1
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Staňo R, Smrek J, Likos CN. Cluster Formation in Solutions of Polyelectrolyte Rings. ACS NANO 2023; 17:21369-21382. [PMID: 37729077 PMCID: PMC10655244 DOI: 10.1021/acsnano.3c06083] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2023] [Accepted: 09/13/2023] [Indexed: 09/22/2023]
Abstract
We use molecular dynamics simulations to explore concentrated solutions of semiflexible polyelectrolyte ring polymers, akin to the DNA mini-circles, with counterions of different valences. We find that the assembly of rings into nanoscopic cylindrical stacks is a generic feature of the systems, but the morphology and dynamics of such a cluster can be steered by the counterion conditions. In general, a small addition of trivalent ions can stabilize the emergence of clusters due to the counterion condensation, which mitigates the repulsion between the like-charged rings. Stoichiometric addition of trivalent ions can even lead to phase separation of the polyelectrolyte ring phase due to the ion-bridging effects promoting otherwise entropically driven clustering. On the other hand, monovalent counterions cause the formation of stacks to be re-entrant with density. The clusters are stable within a certain window of concentration, while above the window the polyelectrolytes undergo an osmotic collapse, disfavoring ordering. The cluster phase exhibits characteristic cluster glass dynamics with arrest of collective degrees of freedom but not the self-ones. On the other hand, the collapsed phase shows arrest on both the collective and single level, suggesting an incipient glass-to-glass transition, from a cluster glass of ring clusters to a simple glass of rings.
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Affiliation(s)
- Roman Staňo
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
- Vienna
Doctoral School in Physics, University of
Vienna, Boltzmanngasse
5, 1090 Vienna, Austria
| | - Jan Smrek
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Christos N. Likos
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
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2
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Staňo R, Likos CN, Egorov SA. Mixing Linear Polymers with Rings and Catenanes: Bulk and Interfacial Behavior. Macromolecules 2023; 56:8168-8182. [PMID: 37900098 PMCID: PMC10601540 DOI: 10.1021/acs.macromol.3c01267] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2023] [Revised: 09/11/2023] [Indexed: 10/31/2023]
Abstract
We derive and parameterize effective interaction potentials between a multitude of different types of ring polymers and linear chains, varying the bending rigidity and solvent quality for the former species. We further develop and apply a density functional treatment for mixtures of both disconnected (chain-ring) and connected (chain-polycatenane) mixtures of the same, drawing coexistence binodals and exploring the ensuing response functions as well as the interface and wetting behavior of the mixtures. We show that worsening of the solvent quality for the rings brings about a stronger propensity for macroscopic phase separation in the linear-polycatenane mixtures, which is predominantly of the demixing type between phases of similar overall particle density. We formulate a simple criterion based on the effective interactions, allowing us to determine whether any specific linear-ring mixture will undergo a demixing phase separation.
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Affiliation(s)
- Roman Staňo
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
- Vienna
Doctoral School in Physics, University of
Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Christos N. Likos
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Sergei A. Egorov
- Department
of Chemistry, University of Virginia, Charlottesville, Virginia 22901, United States
- Erwin
Schrödinger International Institute for Mathematics and Physics, Boltzmanngasse 9, 1090 Vienna, Austria
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3
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Staňo R, Likos CN, Smrek J. To thread or not to thread? Effective potentials and threading interactions between asymmetric ring polymers. SOFT MATTER 2022; 19:17-30. [PMID: 36477247 PMCID: PMC9768673 DOI: 10.1039/d2sm01177h] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/31/2022] [Accepted: 11/22/2022] [Indexed: 06/17/2023]
Abstract
We use computer simulations to study a system of two unlinked ring polymers, whose length and bending stiffness are systematically varied. We derive the effective potentials between the rings, calculate the areas of minimal surfaces of the same, and characterize the threading between them. When the two rings are of the same kind, threading of a one ring through the surface of the other is immanent for small ring-ring separations. Flexible rings pierce the surface of the other ring several times but only shallowly, as compared to the stiff rings which pierce less frequently but deeply. Typically, the ring that is being threaded swells and flattens up into an oblate-like conformation, while the ring that is threading the other takes a shape of an elongated prolate. The roles of the threader and the threaded ring are being dynamically exchanged. If, on the other hand, the rings are of different kinds, the symmetry is broken and the rings tend to take up roles of the threader and the threaded ring with unequal probabilities. We propose a method how to predict these probabilities based on the parameters of the individual rings. Ultimately, our work captures the interactions between ring polymers in a coarse-grained fashion, opening the way to large-scale modelling of materials such as kinetoplasts, catenanes or topological brushes.
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Affiliation(s)
- Roman Staňo
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria.
- Vienna Doctoral School in Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
| | - Christos N Likos
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria.
| | - Jan Smrek
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria.
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4
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Kumari K, Ravi Prakash J, Padinhateeri R. Heterogeneous interactions and polymer entropy decide organization and dynamics of chromatin domains. Biophys J 2022; 121:2794-2812. [PMID: 35672951 PMCID: PMC9382282 DOI: 10.1016/j.bpj.2022.06.008] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2022] [Revised: 04/28/2022] [Accepted: 06/01/2022] [Indexed: 11/02/2022] Open
Abstract
Chromatin is known to be organized into multiple domains of varying sizes and compaction. While these domains are often imagined as static structures, they are highly dynamic and show cell-to-cell variability. Since processes such as gene regulation and DNA replication occur in the context of these domains, it is important to understand their organization, fluctuation, and dynamics. To simulate chromatin domains, one requires knowledge of interaction strengths among chromatin segments. Here, we derive interaction-strength parameters from experimentally known contact maps and use them to predict chromatin organization and dynamics. Taking two domains on the human chromosome as examples, we investigate its three-dimensional organization, size/shape fluctuations, and dynamics of different segments within a domain, accounting for hydrodynamic effects. Considering different cell types, we quantify changes in interaction strengths and chromatin shape fluctuations in different epigenetic states. Perturbing the interaction strengths systematically, we further investigate how epigenetic-like changes can alter the spatio-temporal nature of the domains. Our results show that heterogeneous weak interactions are crucial in determining the organization of the domains. Computing effective stiffness and relaxation times, we investigate how perturbations in interactions affect the solid- and liquid-like nature of chromatin domains. Quantifying dynamics of chromatin segments within a domain, we show how the competition between polymer entropy and interaction energy influence the timescales of loop formation and maintenance of stable loops.
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Affiliation(s)
- Kiran Kumari
- IITB-Monash Research Academy, Indian Institute of Technology Bombay, Mumbai, Maharashtra, 400076, India; Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra 400076, India; Department of Chemical Engineering, Monash University, Melbourne, VIC 3800, Australia.
| | - J Ravi Prakash
- Department of Chemical Engineering, Monash University, Melbourne, VIC 3800, Australia.
| | - Ranjith Padinhateeri
- Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Mumbai, Maharashtra 400076, India.
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5
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Liu G, Rauscher PM, Rawe BW, Tranquilli MM, Rowan SJ. Polycatenanes: synthesis, characterization, and physical understanding. Chem Soc Rev 2022; 51:4928-4948. [PMID: 35611843 DOI: 10.1039/d2cs00256f] [Citation(s) in RCA: 26] [Impact Index Per Article: 13.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Chemical composition and architecture are two key factors that control the physical and material properties of polymers. Some of the more unusual and intriguing polymer architectures are the polycatenanes, which are a class of polymers that contain mechanically interlocked rings. Since the development of high yielding synthetic routes to catenanes, there has been an interest in accessing their polymeric counterparts, primarily on account of the unique conformations and degrees of freedom offered by non-bonded interlocked rings. This has lead to the synthesis of a wide variety of polycatenane architectures and to studies aimed at developing structure-property relationships of these interesting materials. In this review, we provide an overview of the field of polycatenanes, exploring synthesis, architecture, properties, simulation, and modelling, with a specific focus on some of the more recent developments.
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Affiliation(s)
- Guancen Liu
- Department of Chemistry, University of Chicago, Chicago, IL, USA.
| | - Phillip M Rauscher
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA
| | - Benjamin W Rawe
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA
| | | | - Stuart J Rowan
- Department of Chemistry, University of Chicago, Chicago, IL, USA. .,Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA.,Chemical and Engineering Sciences, Argonne National Laboratory, Lemont, IL, USA
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6
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Chubak I, Likos CN, Smrek J. Topological and threading effects in polydisperse ring polymer solutions. Mol Phys 2021. [DOI: 10.1080/00268976.2021.1883140] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Iurii Chubak
- Faculty of Physics, University of Vienna, Vienna, Austria
| | | | - Jan Smrek
- Faculty of Physics, University of Vienna, Vienna, Austria
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7
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Olivi L, Berger M, Creyghton RNP, De Franceschi N, Dekker C, Mulder BM, Claassens NJ, Ten Wolde PR, van der Oost J. Towards a synthetic cell cycle. Nat Commun 2021; 12:4531. [PMID: 34312383 PMCID: PMC8313558 DOI: 10.1038/s41467-021-24772-8] [Citation(s) in RCA: 45] [Impact Index Per Article: 15.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2020] [Accepted: 06/29/2021] [Indexed: 02/08/2023] Open
Abstract
Recent developments in synthetic biology may bring the bottom-up generation of a synthetic cell within reach. A key feature of a living synthetic cell is a functional cell cycle, in which DNA replication and segregation as well as cell growth and division are well integrated. Here, we describe different approaches to recreate these processes in a synthetic cell, based on natural systems and/or synthetic alternatives. Although some individual machineries have recently been established, their integration and control in a synthetic cell cycle remain to be addressed. In this Perspective, we discuss potential paths towards an integrated synthetic cell cycle.
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Affiliation(s)
- Lorenzo Olivi
- Laboratory of Microbiology, Wageningen University, Wageningen, The Netherlands
| | | | | | - Nicola De Franceschi
- Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands
| | - Cees Dekker
- Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands
| | | | - Nico J Claassens
- Laboratory of Microbiology, Wageningen University, Wageningen, The Netherlands
| | | | - John van der Oost
- Laboratory of Microbiology, Wageningen University, Wageningen, The Netherlands.
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8
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Chubak I, Likos CN, Egorov SA. Multiscale Approaches for Confined Ring Polymer Solutions. J Phys Chem B 2021; 125:4910-4923. [PMID: 33938750 PMCID: PMC8279562 DOI: 10.1021/acs.jpcb.1c01953] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2021] [Revised: 04/18/2021] [Indexed: 11/29/2022]
Abstract
We apply a hierarchy of multiscale modeling approaches to investigate the structure of ring polymer solutions under planar confinement. In particular, we employ both monomer-resolved (MR-DFT) and a coarse-grained (CG-DFT) density functional theories for fully flexible ring polymers, with the former based on a flexible tangent hard-sphere model and the latter based on an effective soft-colloid representation, to elucidate the ring polymer organization within slits of variable width in different concentration regimes. The predicted monomer and polymer center-of-mass densities in confinement, as well as the surface tension at the solution-wall interface, are compared to explicit molecular dynamics (MD) simulations. The approaches yield quantitative (MR-DFT) or semiquantitative (CG-DFT) agreement with MD. In addition, we provide a systematic comparison between confined linear and ring polymer solutions. When compared to their linear counterparts, the rings are found to feature a higher propensity to structure in confinement that translates into a distinct shape of the depletion potentials between two walls immersed into a polymer solution. The depletion potentials that we extract from CG-DFT and MR-DFT are in semiquantitative agreement with each other. Overall, we find consistency among all approaches as regards the shapes, trends, and qualitative characteristics of density profiles and depletion potentials induced on hard walls by linear and cyclic polymers.
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Affiliation(s)
- Iurii Chubak
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
- Sorbonne
Université CNRS, Physico-Chimie des
Électrolytes et Nanosystèmes Interfaciaux, F-75005 Paris, France
| | - Christos N. Likos
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Sergei A. Egorov
- Department
of Chemistry, University of Virginia, Charlottesville, Virginia 22901, United States
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9
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Guo F, Li K, Wu J, He L, Zhang L. Effects of Topological Constraints on Penetration Structures of Semi-Flexible Ring Polymers. Polymers (Basel) 2020; 12:E2659. [PMID: 33187232 PMCID: PMC7696204 DOI: 10.3390/polym12112659] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Revised: 11/03/2020] [Accepted: 11/08/2020] [Indexed: 11/24/2022] Open
Abstract
The effects of topological constraints on penetration structures of semi-flexible ring polymers in a melt are investigated using molecular dynamics simulations, considering simultaneously the effects of the chain stiffness. Three topology types of rings are considered: 01-knot (the unknotted), 31-knot and 61-knot ring polymers, respectively. With the improved algorithm to detect and quantify the inter-ring penetration (or inter-ring threading), the degree of ring threading does not increase monotonously with the chain stiffness, existing a peak value at the intermediate stiffness. It indicates that rings interpenetrate most at intermediate stiffness where there is a balance between coil expansion (favoring penetrations) and stiffness (inhibiting penetrations). Meanwhile, the inter-ring penetration would be suppressed with the knot complexity of the rings. The analysis of effective potential between the rings provides a better understanding for this non-monotonous behavior in inter-ring penetration.
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Affiliation(s)
- Fuchen Guo
- Department of Physics, Zhejiang University, Hangzhou 310027, China; (F.G.); (K.L.); (J.W.)
| | - Ke Li
- Department of Physics, Zhejiang University, Hangzhou 310027, China; (F.G.); (K.L.); (J.W.)
| | - Jiaxin Wu
- Department of Physics, Zhejiang University, Hangzhou 310027, China; (F.G.); (K.L.); (J.W.)
| | - Linli He
- Department of Physics, Wenzhou University, Wenzhou 325035, China
| | - Linxi Zhang
- Department of Physics, Zhejiang University, Hangzhou 310027, China; (F.G.); (K.L.); (J.W.)
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10
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Ahmadian Dehaghani Z, Chubak I, Likos CN, Ejtehadi MR. Effects of topological constraints on linked ring polymers in solvents of varying quality. SOFT MATTER 2020; 16:3029-3038. [PMID: 32129365 DOI: 10.1039/c9sm02374g] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We investigate the effects of topological constraints in catenanes composed of interlinked ring polymers on their size in a good solvent as well as on the location of their θ-point when the solvent quality is worsened. We mainly focus on poly[n]catenanes consisting of n ring polymers each of length m interlocked in a linear fashion. Using molecular dynamics simulations, we study the scaling of the poly[n]catenane's radius of gyration in a good solvent, assuming in general that Rg∼mμnν and we find that μ = 0.65 ± 0.02 and ν = 0.60 ± 0.01 for the range of n and m considered. These findings are further rationalized with the help of a mean-field Flory-like theory yielding the values of μ = 16/25 and ν = 3/5, consistent with the numerical results. We show that individual rings within catenanes feature a surplus swelling due to the presence of NL topological links. Furthermore, we consider poly[n]catenanes in solvents of varying quality and we demonstrate that the presence of topological links leads to an increase of its θ-temperature in comparison to isolated linear and ring chains with the following ordering: T > T > T. Finally, we show that the presence of links similarly raises the θ-temperature of a single linked ring in comparison to an unlinked one, bringing its θ-temperature close to the one of a poly[n]catenane.
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11
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Rauscher PM, Schweizer KS, Rowan SJ, de Pablo JJ. Thermodynamics and Structure of Poly[n]catenane Melts. Macromolecules 2020. [DOI: 10.1021/acs.macromol.9b02706] [Citation(s) in RCA: 28] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Affiliation(s)
- Phillip M. Rauscher
- Pritzker School of Molecular Engineering, University of Chicago, 5747 South Ellis Avenue, Chicago, Illinois 60637, United States
| | - Kenneth S. Schweizer
- Department of Materials Science, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801-3028, United States
- Materials Research Laboratory, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801-3028, United States
- Department of Chemical and Biomolecular Engineering, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801-3028, United States
- Department of Chemistry, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801-3028, United States
| | - Stuart J. Rowan
- Pritzker School of Molecular Engineering, University of Chicago, 5747 South Ellis Avenue, Chicago, Illinois 60637, United States
- Department of Chemistry, University of Chicago, 5747 South Ellis Avenue, Chicago, Illinois 60637, United States
- Chemical and Engineering Sciences, Argonne National Laboratory, 9700 Cass Avenue, Lemont, Illinois 60439, United States
- Center for Molecular Engineering, Argonne National Laboratory, 9700 Cass Avenue, Lemont, Illinois 60439, United States
| | - Juan J. de Pablo
- Pritzker School of Molecular Engineering, University of Chicago, 5747 South Ellis Avenue, Chicago, Illinois 60637, United States
- Center for Molecular Engineering, Argonne National Laboratory, 9700 Cass Avenue, Lemont, Illinois 60439, United States
- Materials Science Division, Argonne National Laboratory, 9700 Cass Avenue, Lemont, Illinois 60439, United States
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12
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Zhou R, Gao YQ. Polymer models for the mechanisms of chromatin 3D folding: review and perspective. Phys Chem Chem Phys 2020; 22:20189-20201. [DOI: 10.1039/d0cp01877e] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
In this perspective paper, classical physical models for mammalian interphase chromatin folding are reviewed.
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Affiliation(s)
- Rui Zhou
- Biomedical Pioneering Innovation Center
- Peking University
- 100871 Beijing
- China
| | - Yi Qin Gao
- Biomedical Pioneering Innovation Center
- Peking University
- 100871 Beijing
- China
- Beijing Advanced Innovation Center for Genomics
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13
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Hofmann A, Mäkelä J, Sherratt DJ, Heermann D, Murray SM. Self-organised segregation of bacterial chromosomal origins. eLife 2019; 8:e46564. [PMID: 31397672 PMCID: PMC6701925 DOI: 10.7554/elife.46564] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2019] [Accepted: 08/09/2019] [Indexed: 01/12/2023] Open
Abstract
The chromosomal replication origin region (ori) of characterised bacteria is dynamically positioned throughout the cell cycle. In slowly growing Escherichia coli, ori is maintained at mid-cell from birth until its replication, after which newly replicated sister oris move to opposite quarter positions. Here, we provide an explanation for ori positioning based on the self-organisation of the Structural Maintenance of Chromosomes complex, MukBEF, which forms dynamically positioned clusters on the chromosome. We propose that a non-trivial feedback between the self-organising gradient of MukBEF complexes and the oris leads to accurate ori positioning. We find excellent agreement with quantitative experimental measurements and confirm key predictions. Specifically, we show that oris exhibit biased motion towards MukBEF clusters, rather than mid-cell. Our findings suggest that MukBEF and oris act together as a self-organising system in chromosome organisation-segregation and introduces protein self-organisation as an important consideration for future studies of chromosome dynamics.
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Affiliation(s)
- Andreas Hofmann
- Institute for Theoretical PhysicsHeidelberg UniversityHeidelbergGermany
| | - Jarno Mäkelä
- Department of BiochemistryUniversity of OxfordOxfordUnited Kingdom
| | - David J Sherratt
- Department of BiochemistryUniversity of OxfordOxfordUnited Kingdom
| | - Dieter Heermann
- Institute for Theoretical PhysicsHeidelberg UniversityHeidelbergGermany
| | - Seán M Murray
- Max Planck Institute for Terrestrial Microbiology, LOEWE Centre for Synthetic Microbiology (SYNMIKRO)MarburgGermany
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14
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Dell ZE, Schweizer KS. Intermolecular structural correlations in model globular and unconcatenated ring polymer liquids. SOFT MATTER 2018; 14:9132-9142. [PMID: 30407479 DOI: 10.1039/c8sm01722k] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We employ the field theoretic polymer integral equation theory to construct a segment-level theory for the thermodynamics and pair structure of dense liquids of interpenetrating ring polymers and a simple globule model. The latter is defined by a fractal mass distribution on all internal length scales with an exponent equal to the spatial dimension (dF = ds = 3). In an isochoric ensemble the dimensionless compressibility and pressure is predicted to vary exponentially with macromolecular volume fraction. An intermolecular correlation hole exists down to small length scales. This model appears to be useful for a recently studied experimental soft nanoparticle suspension, and also serves as a reference system for our analysis of ring liquids. Motivated by simulations, a two-fractal exponent ring model is adopted for the intramolecular structure factor. At smaller lengths it describes chain-like macromolecules, while on larger scales it corresponds to a space-filling object in the sense that dF = ds = 3. The crossover between these two regimes is of order the entanglement length of the linear chain analog. Based on a constant compressibility ensemble, the effective volume fraction grows at intermediate values of degree of polymerization (N), and crosses over to a very slow logarithmic growth at large N. A weaker intermolecular correlation hole is predicted. The number of nearest neighbor rings increases dramatically at small N, akin to linear chain melts, but then tends to saturate at large N, in accord with simulations. The tools developed may be relevant for other partially interpenetrating soft objects such as core-shell nanogels or microgels.
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Affiliation(s)
- Zachary E Dell
- Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003, USA
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15
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Affiliation(s)
- Iurii Chubak
- Faculty of Physics, University of Vienna, Vienna, Austria
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16
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Gartner TE, Jayaraman A. Macromolecular 'size' and 'hardness' drives structure in solvent-swollen blends of linear, cyclic, and star polymers. SOFT MATTER 2018; 14:411-423. [PMID: 29251311 DOI: 10.1039/c7sm02199b] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this paper, we apply molecular simulation and liquid state theory to uncover the structure and thermodynamics of homopolymer blends of the same chemistry and varying chain architecture in the presence of explicit solvent species. We use hybrid Monte Carlo (MC)/molecular dynamics (MD) simulations in the Gibbs ensemble to study the swelling of ∼12 000 g mol-1 linear, cyclic, and 4-arm star polystyrene chains in toluene. Our simulations show that the macroscopic swelling response is indistinguishable between the various architectures and matches published experimental data for the solvent annealing of linear polystyrene by toluene vapor. We then use standard MD simulations in the NPT ensemble along with polymer reference interaction site model (PRISM) theory to calculate effective polymer-solvent and polymer-polymer Flory-Huggins interaction parameters (χeff) in these systems. As seen in the macroscopic swelling results, there are no significant differences in the polymer-solvent and polymer-polymer χeff between the various architectures. Despite similar macroscopic swelling and effective interaction parameters between various architectures, the pair correlation function between chain centers-of-mass indicates stronger correlations between cyclic or star chains in the linear-cyclic blends and linear-star blends, compared to linear chain-linear chain correlations. Furthermore, we note striking similarities in the chain-level correlations and the radius of gyration of cyclic and 4-arm star architectures of identical molecular weight. Our results indicate that the cyclic and star chains are 'smaller' and 'harder' than their linear counterparts, and through comparison with MD simulations of blends of soft spheres with varying hardness and size we suggest that these macromolecular characteristics are the source of the stronger cyclic-cyclic and star-star correlations.
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Affiliation(s)
- Thomas E Gartner
- Department of Chemical and Biomolecular Engineering, Colburn Laboratory, University of Delaware, 150 Academy Street, Newark, DE 19716, USA
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17
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Sugawara T, Kimura A. Physical properties of the chromosomes and implications for development. Dev Growth Differ 2017; 59:405-414. [PMID: 28573677 DOI: 10.1111/dgd.12363] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2017] [Revised: 04/13/2017] [Accepted: 04/13/2017] [Indexed: 01/15/2023]
Abstract
Remarkable progress has been made in understanding chromosome structures inside the cell nucleus. Recent advances in Hi-C technologies enable the detection of genome-wide chromatin interactions, providing insight into three-dimensional (3D) genome organization. Advancements in the spatial and temporal resolutions of imaging as well as in molecular biological techniques allow the tracking of specific chromosomal loci, improving our understanding of chromosome movements. From these data, we are beginning to understand how the intra-nuclear locations of chromatin loci and the 3D genome structure change during development and differentiation. This emerging field of genome structure and dynamics research requires an interdisciplinary approach including efficient collaborations between experimental biologists and physicists, informaticians, or engineers. Quantitative and mathematical analyses based on polymer physics are becoming increasingly important for processing and interpreting experimental data on 3D chromosome structures and dynamics. In this review, we aim to provide an overview of recent research on the physical aspects of chromosome structure and dynamics oriented for biologists. These studies have mainly focused on chromosomes at the cellular level, using unicellular organisms and cultured cells. However, physical parameters that change during development, such as nuclear size, may impact genome structure and dynamics. Here, we discuss how chromatin dynamics and genome structures in early embryos change during development, which we expect will be a hot topic in the field of chromatin dynamics in the near future. We hope this review helps developmental biologists to quantitatively investigate the physical natures of chromosomes in developmental biology research.
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Affiliation(s)
- Takeshi Sugawara
- Research Center for the Mathematics on Chromatin Live Dynamics (RcMcD), Hiroshima University, Higashi-Hiroshima, 739-8530, Japan.,Cell Architecture Laboratory, Structural Biology Center, National Institute of Genetics, Mishima, 411-8540, Japan
| | - Akatsuki Kimura
- Cell Architecture Laboratory, Structural Biology Center, National Institute of Genetics, Mishima, 411-8540, Japan.,Department of Genetics, School of Life Science, The Graduate University for Advanced Studies (Sokendai), Mishima, 411-8540, Japan
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18
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Dell ZE, Schweizer KS. Segment-scale, force-level theory of mesoscopic dynamic localization and entropic elasticity in entangled chain polymer liquids. J Chem Phys 2017; 146:134901. [PMID: 28390385 DOI: 10.1063/1.4978774] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We develop a segment-scale, force-based theory for the breakdown of the unentangled Rouse model and subsequent emergence of isotropic mesoscopic localization and entropic elasticity in chain polymer liquids in the absence of ergodicity-restoring anisotropic reptation or activated hopping motion. The theory is formulated in terms of a conformational N-dynamic-order-parameter generalized Langevin equation approach. It is implemented using a universal field-theoretic Gaussian thread model of polymer structure and closed at the level of the chain dynamic second moment matrix. The physical idea is that the isotropic Rouse model fails due to the dynamical emergence, with increasing chain length, of time-persistent intermolecular contacts determined by the combined influence of local uncrossability, long range polymer connectivity, and a self-consistent treatment of chain motion and the dynamic forces that hinder it. For long chain melts, the mesoscopic localization length (identified as the tube diameter) and emergent entropic elasticity predictions are in near quantitative agreement with experiment. Moreover, the onset chain length scales with the semi-dilute crossover concentration with a realistic numerical prefactor. Distinctive novel predictions are made for various off-diagonal correlation functions that quantify the full spatial structure of the dynamically localized polymer conformation. As the local excluded volume constraint and/or intrachain bonding spring are softened to allow chain crossability, the tube diameter is predicted to swell until it reaches the radius-of-gyration at which point mesoscopic localization vanishes in a discontinuous manner. A dynamic phase diagram for such a delocalization transition is constructed, which is qualitatively consistent with simulations and the classical concept of a critical entanglement degree of polymerization.
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Affiliation(s)
- Zachary E Dell
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
| | - Kenneth S Schweizer
- Department of Materials Science, University of Illinois, Urbana, Illinois 61801, USA
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19
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Coil-helix-globule transition for self-attractive semiflexible ring chains. POLYMER 2017. [DOI: 10.1016/j.polymer.2016.12.075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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20
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Poier P, Bačová P, Moreno AJ, Likos CN, Blaak R. Anisotropic effective interactions and stack formation in mixtures of semiflexible ring polymers. SOFT MATTER 2016; 12:4805-4820. [PMID: 27117080 DOI: 10.1039/c6sm00430j] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
By means of extensive computer simulations, we investigate the formation of columnar structures (stacks) in concentrated solutions of semiflexible ring polymers. To characterize the stacks we employ an algorithm that identifies tube-like structures in the simulation cell. Stacks are found both in the real system and in the fluid of soft disks interacting through the effective anisotropic pair potential derived for the rings [P. Poier et al., Macromolecules, 2015, 48, 4983-4997]. Furthermore, we investigate binary mixtures of cluster-forming and non-cluster-forming rings. We find that monodispersity is not a requirement for stack formation. The latter is found for a broad range of mixture compositions, though the columns in the mixtures exhibit important differences to those observed in the monodisperse case. We extend the anisotropic effective model to mixtures. We show that it correctly predicts stack formation and constitutes a significant improvement with respect to the usual isotropic effective description based only on macromolecular centers-of-mass.
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Affiliation(s)
- Peter Poier
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria.
| | - Petra Bačová
- Centro de Física de Materiales (CSIC-UPV/EHU) and Materials Physics Center MPC, Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain and Institute of Applied and Computational Mathematics (IACM), Foundation for Research and Technology Hellas (FORTH), GR-71110 Heraklion, Greece
| | - Angel J Moreno
- Centro de Física de Materiales (CSIC-UPV/EHU) and Materials Physics Center MPC, Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain and Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
| | - Christos N Likos
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria.
| | - Ronald Blaak
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria.
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21
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Poier P, Likos CN, Moreno A, Blaak R. An Anisotropic Effective Model for the Simulation of Semiflexible Ring Polymers. Macromolecules 2015; 48:4983-4997. [PMID: 26240439 PMCID: PMC4519991 DOI: 10.1021/acs.macromol.5b00603] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2015] [Revised: 05/29/2015] [Indexed: 01/27/2023]
Abstract
We derive and introduce anisotropic effective pair potentials to coarse-grain solutions of semiflexible ring polymers of various lengths. The system has been recently investigated by means of full monomer-resolved computer simulations, revealing a host of unusual features and structure formation, which, however, cannot be captured by a rotationally averaged effective pair potential between the rings' centers of mass [Bernabei M.; Soft Matter2013, 9, 1287]. Our new coarse-graining strategy is to picture each ring as a soft, penetrable disk. We demonstrate that for the short- and intermediate-length rings the new model is quite capable of capturing the physics in a quantitative fashion, whereas for the largest rings, which resemble flexible ones, it fails at high densities. Our work opens the way for the physical justification of general, anisotropic penetrable interaction potentials.
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Affiliation(s)
- Peter Poier
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Christos N. Likos
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Angel
J. Moreno
- Centro de Física
de Materiales (CSIC-UPV/EHU) and Materials Physics Center (MPC), Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
- Donostia International
Physics Center, Paseo Manuel de Lardizabal
4, E-20018 San Sebastián, Spain
| | - Ronald Blaak
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
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22
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Narros A, Likos CN, Moreno AJ, Capone B. Multi-blob coarse graining for ring polymer solutions. SOFT MATTER 2014; 10:9601-9614. [PMID: 25356818 DOI: 10.1039/c4sm01904k] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We present a multi-scale molecular modeling of concentrated solutions of unknotted and non-concatenated ring polymers under good solvent conditions. The approach is based on a multi-blob representation of each ring polymer, which is capable of overcoming the shortcomings of single-blob approaches that lose their validity at concentrations exceeding the overlap density of the solution [A. Narros, A. J. Moreno, and C. N. Likos, Soft Matter, 2010, 6, 2435]. By means of a first principles coarse-graining strategy based on analytically determined effective pair potentials between the blobs, computed at zero density, we quantitatively reproduce the single molecule and solution properties of a system with well-defined topological constraints. Detailed comparisons with the underlying, monomer-resolved model demonstrate the validity of our approach, which employs fully transferable pair potentials between connected and unconnected blobs. We demonstrate that the pair structure between the centers of mass of the rings is accurately reproduced by the multi-blob approach, thus opening the way for simulation of arbitrarily long polymers. Finally, we show the importance of the topological constraint of non-concatenation on the structure of the concentrated solution and in particular on the size of the correlation hole and the shrinkage of the rings as melt concentrations are approached.
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Affiliation(s)
- Arturo Narros
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria.
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23
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Sun D, Cho J. Monte Carlo simulations on thermodynamic and conformational properties of catenated double-ring copolymers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062601. [PMID: 25615120 DOI: 10.1103/physreve.90.062601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2014] [Indexed: 06/04/2023]
Abstract
The thermodynamic and conformational properties of catenated double-ring A-B copolymer melts are investigated through lattice Monte Carlo simulations. The topological constraint on the catenated copolymers is shown to suppress demixing of A and B monomers. This action results in their order-to-disorder transition (ODT) at an increased segregation level and the lamellae below ODT with reduced order, when compared to diblock copolymers of linear or single-ring topology. The A and B rings are pulled closer by catenation in the copolymer, which leads to its smaller gyration radius, lamellar domain spacing, and distance between mass centers of the two rings than for the diblock copolymers. With increasing segregation tendencies, the gyration radii of the A rings of the catenated copolymers stretch along the direction normal to lamellae, while the A-block conformations of the single-ring copolymers change their shapes from ellipsoid to sphere.
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Affiliation(s)
- Dachuan Sun
- Department of Polymer Science and Engineering, and Center for Photofunctional Energy Materials, Dankook University, 152 Jukjeon-ro, Suji-gu, Yongin, Gyeonggi-do 448-701, South Korea
| | - Junhan Cho
- Department of Polymer Science and Engineering, and Center for Photofunctional Energy Materials, Dankook University, 152 Jukjeon-ro, Suji-gu, Yongin, Gyeonggi-do 448-701, South Korea
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24
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Montes-Saralegui M, Nikoubashman A, Kahl G. Merging and hopping processes in systems of ultrasoft, cluster forming particles under compression. J Chem Phys 2014; 141:124908. [DOI: 10.1063/1.4896055] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Marta Montes-Saralegui
- Institute of Theoretical Physics and CMS, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
| | - Arash Nikoubashman
- Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
| | - Gerhard Kahl
- Institute of Theoretical Physics and CMS, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
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25
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Haydukivska K, Blavatska V. Ring polymers in crowded environment: conformational properties. J Chem Phys 2014; 141:094906. [PMID: 25194393 DOI: 10.1063/1.4894278] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
Abstract
We analyze the universal size characteristics of flexible ring polymers in solutions in presence of structural obstacles (impurities) in d dimensions. One encounters such situations when considering polymers in gels, colloidal solutions, intra- and extracellular environments. A special case of extended impurities correlated on large distances r according to a power law ~r(-a) is considered. Applying the direct polymer renormalization scheme, we evaluate the estimates for averaged gyration radius ⟨R(g ring)⟩ and spanning radius ⟨R(1/2 ring)⟩ of typical ring polymer conformation up to the first order of double ɛ = 4 - d, δ = 4 - a expansion. Our results quantitatively reveal an extent of the effective size and anisotropy of closed ring macromolecules in disordered environment. In particular, the size ratio of ring and open (linear) polymers of the same molecular weight grows when increasing the strength of disorder according to ⟨R(g ring)(2)⟩/⟨R(g chain)(2)⟩=½(1+(13/48)δ).
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Affiliation(s)
- K Haydukivska
- Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 79011 Lviv, Ukraine
| | - V Blavatska
- Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 79011 Lviv, Ukraine
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26
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Slimani MZ, Bacova P, Bernabei M, Narros A, Likos CN, Moreno AJ. Cluster Glasses of Semiflexible Ring Polymers. ACS Macro Lett 2014; 3:611-616. [PMID: 25083314 PMCID: PMC4111402 DOI: 10.1021/mz500117v] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2014] [Accepted: 06/10/2014] [Indexed: 12/19/2022]
Abstract
We present computer simulations of concentrated solutions of unknotted nonconcatenated semiflexible ring polymers. Unlike in their flexible counterparts, shrinking involves a strong energetic penalty, favoring interpenetration and clustering of the rings. We investigate the slow dynamics of the centers-of-mass of the rings in the amorphous cluster phase, consisting of disordered columns of oblate rings penetrated by bundles of prolate ones. Scattering functions reveal a striking decoupling of self- and collective motions. Correlations between centers-of-mass exhibit slow relaxation, as expected for an incipient glass transition, indicating the dynamic arrest of the cluster positions. However, self-correlations decay at much shorter time scales. This feature is a manifestation of the fast, continuous exchange and diffusion of the individual rings over the matrix of clusters. Our results reveal a novel scenario of glass formation in a simple monodisperse system, characterized by self-collective decoupling, soft caging, and mild dynamic heterogeneity.
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Affiliation(s)
- Mohammed Zakaria Slimani
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
| | - Petra Bacova
- Departamento
de Física de Materiales, Universidad del País Vasco (UPV/EHU), Apartado 1072, E-20080 San Sebastián, Spain
- Materials Physics Center MPC, Paseo
Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
| | - Marco Bernabei
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
- Departament
de Fisica Fonamental, Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
| | - Arturo Narros
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Christos N. Likos
- Faculty
of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Angel J. Moreno
- Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
- Materials Physics Center MPC, Paseo
Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
- Centro de Física de Materiales (CSIC, UPV/EHU), Paseo Manuel de Lardizabal 5, E-20018 San Sebastián, Spain
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27
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Chromosome segregation by the Escherichia coli Min system. Mol Syst Biol 2014; 9:686. [PMID: 24022004 PMCID: PMC3792344 DOI: 10.1038/msb.2013.44] [Citation(s) in RCA: 63] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2013] [Accepted: 08/07/2013] [Indexed: 12/26/2022] Open
Abstract
The mechanisms underlying chromosome segregation in prokaryotes remain a subject of debate and no unifying view has yet emerged. Given that the initial disentanglement of duplicated chromosomes could be achieved by purely entropic forces, even the requirement of an active prokaryotic segregation machinery has been questioned. Using computer simulations, we show that entropic forces alone are not sufficient to achieve and maintain full separation of chromosomes. This is, however, possible by assuming repeated binding of chromosomes along a gradient of membrane-associated tethering sites toward the poles. We propose that, in Escherichia coli, such a gradient of membrane tethering sites may be provided by the oscillatory Min system, otherwise known for its role in selecting the cell division site. Consistent with this hypothesis, we demonstrate that MinD binds to DNA and tethers it to the membrane in an ATP-dependent manner. Taken together, our combined theoretical and experimental results suggest the existence of a novel mechanism of chromosome segregation based on the Min system, further highlighting the importance of active segregation of chromosomes in prokaryotic cell biology.
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28
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Grosberg AY. Annealed lattice animal model and Flory theory for the melt of non-concatenated rings: towards the physics of crumpling. SOFT MATTER 2014; 10:560-565. [PMID: 24652534 DOI: 10.1039/c3sm52805g] [Citation(s) in RCA: 77] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
A Flory theory is constructed for a long polymer ring in a melt of unknotted and non-concatenated rings. The theory assumes that the ring forms an effective annealed branched object and computes its primitive path. It is shown that the primitive path follows self-avoiding statistics and is characterized by the corresponding Flory exponent of a polymer with excluded volume. Based on that, it is shown that rings in the melt are compact objects with overall size proportional to their length raised to the 1/3 power. Furthermore, the contact probability exponent γcontact is estimated, albeit by a poorly controlled approximation, with the result close to 1.1 consistent with both numerical and experimental data.
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Affiliation(s)
- Alexander Y Grosberg
- Department of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, NY 10003, USA.
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29
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Halverson JD, Smrek J, Kremer K, Grosberg AY. From a melt of rings to chromosome territories: the role of topological constraints in genome folding. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2014; 77:022601. [PMID: 24472896 DOI: 10.1088/0034-4885/77/2/022601] [Citation(s) in RCA: 180] [Impact Index Per Article: 18.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
We review pro and contra of the hypothesis that generic polymer properties of topological constraints are behind many aspects of chromatin folding in eukaryotic cells. For that purpose, we review, first, recent theoretical and computational findings in polymer physics related to concentrated, topologically simple (unknotted and unlinked) chains or a system of chains. Second, we review recent experimental discoveries related to genome folding. Understanding in these fields is far from complete, but we show how looking at them in parallel sheds new light on both.
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Affiliation(s)
- Jonathan D Halverson
- Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA
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30
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Narros A, Moreno AJ, Likos C. Architecture-Induced Size Asymmetry and Effective Interactions of Ring Polymers: Simulation and Theory. Macromolecules 2013; 46:9437-9445. [PMID: 24347686 PMCID: PMC3859368 DOI: 10.1021/ma4016483] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2013] [Revised: 11/12/2013] [Indexed: 12/02/2022]
Abstract
We investigate, by means of Monte Carlo simulations, the role of ring architecture and topology on the relative sizes of two interacting polymers as a function of the distance between their centers-of-mass. As a general rule, polymers swell as they approach each other, irrespectively of their topologies. For each mutual separation, two identical linear polymers adopt the same average size. However, unknotted rings at close separations adopt different sizes, with the small one being "nested" within the large one over long time intervals, exchanging their roles in the course of the simulation. For two rings of different architectures and identical polymerization degree, the knotted one is always smaller, penetrating the unknotted one. On the basis of these observations, we propose a phenomenological theory for the effective interactions between rings, modeling them as unequal-sized penetrable spheres. This simple approximation provides a good description of the simulation results. In particular, it rationalizes the non-Gaussian shape and the short-distance plateau observed in the effective potential between unknotted ring polymers and pairs of unequal-sized unknotted/knotted ones. Our results demonstrate the crucial role of the architecture on both the effective interactions and the molecular size for strongly interpenetrating polymers.
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Affiliation(s)
- Arturo Narros
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
| | - Angel J. Moreno
- Centro de Física de Materiales (CSIC, UPV/EHU) and Materials
Physics Center MPC, Paseo Manuel de
Lardizabal 5, E-20018 San Sebastián, Spain
- Donostia International
Physics Center, Paseo Manuel de Lardizabal
4, E-20018 San Sebastián, Spain
| | - Christos
N. Likos
- Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
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31
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DNA double-strand breaks: linking gene expression to chromosome morphology and mobility. Chromosoma 2013; 123:103-15. [DOI: 10.1007/s00412-013-0432-y] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2013] [Revised: 08/06/2013] [Accepted: 08/08/2013] [Indexed: 11/27/2022]
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32
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Li S, Heermann DW. Transcriptional regulatory network shapes the genome structure of Saccharomyces cerevisiae. Nucleus 2013; 4:216-28. [PMID: 23674068 DOI: 10.4161/nucl.24875] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
Among cellular processes gene transcription is central. More and more evidence is mounting that transcription is tightly connected with the spatial organization of the chromosomes. Spatial proximity of genes sharing transcriptional machinery is one of the consequences of this organization. Motivated by information on the physical relationship among genes identified via chromosomal conformation capture methods, we complement the spatial organization with the idea that genes under similar transcription factor control, but possible scattered throughout the genome, might be in physically proximity to facilitate the access of their commonly used transcription factors. Unlike the transcription factory model, "interacting" genes in our "Gene Proximity Model" are not necessarily immediate physical neighbors but are in spatial proximity. Considering the stochastic nature of TF-promoter binding, this local condensation mechanism could serve as a tie to recruit co-regulated genes to guarantee the swiftness of biological reactions. We tested this idea with a simple eukaryotic organism, Saccharomyces cerevisiae. Chromosomal interaction patterns and folding behavior generated by our model re-construct those obtained from experiments. We show that the transcriptional regulatory network has a close linkage with the genome organization in budding yeast, which is fundamental and instrumental to later studies on other more complex eukaryotes.
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Affiliation(s)
- Songling Li
- Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany
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33
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Narros A, Moreno AJ, Likos CN. Effects of Knots on Ring Polymers in Solvents of Varying Quality. Macromolecules 2013; 46:3654-3668. [PMID: 23729865 PMCID: PMC3667624 DOI: 10.1021/ma400308x] [Citation(s) in RCA: 44] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2013] [Revised: 04/03/2013] [Indexed: 11/30/2022]
Abstract
![]()
We employ extensive computer simulations to investigate the conformations
and the interactions of ring polymers under conditions of worsening
solvent quality, in comparison with those for linear polymers. We
determine the dependence of the Θ-temperature on knotedness
by considering ring polymers of different topologies. We establish
a clear decrease of the former upon changing the topology of the polymer
from linear to an unknotted ring and a further decrease of the same
upon introducing trefoil- or 5-fold knots but we find no difference
in the Θ-point between the two knotted molecules. Our results
are based on two independent methods: one considering the scaling
of the gyration radius with molecular weight and one based on the
dependence of the effective interaction on solvent quality. In addition,
we calculate several shape-parameters of the polymers to characterize
linear, unknotted, and knotted topologies in good solvents and in
the proximity of the Θ-point. The shape parameters of the knotted
molecules show an interesting crossover at a degree of polymerization
that depends on the degree of knottedness of the molecule.
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Affiliation(s)
- Arturo Narros
- Faculty of Physics, University of Vienna , Boltzmanngasse 5, A- 1090 Vienna, Austria
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34
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Abstract
In the present article, we review recent computational investigations on the properties of ring polymers in solution. In particular, we focus on effective interactions obtained by means of coarse-graining techniques. We discuss the relative importance of the self-avoidance and the topological contributions in the qualitative features of the effective potential. We extend our previous results on identical rings and determine the effective potential between dissimilar ring polymers of distinct topology and size. The results obtained reveal the dramatic effects of the specific topology on the effective interactions, and hence in the structural correlations, of polymeric systems.
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35
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Li S, Heermann DW. Using chimaeric expression sequence tag as the reference to identify three-dimensional chromosome contacts. DNA Res 2012; 20:45-53. [PMID: 23213109 PMCID: PMC3576657 DOI: 10.1093/dnares/dss032] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Transcription-induced chimaeric transcripts, the potential post-transcriptional processing products, might reflect the spatial proximity of actively transcribed genes co-localized in transcription factories. A growing number of expression data deposited in databases provide us with the raw material for screening such chimaeric transcripts and using them as the probes to identify interactions between genes in cis or in trans. Based on the high-quality chimaeric transcripts gleaned from human expression sequence tag data with selection criteria, we identified the patterns of inter- and intrachromosomal gene–gene interactions. On top the contact pattern from interchromosomal interactions, we also observed an exponential behaviour of the intrachromosomal interactions within a certain length scale, which is consistent with the independent experimental results from Hi-C screening and with the Random Loop Model. A compatible result is found for mouse. Transcription-induced chimaeric transcripts, most of which might be accidental products with trivial functions, shed light on the spatial organization of chromosomes. These inter- and intrachromosomal interactions might contribute to the compaction of chromosomes, their segregation and formation of the chromosome territories, and their spatial distribution within the nucleus.
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Affiliation(s)
- Songling Li
- Theoretical Biophysics Group, Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany
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36
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Coslovich D, Bernabei M, Moreno AJ. Cluster glasses of ultrasoft particles. J Chem Phys 2012; 137:184904. [DOI: 10.1063/1.4765704] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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37
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38
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Yanao T, Yoshikawa K. Chiral selection in supercoiling and wrapping of DNA. POLYMER SCIENCE SERIES C 2012. [DOI: 10.1134/s1811238212070065] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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39
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Mitotic chromosome structure. Exp Cell Res 2012; 318:1381-5. [DOI: 10.1016/j.yexcr.2012.03.027] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2012] [Revised: 03/19/2012] [Accepted: 03/24/2012] [Indexed: 11/18/2022]
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Iyer BVS, Arya G. Lattice animal model of chromosome organization. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011911. [PMID: 23005456 DOI: 10.1103/physreve.86.011911] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2011] [Indexed: 06/01/2023]
Abstract
Polymer models tied together by constraints of looping and confinement have been used to explain many of the observed organizational characteristics of interphase chromosomes. Here we introduce a simple lattice animal representation of interphase chromosomes that combines the features of looping and confinement constraints into a single framework. We show through Monte Carlo simulations that this model qualitatively captures both the leveling off in the spatial distance between genomic markers observed in fluorescent in situ hybridization experiments and the inverse decay in the looping probability as a function of genomic separation observed in chromosome conformation capture experiments. The model also suggests that the collapsed state of chromosomes and their segregation into territories with distinct looping activities might be a natural consequence of confinement.
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Affiliation(s)
- Balaji V S Iyer
- Department of NanoEngineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0448, USA
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Fritsche M, Li S, Heermann DW, Wiggins PA. A model for Escherichia coli chromosome packaging supports transcription factor-induced DNA domain formation. Nucleic Acids Res 2012; 40:972-80. [PMID: 21976727 PMCID: PMC3273793 DOI: 10.1093/nar/gkr779] [Citation(s) in RCA: 69] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2011] [Revised: 09/05/2011] [Accepted: 09/05/2011] [Indexed: 01/07/2023] Open
Abstract
What physical mechanism leads to organization of a highly condensed and confined circular chromosome? Computational modeling shows that confinement-induced organization is able to overcome the chromosome's propensity to mix by the formation of topological domains. The experimentally observed high precision of separate subcellular positioning of loci (located on different chromosomal domains) in Escherichia coli naturally emerges as a result of entropic demixing of such chromosomal loops. We propose one possible mechanism for organizing these domains: regulatory control defined by the underlying E. coli gene regulatory network requires the colocalization of transcription factor genes and target genes. Investigating this assumption, we find the DNA chain to self-organize into several topologically distinguishable domains where the interplay between the entropic repulsion of chromosomal loops and their compression due to the confining geometry induces an effective nucleoid filament-type of structure. Thus, we propose that the physical structure of the chromosome is a direct result of regulatory interactions. To reproduce the observed precise ordering of the chromosome, we estimate that the domain sizes are distributed between 10 and 700 kb, in agreement with the size of topological domains identified in the context of DNA supercoiling.
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Affiliation(s)
- Miriam Fritsche
- Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, D-69120 Heidelberg, Germany.
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42
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Loops determine the mechanical properties of mitotic chromosomes. PLoS One 2011; 6:e29225. [PMID: 22216220 PMCID: PMC3246460 DOI: 10.1371/journal.pone.0029225] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2011] [Accepted: 11/22/2011] [Indexed: 12/23/2022] Open
Abstract
We introduce a new polymer model for mitotic chromosomes. The key assumption of the model is the ability of the chromatin fibre to cross-link to itself due to binding proteins. These protein-chromatin interactions are included by a probabilistic and dynamic mechanism. The hypothesis is motivated by the observation of high repulsive forces between ring polymers. We performed computer simulations to validate our model. Our results show that the presence of loops leads to a tight compaction and contributes significantly to the bending rigidity of chromosomes. Moreover, our qualitative prediction of the force elongation behaviour is close to experimental findings. The Dynamic Loop Model presented here indicates that the internal structure of mitotic chromosomes is based on self-organization of the chromatin fibre rather than attachment of chromatin to a protein scaffold. It also shows that the number and size of loops have a strong influence on the mechanical properties. We suggest that changes in the mechanical characteristics of chromosomes in different stages of the cell cycle, for example, can be explained by an altered internal loop structure.
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Meldi L, Brickner JH. Compartmentalization of the nucleus. Trends Cell Biol 2011; 21:701-8. [PMID: 21900010 PMCID: PMC3970429 DOI: 10.1016/j.tcb.2011.08.001] [Citation(s) in RCA: 66] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2011] [Revised: 08/03/2011] [Accepted: 08/04/2011] [Indexed: 12/14/2022]
Abstract
The nucleus is a spatially organized compartment. The most obvious way in which this is achieved is at the level of chromosomes. The positioning of chromosomes with respect to nuclear landmarks and with respect to each other is both non-random and cell-type specific. This suggests that cells possess molecular mechanisms to influence the folding and disposition of chromosomes within the nucleus. The localization of many proteins is also heterogeneous within the nucleus. Therefore, chromosome folding and the localization of proteins leads to a model in which individual genes are positioned in distinct protein environments that can affect their transcriptional state. We focus here on the spatial organization of the nucleus and how it impacts upon gene expression.
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Affiliation(s)
- Lauren Meldi
- Department of Molecular Biosciences, Northwestern University, Evanston, Illinois, USA
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44
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Reiss P, Fritsche M, Heermann DW. Looped star polymers show conformational transition from spherical to flat toroidal shapes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:051910. [PMID: 22181447 DOI: 10.1103/physreve.84.051910] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2011] [Revised: 11/02/2011] [Indexed: 05/31/2023]
Abstract
Inspired by the topological organization of the circular Escherichia coli chromosome, which is compacted by separate domains, we study a polymer architecture consisting of a central ring to which either looped or linear side chains are grafted. A shape change from a spherical to a toroidal organization takes place as soon as the inner ring becomes large enough for the attached arms to fit within its circumference. Building up a torus, the system flattens, depending on the effective bending rigidity of the chain induced by entropic repulsion of the attached loops and, to a lesser extent, linear arms. Our results suggest that the natural formation of a toroidal structure with a decreased amount of writhe induced by a specific underlying topology could be one driving force, among others, that nature exploits to ensure proper packaging of the genetic material within a rod-shaped, bacterial envelope.
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Affiliation(s)
- Pascal Reiss
- Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany
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Dolgushev M, Berezovska G, Blumen A. Maximum entropy principle applied to semiflexible ring polymers. J Chem Phys 2011; 135:094901. [PMID: 21913779 DOI: 10.1063/1.3631943] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Based on the success of the maximum entropy principle (MEP) in the study of semiflexible treelike polymers [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)], it is of much interest to establish MEP's potential for general semiflexible polymers which contain loops. Here, we embark on this endeavor by considering discrete semiflexible polymer rings in a Rouse-type scheme. Now, for treelike polymers a beads-and-bonds (i.e., a discrete) picture is essential for an easy inclusion of branching points. Moreover, one may envisage (similar to our former work [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)]) to impose for each angle between two bonds a distinct stiffness condition. Working in this way leads already for a polymer ring to a complicated problem. Hence, we follow a reduced variational approach as applied earlier to polymer chains, in which a single Lagrange multiplier is used for each set of identical conditions imposed on topologically equivalent bonds and bonds' orientations. In this way, we obtain for the discrete ring an analytically closed form which involves Chebyshev polynomials. This expression turns out to lead to a series of solutions: Apart from the regular solution, several other solutions appear. One may be tempted to discard the other solutions, since for them the potential energy matrix is not positive definite. A more careful analysis based on topological features suggests, however, that such solutions can be assigned to rings displaying knots. Monte Carlo simulations which take excluded volume interactions into account agree with our interpretation.
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Affiliation(s)
- Maxim Dolgushev
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany.
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46
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Physical nuclear organization: loops and entropy. Curr Opin Cell Biol 2011; 23:332-7. [DOI: 10.1016/j.ceb.2011.03.010] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2011] [Revised: 03/14/2011] [Accepted: 03/14/2011] [Indexed: 01/07/2023]
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Iyer BV, Kenward M, Arya G. Hierarchies in eukaryotic genome organization: Insights from polymer theory and simulations. BMC BIOPHYSICS 2011; 4:8. [PMID: 21595865 PMCID: PMC3102647 DOI: 10.1186/2046-1682-4-8] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/07/2011] [Accepted: 04/15/2011] [Indexed: 12/11/2022]
Abstract
Eukaryotic genomes possess an elaborate and dynamic higher-order structure within the limiting confines of the cell nucleus. Knowledge of the physical principles and the molecular machinery that govern the 3D organization of this structure and its regulation are key to understanding the relationship between genome structure and function. Elegant microscopy and chromosome conformation capture techniques supported by analysis based on polymer models are important steps in this direction. Here, we review results from these efforts and provide some additional insights that elucidate the relationship between structure and function at different hierarchical levels of genome organization.
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Affiliation(s)
- Balaji Vs Iyer
- Department of NanoEngineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0448, USA.
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Tark-Dame M, van Driel R, Heermann DW. Chromatin folding – from biology to polymer models and back. J Cell Sci 2011; 124:839-45. [DOI: 10.1242/jcs.077628] [Citation(s) in RCA: 80] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
There is rapidly growing evidence that folding of the chromatin fibre inside the interphase nucleus has an important role in the regulation of gene expression. In particular, the formation of loops mediated by the interaction between specific regulatory elements, for instance enhancers and promoters, is crucial in gene control. Biochemical studies that were based on the chromosome conformation capture (3C) technology have confirmed that eukaryotic genomes are highly looped. Insight into the underlying principles comes from polymer models that explore the properties of the chromatin fibre inside the nucleus. Recent models indicate that chromatin looping can explain various properties of interphase chromatin, including chromatin compaction and compartmentalisation of chromosomes. Entropic effects have a key role in these models. In this Commentary, we give an overview of the recent conjunction of ideas regarding chromatin looping in the fields of biology and polymer physics. Starting from simple linear polymer models, we explain how specific folding properties emerge upon introducing loops and how this explains a variety of experimental observations. We also discuss different polymer models that describe chromatin folding and compare them to experimental data. Experimentally testing the predictions of such polymer models and their subsequent improvement on the basis of measurements provides a solid framework to begin to understand how our genome is folded and how folding relates to function.
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Affiliation(s)
- Mariliis Tark-Dame
- Swammerdam Institute for Life Sciences, University of Amsterdam, PO Box 94215, 1090GE Amsterdam, The Netherlands
| | - Roel van Driel
- Swammerdam Institute for Life Sciences, University of Amsterdam, PO Box 94215, 1090GE Amsterdam, The Netherlands
| | - Dieter W. Heermann
- Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany
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Bohn M, Heermann DW. Repulsive forces between looping chromosomes induce entropy-driven segregation. PLoS One 2011; 6:e14428. [PMID: 21245914 PMCID: PMC3014947 DOI: 10.1371/journal.pone.0014428] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2010] [Accepted: 11/16/2010] [Indexed: 12/27/2022] Open
Abstract
One striking feature of chromatin organization is that chromosomes are compartmentalized into distinct territories during interphase, the degree of intermingling being much smaller than expected for linear chains. A growing body of evidence indicates that the formation of loops plays a dominant role in transcriptional regulation as well as the entropic organization of interphase chromosomes. Using a recently proposed model, we quantitatively determine the entropic forces between chromosomes. This Dynamic Loop Model assumes that loops form solely on the basis of diffusional motion without invoking other long-range interactions. We find that introducing loops into the structure of chromatin results in a multi-fold higher repulsion between chromosomes compared to linear chains. Strong effects are observed for the tendency of a non-random alignment; the overlap volume between chromosomes decays fast with increasing loop number. Our results suggest that the formation of chromatin loops imposes both compartmentalization as well as order on the system without requiring additional energy-consuming processes.
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Affiliation(s)
- Manfred Bohn
- Institute for Theoretical Physics, Heidelberg University, Heidelberg, Germany.
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50
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Diffusion-driven looping provides a consistent framework for chromatin organization. PLoS One 2010; 5:e12218. [PMID: 20811620 PMCID: PMC2928267 DOI: 10.1371/journal.pone.0012218] [Citation(s) in RCA: 120] [Impact Index Per Article: 8.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2010] [Accepted: 07/02/2010] [Indexed: 01/20/2023] Open
Abstract
Chromatin folding inside the interphase nucleus of eukaryotic cells is done on multiple scales of length and time. Despite recent progress in understanding the folding motifs of chromatin, the higher-order structure still remains elusive. Various experimental studies reveal a tight connection between genome folding and function. Chromosomes fold into a confined subspace of the nucleus and form distinct territories. Chromatin looping seems to play a dominant role both in transcriptional regulation as well as in chromatin organization and has been assumed to be mediated by long-range interactions in many polymer models. However, it remains a crucial question which mechanisms are necessary to make two chromatin regions become co-located, i.e. have them in spatial proximity. We demonstrate that the formation of loops can be accomplished solely on the basis of diffusional motion. The probabilistic nature of temporary contacts mimics the effects of proteins, e.g. transcription factors, in the solvent. We establish testable quantitative predictions by deriving scale-independent measures for comparison to experimental data. In this Dynamic Loop (DL) model, the co-localization probability of distant elements is strongly increased compared to linear non-looping chains. The model correctly describes folding into a confined space as well as the experimentally observed cell-to-cell variation. Most importantly, at biological densities, model chromosomes occupy distinct territories showing less inter-chromosomal contacts than linear chains. Thus, dynamic diffusion-based looping, i.e. gene co-localization, provides a consistent framework for chromatin organization in eukaryotic interphase nuclei.
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