions of different beads is presented and analyzed. The position of a particular bead is updated only at integer multiples of the timesteps associated with its connecting springs. This approach extends the Rouse model to a multiresolution model on both spatial and temporal scales, allowing simulations of localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modeling approach to describe the rest of the polymer chain. A method for changing the model resolution on the fly is developed using the Metropolis--Hastings algorithm. It is shown that this approach maintains key statistics of the end-to-end distance and diffusion of the polymer filament and makes computational savings when applied to a model for the binding of a protein to the DNA filament.
Keywords
MSC codes
Formats available
You can view the full content in the following formats:
References
1.
S. A. Allison, Brownian dynamics simulation of wormlike chains. Fluorescence depolarization and depolarized light scattering, Macromolecules, 19 (1986), pp. 118--124.
2.
S. S. Andrews, Methods for modeling cytoskeletal and DNA filaments, Phys. Biol., 11 (2014), 011001.
3.
M. Barbieri, M. Chotalia, and J. Fraser, et al., Complexity of chromatin folding is captured by the strings and binders switch model, Proc. Natl. Acad. Sci. USA, 109 (2012), pp. 16173--16178.
4.
C. A. Brackley, M. E. Cates, and D. Marenduzzo, Intracellular facilitated diffusion: Searchers, crowders, and blockers, Phys. Rev. Lett., 111 (2013), 108101.
5.
C. A. Brackley, J. Johnson, S. Kelly, et al., Simulated binding of transcription factors to active and inactive regions folds human chromosomes into loops, rosettes and topological domains, Nucleic Acids Res., 44 (2016), pp. 3503--3512.
6.
K. Bystricky, P. Heun, L. Gehlen, et al., Long-range compaction and flexibility of interphase chromatin in budding yeast analyzed by high-resolution imaging techniques, Proc. Natl. Acad. Sci. USA, 101 (2004), pp. 16495--16500.
7.
S. Chib and E. Greenberg, Understanding the Metropolis--Hastings algorithm, Amer. Statist., 49 (1995), pp. 327--335.
8.
T. Cremer, M. Cremer, B. Hübner, et al., The $4$D nucleome: Evidence for a dynamic nuclear landscape based on co-aligned active and inactive nuclear compartments, FEBS Lett., 589 (2015), pp. 2931--2943.
9.
P. De Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979.
10.
J. Dekker, K. Rippe, M. Dekker, and N. Kleckner, Capturing chromosome conformation, Science, 295 (2002), pp. 1306--1311.
11.
J. Dekker, M. A. Marti-Renom, and L. A. Mirny, Exploring the three-dimensional organization of genomes: Interpreting chromatin interaction data, Nat. Rev. Genet., 14 (2013), pp. 390--403.
12.
M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York, 1988.
13.
Z. Duan, M. Andronescu, K. Schutz, et al., A three-dimensional model of the yeast genome, Nature, 465 (2010), pp. 363--367.
14.
R. Erban, From molecular dynamics to Brownian dynamics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140036.
15.
R. Erban, Coupling all-atom molecular dynamics simulations of ions in water with Brownian dynamics, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 472 (2016), 20150556.
16.
R. Erban and S. J. Chapman, Stochastic modeling of reaction--diffusion processes: Algorithms for bimolecular reactions, Phys. Biol., 6 (2009), 046001.
17.
D. L. Ermak and J. A. McCammon, Brownian dynamics with hydrodynamic interactions, J. Chem. Phys., 69 (1978), pp. 1352--1360.
18.
G. Felsenfeld and M. Groudine, Controlling the double helix, Nature, 421 (2003), pp. 448--453.
19.
M. B. Flegg, S. J. Chapman, and R. Erban, The two-regime method for optimizing stochastic reaction-diffusion simulations, J. R. Soc. Interface, 9 (2012), pp. 859--868.
20.
L. R. Gehlen, G. Gruenert, M. B. Jones, et al., Chromosome positioning and the clustering of functionally related loci in yeast is driven by chromosomal interactions, Nucleus, 3 (2012), pp. 370--383.
21.
A. Grosberg, Y. Rabin, S. Havlin, and A. Neer, Crumpled globule model of the three-dimensional structure of DNA, Europhys. Lett., 23 (1993), pp. 373--378.
22.
A. Grosberg, How two meters of DNA fit into a cell nucleus: Polymer models with topological constraints and experimental data, Polym. Sci. Ser. C, 54 (2012), pp. 1--10.
23.
M. G. Guenza, Theoretical models for bridging timescales in polymer dynamics, J. Phys. Condens. Matter, 20 (2008), 033101.
24.
E. Guth and H. Mark, Zur innermolekularen Statistik, insbesondere bei Kettenmolekülen I, Monatsh. Chem., 65 (1934), pp. 93--121.
25.
P. J. Hagerman and B. H. Zimm, Monte Carlo approach to the analysis of the rotational diffusion of wormlike chains, Biopolymers, 20 (1981), pp. 1481--1502.
26.
P. J. Hagerman, Flexibility of DNA, Annu. Rev. Biophys. Biophys. Chem., 17 (1988), pp. 265--286.
27.
S. E. Halford and J. F. Marko, How do site-specific DNA-binding proteins find their targets?, Nucleic Acids Res., 32 (2004), pp. 3040--3052.
28.
A. Hotz, and R. E. Skelton, Covariance control theory, Internat. J. Control, 46 (1987), pp. 13--32.
29.
J. S. Hur, E. S. Shaqfeh, and R. G. Larson, Brownian dynamics simulations of single DNA molecules in shear flow, J. Rheology, 44 (2000), pp. 713--742.
30.
J. S. Hur, E. S. Shaqfeh, H. P. Babcock, et al., Dynamics of dilute and semidilute DNA solutions in the start-up of shear flow, J. Rheology, 45 (2001), pp. 421--450.
31.
S. Jun and B. Mulder, Entropy-driven spatial organization of highly confined polymers: Lessons for the bacterial chromosome, Proc. Natl. Acad. Sci. USA, 103 (2006), pp. 12388--12393.
32.
P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.
33.
R. D. Kornberg, Chromatin structure: A repeating unit of histones and DNA, Science, 184 (1974), pp. 868--871.
34.
N. Korolev, L. Nordenskiöld, and A. P. Lyubartsev, Multiscale coarse-grained modeling of chromatin components: DNA and the nucleosome, Adv. Colloid Interface Sci., 232 (2016), pp. 36--48.
35.
O. Kratky and G. Porod, Röntgenuntersuchung gelöster Fadenmoleküle, Recl. Trav. Chim. Pay. B., 68 (1949), pp. 1106--1122.
36.
K. Kremer and G. S. Grest, Dynamics of entangled linear polymer melts: A molecular-dynamics simulation, J. Chem. Phys., 92 (1990), pp. 5057--5086.
37.
W. Kuhn, Über die Gestalt fadenförmiger Moleküle in Lösungen, Colloid. Polym. Sci., 68 (1934), pp. 2--15.
38.
E. Lieberman-Aiden, N. L. van Berkum, L. Williams, et al., Comprehensive mapping of long-range interactions reveals folding principles of the human genome, Science, 326 (2009), pp. 289--293.
39.
K. Maeshima, K. Kaizu, S. Tamura, et al., The physical size of transcription factors is key to transcriptional regulation in chromatin domains, J. Phys. Condens. Matter, 27 (2015), 064116.
40.
K. Maeshima, R. Rogge, S. Tamura, et al., Nucleosomal arrays self-assemble into supramolecular globular structures lacking 30-nm fibers, EMBO J., 35 (2016), pp. 1115--1132.
41.
L. A. Mirny, The fractal globule as a model of chromatin architecture in the cell, Chromosome Res. 19 (2011), pp. 37--51.
42.
O. Müller, N. Kepper, R. Schöpflin, R. Ettig, K. Rippe, and G. Wedemann, Changing chromatin fiber conformation by nucleosome repositioning, Biophys. J., 107 (2014), pp. 2141--2150.
43.
R. M. Neal, Slice sampling, Ann. Statist., 31 (2003), pp. 705--767.
44.
H. Ochiai, T. Sugawara, and T. Yamamoto, Simultaneous live imaging of the transcription and nuclear position of specific genes, Nucleic Acids Res., 43 (2015), e127.
45.
T. Opplestrup, V. Bulatov, A. Donev, et al., First-passage kinetic Monte Carlo method, Phys. Rev. E, 80 (2009), 066701.
46.
R. Potestio, C. Peter, and K. Kremer, Computer simulations of soft matter: Linking the scales, Entropy, 16 (2014), pp. 4199--4245.
47.
D. Richter, A. Baumgärtner, K. Binder, et al., Dynamics of collective fluctuations and Brownian motion in polymer melts, Phys. Rev. Lett., 47 (1981), pp. 109--113.
48.
A. Rosa and R. Everaers, Structure and dynamics of interphase chromosomes, PLoS Comput. Biol., 4 (2008), e1000153.
49.
R. Robertson, S. Laib, and D. Smith, Diffusion of isolated DNA molecules: Dependence on length and topology, Proc. Natl. Acad. Sci. USA, 103 (2006), pp. 7310--7314.
50.
J. Rotne and S. Prager, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys., 50 (1969), pp. 4831--4837.
51.
P. E. Rouse, A theory of the linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys., 21 (1953), pp. 1272--1280.
52.
R. K. Sachs, G. van den Engh, B. Trask, et al., A random-walk/giant-loop model for interphase chromosomes, Proc. Natl. Acad. Sci. USA, 92 (1995), pp. 2710--2714.
53.
A. L. Sanborn, S. S. P. Rao, S. Huang, et al., Chromatin extrusion explains key features of loop and domain formation in wild-type and engineered genomes, Proc. Natl. Acad. Sci. USA, 112 (2015), pp. E6456--E6465.
54.
Y. Saito and M. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), pp. 2254--2267.
55.
S. Shinkai, T. Nozaki, K. Maeshima, and Y. Togashi, Dynamic nucleosome movement provides structural information of topological chromatin domains in living human cells, PLoS Comput. Biol., 12 (2016), e1005136.
56.
M. Somasi, B. Khomami, N. J. Woo, et al., Brownian dynamics simulations of bead-rod and bead-spring chains: Numerical algorithms and coarse-graining issues, J. Non-Newton. Fluid, 108 (2002), pp. 227--255.
57.
S. Takada, R. Kanada, C. Tan, et al., Modeling structural dynamics of biomolecular complexes by coarse-grained molecular simulations, Acc. Chem. Res., 48 (2015), pp. 3026--3035.
58.
K. Takahashi, S. Tanase-Nicola and P. ten Wolde, Spatio-temporal correlations can drastically change the response of a MAPK pathway, Proc. Natl. Acad. Sci. USA, 107 (2010), pp. 19820--19825.
59.
N. Tokuda, T. P. Terada, and M. Sasai, Dynamical modeling of three-dimensional genome organization in interphase budding yeast, Biophys. J., 102 (2012), pp. 296--304.
60.
E. B. Wilson, Probable inference, the law of succession, and statistical inference, J. Amer. Statist. Assoc., 22 (1927), pp. 209--212.
61.
A. Wolffe, Chromatin, 3rd ed., Academic Press, New York, 2000.
62.
H. Wong, H. Marie-Nelly, S. Herbert, et al., A predictive computational model of the dynamic $3$D interphase yeast nucleus, Curr. Biol., 22 (2012), pp. 1881--1890.
63.
J. Zavadlav, R. Podgornik, and M. Praprotnik, Adaptive resolution simulation of a DNA molecule in salt solution, J. Chem. Theory Comput., 11 (2015), pp. 5035--5044.
64.
B. H. Zimm, Dynamics of polymer molecules in dilute solution: Viscoelasticity, flow birefringence and dielectric loss, J. Chem. Phys., 24 (1956), pp. 269--278.
Information & Authors
Information
Published In
Copyright
© 2017, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
History
Submitted: 28 July 2016
Accepted: 22 May 2017
Published online: 16 November 2017
Keywords
MSC codes
Authors
Funding Information
KAKENHI : 23115007, 16H01408
Japan Agency for Medical Research and Development https://doi.org/10.13039/100009619
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/K032208/1, EP/G03706X/1
Royal Society https://doi.org/10.13039/501100000288
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Rheology of Ring Copolymers in Dilute SolutionsThe Journal of Physical Chemistry B, Vol. 129, No. 1 | 19 December 2024
- Effect of loops on the mean-square displacement of Rouse-model chromatinPhysical Review E, Vol. 109, No. 4 | 9 April 2024
- Asymmetric Periodic Boundary Conditions for All-Atom Molecular Dynamics and Coarse-Grained Simulations of Nucleic AcidsThe Journal of Physical Chemistry B, Vol. 127, No. 38 | 15 September 2023
- On standardised moments of force distribution in simple liquidsPhysical Chemistry Chemical Physics, Vol. 24, No. 9 | 1 January 2022
- Anomalous Dynamics in Macromolecular LiquidsPolymers, Vol. 14, No. 5 | 22 February 2022
- Multiscale Modeling of Chromatin Considering the State and Shape of MoleculesAdvances in Computational Modeling and Simulation | 15 February 2022
- Molecular dynamics analysis of biomolecular systems including nucleic acidsBiophysics and Physicobiology, Vol. 19, No. 0 | 1 Jan 2022
- Static Structure Factor and Viscoelastic Properties of Dendrimer Grafted Nanoparticles in SolutionThe Journal of Physical Chemistry B, Vol. 125, No. 7 | 10 February 2021
- Structure and dynamics of liposomes designed for drug delivery: coarse-grained molecular dynamics simulations to reveal the role of lipopolymer incorporationRSC Advances, Vol. 10, No. 7 | 1 January 2020
- Theory for the Dynamics of Polymer Grafted Nanoparticle in SolutionThe Journal of Physical Chemistry C, Vol. 123, No. 50 | 25 November 2019
- Multi-resolution dimer models in heat baths with short-range and long-range interactionsInterface Focus, Vol. 9, No. 3 | 19 April 2019
- Multi-resolution polymer Brownian dynamics with hydrodynamic interactionsThe Journal of Chemical Physics, Vol. 148, No. 19 | 18 May 2018
View Options
View options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].