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PROTEINS: Structure, Function, and Genetics 28:194–201 (1997)
Roughness of the Globular Protein Surface:
Analysis of High Resolution X-Ray Data
Alexander A. Timchenko, Oxana V. Galzitskaya, and Igor N. Serdyuk*
Institute of Protein Research, Russian Academy of Sciences, Pushchino, Moscow Region, Russian Federation
In an earlier publication
[Serdyuk, I.N. et al., Biofizika, in press, 1997]
we demonstrated that the asymmetry extent of
globular proteins does not change with increasing their sizes, and the observed nontrivial
dependence of the protein accessible surface
area on the molecular mass [Miller, S., J. Mol.
Biol. 196:641–656, 1987] (As 2 M dependence) is
a reflection of the protein surface relief peculiarities. To clarify these peculiarities, an analysis of the molecular surface on the basis of
high-resolution x-ray data has been done for 25
globular proteins not containing prosthetic
groups. The procedure was based on studying
the dependence of the minimal number (N ) of
probe bodies (here cubes) covering the entire
protein surface, both on their size (N 2 R dependence) and on the value of dry protein
volume (N 2 V dependence). Two levels of protein surface organization have been detected
by molecular surface analysis. On the micro
scale (2–7 Å), the surface is characterized by a
D 5 2.1 fractal dimension which is intrinsic to
surfaces with weak deformations and reflects
the local atomic group packing. On the macro
scale, large-scale surface defects are revealed
that are interpreted as the result of secondary
structure elements packing. A simple model of
protein surface representation reflecting largescale irregularities has been proposed. Proteins 28:194–201, 1997. r 1997 Wiley-Liss, Inc.
those results are presented here) that such a dependence results from peculiarities of the protein surface relief. The aim of this work is to elucidate the
features of these peculiarities.
An analysis of the protein surface requires an
appropriate instrument. The use of the accessible
surface area As has a disadvantage because this
surface confines the volume containing both protein
and solvent atoms.5 The concept ‘‘molecular surface’’
first introduced by Richards6 has overcome this
drawback. This surface restricts the volume inaccessible to the solvent (dry volume). The calculation of
such a surface area is more complicated than that of
the accessible one. The procedure of the molecular
surface area (Am) calculation is described in Refs. 7
and 8.
It was demonstrated that the molecular surface
reflects more adequately the protein–solvent interaction.9 At the same time, the analysis of the accessible
surface can qualitatively characterize the properties
of protein packing. So, the power law of As on M can
be considered only as a qualitative characteristic.5,10
The deviation of the power law extent from 2⁄3 in
the As 2 M dependence was considered as an indication of the protein surface fractal structure.11,12
Strictly speaking, a surface is fractal if the dependence of the minimal number of probe bodies (balls,
cubes, etc.) fully covering the surface on the probe
size is a power law:
Key words: accessible and molecular surfaces;
fractal and topological dimensions;
with the extent 2 , D , 3 not coinciding with the
topological dimension (Dtop 5 2) and D being a fractal dimension.13 A strict fractal dimension is determined at r = 0. For self-similar bodies, the relationship between the fractal surface area and the value
of confined volume (V ) has the following power law14:
The technique of protein surface quantitative
analysis using high-resolution x-ray data was first
proposed by Lee and Richards1 where the accessible
surface area (As) was analyzed. Such an analysis
gives a possibility to study overall properties of the
protein surface and its detailed structure. In particular, it has been shown2 from high-resolution x-ray
data of 37 monomeric globular proteins with molecular masses (M ) in the range of 4–35 kDa that the
dependence of As on M is a power law with an extent
of 0.73. For oligomeric proteins, this value was
found3 to be 0.76. It was shown in our paper4 (some of
N (r) 5 const p r2D
Am 5 const p VD/3.
Qualitatively at D . 2 this means that the size of
irregularities increases with the increase of the
particle size.
*Correspondence to: Igor N. Serdyuk, Institute of Protein
Research, Russian Academy of Sciences, 142292, Pushchino,
Moscow Region, Russia.
Received 7 May 1996; Accepted 29 October 1996
The above procedures (with some modifications)
for fractal dimension calculation were used to analyze the surface of some globular proteins.5,12,15,16
Thus, the analysis of the dependence of the accessible surface area on the volume confined by this
surface5 showed the fractal dimension D 5 2.3. The
dependence (Eq. 1) based on the probe body size
variation was not applied to proteins in a strict manner.
Instead, a probe ball was rolled along the surface, and
the molecular surface area15,16 or the number of tightly
packed balls on the surface12 was analyzed. In the first
case, the dependence was more complex than in the last,
but on average the fractal dimension was 2.1, which
differs from the above D value. Calculation of D from
x-ray-scattering patterns of some proteins17 gave a
broad range of D from 2 to 2.8. No systematic
analysis of protein surface fractality was made.
Here we present the systematic analysis of the
molecular surface of 25 globular proteins according
to the dependencies (Eqs. 1 and 2). The usage of both
dependencies is justified by our observation that the
extent of asphericity of proteins does not depend on
their molecular mass.4 As a result, two levels of the
protein surface structural organization have been
detected: fractal on a small-scale level (2–7 Å) and
blocklike on a large-scale level.
Recovery of the Low-Resolution Molecule
Shape With Spherical Harmonics
The detailed integral shape of protein molecules
was obtained by the procedure of envelope function
evaluation and its decomposition into spherical harmonics.18 It is possible to recover the shape of
molecules with different resolutions by using multipole coefficients (harmonics) up to a harmonic order
of L 5 7.19 Such resolution (L 5 7) permits a rather
detailed shape description, particularly, protein domain structure is clearly observed (Fig. 1). The
relationship between the area and the volume of such
bodies was calculated for a set of 25 globular proteins
not containing prosthetic groups with high resolution of
x-ray data (better than 2 Å). The names of these proteins
from the Protein Data Bank20 are given in Table I. Van
der Waals atomic radii were the same as in Ref. 8.
Construction of the Protein Molecular Surface
and Evaluation of Its Fractality
The protein surface fractal properties and the
protein dry volume were evaluated by filling a globule space with cubes having 0.3-Å edges according to
the algorithm.8 The molecular and accessible surfaces of the protein were determined by rolling a ball
(approximating a water molecule) of 2.8 Å in diameter along the protein surface.8 Both surfaces consisted of an array of cubes with a 0.3-Å edge. The
fractal dimension of the protein molecular surface
was evaluated from the minimal number of cubes
fully occupying this surface. Then the log–log depen-
Fig. 1. Recovered protein surface by spherical harmonics up
to the order L 5 7 (see text) for hen egg-white lysozyme (1LZT).
dence of the number of cubes on their edge size (see
Eq. 1) was fitted by a straight line using the leastsquares procedure.21 The slope was considered as an
estimate of the fractal dimension D. The calculations
were performed at different orientations of the protein molecule to avoid accidental deviations. The analysis showed that such a transformation of the protein
molecule coordinates does not affect the results.
The analogous dependence (N 2 R) was analyzed
for varying spheres (with the radii from 11 to 17 Å)
fully occupied by balls 3.2 Å in diameter, corresponding to the minimal size of a protein atom. Such an
analysis permits to elucidate specific features of the
protein chain packing. To design a surface similar to
that of protein, random values of ball radii in the
range of 1.6–2.1 Å were ascribed to balls with
unchanged positions of their centers. It is apparent
that internal balls cannot contribute to the surface
area value due to the inability of a water molecule to
penetrate into the sphere moiety. In general, the
molecular surface was analyzed by the above procedure, and the dependence of its area on the value of
dry volume was estimated according to Equation 2.
Construction of the Protein Moiety Reflecting
the Protein Backbone Folding
The program for construction of protein moiety to
elucidate the influence of the protein backbone folding on the properties of the protein surface has been
developed. For this aim, each Ca atom was surrounded by a sphere of 12 Å in diameter. On the one
hand, such a diameter value is close to the dimension
of secondary structure elements (a helix, b strand,
etc.) and, on the other hand, permits avoidance of
internal cavities in a protein moiety. Such a construction of the internal protein volume excludes direct
influence of side-chain group distribution on the
volume–area relationship. The shape of the constructed particle is represented in Figure 2a for hen
TABLE I. Dependence of the Number of Cubes (N) Surrounding the Molecular Surface on
Their Edge Size (R) for Proteins of Molecular Mass (M) and Spheres of Different
Diameters Filled by Balls of 1.6 Å Radius
Ferredoxin, Paerogenes
Pancreatic trypsin inhibitor, bovine,
form 1
Neurotoxin B, sea snake
Neurotoxin 3, scorpion
Intestinal calcium-binding protein
High potential iron protein
Parvalbumin, carp
Ribonuclease A
Azurin, Alcaligenes denitrificans
Lysozyme, hen egg white
Lysozyme, human
Nuclease, Staphylococcus aureus
Dihydrofolate reductase, E. coli
Lysozyme, bacteriophage T4
Chymotrypsinogen A
Subtilisin BPN8
Carbonic anhydrase B
Carbonic anhydrase, form C
Pepsin, Penicillium
Sphere (D 5 22 Å)
Sphere (D 5 24 Å)
Sphere (D 5 26 Å)
Sphere (D 5 28 Å)
Sphere (D 5 30 Å)
PDB index
Mol. mass (M)
N 2 R dependence
4.15 · 104 · R22.03160.009
5.72 · 104 · R22.11360.017
5.55 · 104 · R22.02860.014
5.90 · 104 · R22.04860.019
6.64 · 104 · R22.08560.024
7.53 · 104 · R22.08760.017
7.18 · 104 · R22.05260.021
7.85 · 104 · R22.03160.013
1.11 · 105 · R22.12760.011
1.17 · 105 · R22.09760.011
1.10 · 105 · R22.09860.014
1.11 · 105 · R22.07960.017
1.21 · 105 · R22.11060.015
1.45 · 105 · R22.11260.016
1.56 · 105 · R22.13560.018
1.64 · 105 · R22.12960.017
1.73 · 105 · R22.10660.015
1.67 · 105 · R22.09960.016
1.97 · 105 · R22.11560.013
1.97 · 105 · R22.11860.014
2.20 · 105 · R22.13360.014
1.95 · 105 · R22.11360.015
1.98 · 105 · R22.09560.009
2.84 · 105 · R22.18360.018
2.38 · 105 · R22.11360.018
2.80 · 104 · R21.96060.016
3.87 · 104 · R21.98760.030
4.24 · 104 · R22.01060.023
5.69 · 104 · R22.02760.033
5.11 · 104 · R21.94860.022
egg-white lysozyme. The external surface area of
such a particle was calculated by division of the ball
surface in pieces of equal area (here 512 pieces) and
counting the number of such pieces on the external
protein surface. The volume was calculated by placing a protein molecule in a rectangular parallelepiped divided in cubes and summation of the cubes
belonging to the protein moiety. The edge of a cube
was 1 Å. The surface properties were analyzed both
in terms of the number of probe bodies (here cubes)
fully surrounding this surface and in terms of the
volume–area relationship for 25 globular proteins.
Similar densely packed particles were constructed
to recognize the contribution of real protein backbone
packing to external surface area–volume relationship.
The particles represent a set of cubic lattices of different
dimensions (from 20 to 36 Å) with 4 Å interatomic
distance filled by balls of 12 Å in diameter and centered
at each atom of the lattice, as it was made for proteins.
The shape of such a particle is represented in Figure 2b.
Low-Resolution Protein Molecule Shape
The observed2 more rapid increase of the protein
accessible surface area (As) on the molecular mass
(M ) as compared with that for isometric particles can
be explained by two reasons. First, this can be due to
the increase in the asymmetry extent of the proteins
studied caused by the increase in their molecular
mass. Second, this can result from the increase of
protein surface roughness caused by the increase of
the protein size. As has been shown in our previous
paper,4 the first reason is not appropriate. Here we
reproduce some results of our paper4 for more clear
understanding of the problem.
The log–log dependence of the area of the approximating ellipsoids of inertia on the protein molecular
mass is represented in Figure 3. The slope of the
straight line is 0.669 with the correlation coefficient
Rc 5 0.989. The corresponding dependence of the
squared radius of gyration on the molecular mass
has the slope 0.674 with Rc 5 0.988. The values
obtained are close to 2⁄3, which is characteristic for
isometric particles. This means that the extent of
molecule asymmetry does not grow with the increase
of protein size.
The shape recovery with spherical harmonics18
described in the Methods section gives a more detailed surface structure. Thus, the protein domain
structure can be easily detected (see Fig. 1). The
Fig. 3. Log–log dependence of area–volume relationship for
the approximating inertia ellipsoids (j) and the constructed
particles (using spherical harmonics up to the order L 5 7, see
text) (d) for proteins indicated in Table I.
Fig. 2. a: The surface constructed with balls of 12 Å in
diameter centered in Ca atoms for hen egg-white lysozyme
(1LZT). b: The same for a cubic lattice with 20 Å edge and 4 Å
interatomic distance.
log–log dependence of the area of recovered particles
(using harmonics up to the order L 5 7) on the
protein molecular mass is shown in Figure 3. The slope
of the straight line is 0.669 6 0.007 with Rc 5 0.9986.
The value coincides with the value 2⁄3 for isometric
particles. This means that globular proteins are isometric on a large-scale level, including the domain structure, and a more rapid increase of the protein accessible
surface area with the growth of the protein size2 could be
explained by peculiarities of the protein surface.
Fractal Properties of the Protein Surface
As noted in the Introduction, the observed As 2 M
dependence was interpreted11 as an indication of the
protein surface fractality. One can judge the fractality more exactly following relationships (1) and (2).
Expression (2) is the most convenient for analysis of
the ‘‘molecular surface,’’ which confines the dry volume V calculated from the partial specific volume or
by the ‘‘cube method.’’8 The V and M values for
proteins are proportional to each other,4 and hence
the power law (Eq. 2) will be the same for the Am 2 M
pair (in contrast to the As 2 M dependence).
Relationship (2) is valid if all the particle surface is
fractal. At the same time, relationship (1) is applicable for analysis of any surface, but the value D in
Equation 1 will depend on the range of probe body
sizes. In other words, expression (1) indicates the
levels of a different surface organization and scans it
with different resolution. In reality, the range of
probe body sizes lies between the atom size and the
overall dimension of a macromolecule. For the globular proteins under study, the range from 2 Å to 10 Å
was chosen (Fig. 4). Beyond this range, the value of D
can be essentially less than the topological dimension (Dtop 5 2). It is clear from the simplest example
that the task is to cover the stick of length L by cubes
with edge d. If L/2 , d , L, two cubes are required
for this. In other words, in this region (a twofold
change in the probe body size) D will be zero.
Dependence (1) was analyzed for the protein molecular surface (representing a set of cubes of 0.3 Å in
size) surrounded by cubes of different sizes as described in the Methods section. The cube edge size
varied from 0.3 to 16 Å, which is sufficient for
investigating the surface fractal properties. As an
example, Figure 4 shows the log–log dependence of
TABLE II. Dependence of the Number of
Cubes (N) Surrounding the Protein Molecular
Surface on the Dry Volume Value (V)
at Different Cube Edge Values (R)
R (Å)
N 2 V dependence
1.52 · V0.76760.018
0.88 · V0.76660.018
0.56 · V0.76660.017
0.37 · V0.77060.016
0.28 · V0.76460.016
0.23 · V0.75660.017
0.17 · V0.76060.015
0.15 · V0.75360.013
0.13 · V0.72560.018
0.074 · V0.74960.015
0.086 · V0.70760.018
0.072 · V0.69660.020
0.062 · V0.68860.020
*Rc is the correlation coefficient.
Fig. 4. Dependence of the number of probe cubes fully
covering a molecular surface on their size for proteins 1FDX (X),
4CHA (d), sphere 24 Å in diameter tightly packed by balls (s). The
same dependence for the surface constructed with balls of 12 Å in
diameter centered in Ca atoms of 1LZT (S). The range for the
straight-line fitting is marked by arrows and the parameters of
fitting are given in Table I.
the number of cubes on their size (the N 2 R dependence) for two proteins with a fivefold difference in
their molecular mass and a sphere 24 Å in diameter
tightly packed by balls. The course of the dependence
is the same for other studied proteins and spheres of
different sizes. One can see a nonlinear character of
the dependence mainly due to the initial points. The
D value calculated from the first points appeared to
be less than the topological dimensionality Dtop 5 2.
The explanation of this behavior has been given
above, and here the stick is a parallelepiped consisting of two or three cubes with 0.3 Å edge size. The
number of such parallelepipeds is not negligible.
Further, the linear part of the curve marked by
arrows in Figure 4 was analyzed. This region corresponds to the probe cube sizes of 2–10 Å and, to avoid
ambiguity, is the same both for the studied proteins
and spheres.
The parameters of obtained dependencies and
their errors for all the studied proteins and spheres
are given in Table I. In all cases the correlation
coefficients are no less than 0.999. The slope values
for proteins are in the range of 2.05–2.15 with a
mean-square error s , 0.02, and in the range of
1.95–2.03 (s , 0.03) for spheres. It is seen that the
slope value for spheres tightly packed by balls virtu-
ally coincides with the topological dimensionality
Dtop 5 2, which should be expected. At the same time
the mean value D 5 2.1 for proteins differs reliably
from Dtop 5 2 and is close to 2.07 observed for a
graphite surface.22 Such a surface is characterized as
one with weak defects. No noticeable dependence of
the slope value on the particle size is observed. A
conclusion is drawn that the protein surfaces are
similar and show ‘‘graphitelike’’ fractality on the
2–10 Å scale. The main reason for the observed
fractality is the packing of atomic groups. This can
be seen from the fact that, for example, the fractality
of human lysozyme (1LZ1) changes from 2.13 to 2.06
upon ascribing the minimal Van der Waals atomic
radius (1.6 Å) and the maximal one (2.1 Å), respectively, to all protein atoms. It is clear that this
circumstance cannot noticeably influence the pattern of large-scale defects. The above fractal analysis, unfortunately, cannot show clearly the dependence of a detailed protein surface pattern on the
protein size (the above example of the stick covering
indicates the reason for this).
Large-Scale Protein Surface Structure
The N 2 V dependence of a number of probe cubes
surrounding the molecular surface on the dry volume value was analyzed at each fixed probe size. The
parameters of this dependence, their errors and
correlation coefficients are presented in Table II. As
follows from the results, the slope of the log-log N 2
V dependence is, on the average, 0.76 at a probe cube
size less than 7 Å. This value coincides with the
analogous one for the log-log Am 2 V dependence. At
larger probe cube sizes a smooth transfer to a 0.69
slope takes place (see Fig. 5 where the fractal
dimension D is given according to Eq. 2). This value
is close to 0.67, which is specific for even self-similar
bodies. The analogous dependence for spheres of
Fig. 5. Fractal dimension D calculated from N 2 V dependence versus probe cube size (R).
different sizes tightly packed with balls has a 0.67
slope, but with a greater error than that for proteins
due to a narrower range of V. A distinct dependence
of this slope on the probe cube size is not observed.
According to Eq. 2, the 0.76 slope for smaller probe
cube sizes corresponds to the fractal dimension D 5
0.76 p 3 5 2.28, and the fractal dimension from
Equation 1 is on the average 2.10 (see Table I). The
noncoincidence of these values can be evidence of
some regular peculiarities of the protein surface.
These peculiarities virtually do not affect the molecule asymmetry,4 but essentially contribute to the
surface area. It is noteworthy that the fractal dimension from Equation 2 is greater than that from
Equation 1. This demonstrates the existence of
large-scale irregularities in the protein surface.
The Influence of the Main Chain Folding on
the Fractal Properties of Protein Surface
The above analysis of fractal properties of molecular surface showed large-scale peculiarities of protein surface. The most simple picture reflecting the
above peculiarities is the case of an even body fully
covered by a set of protuberances. The most possible
candidates in protein for such protuberances are
projections of secondary structure elements on the
protein surface. The volume and the area of such a
particle (e.g., a ball) will be approximated by the
following expressions:
V , R 3 p (1 1 a/R )
A , (aR )2
where R is the sphere radius, a is proportional to the
mean size of a protuberance. If we formally plot log A
versus log V taking a 5 4 Å and R 5 8–26 Å, we
obtain with great accuracy a straight line with a
0.719 slope. A corresponding plot of the area of an
equivalent sphere of volume V gives a straight line
with a 0.667 slope. In both cases the correlation
coefficient differs from 1 by no more than 1024. It
should be stressed that the surface structure is the
same for all the particles, while the area grows faster
than that for even spheres.
The simplest model representing the above picture
is the case of similar cubic lattices of different sizes
filled by large balls as described in the Methods
section. Such a model (see Fig. 2b) permits us to
avoid a change in the asymmetry extent of particles
and demonstrate the dense packing. The fractal
properties of the external surface of such particles
were studied. The log–log dependence of a number of
probe cubes surrounding the surface on their size
gives the slope 1.989 6 0.044 with Rc 5 0.9988,
which shows that the surface is nonfractal. The
log–log dependence of the squared radius of gyration
on the volume value gives the slope 0.677 6 0.001,
which is very close to 2⁄3 for even isometric bodies. In
other words, the extent of asymmetry of such particles does not grow with their size increase. At the
same time, the log–log dependence of the external
surface area on the value of the surrounded volume
has the slope 0.710 6 0.001. This is not surprising
because the particles are really not isometric, but a
crude analysis in terms of the radius of gyration
cannot reveal this fact (the same is observed for balls
with protuberances, see above).
The same procedure as for cubic lattices was used
for real proteins to find a backbone-folding contribution to the properties of protein surface. A protein
was filled by balls of 12 Å in diameter centered in Ca
positions as described in Methods. Figure 2a shows
that such representation reflects well the large-scale
protein morphology. The log–log dependence of the
number of probe cubes on their size is presented in
Figure 4 for hen egg-white lysozyme. The slope of the
straight line is D 5 1.993 6 0.024 with the correlation coefficient Rc 5 0.9994. This value practically
coincides with that (D 5 2) for even bodies. At the
same time, the scaling of this surface on the protein
size shows a different behavior. So, the log–log
dependence of the external surface area on the value
of the surrounded volume gives the slope 0.758 6
0.015 with Rc 5 0.996 essentially different from 2⁄3 for
even isometric bodies. Thus, such a blocklike protein
surface organization can explain the observed dependence of the molecular surface area on the dry
volume (Am 2 V dependence). However, the slope
0.76 differs from the corresponding slope 0.71 for
cubic lattices, thus showing a difference between the
real backbone packing and the regular one.
The above results on low-resolution structures
showed that these structures are scaled as isometric
particles. Particularly, the properties of particles
recovered by the spherical harmonics procedure up
to the harmonic order L 5 7 are interesting. From
the one side, such particles show a clear domain
structure (see Fig. 1). From the other side, they are
scaled as isometric bodies. This means that the size
of domains grows proportionally with the protein
A detailed analysis of protein surface is required to
use a fractal language. It has been shown that fractal
dimensions calculated from Equations 1 and 2 are
different, although, for self-similar fractals, they
should coincide.14 It is interesting that the fractal
dimension calculated according to Equation 1 does
not show a noticeable dependence on the protein
size. In other words, the character of irregularities
on the 2–10 Å scale is the same for all proteins. The
obtained fractal dimension D 5 2.1 is close to the
found values12,15 D 5 2.13 and D 5 2.05.
At the same time, the higher value of D 5 2.28 was
estimated from the dependence A 2 V (Eq. 2). This
value is close to D 5 2.3 calculated from the dependence of the accessible surface area As on the volume
confined by this surface.5 D 5 2.28 can be also
estimated from the As 2 M dependence for oligomeric proteins.3 In the latter case the solvated
volume (Vo) for a globular body will approach that of
the dry volume (V) with increasing molecular mass
(M ), and hence As and M will satisfy the requirements of dependence (Eq. 2).
The higher value of the fractal dimension from the
A 2 V dependence can be interpreted as an increase
of the number of large-scale irregularities on the
protein surface with the increase of the protein size.
The model of even balls with protuberances considered above clearly illustrates this fact. It should be
noted that the analogous model was considered by
Fushman,11 but the molecular mass was postulated
to be proportional to R 3, which is not the case,
judging from formula (3). Respectively, it was decided that such a model does not reflect peculiarities
of the protein surface.
We supposed that the secondary structure elements can be candidates for detected large-scale
irregularities. This hypothesis was confirmed by the
analysis of the backbone packing pattern with large
balls (imitating the averaged dimensions of secondary structure elements) centered in Ca positions.
The surface constructed in this way is nonfractal
(D 5 2 for small probe body sizes) but has many
irregularities (intersections of spheres) (see Fig. 2a).
Such systems are called subfractals.13 Any physical
parameters which are proportional to the surface
area will follow the dependence (Eq. 2) for these
systems. For example, such behavior was detected
for protein hydration.11
The question arises whether the above large-scale
irregularities can be seen on the N 2 R dependence
(Eq. 1). In principle, the answer is positive, but the
probe body size should be comparable to the dimensions of irregularities. However, in this case the
range of probe body sizes will be narrow, and the
number of these bodies will be small, giving poor
statistics. Moreover, transfer should occur from
higher fractality D 5 2.1 to topological dimension
D 5 2 at large-probe body sizes (judging by the
approximating ellipsoids behavior and Fig. 5) that
can screen the transient higher fractality D 5 2.3.
It should be noted that the presence and packing of
secondary structure elements are important for the
observed protein surface properties. The analysis of
a set of cubic lattices filled by large balls centered at
atomic positions (see the Results section) clearly
shows disordered blocklike packing of the protein.
Some literature data also confirm this conclusion.
Thus, two a helices are packed in such a way23 that
the angle between their axes is about 50°, and the b
strands are perpendicular24 or at an angle of about
30°.25 At such packing, virtually each secondary
structure element is exposed to the solvent, and,
since the number of elements in globular proteins is
approximately proportional to the protein chain
length, this results in a fast growth of the protein
surface area. In particular this concerns a-helical
proteins. As reported in Ref. 26, the a helices in these
proteins construct a polyhedron packing surrounding the hydrophobic nucleus, and the protein surface
area will increase faster than the square of the
protein size.
It is interesting to know how such packing of
secondary structure elements influences the chemical composition of the protein surface. It is known
that hydrophobic groups are preferably localized in
the contact interface of a helices.23 The same is
observed for b strands.24,25 In other words, hydrophilic groups are concentrated on the external sides
of secondary structure elements. For small proteins
this effect could be evident. Thus, for small proteins
a little increase of the polar protein surface is
observed with the molecular mass increase.2 The
authors ascribe it to disulfide bridges screening. One
can hope that the better statistics for small proteins
could show the effect more clearly. At the same time,
for larger proteins the effect can be smoothed by
many interblock contacts.
The outlined interpretation of the protein surface
structure shows that this surface has a two-level
organization, on a micro and macro scale. The main
contribution to the observed As 2 M dependence 2 is
due to irregularities namely on the macro scale.
Two levels of the protein surface organization have
been detected by the molecular surface analysis. On
the micro scale (2–7 Å), the surface is characterized
by a D 5 2.1 fractal dimension which is intrinsic to
surfaces with weak deformations and reflects the
local atomic group packing. On the macro scale the
large-scale surface defects are revealed, which are
interpreted as the result of secondary structure
elements packing. A simple model of protein surface
representation reflecting large-scale irregularities
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