Dense Fractal Networks, Trends, Noises and Switches in Homeostasis Regulation of Shannon Entropy for Chromosomes’ Activity in Living Cells for Medical Diagnostics

doi:10.4236/am.2013.411A2006

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Applied Mathematics, 2013, 4, 30-41

Published Online November 2013 (http://www.scirp.org/journal/am)

http://dx.doi.org/10.4236/am.2013.411A2006

Open Access AM

Dense Fractal Networks, Tr ends, Noises and Switches in

Homeostasis Regulation of Shannon Entropy for

Chromosomes’ Activity in Living Cells for Medical

Diagnostics

Nikolay E. Galich

Department of Experimental Physics, Saint-Petersburg State Polytechnic University, Saint-Petersburg, Russia

Email: n.galich@mail.ru

Received July 7, 2013; revised August 7, 2013; accepted August 15, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

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-

meostasis.



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

N. E. GALICH 31

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

Open Access AM

N. E. GALICH

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

P

(see Equations (1)-(3)).

Three examples of frequency distributions of Shannon

entropy i

, based on the frequency distribu-

tions of information l

Ep

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





,EJr



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)

(c)

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











,ln

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).

Open Access AM

N. E. GALICH 33

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-

tropy.

Let

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

the

channel, 1,l



NNl is

intensity

num



;



PI I



 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.

Distributio

n of information

defined as

P (1)

Let us consider the normalized

distribution of infor-

ation

256

pJ J



l



(2)

as the probabilistic measure for frequency distributions

E (3)

Data analysis of all experiments has shown the con-

(4)

Thus, total Shannon entropy E(J) is empirical invariant,

of Shannon entropy l



ln ,

lll l

EppEJIEJ

rvation of total Shannon entropy

256i



const

EJ E







r all neutrophils in all donors [5,7]. Value of total en-

tropy









,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

max

= 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

 

,ln,

Ir PIr . Normal-

ized frequency of informreduction

of range r is

ation l



during

 

,Ln

ll lll

prJrJr JrPr





  (5)

Frequency distribution of entropy





Er for an arbi-

trary rank r is





,ln

EJrpr pr



(6)

Total entropy n is invariant



EJrpp







Open Access AM

N. E. GALICH

id in all cells. Total enentical for given rtropy





,EJr

pends only on rank n Figure 2(c). de r, as it is shown i

Dependence of



,EJr on rank r in Figure

logarithmic

 

2(c) is



, lnln

EJrp pr



 

 (7) ln

p r



Informational homeostasis of total Shannon

for oxidative activity of DNA is observed

various states

tion

no-

fluof fractal

entropy



,lnEJrr

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

tro

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

Open Access AM

N. E. GALICH 35

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



d



~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-

ense

istograms to

according

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)

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

Open Access AM

N. E. GALICH

(a) (b)

Figure 5. Dependences of fractal dimension D(r) on double

logarithm of range r in the variables for ‘exponentiall

small worlds’

(a)











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

Averages

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



,EJr











 





,1, ln

lll ll

erEr EJrErprpr

(10)

Mean characteristics for individual distributions of

noises in a blood sample defined by the central mo

and Holder’s averages for noises of entropy

ments





The central moments of



er for fluctuations of

entropy





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



 (12)









er m and of averages of Hölder





4,er m at

distr

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

Open Access AM

N. E. GALICH 37

(a) (b)

Figure 7. (a) Logarithmic distributions of central statistical

moments









,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

1,3,5,m.

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).

moments



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



provides

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

fluctuatio

u quickgiven person, unlike ofnified and degradation of









er m, as in ure 7(a).

Averages of Hölder

Fig





,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



chromoso

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-

tion



emr ekr











m







,emrof for

formation transfer of DNA activity inside cells. These

m = 46. These are very important characteristics for in-

levels









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









46,rem



, in DNA activity

ensures contrast ination transfer and correlations

inside cells at given scale r for any networks of DNA

activity. Value of

inform









,emr



depends on health status.

In Figure 7(b) maximal noise level









46, 418%emr



 corresponds to strong in-

flammations at asthma.

Distributions of





,emr

s define stable and

clear classification of noise

lation of homeostasis, for given

pe.

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

N. E. GALICH

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





,emr

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

lder



,4mr and central moments



,4Mmre



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

Bifurcation

Let us consider statistical stabil

angin

for different s

ity varied distributions of

pre-

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 0A there is a shunt

fixed point and neutral sy. The introduction of the

source of additive white noise reduquation (13) to

the Langevin equation

tabilit

ces E







 

2,2

tvAvvftftftDt t







  (14)

where





t is the Langevin source,







 is the

Dirac



-function, D



is diffusion coefficient for white

noise. Linearization near the fixed pointduces

linear Langevin eq

deviatio

vA re

uation forEquation (14) to

ns of

the small



,



Axft  (15)

The Fokker-Planck equation that corres to Equa-

tion (15) is

ponds







2,0

tx xx

AD xt







 (16)

Equation (16) determines the

stationary probability

nsity













~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,

tri

attract

appro

0Ation 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

introduce









extr

xI I , where





extr

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

asthma.

The lower parabola in Figure 8 corresponds to unsta-

ble

D

luores

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

mations

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



,4EIr



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

N. E. GALICH 39

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.

obactal

ous

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-

tropy









at rank r

entrop

in the dependence on fluorescence inten-

sity = 256; (b) Dependence of total Shannon in-

formationy





,ln

Jr r on logarithm of range r;

initial histograms at range r = 256; initial histogram shown

in Figure 9.

(a) (b)

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

N. E. GALICH

(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

1,3,5,m.

7. Conclusions

We study various large-scale dist Shannon

blood samples. We

total Shannon entropy



,lnEJrr

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





quences

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

Hölder

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

N. E. GALICH

Open Access AM

exists if [4] N. E. Galich, “Cytometric Distributions and Wavelet

Spectra of Immunofluores

nostics,” World Congress on Medical Physics

correlationsnumber of m for average of Hölder



cence Noise in Medical Diag-

and Bio-

ostics,” Journal of WASET, Vol. 70, 2010,

dative Activity of DNA in Cells for Popu-

nos-

,emr equals to m = 46, when all 46 chromosomes

involved in homeostasis regulation. The levels of satura-

tions are very important characteristics of inner and inter

chromosomal correlations, as asymptotic values the stan-

medical Engineering, Munich, 7-12 September 2009, pp.

1936-1939,

[5] N. E. Galich, “Shannon-Weaver Biodiversity of Neutro-

phils in Fractal Networks of Immunofluorescence for

Medical Diagn

dard deviation of



46,emr



, which characterizes

oise level of entropy for one chromosome, for

averages of Hölder. This level of saturation characterizes

background noise and information transfer of DNA ac-

tivity inside cells in support regulation of homeostasis,

comparisons and us abnormalities for

noises and fluctuations of information entropy of DNA in

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

statistical distributions of information

average n

detections vario

entropy; negative

NCES

tofluorometric Detection of Inflammatory Proc-

Measuring Respiratory Burst Reaction of Pe-

ripheral Blood Neutrophils,” Biochemistry and Molecular

Medicine, Vol-121.

http://dx.doi.org/10.1006/bmme.1995.1041

pp. 504-515.

http://www.waset.org/journals/waset/v45/v45-92.pdf

[6] N. E. Galich, “Complex Networks, Fractals and Topology

Trends for Oxi

lations of Fluorescing Neutrophils in Medical Diag

tics,” Phys i cs Procedia, Vol. 22, 2011, pp. 177-185.

http://dx.doi.org/10.1016/j.phpro.2011.11.028

[7] N. E. Galich, “Informational Homeostasis for Shannon

Entropy in Complex Networks of Oxidative Activit

DNA in Cells; Fractals, Stability and the Sw

y of

itching in

Large-Scale Gene Nets for Fluorescing Neutrophils in

Medical Diagnostics,” World Congress on Medical Phys-

ics and Biomedical Engineering, Beijing, 26-31 May

2012, pp. 542-545.

[8] J. Feder, “Fractals,” Plenum Press, New York, 1988.

http://dx.doi.org/10.1007/978-1-4899-2124-6

[9] T. Gneiting and M.

utral and positive stability for healthy people and do-

minants of autoimmune or inflammation diseases. Chan-

ges of stability, associated with transcritical bifurcation,

are shown in Figure 8.

To be continued.

8. Acknowledgements

Thanks to M. Filatov for kindly providing the experi-

Schlather, “Stochastic Models That

IAM Separate Fractal Dimension and the Hurst Effect,” S

Review, Vol. 46, No. 2, 2004, pp. 269-282.

http://dx.doi.org/10.1137/S0036144501394387

[10] B. Mandelbrot, “The Fractal Geometry of Nature,” W.H.

Freeman, San Francisco, 1977.

[11] D. J. Watts and S. H. Strogatz, “Collective Dynamics of

org/10.1038/30918

Small-World Networks,” Nature, Vol. 393, No. 6684,

1998, pp. 440-442. http://dx.doi.

ntal data.

o. 21,

[12] L. A. N. Amaral, A. Scala, M. Barthélémy and H. E.

Stanley, “Classes of Small-World Networks,” Proceed-

ing s o f the National Academy of Sciences, Vol. 97, N

REFERE

2000, pp. 11149-11152.

http://dx.doi.org/10.1073/pnas.200327197

[13] A. Wagner and D. A. Fell, “The Small World inside

Large Metabolic Network

Soc ie ty of L ondon. Series B, Vol. 268, No. 14

[1] M. V. Filatov, E. Y. Varfolomeeva and E. A. Ivanov,

“Flow Cy

esses by

s,” Proceedings of the Royal

78, 2001, pp.

. 55, No. 2, 1995, pp. 116

nofluo-

1803-1810. http://dx.doi.org/10.1098/rspb.2001.1711

[14] M. E. J. Newman, “The Structure and Function of Com-

plex Networks,” SIAM Review, Vol. 45, No. 2, 2003, pp.

167-256. http://dx.doi.org/10.1137/S003614450342480

[2] N. E. Galich and M. V. Filatov, “Laser Fluorescence

Fluctuation Excesses in Molecular Immunology Experi-

ments,” Proceedings of the Society of Photo-Optical In-

strumentation, Vol. 6597, 2007, Article ID: 65970L.

[3] N. E. Galich, “Bifurcations of Averaged Immu

[15] Y. A. Kuznetsov, “Elements of Applied Bifurcation The-

ory,” Springer, New York, 1995.

[16] N. G. Van Kampen, “Stochastic Processes in Physics an

rescence Distributions Due to Oxidative Activity of DNA

in Diagnostics,” Biophysical Reviews and Letters, Vol. 5,

No. 4, 2010, pp. 227-240.

http://dx.doi.org/10.1142/S1793048010001196

Chemistry,” North-Holland Personal Library, 1984.