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8
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0021810948
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(1985)
Science
, vol.228
, pp. 953
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Bernardi, G.1
Olofsson, B.2
Filipski, J.3
Zerial, M.4
Salinas, J.5
Cuny, F.6
Meunier Rotival, M.7
Rodier, F.8
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13
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84926903890
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Variants of this rule permit studying patchiness arising from other forms of nucleotide heterogeneity. For example, adenine heterogeneity can be studied by forming a walk with 3 steps up for each adenine and 1 step down for a nonadenine.
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14
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84926885259
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To be precise, we randomly divide a sequence of length N into k subregions and assign a compositional bias (purine pyrimidine concentration) to each subregion from a Gaussian distribution of width σ and mean 0.5. The sequences thus generated are patchy by construction and the patch size distribution displays a characteristic length given by N/k. In the case of Fig. 1(a), we divided the whole sequence (105 bp) into 40 subregions with length ranging from 68 bp to 11,679 bp. The characteristic length is 105/40=2500 bp. The width of the bias distribution is σ = 0.1. As expected, the landscape of the typical control sequence from this biased random model is highly heterogeneous.
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15
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0000555231
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To generate long range correlated variables, we start with random uncorrelated variables, taken from a uniform distribution. We then form the Fourier transform and multiply by a power in q space. Finally, we Fourier transform back to real space and thereby obtain long range correlated variables with a correlation function given by a power law. This method was proposed in
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(1988)
Phys. Rev. Lett.
, vol.61
, pp. 1438
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Havlin, S.1
Selinger, R.B.2
Schwartz, M.3
Stanley, H.E.4
Bunde, A.5
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84926885258
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A more general procedure is to apply the same method except, instead of incrementing the box position by a distance ell between each measurement, to increment the position by a distance ell/S, where S is a sliding parameter that can be chosen to give optimal statistics.
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84926903889
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The precise definition of the average / variance on length scale ell is Fd2( ell) equiv 1 over N sumn=1N y ell2(n).
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18
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0027605271
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Note that if α < 1/2, there exist long range correlations that reflect an alternation of different nucleotide types, while if α > 1/2 the long range correlations reflect a persistence of the same nucleotide type. For the ideal case of a random sequence, α=0.5. However, since the length of the sequences is not infinite, we must take into account the statistical fluctuations for a finite size sample. A discussion of the effects of the finite length of DNA sequences analyzed appears in
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(1993)
Phys. Rev. E
, vol.47
, pp. 3730
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Peng, C.K.1
Buldyrev, S.V.2
Goldberger, A.L.3
Havlin, S.4
Simons, M.5
Stanley, H.E.6
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19
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84926903888
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To measure the exponent α, we calculate the slope of log10 Fd( ell) versus log10 ell for ell geq 4, since for very small ell we expect deviation (introduced by the detrending algorithm) from power law behavior. Numerical studies of various types of control sequences show that this deviation is negligible (within the error of statistical fluctuations) if we evaluate the exponent from ell geq4.
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20
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84926923068
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We have tested the DFA method extensively on DNA sequences from genomic and cDNA sequences which we and others had previously studied by other methods, as well as correlated and uncorrelated control sequences. In all cases, the DFA method gave results as convincing as the examples in the text.
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24
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1842314489
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Europhys. Lett. 23, 373 (1993).
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(1993)
Europhys. Lett.
, vol.23
, pp. 373
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