Applied Mathematics, 2013, 4, 30-41
Published Online November 2013 (
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Dense Fractal Networks, Tr ends, Noises and Switches in
Homeostasis Regulation of Shannon Entropy for
Chromosomes’ Activity in Living Cells for Medical
Nikolay E. Galich
Department of Experimental Physics, Saint-Petersburg State Polytechnic University, Saint-Petersburg, Russia
Received July 7, 2013; revised August 7, 2013; accepted August 15, 2013
Copyright © 2013 Nikolay E. Galich. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We analyze correlations and patterns of oxidative activity of 3D DNA at DNA fluorescence in complete sets of chro-
mosomes in neutrophils of peripheral blood. Fluorescence of DNA is registered by method of flow cytometry with
nanometer spatial resolution. Experimental data present fluorescence of many ten thousands of cells, from different
parts of body in each population, in various blood samples. Data is presented in histograms as frequency distributions of
flashes in the dependence on their intensity. Normalized frequency distribution of information in these histograms is
used as probabilistic measure for definition of Shannon entropy. Data analysis shows that for this measure of Shannon
entropy common sum of entropy, i.e. total entropy E, for any histogram is invariant and has identical trends of changes
all values of at reduction of rank r of histogram. This invariance reflects informational homeostasis of
chromosomes activity inside cells in multi-scale networks of entropy, for varied ranks r. Shannon entropy in multi-scale
DNA networks has much more dense packing of correlations than in “small world” networks. As the rule, networks of
entropy differ by the mix of normal D < 2 and abnormal D > 2 fractal dimensions for varied ranks r, the new types of
fractal patterns and hinges for various topology (fractal dimension) at different states of health. We show that all distri-
butions of information entropy are divided on three classes, which associated in diagnostics with a good health or
dominants of autoimmune or inflammatory diseases. This classification based on switching of stability at transcritical
bifurcation in homeostasis regulation. We defined many ways for homeostasis regulation, coincidences and switching
patterns in branching sequences, the averages of Hölder for deviations of entropy from homeostasis at different states of
health, with various saturation levels the noises of entropy at activity of all chromosomes in support regulation of ho-
lnEr r
Keywords: Abnormal Fractals; DNA Activity and Shannon Information Entropy; Fractal Patterns and Fragmentation;
Informational Homeostasis; Saturations of Chromosomal Correlations; Multi-Scale Fractal Networks of
Shannon Entropy
1. Introduction
We oriented on medical diagnostics of health status
based on oxidative activity of DNA in cells for everyday
clinical practice, for given person at given time. Oxida-
tive activity of DNA is visualized in fluorescence. We
analyze experimental data on DNA fluorescence in neu-
trophils of peripheral blood at biochemical reaction of
oxidative burst [1]. This is a high sensitive method for
diagnosing many different and complex diseases, early
diagnostics of illnesses, hidden diseases. Short list of
clinical observations is given in [1-4]. Preparation and
experimental procedures, including small additions of
ethidium bromide for small volumes of blood ~2 ml, ex-
citation and measurements of fluorescence are described
in [1-6]. DNA fluorescence is described by histograms in
flow cytometry method with spatial resolution of meas-
urements at a few nanometers in the flow direction [5,6].
Statistics of fluorescence is presented in histograms for
frequency of flashes P(I) as the functions of fluorescence
intensity I for large populations of many ten thousands of
cells, living in different parts of human body. Chaotic
Brownian motion and rotation of fluorescing cells, flow-
ing through the laser beam in flow cytometry, ensured
good statistics for illustration of real 3D chromosomal
activity inside living cells. Accuracy and reproducibility
of experimental results in histograms of fluorescence
approximately equal to 2% that corresponds to the nor-
mal, usual levels of inevitable and fatal errors and fluc-
tuations of physical and biological nature [1-6]. Three
original histograms, as the illustrations of typical exam-
ples, are shown in Figure 1.
The heterogeneous fluorescence of all chromosomes in
the cells reflects simultaneously the genetic special, indi-
vidual features and immune response to the pathogenic
actions due to oxidative activity of DNA. Detailed accu-
rate statistical analysis of these histograms currently is
absent. Large-scale correlations for distributions of fluo-
rescence flashes of DNA inside living cells differ from
those that we would like to see by abnormal fractal di-
mensions and non-trivial noise level at substantially non-
Gaussian statistics [3-6]. These natural peculiarities of
immunofluorescence are often accompanied by statis-
tical instabilities of local intensity distributions [3-6], i.e.
fast exponential growth of central moments of fluores-
cence intensity. We need to develop a sequence of new
nonlinear statistical methods for data analysis of im-
munofluorescence. Standard smoothing eliminates de-
stroys and removes various peculiarities of fractal net-
works and correlations in the activity of DNA, changes
real statistics, blurs and distorts many aspects of reality.
According to tradition, now and in the latter time,
dominating sciences about DNA based on various ap-
proaches in biochemistry, structural biology, materials
science and combinatorics for lonely DNA. In this case,
(a) (b)
Figure 1. Dependence of normalized frequency distribution
of flashes P(I) on their intensity I (a); for clearest, only cen-
tral part of histogram (b). The area under the final histo-
grams of P(I) normalized to unit; rhombus points corre-
spond to bronchial asthma. Total number of flashes is N0 =
76 623; quadrate points correspond to the healthy donor.
Common number of flashes is N0 = 40 109; triangle points
correspond to the oncology disease. Common number of
flashes is N0 = 40 752.
informational activities of DNA in real life inside living
cell, for full set of 3D chromosomes, their overall com-
munications and information flows, switches in topology
and informational networks in the presence of real
changeability and noises in information transfer are lost,
go off, pushed back far into the background, as the basic
unsolved problems. What it means for information trans-
fer of DNA activity inside living cells, for diagnostics
and comparative analysis of health and diseases?
In Part 2 of this article we defined informational ho-
meostasis of Shannon entropy as the empirical result. In
Part 3 we analyze a density of packing, new types and
classes of fractal networks of DNA entropy. In Part 4 we
analyze deviations of entropy from homeostasis and
various switches for central moments and averages of
Holder at variations regulation of homeostasis for differ-
ent states of health. We observe saturation for averages
of Holder if the number of averages coincides with full
chromosome number; all chromosomes are involved in
support of regulation of informational homeostasis. The
levels of saturations have various switching at changing
the state of health. In Part 5 we show that this switching
connected with stability change in homeostasis regula-
tion. We show that all distributions of information en-
tropy are divided on three classes, which associated in
diagnostics with a good health or dominants of autoim-
mune or inflammatory diseases. This classification based
on switching of stability at transcritical bifurcation in the
nonlinear dynamical system of homeostasis regulation.
2. Information on Oxidative Activity of DNA
inside Cells. Shannon Entropy.
Informational Homeostasis
Oxidizing activity of DNA in the cells activates various
(all possible) correlations with different combinations of
non-coding and coding DNA fragments, both within one
and the same chromosome, and between different chro-
mosomes. Actual, 3D topology of chromosomal correla-
tions in the nuclei of cells has quite notable changes over
time, during only one month, for cells living inside given
human, in the process of human life or at diseases de-
velopment, during medical treatment, etc. [1-6]. Charac-
teristics, changes and deviations various fractal correla-
tions in networks of fluorescing DNA inside cells can be
used for medical diagnostics [6]. How these results may
to use for definition varied structures of informational
activity of DNA?
We all, all our chromosomes and all our cells are the
open systems. How to estimate quality, quantity and
changeability of DNA communications and DNA infor-
mation transfer in life of given person? What about in-
formation ln
and information entropy for DNA
activity inside cells? What need for comparison various
data on informational activity of DNA inside cells for
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one and the same human at various times and for differ-
ent people? Here we have no clear criteria. How to de-
termine the existence and level of information noise,
switches, changeability and stability of information
transfer inside living cells for any being? What it means
for medical diagnostics from the point of view different
types of oxidative metabolism of 3D DNA inside cells,
for inner and inter chromosomal correlations at different
states of health? The answers on these questions now are
absent, that associated with deep and unsolved problems
in fundamental mathematics, information theory, etc.
To diagnostics features oxidizing activity of DNA in
the blood cells for one and the same person at different
times or in blood cells of different people need to com-
pare various distributions of fluorescence, which are very
diverse and changeable, as for three examples in Figure
1. A huge role here is played by the irregularity, infre-
quency and brokenness of histograms, which define basic
information on DNA activity [5-7]. Any artificial smooth-
ing results destroy this information. For comparison
various distributions of fluorescence we must under-
stand the origin and function of these distributions under
what conditions and parameters they need to be com-
pared. For example, at Gaussian distribution of random
variables is important to know only the mean and vari-
ance. If the statistics is very complex, such parameters
and their combinations can be very, very, many. Here
observed non-trivial noises of fluorescence and the ex-
ponential divergence central moments for fluctuations of
intensity at increasing the number of central moments
[3-5]. This is a clear sign of turbulence [3]. In this case,
when comparing different distributions and moments for
fluctuations of intensity the number of corresponding
moments also exponentially quickly grows with increas-
ing the order of diverging statistical moments. The simi-
lar procedures of comparisons haven’t the sense in nature
and in science, excluding examples and illustrations the
growth for rate of statistical instabilities in the interpreta-
tion of complexity. Reproducible results and clear analy-
sis real activity various DNA inside cells produce the
need of clear, stable levels and criteria for comparison
different distributions of DNA fluorescence.
Let us introduce frequency distribution of Shannon
entropy , based on the frequency distribu-
tions of information l
Ep i
(see Equations (1)-(3)).
Three examples of frequency distributions of Shannon
entropy i
, based on the frequency distribu-
tions of information l
P in Figure 2(a), are
shown in Figure 2(b). We present comparison of distri-
butions of immunofluorescence based on the universal,
empirical invariant of information entropy
at given rank r. Rank r is defined by the maximal
number of measuring channels max
ln r
r. Detailed defi-
nition of empirical invariant of information entropy
,lnEJr r is presented below, in Equations (4)-(7).
This empirical invariant shown in Figure 2(c), as only
one¸ overall numerical value of total Shannon entropy
,lnEJr r for given rank r, for DNA fluorescence
inside any neutrophils, living in any people with different
states of health.
This invariant was observed during fluorescence of
DNA in human neutrophils in different samples of blood
[5,7], as in Figure 2(c). We observe only one unified
value of total Shannon entropy , like the
empirical invariant, as the identical sum in each given
distribution of entropy, in given sample of blood, for all
cells and any donor. This invariant has one and the same
value of total Shannon entropy at fluo-
rescence for given rank r of histograms, i.e. given scale
,lnEJr r
,lnEJr r
(a) (b)
Figure 2. (a) Logarithmic dependence LnP(I) for frequency
of flashes P(I) on their intensity I(r = 256); The area under
the initial histograms of P(I) normalized to unit; Original
histograms for P(I) shown in Figure 1; (b) Normalized dis-
tributions of information entropy
in the de-
pendence on fluorescence intensity I(r = 256); rhombuses
correspond to bronchial asthma. Total number of flashes is
N0 = 76 623; squares correspond to the healthy donor, N0 =
40 109; cross correspond to the oncology disease. Common
number of flashes is N0 = 40 752; (c) Dependence of total
Shannon information entropy
Jr r on logarithm
of range r; initial histograms at range r = 256 show n in Fig-
ures 1(a) and 2(b).
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of clusters in networks of DNA entropy. This invariant
a very strong roughness and dif-
defines the informational homeostasis of oxidative activ-
ity of 3D DNA for full set of chromosomes inside living
cells, at any biodiversity of cells for Shannon-Weaver
index [5]. This invariant gives the overall zero level for
countdown of information activity of DNA in cells for
any person, at any state of health and for any genome of
different people.
In Figure 2(a) we see
rences in information
 
IPI over a wide
range of changing the ordeom zero up to
ten; typical level of information is about J ~ 7. High level
of noises instead of a smooth continuity for all local dis-
tributions of information
 
r of values J(I) fr
IPI reflects main
natural properties of DNA provide main
correlations in oxidative activity of DNA inside cells, in
gene’s networks, in metabolism and cell viability. Any
forced smoothing of experimental data here hampering
any our attempts to adequate perception of real life DNA
inside living cells and violates the homeostasis of en-
activity, which
us consider the probability density P(I). i.e. fre-
quency distribution l
P the number of flashes,
PNlN, where s the number of measuring
2,, 256; 0
Nis the total number of
flashes; ber of flashes with the
assignedIl; dimensionless intensity I coin-
cides with the number measuring channel l, i.e. Il
l i
channel, 1,l
NNl is
 is the mean probability value;
max min
symbol denotes the statistical average for number
flashes oorescence in all of r = 256 channels of in-
tensity measurement. The mean value of
f flu P is equal
to 1/256 for r = 256 channels of intensity msurement.
Three examples of frequency distributions of l
P for
different donors with varied states of health are shon in
Figure 1.
n of information
defined as
P (1)
Let us consider the normalized
distribution of infor-
pJ J
as the probabilistic measure for frequency distributions
E (3)
Data analysis of all experiments has shown the con-
Thus, total Shannon entropy E(J) is empirical invariant,
of Shannon entropy l
ln ,
lll l
rvation of total Shannon entropy
r all neutrophils in all donors [5,7]. Value of total en-
,constEJ EJrr depends on given
rank r of histod) and Equation (7)).
Rank r is defined by the maximal number of measuring
channels max
gram (see Figure 2(
. At rank r = 256 all experimental data,
for all dogive one and the same value of nors
,256 5.48EJr with standard deviation ~2% ,
l for flow cytometry experimental
errors ~2% at 256 measuring channels [1,2]. Decreasing
maximal number of channels or rank r leads to decreas-
ing the value of invariant
,EJr .Three illustrations of
informational homeostasistributions of informa-
tion Ln
within limits of typica
for dis
and entropy in Figures 2(a) and (b) are
show 2(c) at different rank r. Other examples
of informational homeostasis for different patients with
various diseases had shown in [5].
Invariance of total entropy
n in Figure
defines special
tropy le of distribution of Shannon en
EJI , as is
for all functions, associated with the conservation laws,
as the main dominant variable to describe the states and
dynamics of informational activity of DNA inside cells.
How will be changing the information J(I) due to re-
duction of rank r or the range r at definition of Shannon
entropy? Here, as everywhere, are used different terms a
rank r and range r for the same value of r. Range of his-
togram r interconnected with the selection of multistage
clusters in networks with structure of bronchial tree; here
range r coincides with the number of columns in a histo-
gram or with the number of channels for measurements
of fluorescence intensity at given maximal value of di-
mensionless intensity, i.e. max
rI. In our experiments
the number of channels is r ariations of range r,
i.e. rank of histogram r, or variations the scale r, when
= 256. V
, provide the changes in irregularity and broken-
frequency distribution of fluorescence for histo-
grams of various rank r. Various examples decreasing of
histograms rank r presented in [3,5,7]. Integer r defines
the total range r for distribution of entropy as maximal
number of columns in reduced histogram. Each reduction
of r leads to the redistribution of probability density
ness of
,PIr and information
Ir. Reduced distribution
bility of proba
,PIr describein [3]. Reduced distri-
bution of information is
 
Ir PIr . Normal-
ized frequency of informreduction
of range r is
ation l
 
ll lll
prJrJr JrPr
Frequency distribution of entropy
Er for an arbi-
trary rank r is
EJrpr pr
Total entropy n is invariant
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id in all cells. Total enentical for given rtropy
pends only on rank n Figure 2(c). de r, as it is shown i
Dependence of
,EJr on rank r in Figure
 
2(c) is
, lnln
EJrp pr
 
(7) ln
p r
Informational homeostasis of total Shannon
for oxidative activity of DNA is observed
various states
fluof fractal
in all cells of blood different old and young patients with
of health and all the time. It means exis-
tence of informational homeostasis of DNA during all
the life time of cells from birth to death.
3. Manifold of Dense Fractal Networks of
Shannon Entropy for Informa
Let us consider some fractal peculiarities of immu
orescence distribution. Different analogies
networks such as bronchial tree, structure of oncology
tumor, arterials tree, etc with networks and distributions
of immunofluorescence are described in [3]. Many histo-
grams of different origin are the similar to the histograms
for fluorescence of neutrophils in Figure 1 [3]. Range of
histogram r interconnected with the selection of multi-
stage clusters in networks with structure of bronchial tree.
Variations of range r, i.e. rank of histogram r, or varia-
tions the scale r provide the changes in irregularity and
brokenness of frequency distribution of fluorescence for
histograms of various rank r. The quantitative measure of
irregularity and brokenness for frequency distribution of
flashes for any rank r, in all histograms may serve a
Hurst index H. может служить.
Hurst exponent H [8] is determined by means of re-
gression equation
LnLn constRS HI  (8)
where R/S is rescaled range (R = S), R is
mal deviation of P(I) from local mean le
ure 3 presented three distributions of fractal di-
mensions D for frequency di
range or maxi-
vel, S is standard
deviation of P(I). Illustration of definition Hurst index
was presented in [5,6]. Hurst index H for frequency of
flashes P(I) corresponds to fractal (Hausdorff) dimension
D [8] if
2DH (9)
In Fig
stributions of Shannon en-
py in Figure 2(b).
As the rule, networks of entropy are characterized by a
mix of normal D < 2 and abnormal D > 2 fractal dimen-
sions, as in Figure 3. Abnormal fractals D > 2 are typical
for entropy’ networks in a good health and for oncology
at different values of rank r and perfectly absent at bron-
chial asthma, where D < 2 for all rank r. Networks of
Figure 3. Dependence of fractal dimension D(r) on loga-
rithm of range r in networks of Shannon entropy with dif-
ferent scales for three different states of health conected
. Normal fractal dimension
corresponds to interval 1 < D < 2 and positive Hurst
of the po
orks. The size of all and any frac-
e of information entropy
with asthma, with good health and at oncology; initial his-
tograms shown in Figure 2(b).
information entropy for DNA fluorescence are formed by
normal and abnormal fractals
index H > 0. Negative Hurst index H < 0 gives the
anomalous fractal dimensions
22DH . Abso-
lute majority of the authors ignore any anomaly fractal
dimensions. Nonetheless, negative Hurst index H < 0
does not contradict of main definitions wer-law
correlations for fractal distributions [9], subject to rejec-
tion from the hypothesis of self-affinity. In the case of H
< 0 an anomalous fractal dimension D > 2 can serve as a
measure of fragmentation for correlations in complex
networks [10].
Abnormal fractal dimensions D > 2 give not only
fragmentation of correlations but also ensure more dense
packing of fractal netw
l clusters d ~ N1/D, for certain number of nodes N, de-
creases at increasing the value of D. We do not know
characteristics of fragmentation of information entropy
of DNA and we do not have the recipes and methods of
its descriptions. Therefore, we guide by clear signs of
new peculiarities and strong expressions of contradic-
tions with the traditional images of modern standards,
diagrams and descriptions of DNA activity inside cell.
For example, we consider different unusual deviations of
typical features from very popular networks of “small
worlds” [11], that are often used to describe global DNA
activity inside cells [12,13].
According to Figure 3 there is no unambiguous and
simple connection of network topology or fractal dimen-
sion D with the certain valu
,lnJrr at informational homeostasis.
Consider an undirected network, and let us define d as
the mean geodesic (i.e., shortest) distance between pairs
odes in a network of flashes of fl
of vertex or nuorescence.
The certain number N of synchronized nodes-flashes in
networks of DNA fluorescence inside cells are charac-
terized by the intensity I ~ N, where N defines a common
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number of correlated nodes in network, if every node in
fluorescence network has the approximately identical
fragment of oxidative activity of DNA with approxi-
mately identical quantity of fluorescing dye. More de-
tailed determination of correlated nodes N in the clusters
of fluorescence networks of DNA inside cells now is
unknown. The correlation length d depends on the net-
work topology. Random networks with a given degree
distribution may be the networks of “small worlds” [14],
as in one from most popular family of complex networks
[11-13]. “Small world” behavior is typically character-
ized by logarithmic scaling for path length tends d ~ lnN
[14]. On the other hand the expression of d ~ N1/D defines
a linear size of D-dimensional lattice or the size of a
fractal cluster d ~ N1/D. Therefore estimation of fractal
dimension D of fluorescence in the networks of “small
worlds” is
~lnlnlnDNN N. Standard definition of
fractal dimension D [8,10]
lim LnLnDNdd (10)
also gives
~lnlnDNNlnN in “small worlds” net-
work. We use various experimental data in h
define Hurst index H and fractal dimension D
(8)-(10). The tr
topology for
) also ex-
istograms to
to Equationsansformation of “small
worlds” due to reduction of range r = I ~ N leads to ex-
pression ln~ln lnrD r. In these variables data in Fig-
ure 3 are transformed to Figure 4(a). Using of variables
for networks of “small worlds” gives varied distributions
of fractal Shannon entropy which have a
view of the correlations presented in Figure 4.
We observe more fast than linear and various growth
of correlations at increasing rank r in Figure 4(a). Rich
diversity of various hinges in Figures 4(b)-(d
oude posibility for an unambiguous and simple identi-
fication any networks of entropy as networks of “small
worlds”. Therefore, hypothesis about “small worlds”
structure for information entropy of DNA, as a common
universal principe, here is no good. Complex hinges in
Figures 4(b)-(d) and their dependence on the state of
health may to serve one of illustration of various frag-
mentations in fractal networks. Strong variations of
hinges in Figures 4(b)-(d) at variations states of health
show that topology or fractal structure of entropy net-
work has strong dependence on the states of health.
Moreover, more dense and more real types of entropy
networks presented in Figure 5. Packing of data in Fig-
ures 5(a) and (b) corresponds to exponentially d
cking of “small worlds”. According to Figure 5 we
never have ideal networks of “exponentially small
worlds” in real life, but we have a very perfect networks
of “exponentially small worlds” without fractals (D = 2)
as a clear simple standard for comparisons of deviations
of various fractal correlations with this ideal standard. In
Figures 5(a) and (b) are observed more ordered situa-
(a) (b)
(c) (d)
Figure 4. Large-scale distri butions of fractal dime nsion D(
for variables corresponded to networks of ‘small worlds’; (a)
ln~ lrDr tal dimension nlnr; various hinges of frac
D(r) in the dependence of
~ln lnln
NNN for (b)
asthma, (c) good health (d) oncology; initial histograms
tions in topology structure of networks than in Figures
4(a)-(d) for networks
shown in Figure 2(b).
of “small worlds”. Therefore net-
orks of “exponentially small worlds” are more suitable
nd rather
for description of information entropy of DNA.
Other type of correlations may be presented in net-
works of “double logarithm scale” in Figure 6.
According to Figures 5 and 6, branching a
table differences in networks corresponding to various
states of health are observed for small rank r = 4 and b
nk r > 32. Rather good coincidence and local univer-
sality of entropy networks for different states of health
are observed at rank r = 8, 16, 32 in Figure 5 for “expo-
nentially small worlds” and in Figure 6 for “double loga-
rithm scale”. Stratification and individual deviations
from common correlations at other values of rank r de-
pend on the states of health; variations of health status
correspond to variations in informational networks of
DNA activity. In Figures 5(b) and 6(b) we observe two
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(a) (b)
Figure 5. Dependences of fractal dimension D(r) on double
logarithm of range r in the variables for ‘exponentiall
small worlds’
lnln~ lnlnrDr r (b)
lnln~ lnlnlnrDrr r in multi-scale networks of “real
worlds” and in nially sm
for information e of “exponentially small worlds”,
without fractals (D = 2), corresponds to violet line with
round dots; initial histogr ams show n in Fi gure 2(b).
etworks of “exponentall worlds”
ntropy of fluorescing DNA inside neutr on-
phils; the ideal network
(a) (b)
Figure 6. Dependences of fractal dimension D(r) on double
logarithm of range r in the variables for “double logarith-
mic scale” (a)
lnln~ ln
rr (b) lnln~lnlnln
rr r in
inside neutrophils; the ideal network
ratification and branching of fractal
ructures for DNA information entropy shape and define
nd Switches for Holder’s
wh uations
duof stability near homeostasis. These
multiscale networks of “real worlds” and in networks of
“double logarithmic scale” for for information entropy of
fluorescing DNA of
“double logarithmic scale”, without fractals (D = 2), corre-
sponds to violet line with round dots; initial histograms
shown in Figure 2(b).
different branches for small and big values of rank r in
the point of r = 16. St
various alternative ways of transmitting information. Dif-
ferent alternative means of information communication
for the global network of DNA inside cells ensure by
local networks of different clusters with the same topol-
ogy (identical D) at different scales of r or at varied
values of entropy (E = lnr) in the dependence the states
of health, as in Figure 3. The same fractal dimension D
can match the clusters of different scales r with a
different number of flashes. Currently we don’t now
other clear details of fragmentation. We have no univer-
sality in informational networks of DNA, only perfect
etalon, as theoretical measure for estimations of informa-
tion communications in the not existing in real life and
ideal cells, which presented here as violet lines in Fig-
ures 5 and 6 for an ideal case D = 2. Reality connected
with variations of homeostasis regulation, i.e. with chang-
ing noises of information entropy for DNA activity dur-
ing life of cell.
4. Noise of Entropy in Homeostasis Support.
Patterns a
We have no of ideal, absolute, correct homeostasis, no-
ere and never. We always observe various fluct
ring regulation
fluctuations also very individual and consist many in-
formation on stability regulation the dynamic equilibrium
in homeostasis. Let us consider various deviations, fluc-
tuations or noise of entropy
er near homeostasis of
total information entropy
 
,1, ln
lll ll
erEr EJrErprpr
Mean characteristics for individual distributions of
noises in a blood sample defined by the central mo
and Holder’s averages for noises of entropy
The central moments of
er for fluctuations of
near homeostasis defined by the statis-
tical averages
 
erMer m, (11)
where m determ
er m. ines the order of moment
Here symbol ... denotes statistical
tuations of entropy
average for fluc-
er. The power means or averages
of Hölder for deviations of entropy
er a
lr m
emr e
Three illustrations for distribution of central moments
er m and of averages of Hölder
4,er m at
rank r = 4 presented in Figure 7. These
rank r = 4 for inf.
Two branch
ibutions are determined on the base of histograms of
ormation entropy in Figure 8
es with even and odd numbers m of central
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(a) (b)
Figure 7. (a) Logarithmic distributions of central statistical
,Mer m
homeostasis for bronchial as
sented in F
for fluctuations of entropy
thma (rhombuses), fo
e(r) near
r a good
health (quadrates), for oncology (triangles); initial histo-
grams preigure 8; lower and upper branches
correspond to the even 2,4,6,m and odd 1,3,5,m;
(b) Distributions for averages of Hölder
,emr with
different number m at fluctuations of entropy e(r) near ho-
meostasis for bronchihombuses),d
health (quadrates), for oncology (triangles)histo-
grams presented in Figure 8; upper and lower branches
correspond to the even 2,4,6,m and odd
al asthma (r for a
; initial goo
Figure 8. Dependence of normalized distributions of Shan-
non entropy E(I, r = 4) on intensity I at rank r = 4. The con-
tinuous lines correspond to the parabolic apps of roximation
probability density distributions, three types of lines cor-
respond to different types of stability for transcritical bi-
furcation in homeostasis regulation, at change the states of
health; initial histograms shown in Figure 2(b).
erm have unique universality with
zero and clear exponential decreasing of
er m,
as in Fi he exponential decreasing gure 7(a). T
er mthe informational stability of
DNA activity. Decreasing rate of
to growth at increasing numbers m and r. A very slow
erages of Hölder
have trends
growth for av,emr in Figure 7(b)
clearly defines individual level of ns and auto-
correlations of entropy
,emr at homeostasis for
central moments
u quickgiven person, unlike ofnified and degradation of
er m, as in ure 7(a).
Averages of Hölder
,emr define various orders
of autocorrelation, at varioers of m, for fluctua-
tions of entropy near the “centre of gravity” in stability
regulation the dybrium in homeostasis.
us numb
namic equiliAc-
co all devrding to Figure 7(b)iations of entropy, for
various individual distributions of
,emr from zero
level, increase with increasing the number of m, with
various saturations of chromosomal correlations of en-
tropy at different states of health until the value of m =
46, which defines the number of mes inside
cells. Therefore, the value of
46,em r is the
largest among all mean. Therefore, overall level of en-
tropy noise defined by the values of
46,em r ,
according to a well known inequality
1, ,em remr for Hes. This
means, also, that all 46 chromosomes involved in support
regulation of homeostasis inside cells, as ont.
All distributions of
older’s averag
e united full se
,emr in F
reasing of m. Two branches
with even and odd numbers m correspond to the negative
and positive values of
igure 7(b) have an
oscillatory behavior at inc
,emr . These oscillations with
pe Figure
r is define
riod 1 not specified in 7(b); the mandatory be-
havior doesn’t needing in comments.
Average noise level of information entropy in DNA
activity at given range d by the standard devia-
emr ekr
,emrof for
formation transfer of DNA activity inside cells. These
m = 46. These are very important characteristics for in-
46,em r
rages of Höld
defined byf
46 for all aveer
all values o m =
,emr , but not only the
lower numbers of 1, 2m
; we have 46 chromosomes in
the cells and many types of cross-correlations. Noise of
entropy orevel for one chromosome, de-
fined by the values of
average noise l
, in DNA activity
ensures contrast ination transfer and correlations
inside cells at given scale r for any networks of DNA
activity. Value of
depends on health status.
In Figure 7(b) maximal noise level
46, 418%emr
 corresponds to strong in-
flammations at asthma.
Distributions of
s define stable and
clear classification of noise
lation of homeostasis, for given
itching observed for distribution
Figure 7(b) depend on
the health status. These distribution
level in networks of entropy
for DNA activity at regu
rson at given time
We observe very strong switching of branches for r =
4 in the averages of Hölder in Figure 7(b) for asthma
with respect the same branches for a good health and
oncology. The same sw
central moments
er m in Figure 7(a). Analy-
sis show that distributions of various other averages of
Open Access AM
Hölder, for other values of rank r are rather close to each
other. This means the similar noise levels of information
entropy near homeocology and in a good
health for given donors at other values of rank r. In the
latter case we observe very notable differences in fractal
structures of networks of information entropy in Figures
3-6 and difference in their stability (see bellow Figure 8).
Therefore, we cannot assume that similarity in the aver-
ages of Hölder linked with the coinciding characteristics
of collective correlations inside and between chromo-
somes in networks of a certain scale r for different states
of health. This means only some similarity in the levels
of noises for ensuring information transfer and chromo-
somal correlations near homeostasis, as for one and the
same noise level in various Brownian motions.
Mean level of experimental errors in original cytomet-
ric histograms for r = 256 is about 2% [6-8]. We observe
much more noticeable and very clear difference between
initial and transformed experimental distribution
stasis at on
s in Fig-
ures 1 and 2 for different states of health and much more
essential difference for averages of Hölder
for deviations of entropy in Figure 7(b). This means that
in all cells exists and maintained a very effective stabili-
zation of homeostasis in various networks of entropy for
various states of health. The origin and mechf
this universal stabilization now are unknown. Various
regulation of homeostasis must be very quick and must
have the general physical origins, as for the dipole and
multiple resonances for excitations in large clusters.
Switching between networks of entropy in Figures 4-
6, corresponds to different states of health, with different
deviations of
er from homeostasis of entropy. Hid-
den switching between branches for averages of Hö
anism o
,4mr and central moments
Figure 7 for a good health, oncology and for asthma here
linked with chg stability due to transcritical bifur-
cation in distributions of information entropy in Figure 8
tatuses of health.
5. Statistical Stability and Transcritical
Let us consider statistical stabil
for different s
ity varied distributions of
n of
en Figure 8. Large-scale distributions
(13) has stationary solutions that correspond to fixed
information entropy following to some approaches
sented in [3]. Reduction of rank r leads to distributio
tropy presented in
of information entropy for rank r = 4 in Figure 8 have
different statistical properties and different stability types
for various states of health. These properties are con-
nected with transcritical bifurcation in homeostasis regu-
lation. Transcritical bifurcation has a normal form [15]
tvAvv (13)
where A is a control parameter. The dynamical system
points. The fixed point 0
depends on the sign and
value of a control param. If
tractive fixed point
eter A
and stable solutions
0A there is an at-
to Equation (13).
If 0A
there is a repelled fixed point and unstable
solutions to Equation (13). If 0A there is a shunt
fixed point and neutral sy. The introduction of the
source of additive white noise reduquation (13) to
the Langevin equation
ces E
 
tvAvvftftftDt t
  (14)
t is the Langevin source,
is the
-function, D
is diffusion coefficient for white
noise. Linearization near the fixed pointduces
linear Langevin eq
vA re
uation forEquation (14) to
ns of
the small
Axft  (15)
The Fokker-Planck equation that corres to Equa-
tion (15) is
tx xx
AD xt
 (16)
Equation (16) determines the
stationary probability
~exp 2
Ax D
. The prob-
ability density distribution
corresponds to at-
tractor if . The statistical determina
tractor is used as the more probable state,
0Ation of the at-
when the dis-
bution function has a maximum as convex parabola in
Figure 8
A similar determination of or is actually equiva-
lent to theach proposed in [16]. If 0A there is
a repelled fixed point and minimum of
. Let us
xI I , where
I corre-
sponds to the position of minimumum for
or maxim
rves in Figure 8. The attractor fixed point corresponds
the upper parabola in Figure 8 if
220Ax . This
parabola characterizes the blood immunofcence at
inflammato onchial
The lower parabola in Figure 8 corresponds to unsta-
ry diseases; in our case this is br
critical point, characealth.
Fixed point of neutral stability with the ideal value of
terizes a good h
was not observed. Instead, a large family of lines
with very small positive and negative curvatures for
If o
rious autoimmune and oncology diseases without in-
flam is observed, as the line passing through tri-
angular points in Figure 8.
ne considers the curvature or curvature radius of
parabolic approximations of entropy
Figure 8 as the bifurcation parameter A, then various
types of statistical distributions of
,4EIr can be
classified in the frame of trPro-
tasis corre
anscritical bifurcation.
sed criteria correspond to informational homeostasis of
entropy. The bifurcation corresponds to stability change
in homeostasis regulation for various health statuses. Thus,
three types of informational homeosspond to
Open Access AM
positive, negative and neutral stability for distributions of
information entropy at oxidative activity of DNA inside
cell. The bifurcation reflects the collective statistics ef-
fects of various cellular processes and switching of tran-
sitions from health to illness and from illness to health.
6. Changeability and Switching of Entropy’s
Noise for a Healthy Donor in Real Time
Let us compare, in real time, during one year, Shannon
entropy for DNA activity inside cells of healthy donor.
Three histograms are shown in Figure 9.
One may to compare histograms in Figures 1 and 9.
stributions of entropy are shown in Figure 10. Di
re 12.
One may to compare histograms in Figures 2 and 10.
Fractal peculiarities of entropy are shown in Figure 11
Dense packing of entropy is shown in Figu
Some features of stability and noises of information
entropy at low rank r = 4 are shown in Figure 13.
One may to compare histograms in Figures 1 and 9, 2
d 10, 4 and 11, 6 and 12 , 7 and 13(b), 8 and 13(a ). We
serve changes in distributions of entropy and fr
ructures, noise level etc. in real time for one and the
same healthy man and in the comparisons with vari
stributions for different unhealthy people. Many of
fractal peculiarities of entropy change in time and depend
on health status. Constancy belongs to homeostasis or to
invariance of total Shannon entropy in Figures 2(c) and
10(b) for any states of health of any human. Constancy
also belongs to statistical instability (positive curvature
of all approximations) for entropy distributions of un-
changeable healthy man in Figures 8 and 13(b) during one
year. This is example of a very good immunity, as and
perfect closeness to the ideal of networks in Figure 12.
(a) (b)
Figure 9. Dependence of normalized frequency distribution
of flashes P(I) on their intensity I (a) and for clearest only
central part of histogram (b), for one and the same invaria-
bly healthy, donor in different times. Triangle green pots
correspond to 832, analysis
the total flashes number N0 = 30
time is 19 July (first year); rhombus yellow points corre-
spond to the total flashes number N0 = 38 758, analysis time
is 11 July (next year); square red points correspond to the
total flashes number N0 = 40 109, analysis time is 03 June,
before 11 July, histogram range r = 256, as in Figure 1.
(a) (b)
Figure 10. (a) Normalized distributions of information en-
at rank r
in the dependence on fluorescence inten-
sity = 256; (b) Dependence of total Shannon in-
Jr r on logarithm of range r;
initial histograms at range r = 256; initial histogram shown
in Figure 9.
(a) (b)
(c) (d)
Figure 11. Distributions of fractal dimension D(r) in the
dependence on rank r ((a)-(d)); ((b)-(d)) various hinges of
fractal dimension D(r) in “sma ll world” network, in the de-
pendence of
~ln lnln
NNN, for a healthy man in rea
time, during on in Figure 9.
e year; initial histograms shown
Open Access AM
(a) (b)
Figure 12. Dependenc es of fractal dimension D(r) on double
logarithm of range r in the variables for “double logarithm-
mic scale” (a)
lnln~ ln
rr (b) lnln~lnlnln
rr r
multi-scale net networks of
“double logarit
ound dots; ihistograms shown
works of “real worlds” and in
hmic scale” for information entropy of fluo-
rescing DNA inside neutrophils; the ideal network of “dou-
ble logarithmic scale”, without fractals (D = 2), corre sponds
to violet line with rnitial in
Figures 9 and 11(a).
(a) (b)
Figure 13. (a) Dependence of normalized distributions of
Shannon entropy E(I, r = 4) on intensity I at rank r = 4 . Th e
continuous lines correspond to the parabolic approxima-
tions; initial histograms shown in Figures 9 and 11(a); (b
Distributions f
or averages of Hölder
,emr with dif-
ributions of
entropy, based on normalized distribution of information
in original histogram for frequency of flashes in different
observe only one unified value of
, for all rank r, like
of 3D DNA
ibe the states and dyrmational
ferent number m at fluctuations of entropy e(r) near ho-
meostasis for a good health man in real time, during one
year; initial histograms shown in Figure 9; upper and lower
branches correspond to the even 2,4,6,m and odd
7. Conclusions
We study various large-scale dist Shannon
blood samples. We
total Shannon entropy
the empirical invariant, as the identical sum in each given
distribution of information entropy, in all samples of
blood and any donor, as in Figures 2(c) and 10(b) in real
time. This invariant defines the informational homeosta-
sis of oxidative activityfor full set of chro-
mosomes inside living neutrophils, in all clusters of all
multi-scale networks for information activity of DNA
inside cells. This invariant gives Shannon entropy as
general characteristics for definition of information ac-
tivity of 3D DNA in cells for any person, at any state of
health, for any genome as the overall zero level for
analysis and comparison chromosomal activity and DNA
networks in cells for different people (Figures 5-8) and
for one and the same human in different times (Figures
12 and 13).
Invariance of total entropy
,lnEJr r defines
special role of distribution of Shannon entropy, as it is
for all functions associated with the conservation laws
(pulse, energy, charge etc.), as the main dominant vari-
able to descrnamics of info
tivity of DNA in cells. Nevertheless, informational
invariant of
,lnEJrr is difficult to compare with
the conservation laws of mass, energy, etc.; we have no
different values of total entropy in different cells. Invari-
ance of entropy reflects the unchangeable measure or
quantitative units of information entropy at certain scale
r of correlations at DNA activity inside all and any cells.
Informational homeostasis reflects a number of fun-
damental phenomena in information and physics of real
life of 3D DNA inside cells. Shannon entropy of 3D
DNA in cells has much more dense packing of correla-
tions than in well known networks of ‘small worlds, than
in all and any technical and computer systems. As the
rule, networks of entropy are characterized by various
mix of normal D < 2 and abnormal D > 2 fractal dimen-
sions (Figures 3 and 11(a)) and new types of fractal pat-
terns and fractal hinges for various fractal topology at
different states of health (Figures 4-6 and 12).
Results generalized in the double logarithmic scales, in
the triple and quadruple logarithmic scales, etc., as a re-
flection of complexity, what permits, also, detect a frag-
mentation, as it is shown in Figures 4-6 and 12.
Deviations or noises of information entropy
om homeostasis level define regulation of informa-
tional homeostasis, information transfer and information
flow inside cells for different states of health. The coin-
cidences and switching patterns in branching se
r central moments
em and for average
s of
,emr of these noises are shown in Figures
7 and 13(b) . We observe various saturations for averages
of Hölder
,emr in the levels of chromosomal cor-
relations of entropy at different states of health in Fig-
ures 7(b) and 13(b), at increasing number of correlations
m. All cmes are interconnected and involved in
the support regulation of homeostasis; saturation of
hro mo s o
Open Access AM
Open Access AM
exists if [4] N. E. Galich, “Cytometric Distributions and Wavelet
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[5] N. E. Galich, “Shannon-Weaver Biodiversity of Neutro-
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various blood sample in diagnostics. For instance, noise
level of entropy near homeostasis may be rather high and
to reach ~20% for strong inflammations, as it observed
for bronchial asthma in Figure 7(b).
We defined manifold of chromosomal networks of en-
tropy in Figures 4-6 and 12 and (or) many ways for ho-
meostasis regulation. We show that regulation of homeo-
stasis belongs to one from three classes of stability for all
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[8] J. Feder, “Fractals,” Plenum Press, New York, 1988.
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utral and positive stability for healthy people and do-
minants of autoimmune or inflammation diseases. Chan-
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are shown in Figure 8.
To be continued.
8. Acknowledgements
Thanks to M. Filatov for kindly providing the experi-
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