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Singh S, Granek R. Active fractal networks with stochastic force monopoles and force dipoles: Application to subdiffusion of chromosomal loci. CHAOS (WOODBURY, N.Y.) 2024; 34:113107. [PMID: 39485136 DOI: 10.1063/5.0227341] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2024] [Accepted: 10/10/2024] [Indexed: 11/03/2024]
Abstract
Motivated by the well-known fractal packing of chromatin, we study the Rouse-type dynamics of elastic fractal networks with embedded, stochastically driven, active force monopoles and force dipoles that are temporally correlated. We compute, analytically-using a general theoretical framework-and via Langevin dynamics simulations, the mean square displacement (MSD) of a network bead. Following a short-time superdiffusive behavior, force monopoles yield anomalous subdiffusion with an exponent identical to that of the thermal system. In contrast, force dipoles do not induce subdiffusion, and the early superdiffusive MSD crosses over to a relatively small, system-size-independent saturation value. In addition, we find that force dipoles may lead to "crawling" rotational motion of the whole network, reminiscent of that found for triangular micro-swimmers and consistent with general theories of the rotation of deformable bodies. Moreover, force dipoles lead to network collapse beyond a critical force strength, which persists with increasing system size, signifying a true first-order dynamical phase transition. We apply our results to the motion of chromosomal loci in bacteria and yeast cells' chromatin, where anomalous sub-diffusion, MSD∼tν with ν≃0.4, was found in both normal and cells depleted of adenosine triphosphate (ATP), albeit with different apparent diffusion coefficients. We show that the combination of thermal, monopolar, and dipolar forces in chromatin is typically dominated by the active monopolar and thermal forces, explaining the observed normal cells vs the ATP-depleted cells behavior.
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Affiliation(s)
- Sadhana Singh
- Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
| | - Rony Granek
- Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
- Department of Physics, and Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
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Lebedev DV, Egorov VV, Shvetsov AV, Zabrodskaya YA, Isaev-Ivanov VV, Konevega AL. Neutron Scattering Techniques and Complementary Methods for Structural and Functional Studies of Biological Macromolecules and Large Macromolecular Complexes. CRYSTALLOGR REP+ 2021. [DOI: 10.1134/s1063774521020103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
Abstract
The review describes the application of small-angle scattering (SAS) of neutrons and complementary methods to study the structures of biomacromolecules. Here we cover SAS techniques, such as the contrast variation, the neutron spin-echo, and the solution of direct and inverse problems of three-dimensional reconstruction of the structures of macromolecules from SAS spectra by means of molecular modeling. A special section is devoted to specific objects of research, such as supramolecular complexes, influenza virus nucleoprotein, and chromatin.
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Kuklin AI, Ivankov OI, Rogachev AV, Soloviov DV, Islamov AK, Skoi VV, Kovalev YS, Vlasov AV, Ryzykau YL, Soloviev AG, Kucerka N, Gordeliy VI. Small-Angle Neutron Scattering at the Pulsed Reactor IBR-2: Current Status and Prospects. CRYSTALLOGR REP+ 2021. [DOI: 10.1134/s1063774521020085] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
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4
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Metze K, Adam R, Florindo JB. The fractal dimension of chromatin - a potential molecular marker for carcinogenesis, tumor progression and prognosis. Expert Rev Mol Diagn 2019; 19:299-312. [DOI: 10.1080/14737159.2019.1597707] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Affiliation(s)
- Konradin Metze
- Department of Pathology, Faculty of Medical Sciences, State University of Campinas (UNICAMP), Campinas, Brazil
| | - Randall Adam
- Department of Pathology, Faculty of Medical Sciences, State University of Campinas (UNICAMP), Campinas, Brazil
| | - João Batista Florindo
- Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computing, State University of Campinas, Campinas, Brazil
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Woringer M, Darzacq X. Protein motion in the nucleus: from anomalous diffusion to weak interactions. Biochem Soc Trans 2018; 46:945-956. [PMID: 30065106 PMCID: PMC6103463 DOI: 10.1042/bst20170310] [Citation(s) in RCA: 38] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2018] [Revised: 05/02/2018] [Accepted: 05/14/2018] [Indexed: 12/12/2022]
Abstract
Understanding how transcription factors (TFs) regulate mammalian gene expression in space and time is a central topic in biology. To activate a gene, a TF has first to diffuse in the available space of the nucleus until it reaches a target DNA sequence or protein (target site). This eventually results in the recruitment of the whole transcriptional machinery. All these processes take place in the mammalian nucleoplasm, a highly organized and dynamic environment, in which some complexes transiently assemble and break apart, whereas others appear more stable. This diversity of dynamic behaviors arises from the number of biomolecules that make up the nucleoplasm and their pairwise interactions. Indeed, interactions energies that span several orders of magnitude, from covalent bounds to transient and dynamic interactions, can shape nuclear landscapes. Thus, the nuclear environment determines how frequently and how fast a TF contacts its target site, and it indirectly regulates gene expression. How exactly transient interactions are involved in the regulation of TF diffusion is unclear, but are reflected by live cell imaging techniques, including single-particle tracking (SPT). Overall, the macroscopic result of these microscopic interactions is almost always anomalous diffusion, a phenomenon widely studied and modeled. Here, we review the connections between the anomalous diffusion of a TF observed by SPT and the microscopic organization of the nucleus, including recently described topologically associated domains and dynamic phase-separated compartments. We propose that anomalous diffusion found in SPT data result from weak and transient interactions with dynamic nuclear substructures, and that SPT data analysis would benefit from a better description of such structures.
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Affiliation(s)
- Maxime Woringer
- Department of Molecular and Cell Biology, Li Ka Shing Center for Biomedical and Health Sciences, CIRM Center of Excellence, University of California, Berkeley, CA 94720, U.S.A.
- Unité Imagerie et Modélisation, Institut Pasteur, 25 rue du Docteur Roux, 75015 Paris, France
- Sorbonne Universités, CNRS, F-75005 Paris, France
| | - Xavier Darzacq
- Department of Molecular and Cell Biology, Li Ka Shing Center for Biomedical and Health Sciences, CIRM Center of Excellence, University of California, Berkeley, CA 94720, U.S.A.
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Cherkezyan L, Zhang D, Subramanian H, Capoglu I, Taflove A, Backman V. Review of interferometric spectroscopy of scattered light for the quantification of subdiffractional structure of biomaterials. JOURNAL OF BIOMEDICAL OPTICS 2017; 22:30901. [PMID: 28290596 PMCID: PMC5348632 DOI: 10.1117/1.jbo.22.3.030901] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/13/2016] [Accepted: 02/20/2017] [Indexed: 05/19/2023]
Abstract
Optical microscopy is the staple technique in the examination of microscale material structure in basic science and applied research. Of particular importance to biology and medical research is the visualization and analysis of the weakly scattering biological cells and tissues. However, the resolution of optical microscopy is limited to ? 200 ?? nm due to the fundamental diffraction limit of light. We review one distinct form of the spectroscopic microscopy (SM) method, which is founded in the analysis of the second-order spectral statistic of a wavelength-dependent bright-field far-zone reflected-light microscope image. This technique offers clear advantages for biomedical research by alleviating two notorious challenges of the optical evaluation of biomaterials: the diffraction limit of light and the lack of sensitivity to biological, optically transparent structures. Addressing the first issue, it has been shown that the spectroscopic content of a bright-field microscope image quantifies structural composition of samples at arbitrarily small length scales, limited by the signal-to-noise ratio of the detector, without necessarily resolving them. Addressing the second issue, SM utilizes a reference arm, sample arm interference scheme, which allows us to elevate the weak scattering signal from biomaterials above the instrument noise floor.
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Affiliation(s)
- Lusik Cherkezyan
- Northwestern University, Department of Biomedical Engineering, Evanston, Illinois, United States
| | - Di Zhang
- Northwestern University, Department of Biomedical Engineering, Evanston, Illinois, United States
| | - Hariharan Subramanian
- Northwestern University, Department of Biomedical Engineering, Evanston, Illinois, United States
| | - Ilker Capoglu
- Northwestern University, Department of Biomedical Engineering, Evanston, Illinois, United States
| | - Allen Taflove
- Northwestern University, Department of Electrical Engineering, Evanston, Illinois, United States
| | - Vadim Backman
- Northwestern University, Department of Biomedical Engineering, Evanston, Illinois, United States
- Address all correspondence to: Vadim Backman, E-mail:
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Yi J, Stypula-Cyrus Y, Blaha CS, Roy HK, Backman V. Fractal Characterization of Chromatin Decompaction in Live Cells. Biophys J 2016; 109:2218-26. [PMID: 26636933 DOI: 10.1016/j.bpj.2015.10.014] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2015] [Revised: 09/15/2015] [Accepted: 10/08/2015] [Indexed: 10/22/2022] Open
Abstract
Chromatin organization has a fundamental impact on the whole spectrum of genomic functions. Quantitative characterization of the chromatin structure, particularly at submicron length scales where chromatin fractal globules are formed, is critical to understanding this structure-function relationship. Such analysis is currently challenging due to the diffraction-limited resolution of conventional light microscopy. We herein present an optical approach termed inverse spectroscopic optical coherence tomography to characterize the mass density fractality of chromatin, and we apply the technique to observe chromatin decompaction in live cells. The technique makes it possible for the first time, to our knowledge, to sense intracellular morphology with length-scale sensitivity from ∼30 to 450 nm, thus primarily probing the higher-order chromatin structure, without resolving the actual structures. We used chromatin decompaction due to inhibition of histone deacytelases and measured the subsequent changes in the fractal dimension of the intracellular structure. The results were confirmed by transmission electron microscopy and confocal fluorescence microscopy.
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Affiliation(s)
- Ji Yi
- Department of Biomedical Engineering, Northwestern University, Evanston, Illinois; Boston Medical Center, Department of Medicine, Boston University, Boston, Massachusetts
| | | | - Catherine S Blaha
- Department of Biomedical Engineering, Northwestern University, Evanston, Illinois
| | - Hemant K Roy
- Boston Medical Center, Department of Medicine, Boston University, Boston, Massachusetts
| | - Vadim Backman
- Department of Biomedical Engineering, Northwestern University, Evanston, Illinois.
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Cherkezyan L, Stypula-Cyrus Y, Subramanian H, White C, Dela Cruz M, Wali RK, Goldberg MJ, Bianchi LK, Roy HK, Backman V. Nanoscale changes in chromatin organization represent the initial steps of tumorigenesis: a transmission electron microscopy study. BMC Cancer 2014; 14:189. [PMID: 24629088 PMCID: PMC3995586 DOI: 10.1186/1471-2407-14-189] [Citation(s) in RCA: 55] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2013] [Accepted: 03/05/2014] [Indexed: 12/16/2022] Open
Abstract
Background Nuclear alterations are a well-known manifestation of cancer. However, little is known about the early, microscopically-undetectable stages of malignant transformation. Based on the phenomenon of field cancerization, the tissue in the field of a tumor can be used to identify and study the initiating events of carcinogenesis. Morphological changes in nuclear organization have been implicated in the field of colorectal cancer (CRC), and we hypothesize that characterization of chromatin alterations in the early stages of CRC will provide insight into cancer progression, as well as serve as a biomarker for early detection, risk stratification and prevention. Methods For this study we used transmission electron microscopy (TEM) images of nuclei harboring pre-neoplastic CRC alterations in two models: a carcinogen-treated animal model of early CRC, and microscopically normal-appearing tissue in the field of human CRC. We quantify the chromatin arrangement using approaches with two levels of complexity: 1) binary, where chromatin is separated into areas of dense heterochromatin and loose euchromatin, and 2) grey-scale, where the statistics of continuous mass-density distribution within the nucleus is quantified by its spatial correlation function. Results We established an increase in heterochromatin content and clump size, as well as a loss of its characteristic peripheral positioning in microscopically normal pre-neoplastic cell nuclei. Additionally, the analysis of chromatin density showed that its spatial distribution is altered from a fractal to a stretched exponential. Conclusions We characterize quantitatively and qualitatively the nanoscale structural alterations preceding cancer development, which may allow for the establishment of promising new biomarkers for cancer risk stratification and diagnosis. The findings of this study confirm that ultrastructural changes of chromatin in field carcinogenesis represent early neoplastic events leading to the development of well-documented, microscopically detectable hallmarks of cancer.
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Affiliation(s)
| | | | | | | | | | | | | | | | | | - Vadim Backman
- Department of Biomedical Engineering, Northwestern University, Evanston, Illinois 60208, USA.
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Relevance and limitations of crowding, fractal, and polymer models to describe nuclear architecture. INTERNATIONAL REVIEW OF CELL AND MOLECULAR BIOLOGY 2014; 307:443-79. [PMID: 24380602 DOI: 10.1016/b978-0-12-800046-5.00013-8] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/19/2023]
Abstract
Chromosome architecture plays an essential role for all nuclear functions, and its physical description has attracted considerable interest over the last few years among the biophysics community. These researches at the frontiers of physics and biology have been stimulated by the demand for quantitative analysis of molecular biology experiments, which provide comprehensive data on chromosome folding, or of live cell imaging experiments that enable researchers to visualize selected chromosome loci in living or fixed cells. In this review our goal is to survey several nonmutually exclusive models that have emerged to describe the folding of DNA in the nucleus, the dynamics of proteins in the nucleoplasm, or the movements of chromosome loci. We focus on three classes of models, namely molecular crowding, fractal, and polymer models, draw comparisons, and discuss their merits and limitations in the context of chromosome structure and dynamics, or nuclear protein navigation in the nucleoplasm. Finally, we identify future challenges in the roadmap to a unified model of the nuclear environment.
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Abstract
Fractal characteristics of chromatin, revealed by light or electron microscopy, have been reported during the last 20 years. Fractal features can easily be estimated in digitalized microscopic images and are helpful for diagnosis and prognosis of neoplasias. During carcinogenesis and tumor progression, an increase of the fractal dimension (FD) of stained nuclei has been shown in intraepithelial lesions of the uterine cervix and the anus, oral squamous cell carcinomas or adenocarcinomas of the pancreas. Furthermore, an increased FD of chromatin is an unfavorable prognostic factor in squamous cell carcinomas of the oral cavity and the larynx, melanomas and multiple myelomas. High goodness-of-fit of the regression line of the FD is a favorable prognostic factor in acute leukemias and multiple myelomas. The nucleus has fractal and power-law organization in several different levels, which might in part be interrelated. Some possible relations between modifications of the chromatin organization during carcinogenesis and tumor progression and an increase of the FD of stained chromatin are suggested. Furthermore, increased complexity of the chromatin structure, loss of heterochromatin and a less-perfect self-organization of the nucleus in aggressive neoplasias are discussed.
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Affiliation(s)
- Konradin Metze
- Department of Pathology, Faculty of Medical Sciences Research Group, 'Analytical Cellular Pathology' and National Institute of Photonics Applied to Cell Biology, University of Campinas, Campinas, Brazil +55 19 32893897 kmetze.at.fcm.unicamp.br
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Bancaud A, Lavelle C, Huet S, Ellenberg J. A fractal model for nuclear organization: current evidence and biological implications. Nucleic Acids Res 2012; 40:8783-92. [PMID: 22790985 PMCID: PMC3467038 DOI: 10.1093/nar/gks586] [Citation(s) in RCA: 74] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/07/2023] Open
Abstract
Chromatin is a multiscale structure on which transcription, replication, recombination and repair of the genome occur. To fully understand any of these processes at the molecular level under physiological conditions, a clear picture of the polymorphic and dynamic organization of chromatin in the eukaryotic nucleus is required. Recent studies indicate that a fractal model of chromatin architecture is consistent with both the reaction-diffusion properties of chromatin interacting proteins and with structural data on chromatin interminglement. In this study, we provide a critical overview of the experimental evidence that support a fractal organization of chromatin. On this basis, we discuss the functional implications of a fractal chromatin model for biological processes and propose future experiments to probe chromatin organization further that should allow to strongly support or invalidate the fractal hypothesis.
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Affiliation(s)
- Aurélien Bancaud
- CNRS, LAAS, 7 avenue du colonel Roche, Toulouse F-31077, France.
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Soloviev AG, Murugova TN, Islamov AH, Kuklin AI. FITTER. The package for fitting a chosen theoretical multi-parameter function through a set of data points. Application to experimental data of the YuMO spectrometer. ACTA ACUST UNITED AC 2012. [DOI: 10.1088/1742-6596/351/1/012027] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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13
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Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons. Symmetry (Basel) 2011. [DOI: 10.3390/sym3040750] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
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Cherny AY, Anitas EM, Osipov VA, Kuklin AI. Deterministic fractals: extracting additional information from small-angle scattering data. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:036203. [PMID: 22060471 DOI: 10.1103/physreve.84.036203] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2011] [Indexed: 05/31/2023]
Abstract
The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.
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Affiliation(s)
- A Yu Cherny
- Joint Institute for Nuclear Research, Dubna 141980, Moscow region, Russia.
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15
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Meyer B, Chevalier C, Voituriez R, Bénichou O. Universality classes of first-passage-time distribution in confined media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051116. [PMID: 21728499 DOI: 10.1103/physreve.83.051116] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2010] [Indexed: 05/31/2023]
Abstract
We study the first-passage time (FPT) distribution to a target site for a random walker evolving in a bounded domain. We show that in the limit of large volume of the confining domain, this distribution falls into universality classes indexed by the walk dimension d(w) and the fractal dimension d(f) of the medium, which have been recently identified previously [Bénichou et al., Nat. Chem. 2, 472 (2010)]. We present in this paper a complete derivation of these universal distributions, discuss extensively the range of applicability of the results, and extend the method to continuous-time random walks. This analysis puts forward the importance of the geometry, and in particular the position of the starting point, in first-passage statistics. Analytical results are validated by numerical simulations, applied to various models of transport in disordered media, which illustrate the universality classes of the FPT distribution.
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Affiliation(s)
- B Meyer
- Laboratoire de Physique Théorique de la matière Condensée (UMR 7600), Université Paris 6, Paris, France
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Bénichou O, Chevalier C, Meyer B, Voituriez R. Facilitated diffusion of proteins on chromatin. PHYSICAL REVIEW LETTERS 2011; 106:038102. [PMID: 21405302 DOI: 10.1103/physrevlett.106.038102] [Citation(s) in RCA: 46] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2010] [Indexed: 05/30/2023]
Abstract
We present a theoretical model of facilitated diffusion of proteins in the cell nucleus. This model, which takes into account the successive binding and unbinding events of proteins to DNA, relies on a fractal description of the chromatin which has been recently evidenced experimentally. Facilitated diffusion is shown quantitatively to be favorable for a fast localization of a target locus by a transcription factor and even to enable the minimization of the search time by tuning the affinity of the transcription factor with DNA. This study shows the robustness of the facilitated diffusion mechanism, invoked so far only for linear conformations of DNA.
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Affiliation(s)
- O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée CNRS-UPMC, 4 Place Jussieu, 75255 Paris Cedex, France
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Cherny AY, Anitas EM, Kuklin AI, Balasoiu M, Osipov VA. Scattering from generalized Cantor fractals. J Appl Crystallogr 2010. [DOI: 10.1107/s0021889810014184] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
Abstract
A fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set, is considered. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from zero to one in one dimension and from zero to three in three dimensions. The intensity profile of small-angle scattering from the generalized Cantor fractal in three dimensions is calculated. The system is generated by a set of iterative rules, each iteration corresponding to a certain fractal generation. Small-angle scattering is considered from monodispersive sets, which are randomly oriented and placed. The scattering intensities represent minima and maxima superimposed on a power law decay, with the exponent equal to the fractal dimension of the scatterer, but the minima and maxima are damped with increasing polydispersity of the fractal sets. It is shown that, for a finite generation of the fractal, the exponent changes at sufficiently large wave vectors from the fractal dimension to four, the value given by the usual Porod law. It is shown that the number of particles of which the fractal is composed can be estimated from the value of the boundary between the fractal and Porod regions. The radius of gyration of the fractal is calculated analytically.
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Abstract
It has long been appreciated that the transport properties of molecules can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target-the first-passage time (FPT). Determining the FPT distribution in realistic confined geometries has until now, however, seemed intractable. Here, we calculate this FPT distribution analytically and show that transport processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes. Beyond the theoretical aspect, this result changes our views on standard reaction kinetics and we introduce the concept of 'geometry-controlled kinetics'. More precisely, we argue that geometry-and in particular the initial distance between reactants in 'compact' systems-can become a key parameter. These findings could help explain the crucial role that the spatial organization of genes has in transcription kinetics, and more generally the impact of geometry on diffusion-limited reactions.
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Montel F, Menoni H, Castelnovo M, Bednar J, Dimitrov S, Angelov D, Faivre-Moskalenko C. The dynamics of individual nucleosomes controls the chromatin condensation pathway: direct atomic force microscopy visualization of variant chromatin. Biophys J 2009; 97:544-53. [PMID: 19619469 DOI: 10.1016/j.bpj.2009.04.042] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2009] [Revised: 04/20/2009] [Accepted: 04/21/2009] [Indexed: 10/20/2022] Open
Abstract
Chromatin organization and dynamics is studied at scales ranging from single nucleosome to nucleosomal array by using a unique combination of biochemical assays, single molecule imaging technique, and numerical modeling. We show that a subtle modification in the nucleosome structure induced by the histone variant H2A.Bbd drastically modifies the higher order organization of the nucleosomal arrays. Importantly, as directly visualized by atomic force microscopy, conventional H2A nucleosomal arrays exhibit specific local organization, in contrast to H2A.Bbd arrays, which show "beads on a string" structure. The combination of systematic image analysis and theoretical modeling allows a quantitative description relating the observed gross structural changes of the arrays to their local organization. Our results suggest strongly that higher-order organization of H1-free nucleosomal arrays is determined mainly by the fluctuation properties of individual nucleosomes. Moreover, numerical simulations suggest the existence of attractive interactions between nucleosomes to provide the degree of compaction observed for conventional chromatin fibers.
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Affiliation(s)
- Fabien Montel
- Université de Lyon, Laboratoire Joliot-Curie, CNRS USR 3010, Ecole Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France
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