FRACTAL HYBRID ORBITALS ANALYSIS OF TERTIARY STRUCTURE OF PROTEIN
MOLECULE
Francisco Torrens
Institut Universitari de Ciència Molecular, Universitat de València, Dr. Moliner 50, E-46100 Burjassot
(València), Spain
The structure and shape of the polypeptide chain of proteins are determined by the hybridized states of
atomic orbitals in the molecular chain. The calculated s ratio in the spn hybrid orbitals is computed from
the fractal dimension D of the tertiary structures of 43 proteins selected so as to cover the five structural
classes of protein molecule. A brief introduction to the fractal theory is given in the text. It is
demonstrated that the principles dictating the folding of the local backbone structure and the global
backbone structure are well characterized in terms of the representation of the fractal theory. Comparison
of the fractal character of protein molecules with that of the ideal Gaussian chain revealed several
characters of these principles. It is also shown that the proteins in the structural class of  type are
distinguished quantitatively from other classes with this representation. They show a higher s ratio (ca.
0.32) in sp2.20 hybrids, rather similar to planar sp2. Comparison of the proteins with a Gaussian chain is
interpreted in terms of steric repulsion.
Index Terms- Fractal hybrid orbital, fractal dimension, fractal theory, protein conformation, structural
class.
INTRODUCTION
Several kinds of regular backbone conformation of protein have been discovered commonly in various
proteins by careful observations of the tertiary structures of proteins determined by X-ray (crystals) and
nuclear magnetic resonance (NMR, in solutions) analyses. These regular arrangements are the -helix, the
-pleated sheet and the reverse turns, and are called the secondary structures of proteins [1,2]. Careful
observations have revealed further that the aggregates of the secondary structures, which are commonly
observed in various proteins make other class of arrangements of the backbone chain, and are called the
super-secondary structures. The coiled-coil -helix [3], the  -unit [4] and the -meanders [5] are the
well-known members of the super-secondary structures. In addition to the secondary and the
super-secondary structures, the structural domains referring to those parts of the proteins, which form
1
well-separated globular regions have been revealed to be also constructive units of the protein
conformation [6]. These local arrangements of the backbone chain of protein seem not only to make
structural units in completely folded conformation but also, even though not proved yet completely, to be
folding units, which fold almost independently each other when the protein molecule folds from the
denatured conformation to the native conformation.
These local arrangements of backbone chain have been discovered mainly by careful, but
subjective and qualitative, observations of the tertiary structures of proteins resorting to the pattern
recognition ability of man. The objective and quantitative analysis of the conformation of proteins is
desirable, but is hindered by the irregularity of the structures of protein molecules. The classical Euclidean
geometry and the differential geometry have been the common tools for dealing with the forms, but their
ability has been restricted within narrow realm of forms, e.g. circle, ellipse, parabola, cubic, sphere and
the curves or surfaces, which are differentiable almost everywhere etc. While the extremely irregular
forms, such as the trajectory of a particle under the Brownian motion and also protein conformation have
been beyond the traditional geometries. Applying the differential geometry to the analysis of protein
structure, Rackovsky and Scheraga [7] have demonstrated that various types of ordered backbone
structures are well characterized in terms of the differential geometric representation. Bends had been
classified in a very natural way in their picture. Availability of their method, however, is strictly restricted
to the ordered structures or to the very short local structures like bends. A new mathematical tool, which
deals with the irregular form is desirable for the analysis of protein conformation.
A newly devised mathematical theory called the fractal theory has been developed rapidly in the
last decade [8-10]. The object of this theory is to deal with the irregular forms, e.g. the shape of cloud in
the sky or the coast line on the map etc., which have been out of the vision of the Euclidean geometry and
of the differential geometry. This theory provides means to extract a rule or a regularity hidden in the
irregular forms. If this is the case, this fractal theory seems to be of use in the analysis of the tertiary
structures of proteins, which have extremely complex irregular form.
We report here the results of an attempt to apply the fractal theory to the analysis of the tertiary
structure of protein. In the second section a brief introduction to the fractal theory will be given. The
fractal dimension is related to the bonded state of atomic orbitals and the fractal hybrid orbitals in the
third section. In the fourth section the results and discussion are given. The last section is devoted to our
conclusions.
2
FRACTAL DIMENSION OF PROTEINS
Structures of fractional dimension are known. Mandelbrot [8-10] has pioneered the theoretical concepts
and physical applications of this relatively new field of geometry [11,12] and has popularized the term
fractal for a structure characterized by a fractional dimension. By definition, any structure possessing a
self-similar or repeating motif that is invariant under a transformation of scale may be represented by a
fractal dimension. Self-similarity (scaling) is geometric in regular structures; in random or irregular
objects, self-similarity is primarily statistical in nature. The average (root mean square, RMS) end-to-end
length L of an unbranched polymer chain constitutes a statistically self-similar property. The fundamental
relationship in fractals predicts that the number of monomer segments N of length  is related to L by the
fractal dimension D
1
D
L 
L 
N    F   
 
 
(1)
where the exponent is also equal to the inverse Flory constant F in polymer theory [13-16]. Theoretical
considerations provide limits of ( 1  D  2 ) in Equation (1) that correspond to a linear structure,
L   N , and a structure represented by an unrestricted random walk, i.e., Brownian motion in D  2 ,
where L   N 1 2 [17].
Obeying the precept of the fractal theory, I define the length of a protein molecule (L) as the
function of fineness or coarseness of the scale () as follows [8-10]. At first a zigzag line is drawn by
connecting the C atoms of the protein step by step at intervals of  residues starting from the C atom of
the N-terminal residue. If there remains no enough  residues to span at the final step, leave the remaining
residues as they are and stop the drawing at the point. The parameter , the intervals at which the zigzag
line is drawn, represents the coarseness of the scale, because small  corresponds to closer observation of
the protein molecule.
The length of the molecule with a scale of  is defined as the sum of the length of the zigzag line
and a correction term, which takes account of the contribution from the residues left at the side of
C-terminal and is to be discussed in detail below. Namely the length of the molecule with a scale of  is
expressed as follows:

L   length of the zigzag line  correction term.
3
The correction term, which takes account of the contribution of the residues left unconnected at
the C-terminal side is evaluated with the equation
correction term 
N
 mean length,
 1
where N and  are the number of residues left unconnected and the scale, respectively, and the mean
length is the mean length of the fractional lines of the zigzag line, i.e. the length of the zigzag line/(the
folding number of the zigzag line 1 ).
Plotting the common logarithms of the scale  and the length of the molecule on the abscissa and
the ordinate, respectively, the fractal diagrams were drawn for 43 proteins. The 43 proteins adopted in this
work were selected from the proteins in the data base of the Protein Data Bank [18] so as to cover the five
structural classes of protein [19,20]. The names of the proteins adopted here and their classes are listed in
Table 1. Five structural classes of protein, i.e. the class of only--helices, the class of almost exclusively
-sheet, the class of -helix and -sheet tend to be segregated along the chain, the class of -helix and
-sheet tend to alternate along the chain and the class of no -helix nor -sheet, are abbreviated hereafter
, ,    ,   and miscellaneous classes, respectively.
Table 1. s ratio in the hybrid orbitals from the fractal dimension of proteins.
Protein
Res.
Da
No.
calcium binding parvalbumin B
108
1.42
Only -helices
cytochrome C (albacore, reduced)
103
1.42
hemoglobin (deoxy)
141
1.40
myoglobin (sperm whale, deoxy)
153
1.42
myohemerythrin
118
1.40
virus coat protein
219
1.28
hemerythrin
113
1.38
Mean
136
1.39
Almost exclusively
139
1.27
-chymotripsin A
concanavalin A
237
1.26
-sheet
immunoglobulin
208
1.26
prealbumin (human, plasma)
228
1.27
superoxide dismutase
151
1.30
trypsin (native, pH 8)
223
1.30
acid protease
324
1.31
rubredoxin
53
1.38
streptomyces subtilisin inhibitor
107
1.30
tosyl-elastase
240
1.29
Mean
191
1.29
carbonic anhydrase B (human)
254
1.31
-helix and -sheet
tend to be segreoxidized high potential iron
85
1.41
gated along the
protein
chain
lysozyme (hen egg-white)
129
1.42
ribonuclease A
124
1.33
thermolysin
316
1.39
papain (native)
212
1.32
trypsin inhibitor (bovine
58
1.31
pancreas)
Structural class
4
s
ratiob
0.25
0.25
0.26
0.25
0.26
0.32
0.27
0.26
0.33
0.33
0.34
0.33
0.31
0.31
0.31
0.27
0.31
0.32
0.32
0.31
0.25
nc
0.25
0.30
0.26
0.30
0.31
3.0
2.4
2.8
2.3
2.3
3.0
3.0
2.9
3.1
2.9
2.1
2.7
2.8
2.0
2.0
2.0
2.0
2.2
2.2
2.3
2.7
2.2
2.1
2.2
2.2
3.0
-helix and -sheet
tend to alternate
along the chain
5
cytochrome B5
bacteriochlorophyl-A protein
Mean
adenylate kinase (porcine
muscle)
L-arabinose-binding protein
carboxypeptidase A (bovine)
flavodoxin
glyceraldehyde-3-phosphate
dehydrogenase (lobster)
phosphoglycerate kinase (horse)
pyruvate kinase (cat)
rhodanase
subtilisin novo
thioredoxin (E. coli, oxidized)
85
358
180
194
1.35
1.25
1.34
1.36
0.29
0.34
0.29
0.28
2.5
1.9
2.5
2.6
306
307
138
333
1.34
1.34
1.31
1.34
0.29
0.29
0.30
0.29
2.5
2.5
2.3
2.5
408
432
293
275
108
1.33
1.34
1.34
1.34
1.36
0.29
0.29
0.29
0.29
0.28
2.4
2.5
2.4
2.4
2.6
triose phosphate isomerase
247
1.33
0.29
2.4
hexoki
nase A
457
1.35
0.28
2.5
yeast
phosp
hoglyc
erate
mutas
e
218
1.34
0.29
2.4
dihydr
ofolat
e
reduct
ase
148
1.30
0.31
2.2
d-gluc
ose-6phosp
hate
isomer
ase
514
1.35
0.28
2.6
Mean
292
1.34
0.29
2.5
No
-heli
x
aggluti
nin
(wheat
germ)
164
6
1.48
0.22
3.6
nor -sheet
ferredoxin (Peptococcus
aerogenes)
Mean
Mean of the five structural classes
Gaussian Chain
a Fractal dimension taken from Reference 28.
b Ratio of containing s orbital in the spn hybrid orbitals.
c n index in the spn hybrid orbitals.
54
1.33
0.30
2.4
109
211
-
1.40
1.34
1.50
0.26
0.29
0.21
3.0
2.5
3.8
In this work, we use the BABEL program, which implements a general framework for converting
between 37 file formats used for molecular modelling [21], including the Protein Data Bank (PDB) [18].
We have written a version of this program, called BABELPDB, for search, retrieval, analysis and display
of information from database PDB. Several options are allowed: (1) convert from PDB to other formats;
(2) add hydrogen atoms; (3) strip water molecules; and (4) separate -carbons. The -carbons skeleton
extracted with BABELPDB allows for drawing the ribbons image, which determines the secondary
structure of proteins. The coordinates obtained with BABELPDB have allowed for characterizing the
presence of hydrogen bonds. An algorithm for detecting hydrogen bonds has been implemented in the
TOPO program for the theoretical simulation of the molecular shape [22].
FRACTALS FOR HYBRID ORBITALS IN PROTEIN MODELS
The concept of fractal has recently been applied to a number of properties of proteins [23]. A protein
consists of a polypeptide chain that is made up of residues of amino acids linked together by peptide
bonds. An enzyme is, in general, a kind of protein with catalytic activity and long chain, and its structure
and shape are determined by the hybridized states of atomic orbitals in the molecular chain.
The polypeptide chain of proteins and enzymes resemble the Koch curve, whose shape and
conformation are related to the bond angle of atomic orbitals [24-29]. The bond angle may be regarded as
a generator. Assuming AO  OB ,  A OB   , then, the number of intervals N  2 , the similarity
ratio   1
D 


2 1  cos  , and the fractal dimension is given by
ln N
2 ln 2

1
ln 2 1  co s 
ln



For a given molecular chain, basing on the principles of the orthogonality of the hybrid molecular orbitals,
the bond angle ij between i and j orbitals is given by
cos  ij  
7
si s j
1  s i 1  s j 
(2)
where si and sj denote the ratio of containing s orbital in the hybrid orbitals i and j, respectively.
Equation (2) can be simplified as follows
cos   
s
1s
For the equivalent hybrid orbital ( s i  s j  s ), therefore,
D ij 
2 ln 2
 
l n 2 1 
 
 


1  s i 1  s j 


si s j

(3)

and
D 
2 ln 2
 
s 
ln 2 1 

  1  s 
Obviously, D depends on the bonded state of atomic orbitals, e.g., D  1.262 for sp2 hybridization.
The relation of hybridization with structural properties as the 13C NMR carbon-proton coupling
constant was reported elsewhere [30-33]. The method was successfully applied to iron proteins and other
biopolymers [34,35]. A version of the hybrid orbiltal fractal model has been implemented in the TOPO
program for the theoretical simulation of molecular shape. Another version of the algorithm has been
implemented in the GEPOL program [36-39] for the calculation of molecular volume and surface.
CALCULATION RESULTS AND DISCUSSION
Isogai et al. [40] analyzed the tertiary structures of 43 proteins selected so as to cover the five structural
classes of protein molecules: proteins with only -helices (), almost exclusively -sheet (), -helix and
-sheet tending to be segregated along the chain (    ), -helix and -sheet tending to alternate along
the chain (   ) and no -helix nor -sheet (miscellaneous). They applied the fractal theory with the
intention of devising a new tool for quantitative description of the tertiary structure of the protein. Their
results for the fractal dimension are summarized in Table 1.
The fractal dimension of the Gaussian chain is 1.5. The Gaussian chain is composed of serially
aligned identical elements, which are connected in the manner that the orientation of any element is
completely random, i.e., independent of the orientations of other elements. In other words no repulsive nor
attractive force acts between the elements. The mean value of the fractal dimension of real protein
( D  1.34 ), which reflects the local conformation of protein molecule is smaller than that of Gaussian
8
chain. This demonstrates that the local conformation of protein molecule is more extended or swollen than
that of Gaussian chain. This might be owing to the steric hindrance between the atoms, which keeps
atoms away from each other.
The fractal dimension D of various structural classes are examined separately next (see Table 1).
The magnitude of the mean values of the fractal dimension D of the five structural classes decreases in
the order of miscellaneous, ,    ,   and  classes, and the class of  type shows significant
differences between the other classes except that of miscellaneous type. The miscellaneous class shows
the largest value of the fractal dimension D, but this should not be stressed so much because this class
includes only two sample proteins in it. It is a natural consequence that the  and  classes show the
largest and the smallest value of the fractal dimension D, respectively, among the classes except the
miscellaneous class, because the local structure of  class is extended more than that of  class and those
of    and   classes fall in between. There might be several possible explanations for the
observation that only the  class takes significantly different value of the fractal dimension D from those
of other classes and no significant difference is observed between other pairs of classes statistically. One
is that the numbers of sample proteins in each class adopted in this research are too small to distinguish
the difference of the local structures of the classes, and that if the number of sample proteins are increased
the fractal analysis could distinguish these differences. Another is that the ability of the fractal analysis is
restricted with narrow limits not to be able to distinguish minute difference of the local structures of
   and   types. The fractal dimension D is defined with the local structures within the range of 10
amino acid residues. This range is too short to distinguish minute difference of the local structures of
   and   types whose remarkable local structures are characterized beyond this range. Thus the
latter explanation seems to be pertinent.
The values of fractal dimension D of various proteins are distributed within the range of 1.25 to
1.48. Isogai et al. showed that the fractal dimension D is independent of the chain length.
The calculated ratio of containing s orbital in the spn hybrid orbitals is computed from the
fractal dimension. A mean value of 0.29 predicts sp2.5 hybrid orbitals, half-way between planar sp2
hybrid orbitals and tetrahedral sp3 orbitals. In particular, -helix proteins show the lowest s ratio (0.26),
which predicts sp2.8 hybrid orbitals, rather similar to tetrahedral sp3 orbitals. Remarkably, -sheet
proteins show the greatest s ratio (0.32), which predicts sp2.2 hybrid orbitals, rather similar to planar sp2
orbitals.
9
The fractal dimensions of all proteins in Table 1 are summarized in Figure 1. The dotted line
corresponds to the Gaussian chain model. The proteins in the structural class of  type are distinguished
quantitatively from other classes and correspond to big steric hindrance.
1.5
no -helix nor -sheet
1.45
-helix
1.4
D
separate  -helix and -sheet
1.35
alternate  -helix and -sheet
-sheet
1.3
Gaussian chain
1.25
0
2
4
6
8
10
12
14
16
protein number
Figure 1. Fractal dimension of proteins. The dotted line corresponds to the Gaussian chain.
The s ratio in the spn hybrid orbitals of proteins are summed up in Figure 2. The proteins in the
structural class of  type correspond to big s ratios.
10
0.35
-sheet
alternate -helix and -sheet
0.3
s
sep arate -helix and -sheet
-helix
0.25
no -helix nor -sheet
Gaussian chain
0.2
0
2
4
6
8
10
12
14
16
protein number
Figure 2. s ratio in the spn hybrid orbitals of proteins. The dotted line corresponds to the Gaussian chain.
The n index in the spn hybrid orbitals of proteins are summarized in Figure 3. It should be noted
that the order of the indicated hybrid orbitals is reversed from that in Figure 2. The  type proteins
correspond to small n values.
3.6
not -helix nor -sheet
-helix
3.2
n
sep arate
2.8
alternate
-helix and
-helix and
-sheet
-sheet
-sheet
2.4
Gaussian chain
2
0
2
4
6
8
10
12
14
16
protein number
Figure 3. n index in the spn hybrid orbitals of proteins. The dotted line corresponds to the Gaussian
chain.
11
CONCLUSIONS
Defining a measure of the length of protein molecule as a function of coarseness of the scale to see the
molecule, the fractal analysis of the tertiary structure of protein was carried out. The fractal dimension D
was calculated for 43 proteins. The fractal dimension D
represents the complexity of the local
conformation (within the range of ten amino acid residues).
Comparison of this dimension D with that of Gaussian chain revealed that the steric hindrance
between atoms is the major factor determining the local conformation. There is nothing new about this
conclusion, but there has been no quantitative description of the major factors that dictate the formation of
local conformation. Namely, the more the steric hindrances dominate for the construction of the
conformation of protein molecule, the smaller the fractal dimension is. Conversely, the more the attractive
forces dominate for it, the greater the fractal dimension is. Thus the fractal dimension can be used as an
indicator of the average strength of the forces acting between the atoms in the protein molecule. The
stability of the structure of protein molecule, which has strong connection with the forces maintaining the
molecular structure might possibly be a good object for the fractal theory.
The fractal dimension D of various proteins were examined separately for the five structural
classes of proteins. The classes of  and  types showed the smallest and largest fractal dimension D,
respectively, reflecting the extended character of -structure and the closely packed character of -helix,
respectively. The class of  type was shown to be significantly different from other classes in terms of the
fractal dimension D, but Isogai et al. failed to discriminate the differences between the pairs of other
classes with the fractal dimension D, showing the availability and the limitation of the fractal dimension
D.
The results of an attempt to apply the fractal theory to the analysis of the tertiary structure of
protein were reported in this article. The analysis was carried out focusing on the relation between the
fractal dimension and the structural classes, which had been accepted widely. Availability and also
limitation of the theory were made clear. Detecting the minute difference of protein conformations is not
in its line, but grasping the general character of global conformation is in its line. Phenomena deeply
connected with global conformation, e.g. the stability of tertiary structure of protein, might possibly be
good objects of the fractal theory.
The structure and shape of the polypeptide chain of proteins are determined by the hybridized
states of atomic orbitals in the molecular chain. The calculated s ratio in the spn hybrid orbitals is
12
computed from the fractal dimension. The proteins in the structural class of  type are distinguished
quantitatively from other classes with this representation.
Further work is in progress dealing with the assignment of a fractal dimension to each individual
atom of the proteins using the TOPO program. A particular interest is the characterization atom-by-atom
of the ratio of containing s orbital because linear relationships exist between the ratio of containing s
1
orbital and the carbon-proton coupling constant, J 13 C H , C-H bond energy, C-H bond length, bond
angle and pKa.
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