1
|
Bogachev MI, Markelov OA, Kayumov AR, Bunde A. Superstatistical model of bacterial DNA architecture. Sci Rep 2017; 7:43034. [PMID: 28225058 PMCID: PMC5320525 DOI: 10.1038/srep43034] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2016] [Accepted: 01/18/2017] [Indexed: 12/15/2022] Open
Abstract
Understanding the physical principles that govern the complex DNA structural organization as well as its mechanical and thermodynamical properties is essential for the advancement in both life sciences and genetic engineering. Recently we have discovered that the complex DNA organization is explicitly reflected in the arrangement of nucleotides depicted by the universal power law tailed internucleotide interval distribution that is valid for complete genomes of various prokaryotic and eukaryotic organisms. Here we suggest a superstatistical model that represents a long DNA molecule by a series of consecutive ~150 bp DNA segments with the alternation of the local nucleotide composition between segments exhibiting long-range correlations. We show that the superstatistical model and the corresponding DNA generation algorithm explicitly reproduce the laws governing the empirical nucleotide arrangement properties of the DNA sequences for various global GC contents and optimal living temperatures. Finally, we discuss the relevance of our model in terms of the DNA mechanical properties. As an outlook, we focus on finding the DNA sequences that encode a given protein while simultaneously reproducing the nucleotide arrangement laws observed from empirical genomes, that may be of interest in the optimization of genetic engineering of long DNA molecules.
Collapse
Affiliation(s)
- Mikhail I. Bogachev
- Biomedical Engineering Research Centre, St. Petersburg Electrotechnical University, St. Petersburg, 197376, Russia
- Molecular Genetics of Microorganisms Lab, Institute of Fundamental Medicine and Biology, Kazan (Volga Region) Federal University, Kazan, Tatarstan, 420008, Russia
| | - Oleg A. Markelov
- Biomedical Engineering Research Centre, St. Petersburg Electrotechnical University, St. Petersburg, 197376, Russia
| | - Airat R. Kayumov
- Molecular Genetics of Microorganisms Lab, Institute of Fundamental Medicine and Biology, Kazan (Volga Region) Federal University, Kazan, Tatarstan, 420008, Russia
| | - Armin Bunde
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
| |
Collapse
|
2
|
Bénichou O, Krapivsky PL, Mejía-Monasterio C, Oshanin G. Temporal Correlations of the Running Maximum of a Brownian Trajectory. PHYSICAL REVIEW LETTERS 2016; 117:080601. [PMID: 27588841 DOI: 10.1103/physrevlett.117.080601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2016] [Indexed: 06/06/2023]
Abstract
We study the correlations between the maxima m and M of a Brownian motion (BM) on the time intervals [0,t_{1}] and [0,t_{2}], with t_{2}>t_{1}. We determine the exact forms of the distribution functions P(m,M) and P(G=M-m), and calculate the moments E{(M-m)^{k}} and the cross-moments E{m^{l}M^{k}} with arbitrary integers l and k. We show that correlations between m and M decay as sqrt[t_{1}/t_{2}] when t_{2}/t_{1}→∞, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient ρ(m,M) and the power spectrum of M_{t}, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.
Collapse
Affiliation(s)
- Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, Sorbonne Universités, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - P L Krapivsky
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Carlos Mejía-Monasterio
- Laboratory of Physical Properties, Technical University of Madrid, Avenida Complutense s/n, 28040 Madrid, Spain
| | - Gleb Oshanin
- Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, Sorbonne Universités, 4 Place Jussieu, 75252 Paris Cedex 05, France
| |
Collapse
|
3
|
Godrèche C, Majumdar SN, Schehr G. Exact Statistics of Record Increments of Random Walks and Lévy Flights. PHYSICAL REVIEW LETTERS 2016; 117:010601. [PMID: 27419552 DOI: 10.1103/physrevlett.117.010601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2016] [Indexed: 06/06/2023]
Abstract
We study the statistics of increments in record values in a time series {x_{0}=0,x_{1},x_{2},…,x_{n}} generated by the positions of a random walk (discrete time, continuous space) of duration n steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of n for large n, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability Q(n) that the record increments decrease monotonically up to step n. Remarkably, Q(n) is universal (i.e., independent of the jump distribution) for each n, decaying as Q(n)∼A/sqrt[n] for large n, with a universal amplitude A=e/sqrt[π]=1.53362….
Collapse
Affiliation(s)
- Claude Godrèche
- Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
4
|
Ramola K, Majumdar SN, Schehr G. Spatial extent of branching Brownian motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042131. [PMID: 25974462 DOI: 10.1103/physreve.91.042131] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2015] [Indexed: 06/04/2023]
Abstract
We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost (X(max)≥0) and leftmost (X(min)≤0) visited sites up to time t. At each time step the existing particles in the system either diffuse (with diffusion constant D), die (with rate a), or split into two particles (with rate b). We focus on the regime b≤a where these two extreme values X(max) and X(min) are strongly correlated. We show that at large time t, the joint probability distribution function (PDF) of the two extreme points becomes stationary P(X,Y,t→∞)→p(X,Y). Our exact results for p(X,Y) demonstrate that the correlation between X(max) and X(min) is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF p(ζ) of the (dimensionless) span ζ=(X(max)-X(min))/√[D/b], which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior p(ζ)∼1/2(1+Δ)ζ for small spans, with Δ=(a/b-1). In the critical case (Δ=0) this distribution has a nontrivial power law tail p(ζ)∼8π√[3]/ζ(3) for large spans. On the other hand, in the subcritical case (Δ>0), we show that the span distribution decays exponentially as p(ζ)∼(A(2)/2)ζexp(-√[Δ]ζ) for large spans, where A is a nontrivial function of Δ, which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between X(max) and X(min). Finally we verify our results via direct Monte Carlo simulations.
Collapse
Affiliation(s)
- Kabir Ramola
- Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626, Université Paris-Sud 11 and CNRS, Bâtiment 100, Orsay F-91405, France
| | - Satya N Majumdar
- Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626, Université Paris-Sud 11 and CNRS, Bâtiment 100, Orsay F-91405, France
| | - Grégory Schehr
- Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626, Université Paris-Sud 11 and CNRS, Bâtiment 100, Orsay F-91405, France
| |
Collapse
|
5
|
Perret A, Comtet A, Majumdar SN, Schehr G. Near-extreme statistics of Brownian motion. PHYSICAL REVIEW LETTERS 2013; 111:240601. [PMID: 24483638 DOI: 10.1103/physrevlett.111.240601] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2013] [Revised: 09/18/2013] [Indexed: 06/03/2023]
Abstract
We study the statistics of near-extreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states near the maximum, ρ(r,t), which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function of ρ(r,t) and obtain an explicit expression for the moments <[ρ(r,t)]k> for arbitrary integer k. We also study near extremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.
Collapse
Affiliation(s)
- Anthony Perret
- Université Paris-Sud-Paris 11, CNRS, LPTMS, 91405 Orsay Cedex, France
| | - Alain Comtet
- Université Paris-Sud-Paris 11, CNRS, LPTMS, 91405 Orsay Cedex, France and Université Pierre et Marie Curie-Paris 6, 75252 Paris Cedex 05, France
| | - Satya N Majumdar
- Université Paris-Sud-Paris 11, CNRS, LPTMS, 91405 Orsay Cedex, France
| | - Grégory Schehr
- Université Paris-Sud-Paris 11, CNRS, LPTMS, 91405 Orsay Cedex, France
| |
Collapse
|