Fractals and Mathematical Morphology are immensely used to study many problems in different branches of science and technology including the domain of Biology. There are many more unrevealed facts and figures of genes and genome in Computational Biology. In this paper, our objective is to explore how the evolutionary network is associated among Human, Chimpanzee and Mouse with regards to their genomic information. We are about to explore their genomic evolution through the quantitative measures of fractals and morphology. We have considered olfactory receptors for our case study. These olfactory receptors do function in different species with subtle differences in the structures of DNA sequences. Those subtle differences can be exposed through intricate details of Fractals and Mathematical Morphology

Arunava Goswami

B. S. Dayasagar

Pabitra Pal Choudhury

Ranita  Guha

Shantanav Chakraborty

Sk. Sarif Hassan

English

Nature Precedings

1 
 
Understanding Genomic Evolution of Olfactory Receptors through 
Fractal and Mathematical Morphology 
Sk. Sarif Hassan
1, 2
, Pabitra Pal Choudhury
1
, B. S. Daya Sagar
2 
, Shantanav Chakraborty
1
, Ranita Guha
 1
, Arunava 
Goswami
3
 
1
Applied Statistics Unit, Indian Statistical Institute, Kolkata, India 
2
System Science and Informatics Unit, Indian Statistical Institute, Bangalore, India 
3
Biological Sciences Division, Indian Statistical Institute, Kolkata, India 
Emails according to authors ordering above: sarimif@isical.ac.in, pabitra@isical.ac.in , bsdsagar@isibang.ac.in, 
shantanav89@gmail.com, ranita1990@gmail.com. agoswami@isical.ac.in 
Corresponding author email: sarimif@isical.ac.in  
 
Abstract 
 
Fractals and Mathematical Morphology are immensely used to study many problems in different branches of science 
and technology including the domain of Biology. There are many more unrevealed facts and figures of genes and 
genome in Computational Biology. In this paper, our objective is to explore how evolutionary network is associated 
among Human, Chimpanzees and Mouse with regards to their genomic information. We are about to explore their 
genomic evolution through the quantitative measures of fractals and morphology. We have considered olfactory 
receptors for our case study. These olfactory receptors do function in different species with the subtle differences in 
between the structures of DNA sequences.  Those subtle differences could be exposed through intricate details of 
Fractals and Mathematical Morphology.  
 
Keywords: Olfactory Receptor, Succolarity, Fractal dimension, Morphological Skeleton, Bifurcation Dimension. 
Hurst Exponent. 
 
1. Introduction: Without loss of generality, let us consider the Olfactory Receptors (ORs) OR1D2, 
CONTIG3463.6-1888, GA_x5J8B7W3YLM-7052533-7051808 of Human, Chimpanzee, and Mouse respectively 
for our case study. It is to be noted that first we have selected the olfactory receptor OR1D2 from HORDE database 
and it was blasted in the NCBI database to get highly similar OR sequences in Chimpanzee, and Mouse and we 
found the CONTIG3463.6-1888, GA_x5J8B7W3YLM-7052533-7051808 of Chimpanzee, and Mouse respectively. 
In this paper, we have captured the evolution in ORs with the help their textural quantitative views with regards to 
the fractals and morphological parameters. Also, we have shown that OR1D2 and CONTIG3463.6-1888, 
GA_x5J8B7W3YLM-7052533-7051808 are associated with two other human ORs namely OR4D2 and OR3A3.  
2. Some Basics and Fundamentals 
In this article we are about to use some standard techniques from Mathematical Morphology and Fractals. So let us 
warm up about some of the definitions from Mathematical morphology and Fractals. 
 
 
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2.1 Basics on Mathematical Morphology 
Mathematical Morphology can be used as very fundamental tool for extracting image components that are useful for 
representation and description. This field was originally developed by J. Serra [1]. The Mathematical morphology is 
based on axiomatic set theory and more relevantly lattice theory. This technique is used for image analysis which 
provides a quantitative description of geometrical structures. Morphology can provide boundaries of images, their 
skeletons, convex hulls, watershed for segmentation and many more [2, 3, 4]. The ultimate aim is to extract 
important features from image data, from which a quantitative significant understanding of the topology of the 
image can be drawn. In this article, we are emphasizing on morphological skeleton of the images and consequently 
bifurcation dimension which could be sketched out using Horton law [5, 6]. For morphological transformations we 
may refer any text book on Mathematical Morphology or some papers as given in the references [7, 8]. Also we 
have emphasized some other quantitative parameters as reviewed from a well written article [9]. 
2.2 Basics on Fractals 
The word Fractal is derived from the Latin adjective fractus. The corresponding Latin verb frangere means „to 
break‟ to create irregular fragments. In I975, B. Mandelbrot coined the subject Fractals. The precise definition of 
“Fractal” according to Benoit Mandelbrot is, a set for which the Hausdroff Besicovitch dimension strictly exceeds 
the topological dimension [10, 11, 12].  
2.2.1 Measuring Fractal dimension 
Mandelbrot founded his insights in the idea of self similarity, requiring that a true fractal “fracture” or break apart 
into smaller pieces that resemble the whole. This is a special case of the idea that there should be a dynamical 
system underlying the geometry of the set. This is partly why the idea of fractals have become so popular throughout 
science; it is a fundamental aim of science to seek to understand the underlying dynamical properties of any natural 
phenomena. It has now become apparent that relatively simple dynamics, more precisely dynamical system can 
produce the fantastically intricate shapes and behavior that occur throughout nature. Let us now talk about one 
fundamental fractal parameter “Fractal Dimension” for a self similar object. There are several methods like box 
counting method, perimeter area dimension method and so on to compute fractal dimension of an object. Let us 
focus on the self-similarity dimension as the following. 
 Given a self-similar structure [10], there is a relation between the reduction factor (scaling factor) „S‟ and the 
number of pieces „N‟ into which the structure can be divided; and that relation is… 
                       N =1/S
D
, equivalently, D =log (N)/log (1/S) 
This „D‟ is called the Fractal dimension (Self-similarity dimension)    
The fractal dimension alone does not give an idea of what “fractals” are really about. So there was a real need of 
defining some other fractal parameters. One of the important parameter is Succolarity which is really meant for the 
continuous density of the image/ fractal. The primary notion of succolarity was given by Mandelbrot and later R. H. 
C. de Melo and A. Conci described the method to compute the succolarity of an image/object [13].  
It should be noted that another important fractal parameter is known as Hurst Exponent (H) to have fractal 
dimension for a one dimensional data as explained below.  
Hurst Exponent: The concept of Hurst Exponent was introduced by Harold Edwin Hurst and later in Fractal 
Geometry, B. Mandelbrot had modified it as a parameter of relative tendency of a time series to either strongly 
regress to the mean or 'cluster' in a direction [14]. In statistical terms, it is sometime referred to long range 
correlation of a one dimensional time series. 
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Let us consider a one dimensional finite sequence , and then we can define readily the following 
entities regarding the sequence as follows: 
 
 
 
 
Then Hurst Exponent (H) is defined as  
The relation between Hurst exponent (H) and fractal dimension (D) is . 
3. Results and Discussion 
Let us see that how these Human, Mouse and Chimpanzee ORs are evolutionarily connected through Fractals and 
Mathematical Morphology.   
3.1 Evolutionary Connection of ORs of Mouse and Chimpanzee with Human ORs  
We have considered a DNA as a one dimensional nucleotide sequence and let us define a map 
 So corresponding to a DNA sequence we now have a binary string. We then 
calculate the Hurst exponent for the binary string. The result is as shown below: 
 
Olfactory Receptors Hurst Exponent (H) Fractal Dimension (D) 
OR1D2 0.598911 1.401089 
GA_x5J8B7W3YLM-
7052533-7051808 
0.645594 1.354406 
CONTIG3463.6-1888 0.539152 1.460848 
Table-I:  Hurst Exponent of ORs 
We have classified all the human ORs based on our own classification methodology on the poly-string mean and 
standard deviation as proposed in [15]. Using the same we have classified GA_x5J8B7W3YLM-7052533-7051808 
(Mouse) and CONTIG3463.6-1888(Chimpanzee) and the results are as follows: 
 
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Olfactory Receptors Class According to Poly-
String Mean 
Class According to Poly-
String SD 
Maps to (With respect to 
Hurst Exponent ) 
OR1D2 CGTA CGAT OR1D2 (Trivially) 
GA_x5J8B7W3YLM-
7052533-7051808 
GCTA GCAT OR4D2 
CONTIG3463.6-1888 CGAT ACTG OR3A3 
Table-II:  Evolutionary Connection of ORs with Human 
The Mouse OR (GA_x5J8B7W3YLM-7052533-7051808) maps to a human OR OR4D2  based on classification and 
closest Hurst exponent. But it is to be noted that GA_x5J8B7W3YLM-7052533-7051808 is more similar to OR1D2. 
But as far as Hurst exponent is concerned (amount of long range correlation in the sequence) the mouse OR maps to 
OR4D2. In this connection, it is our strong conviction that, OR4D2 is very much similar to OR1D2 in the sense of 
structural similarity in sequence, although they belong to different families as per HORDE qualitative classification. 
Also we could validate that mouse and human ORs are almost similar in structure and consequently in function too.  
The Chimpanzee OR (CONTIG3463.6-1888) maps to a human OR OR3A3 according to the classification as shown 
in Table –I and II. Although OR3A3 and OR1D2 belong to different family but with respect to evolution in 
connection with Chimpanzee OR CONTIG3463.6-1888, they are structurally almost same as per quantification 
shown above.  
 
Figure-I: Evolutionary connect ion among Human, Mouse and Chimpanzee ORs 
OR1D2, GA_x5J8B7W3YLM-7052533-7051808 and CONTIG3463.6-1888 are also most similar to OR4D2 and 
OR3A3 as shown above. They are evolutionarily connected and hence through biological evolution 
CONTIG3463.6-1888 and GA_x5J8B7W3YLM-7052533-7051808 are updated as OR3A3 and OR4D2 
respectively.  
 
 
 
 
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3.2. Fractal and Morphological Quantification of ORs 
Now, we are about to find the same fractal dimension from a different look as stated below. 
3.2.1 Fractal Dimension of Binary Image Matrix  
We consider a DNA nucleotide sequence and plot the sequence in two axes and we define a mapping as follows: 
 
 
 
Consequently, we would be able to have a binary square matrix. Now let us consider the DNA sequences of OR1D2 
CONTIG3463.6-1888, GA_x5J8B7W3YLM-7052533-7051808 of Human, Chimpanzee, and Mouse respectively. 
The corresponding matrices of the same are shown in figure-II. 
   
   [Matrix for OR1D2]                                 [Matrix for CONTIG3463.6-1888]                  [Matrix for GA_x5J8B7W3YLM-7052533-7051808] 
Figure-II: Binary Image matrices 
The fractal dimension corresponding to each of the above matrix is given in the table below: 
Olfactory Receptors Fractal Dimension (D) 
OR1D2 1.77687 
GA_x5J8B7W3YLM-
7052533-7051808 
1.81916 
CONTIG3463.6-1888, 1.82463 
Table-III:  Fractal dimensions of ORs 
Here we see that fractal dimensions of ORs of Chimpanzee and Mouse are almost same. Through genomic evolution 
they got updated into OR1D2 in human and fractal dimension is also reduced by a small amount 0.04 i.e. through 
genomic evolution amount of complexity or disorderliness got decremented.  
 
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3.2.2. Results on Succolarity  
We have taken three olfactory receptor DNA sequences corresponding to Human, Mouse and Chimpanzee. A DNA 
sequence can be thought of as a texture of four disjoint template of A, T, C and G. So we have four different 
templates of each DNA sequence. We found the succolarity for each of those three sequences as shown in the table 
below: 
Olfactory Receptors Succolarity Results 
OR1D2 Template of A 0.001026 
Template of T 0.001690 
Template of C 0.001482 
Template of G 0.000522   
 
GA_x5J8B7W3YLM-
7052533-7051808 
Template of A .0008360 
Template of T .0004520 
Template of C .0019240 
Template of G .0003520 
 
CONTIG3463.6-1888, Template of A .0018160 
Template of T .0020440 
Template of C .0026380 
Template of G .0016020 
 
Table-IV:  Succolarity of ORs 
The succolarity of all the textures of A, T, C, and G are almost same for Mouse and Chimpanzee ORs but in case of 
Human OR OR1D2 are less than same of other two ORs. It is seen that over genomic evolution the succolarity 
(amount of continuous density) in sequence structure in Human OR gets reduced than other similar sequences in 
Mouse and Human.  
3.2.3 Results on Bifurcation Dimension of Skeleton 
Let us think of a DNA sequence in terms of a four colored image as (A=Red, T=Blue, G=Green and C=Yellow) as 
shown below for CONTIG3463.6-1888.  
 
Figure-III: Colored template of CONTIG3463.6-1888 
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We then decompose the four colored image into four binary images  
 using the transformation  
 
 
Corresponding to each of the binary image we have obtained their respective skeletons of which the one for 
CONTIG3463.6-1888 is shown below: 
 
 
 
Figure-IV: Skeletons of OR Chimpanzee 
Hence, using the technique as explained by Dayasagar et al [3, 4] in for computing Bifurcation Dimension for the 
skeletons mentioned above, we have found the same as stated below: 
Olfactory Receptors Bifurcation Dimension (BD) 
OR1D2   (3.8699    1.4512    3.2358    3.5306) 
GA_x5J8B7W3YLM-
7052533-7051808 
  (3.9792    4.8959    2.3225    3.3991) 
CONTIG3463.6-1888   (1.0337    2.1623    3.7322    2.8787) 
 
Table-V: Bifurcation Dimension of the skeleton of ORs 
 
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Table-VI: Histogram of Bifurcation Dimension according to Table-V 
In table-VI the bifurcation dimensions are shown corresponding to each of OR of three different species. Typically it 
seems they do not follow a strict order. We believe this parameter would make them distinguished from each other 
in olfaction functioning. 
4. Conclusion and Future Endeavors: In this paper we have shown an evolutionary connection among Human, 
Mouse and Chimpanzee ORs. These sequences have very close sequential similarity but they do function in different 
species due to their intricate details of the structures in the DNA sequence. Those intricate details are illustrated 
here. In near future we are about to report a quantitative classification based on Fractals and Mathematical 
Morphology with some more details about all the ORs of Human, Chimpanzee and Mouse. Also we are about to 
publish all the data we generated through a Web-Server and a platform as a national facility in Mathematical 
Genomics. 
 
Acknowledgement: The authors are grateful to Prof. Jean Serra, Emeritus Professor, ESIEE-Engineering, 
University Paris-Est Former Director of Centre for Mathematical Morphology, France, and Prof. B. S. Dayasagar, 
Indian Statistical Institute, Bangalore for their kind advice and suggestions and express their gratitude to Pratap 
Vardhan, MANIT, Bhopal, for his technical help in writing programs. 
 
References: 
[1] G Matheron and J Serra, History of Mathematical Morphology, 
http://cmm.ensmp.fr/~serra/pdf/birth_of_mm.pdf  
[2] B. S. Daya Sagar, 1996, Fractal relations of a morphological skeleton, Chaos, Solitons & Fractals, v. 7, no. 11, 
p. 1871-1879.  
[3] B. S. Daya Sagar, et al 1998, Morphometric relations of fractal-skeletal based channel network model, Discrete 
Dynamics in Nature and Society, v. 2, no. 2, p. 77-92. 
[4] P. Radhakrishnan, B. S. Daya Sagar, and Teo Lay Lian, (2004) Estimation of fractal dimension through 
morphological decomposition, Chaos, Solitons & Fractals, v. 21, no. 3, p. 563-572.  
[5] B. S. Daya Sagar and Tay Lea Tien, (2004) Allometric power-law relationships in a Hortinian fractal digital 
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[6] Shih and Mitchell Threshold Decomposition of Gray -Scale Morphology into Binary Morphology, IEEE 
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Allometric power-law relationships in a Hortinian fractal digital elevation model,

Estimation of fractal dimension through morphological decomposition,

et al 1998, Morphometric relations of fractal-skeletal based channel network model,

Fractal relations of a morphological skeleton,

Image Analysis and Mathematical Morphology,

N a t u r e P r e c e d i n g s : h d l :

The fractal geometry of nature.

Value Transformation (CVT): It's Application in Fractal formation,

Zu-Guo et al Fractals in DNA sequence analysis,

Understanding Genomic Evolution of Olfactory Receptors through Fractal and Mathematical Morphology

http://precedings.nature.com/documents/5674/version/1

Understanding Genomic Evolution of Olfactory Receptors through Fractal and Mathematical Morphology

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