Abstract
In this paper, the gene sequences involved in Coronary Artery Disease (CAD) were analyzed by fractal analysis. In order to obtain a quantitative characterization of nucleotide patterns and to identify anomalies and singularities belonging to genes, the fractal dimension (FD) and lacunarity were studied as fractal parameters. To obtain accurateparametric values, the Binary Image Indicator Matrix (BIIM) was used. This method converts unidimensional sequences into 2 D images preserving the autocorrelation.
The results show like the fractal values of genes CAD are very similar to the random sequences. In particular, all FD values fall in the range 1.20  1.22 highlighting that the complexity/fractality is the same and suggesting that all genes equally contribute to the pathogenesis of atherosclerosis from statistical point a view. These results are in accordance with the biological studies that indicate CAD as a multifactorial process, so that these values could be considered as characteristics for these genes.
Key words
atherosclerosis, binary image indicator matrix, fractal dimension, lacunarity
Introduction
In this work, 60 genes involved in atherosclerosis process are studied in order to investigate a possible rule that relationship the gene sequences anomalies with possible complexity/fractal nature of the distribution of nucleotides. Thus, we are assuming that biological activity of biosystem could be correlate with fractal proprieties. Starting from this assumption, we could detect the physiological state from not physiological state by fractal values. In recent past, several studies were focused on the DNA complexity and on the complexity of some important diseases such as tumors, neurodegenerative and multifactorial pathologies [14].
Some authors suggest that the fractality and complexity are involved in the degeneration of biological system that lead to origin of pathologies, for example the structural changes in the morphology of cells and tissues could cause the shift from a healthy to a pathological state, but this idea is still under investigation [59].So that seems to be a link between fractal nature and biological function and the FD and lacunarity could be consider as parametric markers of these changes.
In order to identify the characteristics values of FD and lacunarity, a fractal analysis on gene sequences belonging to atherosclerosis was conducted and the results were compared with random sequences. The apparently random distribution is considered as sign of complexity of DNA and in recent papers the existence of a fractal organizationin some diseases was investigated [10,11]. The mean idea consider the biological activity correlated with fractal proprieties [1215] and in particular, the normal and altered states could be identified by fractal values.
Atherosclerosis
The atherosclerosis is one of the most common chronic noninfectious unhealthy condition and can be considered as one of the main factor responsible for the development of coronary artery disease. Atherosclerosis is followed by the overthickness of the arterial wall as the result of many factors, some of them include deposition of oxidized Low Density Lipoprotein (LDL) and cholesterol and shear stress of the wall induced by high blood pressure. Many diseases such as diabetes mellitus, or the unhealthy lifestyle and nutrition may speed up the process of atherosclerosis. The evolution of atherosclerosis and CAD is thought to have genetic basis. From physiological point of view, populationbased epidemiological studies have confirmed that there isnot a single gene that can be considered as the main responsible for CAD pathogenesis. CAD is rather a multifactorial disease so that multiple genes, located on various chromosomes, are proposed to havea role in its pathophysiology. Recently, many efforts have been made to identify these CADrelated genes and to design an adequate database for future research. Although today much is known about the location and structure of these genetic sequences, many aspects related to their biophysical properties still remain unknown [1618].
BIIM method
The fractal values of CAD’s genes have been obtained by BIIM method. This method allows the display of the nucleotide patterns in 2 dimensions with preservation of autocorrelation between symbols of sequence [19]. In this studies we have considered only 4 symbols, in particular Adenine (A), Cytosine (C), Thymine (T) and Guanine (G) that form the DNA sequences.
The BIIM method was usually used in the studies on alignment of proteins and sequences of DNA, to evaluate the degree of homology between biological sequences and for the analysis of DNA regions [2023]. In this paper, the BIIM was used to convert uni dimensional symbolic sequences (such as DNA sequences) in 2D images and to compute the FD and lacunarity. This method allows to visualize a typical patterns of nucleotide that looks like a fractal arrangement such as demonstrated by Cattani in recent works and in the study of complexity of Caenorhabditis elegans [2,19,2426].
Fractal
According to the most popular definition, we can consider a fractal as a geometric object that is characterized by the selfsimilarity, with its structure cyclically nested at different scales. In particular, the fractal is characterized from at least four properties: selfsimilarity, fine structure, irregularities and noninteger dimension. The fractal dimension is a parameter used to describe the degree of disorder within the object. In addition, this value is the measure of information contained in the sequence, so that the higher value corresponds to a higher information content [27,28].
Lacunarity
The lacunarity is a parameter which describes the gaps present within a structure or fractal object and in general the high lacunarity means a texture with many gaps (heterogeneous distribution), instead the low lacunarity corresponds to a texture with few gaps (homogeneous distribution) [29]. In the recent past, the analysis of lacunarity was applied on medical images and in particular in the studies of pathological and normal tissue [30,31].
FD and lacunarity correlation
As a statistical parameter to verify the relation between FD and lacunarity, the Pearson product moment correlation coefficient was used.
Materials & Methods
Materials
A representative number of genes sequences of Coronary Artery Disease Gene Database (CADGD)has been taken in account [32,33]. Indetail, we have selected 60 genes thatare shown in table 1. In particular, we have 12 different categories: Vascular smooth muscle cell abnormalities, rennin –angiotensin system, oxidationreduction state, lipid and lipoprotein metabolism, endothelial integrity, immune and inflammation, gender difference, glucose metabolism, thrombosis, homocysteine metabolism, metalloproteinase  ECM and others generic genes involved in CAD.
The 60 genes are divided into 12 categories according to CAD database as follows:
Category

Gene ID

Symbol Gene

Length bp

Category

Gene ID

Symbol Gene

Length bp

Vasc. Smooth. Mus.

84159

ARID5B

195696

Gender difference

367

AR

186589


339479

FAM5C

379964


1586

CYP17A1

7004


3456

IFNB1

841


1588

CYP19A1

130543


4045

LSAMP

643177


2099

ESR1

412780


9927

MFN2

33336


2100

ESR2

111519

Reninangiot.system

118

ADD1

86221

Glucose metabolism

3938

LCT

49337


4878

NPPA

2076


11132

CAPN10

12395


1636

ACE

21321


2645

GCK

45154


185

AGTR1

45134


231

AKR1B1

16783


133

ADM

2283


387082

SUM04

689

OxRed. state

84735

CNDP1

50571

Thrombosis

7450

VWF

175798


5444

PON1

26217


5175

PECAM1

8083


6648

SOD2

14207


2149

F2R

19729


2730

GCLM

22424


2266

FGG

8618


23564

DDAH2

3224


3690

ITGB3

58871

LipidLipoMetab.

5465

PPARA

93156

Homocyst. Metab.

875

CBS

23171


350

APOH

17411


249

ALPL

69049


1581

CYP7A1

9985


2524

FUT2

9965


6720

SREBF1

25664


4524

MTHFR

20375


126

ADH1C

16270


4507

MTAP

63337

Endothelial integrity

4205

MEF2A

150499

Metalloprot. ECM

7077

TIMP2

72415


1573

CYP2J2

33445


7078

TIMP3

62228


3082

HGF

68010


1471

CST3

4282


5328

PLAU

6398


4314

MMP3

7816


7035

TFPI

90264


4313

MMP2

27507

Imm. Inflamm.

4790

NFKB1

115975

Others genes

2247

FGF2

71529


4048

LTA4H

42769


54658

UGT1A1

13028


8600

TNFSF11

45279


6546

SLC8A1

400291


10135

NAMPT

36909


10268

RAMP3

26484


6868

ADAM17

66527


3791

KDR

47338










Table 1. Genes.
Methods
The method consist in the computation of fractal dimension and lacunarity by BIIM. In particular,this method converts the un dimensional sequences in 2D imageswhereby is possible to compute the fractal parameters. Concerning the parameters, we should notice that all sequences have different lengths in bp and so that in order to make some reasonable comparisons, we havedivided the gene sequences in parts of 500 bp in length. The results shown in table 2 are the meanof values on 500 bp.
Fractal Dimension on correlation matrix
Fractal dimension iscomputed on the BIIM (correlation matrix) [13,19,20] as follows:
Let
be the finite alphabet of nucleotides and any member of the 4 symbols alphabet and let
be two Nlength DNA sequences, the indicator functionis the map[34]:
(2.1)
such that the indicator (correlation) matrix:
(2.2)
is a matrix of 0’s and 1’s showing the existence of correlation.
When the sequences are the same, the indicator function shows the existence of autocorrelation on the sequence. The indicator matrix can be used to obtain the binary image of DNA as a two dimensional dotplot, which can be used to visualize some correlations between the two sequences [13]. It can be easily drawn by assigning a black dot to 1 and a white dot to 0 [19,20] (Figure 1).
(2.2)
Figure 1.Indicator Matrix for a given sequence of nucleotides.
So that the value of fractal dimension is given by
(2.3)
Lacunarity on correlation matrix.
Let S_{n} be a given Nlength DNA sequence and v*_{hk} (h=1,...N; k=1,...4) be the 4 sequences of the Voss indicator map. On each indicator sequence, for the h^{th} nucleotide, we can consider a gliding box of rlength, so that
(2.4)
is the frequency of “1” within the box. The corresponding probability is
(2.5)
In two dimensions a natural generalization of the lacunarity will be computed as follows: let u_{hk}(h,k=1,….N) be the indicator matrix which gives rise to the binary image of DNA, the lacunarity is defined as a function of the square side, by the ratioof the second and first moment of the distribution.
(2.6)
Pearson product moment correlation coefficient
The Pearson product moment correlation coefficient is a statistical method used to describe the correlation between two or more data sets [35]. In particular, the coefficient value falls into the range [1; 1], in particular (1) for negative correlation, (+1) for positive correlation and (0) for no correlation.
Results
Fractal and Lacunarity values
In the following Table 2 the mean values with standard deviation of fractal dimension and lacunarity are given.
Symbol Gene

Fractal Value

Lacunarity Value

Symbol Gene

Fractal Value

Lacunarity Value

ARID5B

1,219 ± 0,002

0,00248 ± 0,001

AR

1,218 ± 0,002

0,002308 ± 0,003

FAM5C

1,216 ± 0,0004

0,00166 ± 0,001

CYP17A1

1,220 ± 0,0005

0,000166 ± 0,0001

IFNB1

1,220 ± 0,0001

0,00019± 0,0002

CYP19A1

1,220 ± 0,0003

0,001178 ± 0,001

LSAMP

1,219 ±0,0009

0,00337 ± 0,002

ESR1

1,218 ± 0,0006

0,00008 ± 0,0001

MFN2

1,216 ± 0,003

0,00018± 0,0002

ESR2

1,219 ± 0,001

0,00204 ± 0,002

ADD1

1,215± 0,008

0,00657± 0,006

LCT

1,220 ± 0,0002

0,00039 ± 0,0003

NPPA

1,220± 0,0006

0,00126 ± 0,001

CAPN10

1,216 ± 0,002

0,001188 ± 0,001

ACE

1,217± 0,004

0,00084 ± 0,0007

GCK

1,218 ± 0,001

0,00059 ± 0,0008

AGTR1

1,218± 0,001

0,00033± 0,0003

AKR1B1

1,218 ± 0,003

0,002308 ± 0,001

ADM

1,218± 0,001

0,00031± 0,0002

SUM04

1,220 ± 0,0003

0,000308 ± 0,0002

CNDP1

1,218 ± 0,003

0,00006 ± 0,00008

VWF

1,217± 0,004

0,005522 ± 0,004

PON1

1,219 ± 0,001

0,00033 ± 0,0003

PECAM1

1,220 ± 0,0002

0,001404 ± 0,001

SOD2

1,217 ± 0,003

0,00061 ± 0,0007

F2R

1,218 ± 0,002

0,00070 ± 0,0006

GCLM

1,215 ± 0,004

0,00008 ± 0,0001

FGG

1,216 ± 0,003

0,000167 ± 0,0002

DDAH2

1,218 ± 0,001

0,002351 ± 0,002

ITGB3

1,217 ± 0,005

0,00280 ± 0,003

PPARA

1,216± 0,007

0,001247 ± 0,001

CBS

1,217 ± 0,004

0,00067 ± 0,0007

APOH

1,220± 0,0005

0,00060± 0,0006

ALPL

1,218 ± 0,004

0,00519 ± 0,005

CYP7A1

1,218± 0,002

0,002124± 0,002

FUT2

1,217 ± 0,002

0,00023 ± 0,0002

SREBF1

1,217± 0,003

0,00150 ± 0,001

MTHFR

1,218 ± 0,002

0,000204 ± 0,0002

ADH1C

1,215± 0,002

0,00116± 0,001

MTAP

1,219 ± 0,002

0,001252 ± 0,001

MEF2A

1,214± 0,006

0,00006±0,00006

TIMP2

1,215 ± 0,006

0,000987 ±0,0008

CYP2J2

1,219± 0,0009

0,00063± 0,0008

TIMP3

1,220 ± 0,0005

0,000976 ± 0,001

HGF

1,217± 0,001

0,00026± 0,0002

CST3

1,217 ± 0,004

0,00082 ± 0,001

PLAU

1,217± 0,004

0,00057± 0,0007

MMP3

1,218 ± 0,002

0,00032 ± 0,0004

TFPI

1,217± 0,002

0,000132 ± 0,0001

MMP2

1,217 ± 0,002

0,00074 ± 0,0008

NFKB1

1,216 ± 0,005

0,001497 ± 0,001

FGF2

1,215 ± 0,006

0,002090 ± 0,001

LTA4H

1,219 ± 0,002

0,00063 ± 0,0004

UGT1A1

1,220 ± 0,0001

0,000705 ± 0,0009

TNFSF11

1,219 ± 0,0008

0,00036 ± 0,0004

SLC8A1

1,219 ± 0,001

0,000445 ± 0,0003

NAMPT

1,216 ± 0,003

0,00406 ± 0,005

RAMP3

1,217 ± 0,003

0,001580 ± 0,001

ADAM17

1,218 ± 0,0009

0,00071 ± 0,0009

KDR

1,218 ± 0,0005

0,000671 ± 0,0008

#Random 1

1,215 ± 0,002

0,00401± 0,0053

#Random 11

1,219± 0,0008

0,00101 ± 0,0008

#Random 2

1,214 ± 0,001

0,00064 ± 0,0006

#Random 12

1,214± 0,0008

0,000844 ± 0,0007

#Random 3

1.213 ± 0,001

0,00267 ± 0,003

#Random 13

1,216± 0,0002

0,00041 ± 0,0004

#Random 4

1,220 ± 0,0001

0,00065 ± 0,0006

#Random 14

1,197± 0,001

0,00717 ± 0,008

#Random 5

1,215 ± 0,002

0,00023 ± 0,0002

#Random 15

1,216 ± 0,0007

0,00105 ± 0,0009

#Random 6

1,210 ± 0,001

0,00072 ± 0,0007

#Random 16

1,213 ± 0,0006

0,00052 ± 0,0004

#Random 7

1,211 ± 0,001

0,00965± 0,009

#Random 17

1,217 ± 0,001

0,00235 ± 0,002

#Random 8

1,218 ± 0,0004

0,000235 ± 0,0003

#Random 18

1,235 ± 0,001

0,00363 ± 0,003

#Random 9

1,203 ± 0,006

0,00584 ± 0,0076

#Random 19

1,218 ± 0,001

0,00363 ± 0,003

#Random 10

1,217 ± 0,0003

0,00088 ± 0,001

#Random 20

1,219 ± 0,0004

0,00023 ± 0,0002

Table 2. Fractal dimension and lacunarity values of genes.
In table 2 are shown the higher (in yellow) and the lower (in red) values for gene and random sequences. For CAD’s genes, the FDvaluesfallinto the range1.20 to 1.22. The values of random sequences fall into the range 1.19 to 1.23 and this result suggest us that FD values for genes are very close to random.
For the lacunarity, the range of values between the higher (0.0065) and the lower (0.00019) is very small (referred to gene sequences). Similar result was obtained for the random sequences. In detail, the higher value is shown for the random sequence number 7 and the lower value belongs to the random sequence number 13. Also in this case, the difference of values is very low (0.00924).
Some notes on genes with extreme values are shown as follow:
PPARA (peroxisome proliferator activated receptor alpha) encodes for PPARA protein involved in lipid metabolism. It is situated between 46,546,424 bp and 46,639,653 bp on chromosome 22. The main pathologies associated for PPARA are: hyperapobetalipoproteinemia and tularemia [36].
MEF2A (myocyte enhancer factor 2A) encodes for a protein involved in most important cellular processes such as muscle development, neuronal differentiation, cell growth control and apoptosis. This gene belongs to chromosome 15 and it is localized between 100,017,370 bp and 100,256,671 bp. The coronary artery disease and in particular the autosomal dominant 1 typeare associated to MEF2A gene dysfunction [36].
IFNB1 (interferon beta 1fibroblast) encodes to interferon beta protein. This protein has multiple activities as: antiviral, anticancer and antibacterial. This gene is positioned on chromosome 9 in particular between 21,077,104 bp and 21,077,943bp. A oral erosive lichen and cerebral primitive neuroectodermal tumor are associated with IFNB1 dysfunction[36].
ADD1 (adducing 1 alpha) encodes for a cytoskeleton protein (adducin alpha) that acts binding to calmodulin. The gene is situated on chromosome 4 and in particular between 2,845,584 bp and 2,931,803 bp. The main disease associated to ADD1 is hypertension in particular the type salt sensitive [36].
Statistical analysis
The Pearson product moment correlation coefficient is: 0,376 showing a negative correlation between FD and lacunarity.
Discussion
In recent past, the interest for efficient diagnosis and prognosis of tumors, inflammation of tissues, multi factorial and degenerative disorders is increased.In order to better diagnosis many biological methods andcomputational tools have been used. In the last period these analytical methods were also supported by mathematical and statistical methods, because it is thought that behind the organization of cells, tissues and nucleotide distribution, seem to be some types of recurrences as fractals [3740]. Concerning this conjecture, some authors have discussed the possibility to distinguish the physiological state from pathological state by fractal/ complexity values that biological systems assume when mutate their physiological conditions [4144]. Thus, the idea that fractality is a valuable tool that allows to discriminate altered and normal states, it has been previously noticed in some diseases [4547]. Therefore, the complexity and fractal nature seem to be involved in degeneration of biological system and the authors are searching by mathematical and statistical methods to better understanding the main features of bio systems [4850].
In this work, the multifractality, as an indicator of the degree of disorder/heterogeneity of the system, has been considered in the statistical study of atherosclerosis by parameters of FD and lacunarity (inversely related).
In order to find some singularities that enable us to characterized the nucleotide patterns within the gene sequences, the values of FD and lacunarity were achieved by BIIM. In the past, this method was applied for the alignments of nucleotide sequences, analysis of different regions of DNA [1417], while in this paperit was used to obtain fractal values.
In particular, the findings shown that all mean values tend to be very close both FD than lacunarity. The results lead to consider these values as characteristics for CAD’s genes. To verify the correlation as well as the reliability of two parameters, the Pearson product moment correlation coefficient was applied. According to the negative correlation, if the value ofFDis high,it is expected that the value of lacunarity is low and viceversa. The statistical hypothesis is that exists a negative correlation between FD and lacunarity. The analysis shows that between the two parameters there is a negative value (0,376) and this means that two parameters can be assumed as reliable from statistical point of view.
The statistical study has shown that no genes seems to play a leading role in atherosclerosis and all genes considered seem to have the same fractal characteristics. This result, from a biological point a view, lead to assume that the genes have similar function in the origin of atherosclerosis confirming a multifactorial nature of pathology.
Conclusion
In this paper, the FD and lacunarity values of CAD’s genes and those of random sequences were calculated by BIIM. The analysis of fractal dimension and lacunarity showed values much similar between all sequences. In particular, the values of FD for all gene sequences fall in the range1.20 to 1.22, while the difference of values for lacunarity is only (0.00638). These results suggest that all the genes are involved equally in pathogenesis of CAD confirming the multi factorial nature of pathology from biological point a view. Therefore, the values could be assumed as characteristics for CAD’s genes and easily calculated by BIIM method. From statistical point a view, these results show that is also possible to conjecture a distribution of nucleotide very similar to a random distribution.
Conflict of Interests
The author declare that there is no conflict of interests regarding the publication of this paper.
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