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 ABSTRACT 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; yardstick 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: INTRODUCTION 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 r 1997 WILEY-LISS, INC. N (r) 5 const p r2D Am 5 const p VD/3. (1) (2) 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 ROUGHNESS OF THE GLOBULAR PROTEIN SURFACE 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. METHODS 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- 195 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 196 A.A. TIMCHENKO ET AL. 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 Protein Crambin Ferredoxin, Paerogenes Pancreatic trypsin inhibitor, bovine, form 1 Neurotoxin B, sea snake Neurotoxin 3, scorpion Intestinal calcium-binding protein High potential iron protein Plastocyanin Parvalbumin, carp Ribonuclease A Azurin, Alcaligenes denitrificans Lysozyme, hen egg white Lysozyme, human Nuclease, Staphylococcus aureus Dihydrofolate reductase, E. coli Lysozyme, bacteriophage T4 b-Trypsin Actinidin Elastase Chymotrypsin 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 1CRN 1FDX 4710 5400 4.15 · 104 · R22.03160.009 5.72 · 104 · R22.11360.017 4PTI 1NXB 1SN3 1ICB 1HIP 1PCY 3CPV 1RN3 2AZA 1LZT 1LZ1 2SNS 4DFR 2LZM 1TPO 2ACT 3EST 4CHA 2CGA 1SBT 2CAB 1CAC 2APP 6490 6840 7050 8470 8880 10450 11400 13670 13950 14280 14670 15980 17960 18610 23190 23380 24840 25000 25620 27480 28350 28720 33380 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. RESULTS 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 ROUGHNESS OF THE GLOBULAR PROTEIN SURFACE 197 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 198 A.A. TIMCHENKO ET AL. 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 Rc* 1.67 2.23 2.79 3.35 3.91 4.46 5.02 5.58 6.70 7.81 8.93 10.0 11.2 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 0.9934 0.9933 0.9942 0.9948 0.9951 0.9939 0.9955 0.9962 0.9927 0.9955 0.9921 0.9903 0.9901 *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 ROUGHNESS OF THE GLOBULAR PROTEIN SURFACE 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 ) (3) A , (aR )2 (4) 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 199 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. DISCUSSION The above results on low-resolution structures showed that these structures are scaled as isometric 200 A.A. TIMCHENKO ET AL. 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 size. 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. CONCLUSIONS 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 ROUGHNESS OF THE GLOBULAR PROTEIN SURFACE 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 has been proposed. REFERENCES 1. Lee, B., Richards, F.M. The interpretation of protein structures: Estimation of static accessibility. J. Mol. Biol. 55:379– 400, 1971. 2. Miller, S., Janin, J., Lesk, A.M., Chothia, C. Interior and surface of monomeric proteins. J. Mol. Biol. 196:641–656, 1987. 3. Miller, S., Lesk, A.M., Janin, J., Chothia, C. The accessible surface area and stability of oligomeric proteins. 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