The important result of the present study is that the observed

What distinguishes one type of cell from another and one organism from another is the protein which it contains. And it is DNA which dictates to the cell how many and what types of protein it shall make. Twenty different chemicals called

The genes of higher organisms are seldom 'recorded' in the chromosomes intact, but are scattered in fragmentary fashion along a stretch of DNA, broken up by chunks of DNA which seem at first sight to carry no message at all. All the useless or "junk" DNA, the intervening sequences are known as

Historically, Watson and Crick (1953) put together all the experimental data concerning DNA and decided that the only structure that fitted all the facts was the double helix and postulated that DNA is composed of two ribbonlike "backbones" composed of alternating deoxyribose and phosphate molecules. They surmised that nucleotides extend out from the backbone chains and that the

A summary of recent results relating to long-range correlation (LRC) in DNA sequences is given in the following. Based on spectral analyses, Li

The long-range correlation does not necessarily imply a deviation from Gaussianity. For example, the fractional Brownian motion which has Gaussian statistics shows an inverse power law spectrum. According to Allegrini

In visualizing very long DNA sequences, including the complete genomes of several bacteria, yeast and segments of human genes, it is seen that

Continuous periodogram power spectral analyses of the frequency distribution of bases A, C, G, T in Drosophila DNA base sequence agree with model prediction, namely, the power spectra follow the universal inverse power law form of the statistical normal distribution. The geometrical distribution of the DNA bases therefore exhibit

(1)

Since the large
eddy is but the average of the enclosed smaller eddies, the eddy energy
spectrum follows the statistical normal distribution according to the *Central
Limit Theorem* (Ruhla, 1992). Therefore, the variance represents the
probability densities. Such a result that the additive amplitudes of the
eddies, when squared, represent the probabilities is an observed feature
of the subatomic dynamics of quantum systems such as the electron or photon
(Maddox 1988a, 1993; Rae, 1988). The *fractal* space-time fluctuations
exhibited by dynamical systems are signatures of quantumlike mechanics.
The cell dynamical system model provides a unique quantification for the
apparently chaotic or unpredictable nature of such *fractal* fluctuations
( Selvam and Fadnavis, 1998). The model predictions for quantumlike chaos
of dynamical systems are as follows.

(a) The observed *fractal* fluctuations of dynamical
systems are generated by an overall logarithmic spiral trajectory with
the quasiperiodic *Penrose tiling pattern* (Nelson, 1986; Selvam and
Fadnavis, 1998) for the internal structure.

(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories will reveal a continuum of periodicities with progressive increase in phase.

(c) The broadband power spectrum will have embedded
dominant wave-bands, the bandwidth increasing with period length. The peak
periods (or length* *scales) *E _{n}* in the dominant
wavebands will be given by the relation

*E _{n}=T_{s}(2+*t
)t

(2)

where t
is the *golden mean* equal to *(1+*Ö
5)/2 [@ 1.618] and
*T _{s}*
, the primary perturbation length scale. Considering the most representative
example of turbulent fluid flows, namely, atmospheric flows, Ghil (1994)
reports that the most striking feature in climate variability on all time
scales is the presence of sharp peaks superimposed on a continuous background.

The model predicted periodicities
(or length scales) in terms of the primary perturbation length scale units
are *2.2*,
*3.6*,
*5.8*,
*9.5*,
*15.3*,
*24.8*, *40.1*,
*64.9*,
*105.0
*respectively
for values of *n* ranging from
*-1 to 7*. Periodicities (or length
scales) close to model predicted have been reported in weather and climate
variability (Burroughs, 1992; Kane, 1996), prime number distribution
(Selvam, 2001a), Riemann zeta zeros (non-trivial) distribution (Selvam,
2001b).

Sornette
*et
al*. (1995) also conclude that the observed power law represents
structures similar to '*Elliott waves*' of technical analysis first
introduced in the 1930s. It describes the time series of a stock price
as made of different
*waves*, these *waves* are in relation
to each other through the *Fibonacci* series. Sornette
*et al*.
(1995) speculate that '*Elliott waves*' could be a signature of an
underlying critical structure of the stock market.

(d) The length scale ratio *r/R* also represents
the increment
*d*q
in phase angle q
(Equation 1 ). Therefore the phase angle* *q
represents the variance. Hence, when the logarithmic spiral is resolved
as an eddy continuum in conventional spectral analysis, the increment in
wavelength is concomitant with increase in phase (Selvam and Fadnavis,
1998). Such a result that increments in wavelength and phase angle are
related is observed in quantum systems and has been named *'Berry's phase'*
(Berry 1988; Maddox 1988b; Simon *et al*., 1988; Anandan, 1992). The
relationship of angular turning of the spiral to intensity of fluctuations
is seen in the tight coiling of the hurricane spiral cloud systems.

The overall logarithmic spiral flow structure is given by the relation

(3)

where the constant *k*** **is the steady
state fractional volume dilution of large eddy by inherent turbulent eddy
fluctuations . The constant *k*** **is equal to *1/*t^{2}*(**@0.382*)
and is identified as the *universal constant* for deterministic chaos
in fluid flows (Selvam and Fadnavis, 1998).The steady state emergence of
*fractal*
structures is therefore equal to

*1/k @2.62*

(4)

The model predicted logarithmic
wind profile relationship such as Equation 3 is a long-established (observational)
feature of atmospheric flows in the atmospheric boundary layer, the constant
*k*,
called the *Von Karman* ’s constant has the value equal to
*0.38*
as determined from observations (Wallace and Hobbs, 1977).

In Equation 3, *W*
represents the standard deviation of eddy fluctuations, since *W*
is computed as the instantaneous r.m.s. ( root mean square) eddy perturbation
amplitude with reference to the earlier step of eddy growth. For two successive
stages of eddy growth starting from primary perturbation *w _{*}*
the ratio of the standard deviations

*statistical normalized standard deviation
t=0,1,2,3, etc.*

(5)

The conventional
power spectrum plotted as the variance versus the frequency in log-log
scale will now represent the eddy probability density on logarithmic scale
versus the standard deviation of the eddy fluctuations on linear scale
since the logarithm of the eddy wavelength represents the standard deviation,
i.e., the r.m.s. value of eddy fluctuations (Equation 3). The r.m.s. value
of eddy fluctuations can be represented in terms of statistical normal
distribution as follows. A normalized standard deviation *t=0* corresponds
to cumulative percentage probability density equal to *50* for the
mean value of the distribution. Since the logarithm of the wavelength represents
the r.m.s. value of eddy fluctuations the normalized standard deviation
*t*
is defined for the eddy energy as

(6)

where *L* is the wavelength (or period)
and
*T _{50}* is the wavelength (or period) up to which the
cumulative percentage contribution to total variance is equal to

The periodicities (or length scales)
*T _{50}*
and

The power spectrum, when plotted
as normalised standard deviation *t* versus cumulative percentage
contribution to total variance represents the statistical normal distribution
(Equation 6), i.e., the variance represents the probability density. The
normalised standard deviation values *t* corresponding to cumulative
percentage probability densities *P* equal to
*50* and *95*
respectively are equal to *0* and *2* from statistical normal
distribution characteristics. Since *t* represents the eddy growth
step *n* (Equation 5) the dominant periodicities (or length scales)
*T _{50}*
and

*T _{50} = (2+t
)t^{0 }@
3.6 unit length segment of 50 bases*

(7)

(8)

The above model
predictions are applicable to all real world and computed model dynamical
systems. Continuous periodogram power spectral analyses of number frequency
(per 50 bases) of occurrence of bases A, C, G, T in Drosophila DNA base
sequence at different locations along its length give results in agreement
with the above model predictions.

Figure 1: Representative example for *fractal*
fluctuations exhibited by frequency distribution of base A in 10 to 4500
data sets

The cumulative frequency
of occurrence *p _{j}* of base (A, C, G or T) for class intervals

The cumulative percentage
frequency of occurrence *p _{c}* of base (A, C, G or T) for
class intervals

it is seen that the length scale ratio
*r/R*
(or frequency ratio) represents the variance spectrum (*W ^{2}/w_{*}^{2}*)
and therefore the cumulative frequency distribution follows closely the
cumulative normal distribution as shown in Figure 2.

Figure 2: The cumulative percentage frequency of occurrence of bases A, C, G, T in Drosophila DNA sequence follow closely the statistical normal distribution

*t _{m} = (log L_{m} /
log T_{50})-1*

The cumulative percentage
contribution to total variance, the cumulative percentage normalized phase
(normalized with respect to the total phase rotation) and the corresponding
*t _{m}*
values were computed. The power spectra were plotted as cumulative percentage
contribution to total variance versus the normalized standard deviation

Figure 3: Average
variance (continuous line) and phase (dashed line) spectra for the bases
A, C, G, T for the *50* data sets used in the study. The statistical
normal distribution ( open circles) is also shown.

Figure 4: Average wavelength class interval-wise
distribution of dominant wavebands for the four bases A, C, G, T in the
*50*
data sets (a total of *225000* bases) of Drosophila DNA base sequence
used for the study

Figure 5: Average wavelength class interval-wise
distribution of dominant significant wavebands for the four bases A, C,
G, T in the *50* data sets (a total of *225000* bases) of Drosophila
DNA base sequence used for the study

Figure 6: Average wavelength class interval-wise
distribution of dominant wavebands exhibiting *Berry's phase*
for the four bases A, C, G, T in the *50* data sets (a total of *225000*
bases) of Drosophila DNA base sequence used for the study

Considering eddy growth with overall logarithmic spiral trajectory

The eddy circulation speed is related to eddy radius as

The relative peak wavelength is given in terms of eddy circulation speed as

From Equation (1) the relationship between eddy bandwidth and peak wavelength is obtained as

(9)

A log-log plot of peak
wavelength versus bandwidth will be a straight line with a slope
(bandwidth/peak wavelength) equal to

Figure 7: Log-log plot of average values
of bandwidth versus peak wave length for the four bases A, C, G, T. The
slope (bandwidth/peak wavelength) of this graph, also plotted in the above
figure shows an approximately constant value equal to about *2*.

Figure 8

Figure 9: The periodicities T* _{50}*
up to which the cumulative percentage contribution to total variance is
equal to

Incidentally physics at the atomic scale is determined by the rules of quantum mechanics, which tells us that particles propagate like waves, and so can be described by a quantum mechanical wave function (Rae, 1999). As an immediate consequence, a particle can be in two or more states at the same time - a so-called superposition of states. This curious behaviour has been hugely successful in describing physical systems at the microscopic level. For example, under the rules of quantum mechanics two atoms sharing an electron form a chemical bond, whereas in classical theory the electron remains confined to one atom and the bond cannot form (Blatter, 2000).

Inverse power-law form for power spectra generic to

Inverse power-law form for power spectra of fluctuations in spatial distribution of bases A, C, G, T imply long-range spatial correlations, or in other words, persistence or long-term (length scale) memory of short-term fluctuations. The fine scale structure of longer length scale fluctuations carry the signature of shorter length scale fluctuations. The cumulative integration of shorter length scale fluctuations generates longer length scale fluctuations (eddy continuum) with two-way ordered energy feedback between the fluctuations of all length scales (Equation 1). The eddy continuum acts as a robust unified whole fuzzy logic network with global response to local perturbations. Increase in random noise or energy input into the short-length scale fluctuations creates intensification of fluctuations of all other length scales in the eddy continuum and may be noticed immediately in shorter length scale fluctuations. Noise is therefore a precursor to signal.

Real world examples of noise enhancing signal has been reported in electronic circuits (Brown, 1996). Man-made, urbanisation related, greenhouse gas induced global warming (enhancement of small-scale fluctuations) is now held responsible for devastating anomalous changes in regional and global weather and climate in recent years (Selvam and Fadnavis, 1998). Noise and fluctuations are at the seat of all physical phenomena. It is well known that, in linear systems, noise plays a destructive role. However, an emerging paradigm for nonlinear systems is that noise can play a constructive role—in some cases information transfer can be optimized at nonzero noise levels. Another use of noise is that its measured characteristics can tell us useful information about the system itself. Problems associated with fluctuations have been studied since 1826 (Abbott, 2001).

The apparently irregular

The quasicrystalline structure of the quasiperiodic

The important result of the present study is that the observed