1
|
West BJ, Allegrini P, Buiatti M, Grigolini P. Non-normal Statistics of DNA Sequences of Prokaryotes. J Biol Phys 2013; 26:17-25. [PMID: 23345709 DOI: 10.1023/a:1005284418550] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The √ n-rule of Schrödinger in his discussion of DNA is based onnormal statistics and equilibrium physics. Herein the kurtosis is used tomeasure the deviation from normality of the stistics of non-equilibrium DNAsequences. A pattern for this deviation from normality is identified andthis signature is found in prokaryotes. The signature is explained by atheory of DNA sequences that involves finite length DNA walks withdynamically generated long-range correlations.
Collapse
Affiliation(s)
- B J West
- Army Research Office, Research Triangle Park, NC
| | | | | | | |
Collapse
|
2
|
Balankin AS, Mena B, Martínez-González CL, Matamoros DM. Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:052101. [PMID: 23214828 DOI: 10.1103/physreve.86.052101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Revised: 07/31/2012] [Indexed: 06/01/2023]
Abstract
We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.
Collapse
Affiliation(s)
- Alexander S Balankin
- Grupo Mecánica Fractal, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
| | | | | | | |
Collapse
|
3
|
|
4
|
Scafetta N, Latora V, Grigolini P. Lévy scaling: the diffusion entropy analysis applied to DNA sequences. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:031906. [PMID: 12366151 DOI: 10.1103/physreve.66.031906] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2002] [Indexed: 05/23/2023]
Abstract
We address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis (DEA) [N. Scafetta, P. Hamilton, and P. Grigolini, Fractals 9, 193 (2001)]. This method is based on the evaluation of the Shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations, rather than on the measurement of the variance of this diffusion process, as done with the traditional methods. We compare the DEA to the traditional methods of scaling detection and prove that the DEA is the only method that always yields the correct scaling value, if the scaling condition applies. Furthermore, DEA detects the real scaling of a time series without requiring any form of detrending. We show that the joint use of DEA and variance method allows to assess whether a time series is characterized by Lévy or Gauss statistics. We apply the DEA to the study of DNA sequences and prove that their large-time scales are characterized by Lévy statistics, regardless of whether they are coding or noncoding sequences. We show that the DEA is a reliable technique and, at the same time, we use it to confirm the validity of the dynamic approach to the DNA sequences, proposed in earlier work.
Collapse
Affiliation(s)
- Nicola Scafetta
- Pratt School EE Department, Duke University, P.O. Box 90291, Durham, North Carolina 27708, USA
| | | | | |
Collapse
|
5
|
Dräger J, Klafter J. Strong anomaly in diffusion generated by iterated maps. PHYSICAL REVIEW LETTERS 2000; 84:5998-6001. [PMID: 10991108 DOI: 10.1103/physrevlett.84.5998] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2000] [Indexed: 05/23/2023]
Abstract
We investigate the diffusion generated deterministically by periodic iterated maps that are defined by x(t+1) = x(t)+ax(z)(t)exp[-(b/x(t))(z-1)], z>1. It is shown that the obtained mean squared displacement grows asymptotically as sigma(2)(t) approximately ln (1/(z-1))(t) and that the corresponding propagator decays exponentially with the scaling variable |x|/square root of (sigma(2)(t))]. This strong diffusional anomaly stems from the anomalously broad distribution of waiting times in the corresponding random walk process and leads to a behavior obtained for diffusion in the presence of random local fields. A scaling approach is introduced which connects the explicit form of the maps to the mean squared displacement.
Collapse
Affiliation(s)
- J Dräger
- School of Chemistry, Tel-Aviv University, Tel-Aviv 69978, Israel
| | | |
Collapse
|
6
|
Havlin S, Buldyrev SV, Bunde A, Goldberger AL, Peng CK, Stanley HE. Scaling in nature: from DNA through heartbeats to weather. PHYSICA A 1999; 273:46-69. [PMID: 11543356 DOI: 10.1016/s0378-4371(99)00340-4] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
The purpose of this report is to describe some recent progress in applying scaling concepts to various systems in nature. We review several systems characterized by scaling laws such as DNA sequences, heartbeat rates and weather variations. We discuss the finding that the exponent alpha quantifying the scaling in DNA in smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the scaling exponent alpha is smaller during sleep periods compared to wake periods. We also discuss the recent findings that suggest a universal scaling exponent characterizing the weather fluctuations.
Collapse
Affiliation(s)
- S Havlin
- Center for Polymer Studies and Department of Physics, Boston University, MA 02215, USA.
| | | | | | | | | | | |
Collapse
|
7
|
Havlin S, Buldyrev SV, Stanley HE, Weiss GH. Probability distribution of the interface width in surface roughening: analogy with a Levy flight. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/24/16/008] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
|
8
|
|
9
|
Havlin S, Buldyrev SV, Goldberger AL, Mantegna RN, Ossadnik SM, Peng CK, Simons M, Stanley HE. Fractals in biology and medicine. CHAOS, SOLITONS, AND FRACTALS 1995; 6:171-201. [PMID: 11539852 DOI: 10.1016/0960-0779(95)80025-c] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
Our purpose is to describe some recent progress in applying fractal concepts to systems of relevance to biology and medicine. We review several biological systems characterized by fractal geometry, with a particular focus on the long-range power-law correlations found recently in DNA sequences containing noncoding material. Furthermore, we discuss the finding that the exponent alpha quantifying these long-range correlations ("fractal complexity") is smaller for coding than for noncoding sequences. We also discuss the application of fractal scaling analysis to the dynamics of heartbeat regulation, and report the recent finding that the normal heart is characterized by long-range "anticorrelations" which are absent in the diseased heart.
Collapse
Affiliation(s)
- S Havlin
- Department of Physics, Boston University, MA, USA
| | | | | | | | | | | | | | | |
Collapse
|
10
|
Statistical and linguistic features of noncoding DNA: A heterogeneous «Complex system». ACTA ACUST UNITED AC 1994. [DOI: 10.1007/bf02462019] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
|
11
|
Stoop R. Bivariate thermodynamic formalism and anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:4913-4918. [PMID: 9961811 DOI: 10.1103/physreve.49.4913] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|
12
|
Stanley HE, Buldyrev SV, Goldberger AL, Goldberger ZD, Havlin S, Mantegna RN, Ossadnik SM, Peng CK, Simons M. Statistical mechanics in biology: how ubiquitous are long-range correlations? PHYSICA A 1994; 205:214-53. [PMID: 11541307 DOI: 10.1016/0378-4371(94)90502-9] [Citation(s) in RCA: 44] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
The purpose of this opening talk is to describe examples of recent progress in applying statistical mechanics to biological systems. We first briefly review several biological systems, and then focus on the fractal features characterized by the long-range correlations found recently in DNA sequences containing non-coding material. We discuss the evidence supporting the finding that for sequences containing only coding regions, there are no long-range correlations. We also discuss the recent finding that the exponent alpha characterizing the long-range correlations increases with evolution, and we discuss two related models, the insertion model and the insertion-deletion model, that may account for the presence of long-range correlations. Finally, we summarize the analysis of long-term data on human heartbeats (up to 10(4) heart beats) that supports the possibility that the successive increments in the cardiac beat-to-beat intervals of healthy subjects display scale-invariant, long-range "anti-correlations" (a tendency to beat faster is balanced by a tendency to beat slower later on). In contrast, for a group of subjects with severe heart disease, long-range correlations vanish. This finding suggests that the classical theory of homeostasis, according to which stable physiological processes seek to maintain "constancy," should be extended to account for this type of dynamical, far from equilibrium, behavior.
Collapse
Affiliation(s)
- H E Stanley
- Department of Physics, Boston University, MA 02215, USA
| | | | | | | | | | | | | | | | | |
Collapse
|
13
|
Schnitzer MJ. Theory of continuum random walks and application to chemotaxis. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:2553-2568. [PMID: 9960890 DOI: 10.1103/physreve.48.2553] [Citation(s) in RCA: 213] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
14
|
Zumofen G, Klafter J, Blumen A. Lévy walks and propagators in intermittent chaotic systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:2183-2186. [PMID: 9960240 DOI: 10.1103/physreve.47.2183] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
15
|
Zumofen G, Klafter J. Scale-invariant motion in intermittent chaotic systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:851-863. [PMID: 9960080 DOI: 10.1103/physreve.47.851] [Citation(s) in RCA: 310] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
|