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Journal of Proteomics & Bioinformatics

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Proteomics in the Light of Integral Value Transformations

Sk. S. Hassan1, P. Pal Choudhury1 and S. Chakraborty2*

1Applied Statistics Unit, Indian Statistical Institute, Kolkata, India

2Indian Institute of Technology, Rajasthan, India

*Corresponding Author:
S. Chakroborty
Indian Institute of Technology
Rajasthan, India
E-mail: [email protected]

Received Date: August 20, 2012; Accepted Date: September 09, 2012; Published Date: September 11, 2012

Citation: Hassan SS, Choudhury PP, Chakraborty S (2012) Proteomics in the Light of Integral Value Transformations. J Proteomics Bioinform 5: 208-213. doi: 10.4172/jpb.1000237

Copyright: © 2012 Hassan SS, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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In this paper, proteomics have been studied in the light of Integral Value Transformations (IVTs) which was introduced in. For case study, a Human olfactory receptor OR1D2 protein sequence has been considered as the initial sequence and then different IVTs have been used to evolve OR1D2 into some other proteomic sequences. As ensued, it has been found that some of the generated sequences have been mapped to another olfactory receptor in Human or in some other species. Also it has been corroborated through fractal dimension that some of the fundamental protein properties have been nearly intact. Thus, we propose a methodology via which proteins having the same properties can be connected or grouped. This study will help to comprehend proteomic evolutionary network through IVTs.


Olfactory receptors (ORs); Box-counting dimension; Proteomics


The study of proteins such as structures, functions and evolutions is universally known to as Proteomics, was first coined in 1997 to make an analogy with Genomics, the study of the genes [1]. After genomics, proteomics is considered to be the next step in the study of biological systems. While we humans probably have only some 21 thousand genes, we possess at least 10 times that number of different proteins. The study of proteomics is important because proteins are responsible for both the structure and the functions of all living things. Genes are simply the instructions for making proteins. Therefore, a proper quantitative understanding of proteins characteristics and their inter-network are required. In this paper, an olfactory receptor OR1D2 has been considered for our analysis. Interestingly, on applying IVT systematically, we have been able to show that each DNA sequence at various discrete time instances in IVT evolutions can be directly mapped to another specific proteomic sequences existing in different species. A number of the fundamental properties namely percentage of accessible residues, alpha helix (Chou & Fasman), amino acid composition (%), beta sheet (Chou & Fasman), beta turn (Chou & Fasman), coil (Deleage & Roux), hydrophobicity (Aboderin) and total beta strand have been considered for the protein properties of the IVT generated sequences. All protein plots for all the IVT generated sequences including OR1D2 (the primitive sequence) have been generated using MATLAB (bioinformatics toolbox). Then box-counting dimension for each of the protein plots have been calculated through BENOITTM. This study will help us to ascertain potential new drugs for the treatment of various diseases.

Some Reviews and Fundamentals

In this section, we describe very briefly about IVTs, fractal and proteins.

Notion of integral value transformation (IVT)

Let us define the Integral Value Transformations (IVTs) in N0K as the following [2-5]:






m is the decimal conversion from the p adic number.

Obviously for equation system there are equation number of possible rules and out of them the function number that we select is indicated by j.

Let us fix the domain of IVTs as N0 (k=1) and thus the above definition boils down to the following:


where m is the decimal conversion from the p adic number, and


Now, let us denote the set of equation as



Where m is the decimal conversion from the P adic number and equation

Let us define the IVT in N0 in 4-adic number systems. There are 256equation one variable four state CA rules. Corresponding to each of those CA rules there are 256 IVTs in 4 adic system in one dimension.

IVT4,1 # is mapping a non-negative integer to a non-negative integer.


Where ‘a’ is a non-negative integer and equation and ‘b’ is the decimal value corresponding to the 4-adic number.

For an example, let us consider a = 225 = (3201)4 and equation

Therefore, equation

Consequently, equation

Let us denote T4,1# as set of all IVTp,k # transformations. It is worth noting that there are 4! = 24 number of bijective functions in T4,1# . So out of the 256 equation transformations in T4,1 # four are linear and rest is nonlinear [6].

Fractal and fractal dimension

Our artificial world can be described easily through Euclidean geometric shapes but there are many things in nature such as shape of cloud, geometry of lightening etc. could not be described through Euclidean geometry. Many mathematicians descended the challenge for a fair enough description of natural objects but after a long period in 1975, B. Mandelbrot took up the challenge and gave the birth of a new geometry to describe nature which is known to us as ‘Fractal Geometry’ (in short ‘Fractal’). The precise definition of "Fractal" according to Benoit Mandelbrot is a set for which the Hausdroff Besicovitch dimension strictly exceeds the topological dimension. To gain a quantitative insight of Fractal, some fractal parameters namely Fractal dimension, Hurst exponent, succolarity, lacunarity etc. are also introduced in the literature. A brief discussion follows about one of the well-known methods of calculating fractal dimension namely ‘Box- Counting method’.

Box-Counting Method: This method computes the number of cells required to entirely cover an object, with grids of cells of varying size. Practically, this is performed by superimposing regular grids over an object and by counting the number of occupied cells. The logarithm of N(r), the number of occupied cells, versus the logarithm of 1/r, where r is the size of one cell, gives a line whose gradient corresponds to the box dimension [7].

Problem in protein structures

Proteins are an important class of biological macromolecules present in all organisms. After the structure of DNA was discovered by James Watson and Francis Crick, who used the experimental evidence of Maurice Wilkins and Rosalind Franklin (among others), serious efforts to understand the nature of the encoding of proteins began. George postulated that a three-letter code must be employed to encode the 20 standard amino acids used by living cells to encode proteins, because 3 is the smallest integer n such that 4n is at least 20 [8]. The three-dimensional structures of proteins were first determined by X-ray diffraction analysis; Perutz and Kendrew shared the 1962 Nobel Prize in Chemistry for these discoveries. At present, more than ten thousand protein structures were found with their atomic details. The structure of the protein is ultimately defined by its primary structure, or amino acid sequence. There are no theories or computational techniques at the moment which will allow us to predict the new protein folding by its sequence. Even, how protein sequences and their tertiary structures are evolved during evolution remains unclear. Therefore proper understanding is required at the primary structure level i.e. in the amino acids sequence level of proteins.

Methods and Results

Method of sequence generation through IVTs

The domain of action of IVTs is a set of non-negative positive integers. So it is required to have a numeric sequence corresponding to each of the proteomic sequence. A simple mapping f is defined below:

Let P = {A,C,D,E,F,G,H, I,K,L,Q,N,P,Q,R,S,T,V,W,Y} be the set of amino acid codes and


Therefore, a protein sequence is now simply a string of twenty variables namely 0, 1, 2…19 as per coding scheme f.

Starting from a protein sequence to generate another proteomic like sequences, it is required to have all the IVTs in a particular T(p,1)# ,which maps N to itself (bijective rules).

The list of some such IVTs in T(p,1)# is given below in Table 1.

P-adicTable Table Table Table Table Table Table Table Table
# 1 2 5 11 21 99 114 147 177 180 210 225 228 194 214 294 334 414 434 694 714 894 28565 28595 28745 28805 28955 28985 29860 29890 30040 30100 297051 297093 297393 297435 299109 299151 299793 299835 5135375 5135431 5135886 5135942 5138959 5139015 5139981 102907844 102907916 102908572 102908644 102913676 102913748 102915132

Table 1: IVTs in Table.

Now we apply Integral Value Transformations ,equation systematically [3-10]:-

Firstly, divide the whole one dimensional initial sheet of proteomic sequence (numeric sequence) of length n into multiple blocks. We designate the initial sequence as S(t0 ) .

Secondly, we apply bijective domain preservative transformations (need not to be all distinct) taken from ,equation for different p starting from 2 to 19) over each of the r different blocks in t0. We call such application of different rules to different blocks as Hybrid Application of IVTs. In other words, we are getting S(t1) from S(t0)through hybrid application of IVTs. Next, we follow this step successively as long as we wish to iterate. The results, on applying the proposed systematic technique of application of IVTs on OR1D2 are enumerated in the following subsections.


Here we discuss the results on applying different IVTs in two following cases.

On applying ,equation: The proteomic sequence of OR1D2 is of length 312 (sequence shown below in Text-1). Choose r=50, so there are 7 blocks are there. The following two IVTs are used to generate S(t1) as shown below in Table 2.

BLOCK Sequence-1 in 2 adic IVT Sequence-1 in 3 adic IVT Sequence-1 in 4 adic IVT
Block-1 IVT2,11 IVT3,15 IVT4,199
Block-2 IVT2,11 IVT3,15 IVT4,1114
Block-3 IVT2,12 IVT3,111 IVT4,1147
Block-4 IVT2,11 IVT3,111 IVT4,1177
Block-5 IVT2,12 IVT3,121 IVT4,1180
Block-6 IVT2,12 IVT3,121 IVT4,1110
Block-7 IVT2,12 IVT3,121 IVT4,1225
BLOCK Sequence-1 in 5 adic IVT Sequence-1 in 6 adic IVT Sequence-1 in 7 adic IVT
Block-1 IVT5,1194 IVT6,128565 IVT7,1297051
Block-2 IVT5,1214 IVT6,128595 IVT7,1297093
Block-3 IVT5,1294 IVT6,128745 IVT7,1297393
Block-4 IVT5,1334 IVT6,128805 IVT7,1297435
Block-5 IVT5,1414 IVT6,128985 IVT7,1297109
Block-6 IVT5,1434 IVT6,128955 IVT7,1297151
Block-7 IVT5,1694 IVT6,128960 IVT7,1297793
BLOCK Sequence-1 in8 adic IVT Sequence-1 in 9 adic IVT  
Block-1 IVT8,15135375 IVT9,1102907844  
Block-2 IVT8,15135431 IVT9,1102907916  
Block-3 IVT8,15135886 IVT9,1102908572  
Block-4 IVT8,15135942 IVT9,1102908644  
Block-5 IVT8,15138959 IVT9,1102913676  
Block-6 IVT8,15139015 IVT9,1102913748  
Block-7 IVT8,15139981 IVT9,1102915132  

Table 2: IVTs from Table used for generation of Table


Text 1: Protein Sequence of OR1D2.

Similarly, other S(ti)can be generated applying the IVTs in different blocks of the S(ti-1)as tabulated in supl.met-I. We have generated 90 such S(ti)s corresponding to OR1D2 in each ,equation system (for p=2, 3… 20) (available in supl. met.-II).

All these generated sequences have been blast in the NCBI database for significant similarity. The blast result is shown in supl. met.-III.

Most of the generated sequences are mapped to olfactory receptors (specifically close to OR1D2) in different organisms like homo sapiens, pan troglodytes, lagothrixlagotricha etc. Some of the sequences are not mapped due to the fact that they are more conserved sequence than OR1D2.

Also we have been observed that some of the protein primary structural properties (listed below) are intact with respect to the two dimensional protein plot graphs (using bioinformatics toolbox of Matlab-R2010b) for each of the generated sequences.

The protein properties which we have considered here are as follows:

• Prop-1: Accessible residues (%)

• Prop-2: Alpha helix (Chou &Fasman)

• Prop-3: Amino acid composition (%)

• Prop-4: Beta sheet (Chou &Fasman)

• Prop-5: Beta turn (Chou &Fasman)

• Prop-6: Coil (Deleage& Roux)

• Prop-7: Hydrophobicity (Aboderin)

• Prop-8: Total beta strand

Corresponding to each property of the S(ti), we have had eight protein plot graphs from which we have calculated box counting dimensions using BENOIT™.

The data for OR1D2 sequence are stated below in the Table 3. The rest of the data are available in the supl. met-IV. We have observed that the box-counting dimensions for all the eight protein plots corresponding to each of the protein property for all the generated sequences S(ti)s are almost same to the same of OR1D2. The data for all the box counting dimension of protein plots for the S(ti) generated through the equationsystem is shown below. Hereby we can come to a conclusion that these IVTs preserve the protein properties of the strings. It is to be noted that all these IVTs are bijective; therefore one can switch from one protein to another protein through the IVTs without encumbering the protein properties. Most of the S(ti) (IVT generated sequences) preserve all the eight protein properties. It is to be noted that in the case equation system, S(t1) and S(t2) are both mapped to G-protein-coupled receptor in OR1D2 in human. Also they follow all the protein properties as in OR1D2 (Table 4).

Sequence Property Box-counting dimension
OR1D2 Prop1 1.91092
  Prop2 1.91103
  Prop3 1.90855
  Prop4 1.91141
Sequence Property Box-counting dimension
  Prop5 1.91095
  Prop6 1.91348
  Prop7 1.90989
  Prop8 1.91071

Table 3: Box-counting dimension for protein plots of OR1D2.

Sequence Property Box-counting dimension
S(t1) Prop1 1.92694
  Prop2 1.91117
  Prop3 1.90976
  Prop4 1.91111
  Prop5 1.9113
  Prop6 1.93038
  Prop7 1.91021
  Prop8 1.91144
S(t2) Prop1 1.91124
  Prop2 1.91099
  Prop3 1.91389
  Prop4 1.90948
  Prop5 1.91064
  Prop6 1.93051
  Prop7 1.91398
  Prop8 1.90983
S(t3) Prop1 1.91045
  Prop2 1.91049
  Prop3 1.90994
  Prop4 1.91299
  Prop5 1.92765
  Prop6 1.91648
  Prop7 1.92813
  Prop8 1.91448
S(t4) Prop1 1.91294
  Prop2 1.91495
  Prop3 1.91084
  Prop4 1.9108
  Prop5 1.91155
  Prop6 1.91577
  Prop7 1.9281
  Prop8 1.93043
S(t5) Prop1 1.91443
  Prop2 1.91431
  Prop3 1.91259
  Prop4 1.93055
  Prop5 1.92909
  Prop6 1.91638
  Prop7 1.92901
  Prop8 1.91676
S(t6) Prop1 1.92863
  Prop2 1.928
  Prop3 1.91431
  Prop4 1.9295
  Prop5 1.91133
  Prop6 1.91751
  Prop7 1.91379
  Prop8 1.91292
S(t7) Prop1 1.91421
  Prop2 1.928
  Prop3 1.9142
  Prop4 1.91614
  Prop5 1.9101
  Prop6 1.91402
  Prop7 1.9108
  Prop8 1.91314
S(t8) Prop1 1.9104
  Prop2 1.91378
  Prop3 1.91039
  Prop4 1.91287
  Prop5 1.91177
  Prop6 1.91392
  Prop7 1.90987
  Prop8 1.91378
S(t9) Prop1 1.91428
  Prop2 1.91129
  Prop3 1.91367
  Prop4 1.91337
  Prop5 1.91263
  Prop6 1.91431
  Prop7 1.91084
  Prop8 1.91413
S(t10) Prop1 1.91082
  Prop2 1.9108
  Prop3 1.91081
  Prop4 1.91337
  Prop5 1.91263
  Prop6 1.91514
  Prop7 1.91084
  Prop8 1.9176

Table 4: Box-counting dimension for all protein plots of Table in Table

But interestingly, there are many S(ti ) in different ,equation systems, do not map significantly in any organisms but they possess the protein properties as in OR1D2. One of the main reasons for this is that most of the sequences are conserved whereas OR1D2 is not so. Some of the S(ti ) are not mapped to any of the ORs in any organism although the box-counting dimension for all the protein plots are intact as it is in OR1D2. It is our strong conviction that these S(ti ) serve the purpose for replacement of OR1D2 in the genetic evolutionary future. In the next section we are about to discuss the case on applying the bijective IVTs from equation(Table 5).

Sequence Property Box-counting dimension
S(t1) Prop1 1.90836
  Prop2 1.91371
  Prop3 1.92937
  Prop4 1.91313
  Prop5 1.92746
  Prop6 1.9128
  Prop7 1.91234
  Prop8 1.91291
S(t2) Prop1 1.91418
  Prop2 1.91204
  Prop3 1.91182
  Prop4 1.91205
  Prop5 1.91418
  Prop6 1.92998
  Prop7 1.9099
  Prop8 1.91351
S(t3) Prop1 1.91459
  Prop2 1.91308
  Prop3 1.91151
  Prop4 1.91464
  Prop5 1.91434
  Prop6 1.91216
  Prop7 1.91306
  Prop8 1.91321
S(t4) Prop1 1.91087
  Prop2 1.91468
  Prop3 1.90957
  Prop4 1.90991
  Prop5 1.92755
  Prop6 1.9159
  Prop7 1.9104
  Prop8 1.91369
S(t5) Prop1 1.91448
  Prop2 1.91485
  Prop3 1.92691
  Prop4 1.914
  Prop5 1.9123
  Prop6 1.91203
  Prop7 1.92751
  Prop8 1.92845
S(t6) Prop1 1.91315
  Prop2 1.91176
  Prop3 1.91169
  Prop4 1.91317
  Prop5 1.91348
  Prop6 1.91507
  Prop7 1.91141
  Prop8 1.92879
S(t7) Prop1 1.91258
  Prop2 1.91057
  Prop3 1.91388
  Prop4 1.91508
  Prop5 1.92907
  Prop6 1.91605
  Prop7 1.91244
  Prop8 1.91098
S(t8) Prop1 1.92725
  Prop2 1.92767
  Prop3 1.91331
  Prop4 1.91074
  Prop5 1.91459
  Prop6 1.91608
  Prop7 1.90883
  Prop8 1.91143
S(t9) Prop1 1.90984
  Prop2 1.92917
  Prop3 1.9154
  Prop4 1.91098
  Prop5 1.91336
  Prop6 1.91545
  Prop7 1.91013
  Prop8 1.92845
S(t10) Prop1 1.91286
  Prop2 1.91425
  Prop3 1.91506
  Prop4 1.91402
  Prop5 1.92938
  Prop6 1.91632
  Prop7 1.91337
  Prop8 1.9125

Table 5: Box-counting dimension for all protein plots of Table in Table

On Applying equation : We have chosen a few bijective IVTs (available in supl. met.-I) from system to generate S(ti) from equation system to generate S(ti) from the protein code for OR1D2 (methodology is discussed in 3.1). Here all the S(ti) have been blasted in NCBI and they all are mapped to G protein-coupled receptor, or MOR30-1, hypothetical protein and conserved hypothetical protein in different organisms ranging from human to plasmodium species (data shown in supl. met.-III). The box counting dimension is still intact for all the protein plots for all the IVT generated sequence in equation system as shown in (Figure 1) (raw data shown in supl. met-IV). It is noted that the number of bijective, domain preservative IVTs is increased as p increased in T(p,1)# . Consequently the sequential conservation is inversely proportional to p.


Chart 1: Flow Chart for our methodology for a generalith iteration.

Summary and Conclusion

In summary, we have seen that IVTs steer a given OR sequence of a species to another of the same or different (most likely) species, preserving the protein properties of the original sequence. This methodology will be helpful to mimic the genomic evolution procedure artificially, which is required for genetic replacement therapy. IVTs may also be considered to be a platform to comprehend the morphological connections among the various species. A naïve question to the biologists, which rose amongst us:

Suppose, we are given an olfactory receptor or1 of a species s1 which help it to identify the odors x1, x2,…

Now, we apply the proposed methodology to or1 and obtain a new olfactory receptor or2 (supposedly) of species s2.

So, does or2 help s2 in identifying the same odors x1, x2,...?

In the near future, we are really interested to explore the underlying biological methodology that governs the entire process.


Authors would like to thank to their visiting students Anjan Pal and Snehasish Banerjee for their enormous help in writing computer programs.


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