Access the full text.
Sign up today, get DeepDyve free for 14 days.
The applicability of the model for protein folding process simulation is presented using as the test two homologous proteins of different fold: helical in 3BD1 and structural form in 2PIJ [L. van Dorn, T. Newlove, S. Chang, W. Ingram, M. Cordes. Biochemistry 45, 10542 (2006)]. The folding process is assumed to be directed by hydrophobic core directing the hydrophobic residues toward the center of the molecule and exposing the hydrophilic residues on the surface. The "fuzzy oil drop" model is expressed by the 3-dimensional Gauss function which mimics the external force field. The value of Gauss function is interpreted as the hydrophobicity density calculated in any point of the space of the protein body. The accordance of idealized and observed hydrophobicity distributions (calculated according to Levitt function) measured using the Kullback-Leibler divergence entropy reveals good accordance in two homological proteins of different folds. The structural differences appeared to be easily explainable on the basis of "fuzzy oil drop" model. KEYWORDS: protein structure, homologous proteins, hydrophobicity, information theory Introduction The relationship between sequence determinants of stability for two natural homologous proteins with different folds was analysed in detail in [1,2]. These two proteins belonging to the Cro family differ in terms of secondary structure, which evolved from an -helical fold to a mixture of -helix and structure. The major structural differences are observed in the C-terminal fragment. The structural analysis of these two proteins is based here on a model of hydrophobic core, the structure of which is assumed to be described as 3dimensional Gauss function differentiating the hydrophobic density: the highest hydrophobic density localized in the center of the protein decreasing when moving away from the center, reaching zero values on the protein surface . This idealized hydrophobic density distribution is compared with the experimentally observed one, expressed as the sum of hydrophobic interactions between residues in the structure of a particular protein. It is shown that the hydrophobic density distribution is consistent with the idealized one in the proteins under consideration. The accordance is measured by applying the Kulback-Leibler deficiency entropy  expressing the distance between the two distributions, i.e. the idealized and the observed one. The structural differences between the two proteins are shown in Figure 1. Figure 1. The structure and sequence of 2PIJ and 3BD1 homologous proteins with the C-terminal fragment in 2PIJ evolved from the ancestral structural form in 3BD1, together with secondary structure plot. Picture according to PDBSum . Materials and methods "Fuzzy oil drop" model. The structures are compared using the "fuzzy oil drop" model assuming the distribution of hydrophobicity density accordant to the three-dimensional Gauss function the highest value of which is localized in the center of the ellipsoid with zero level of hydrophobicity on the surface. This distribution is assumed to express the idealized hydrophobic core in protein molecule expressed by the three-dimensional Gauss function as follows: xi x 2 ~ 1 H it ~ t exp 2 H sum 2 x 2 exp yi y 2 2 y 2 exp zi z 2 2 z ~ where H t denotes the theoretical (idealized) hydrophobic density at the point with coordinates (x,y,z), the point ( ) describes the position of the geometric center of the protein molecule in the coordinate system (usually the molecule is located in the origin of the coordinate center, making these values equal to zero). The parameters represent the size of protein molecule (in the traditional interpretation of the Gauss function this would be ~t understood as standard deviation). The parameter H sum represents the sum of ~ ~ all H t values, making the H t value standardised2. The observed (empirical) hydrophobic density in the protein molecule is calculated according to the Levitt function : r H ir H r 1 1 7 ij 1 ~e j 2 c Hi ~ e H sum j 0, forrij c r 9 ij c r 5 ij c rij c , forr c , ij where H ie denotes the empirical hydrophobic interactions of the i-th residue with all other residues, satisfying the condition of distance below the cut-off value (according to Levitt c=9Å), ( H ir H r ) expresses the sum of j hydrophobicity (hydrophobic scale according to  of interacting i-th and j-th residues and rij expresses the distance between the effective atoms representing side chains or residues. Divergence entropy To evaluate quantitatively the accordance between the idealized and empirically observed hydrophobic density distribution, divergence entropy (also known as Kullback-Leibler entropy ) was calculated: DKL ( P || Q) P(i) log 2 P(i) , Q(i) where DKL ( P || Q) denotes the distance entropy (also called deficiency/ divergence entropy), which is a measure of the distance between P(i) and Q(i) distributions (probabilities), where Q(i) plays the role of target distribution. Results Hydrophobic core structure. The distributions of idealized and empirical hydrophobic density in 2PIJ and 3BD1 are shown in the form of a profile in Figure 2.A. and 2.B. Figure 2. Idealized (T) and empirical (O) hydrophobic density distributions in A 2PIJ and B - 3BD1, respectively. The yellow line represents the random distribution (R) (all residues of equal hydrophobicity). The D KL profile for both proteins. C 3BD1, D- 2PIJ. The converted fragments are marked in order to visualize the characteristics of these fragments. Figure 2.A and 2.B show that the differences between the empirical and idealized distributions of hydrophobic density are lower than those between the empirical and random distributions. Particularly the theoretical and observed hydrophobicity density profiles in 3BD1 appears very similar. Comparison of the 3BD1 profiles with the analogical ones for 2PIJ reveals also the hydrophobic core changes accompanied the sequence differences and in consequence the structural change on C-terminal fragment. It may be seen that the differences are localized besides the C-terminal fragment also in the fragment 15-25. Kullback-Leibler entropy The idealized hydrophobic density distribution (Q(i)) is used as a reference so that the relative distance from the empirical distribution may be measured. Since the value of DKL may be interpreted only in relation to an opposite solution, the random distribution was used. If DKL for the idealized (T) versus observed (O) distributions is smaller than the distance between the observed (O) versus random (R) distributions, the observed distribution is considered as consistent with the idealized one. The protein of the DKL relation O/T < O/R is treated as accordant with the "fuzzy oil drop" model. It means that the protein represents the structure stabilized by hydrophobic core which is generally treated as the interaction responsible for stabilization of proteins' tertiary structure. The quantitative comparison of the differences between theoretical and observed hydrophobicity distribution expressed by DKL may be found in Table 1. The values shown there reveal the accordance between the observed and idealized hydrophobic density distributions (the theoretical distribution treated as target one) in both proteins. This accordance is also seen for both proteins deprived of fragments of different secondary structure. One of the two fragments of different secondary structure the helical one in 3BD1 appeared to be also accordant as expected. The -structural fragment in 2PIJ appeared to be not accordant with the idealized one. A graphic representation of Kullback-Leibler results is shown in Figure 2.C. and 2.D. The profiles of DKL values in both proteins are shown. The helical fragments converted to -fragments are marked in order to visualize the characteristics of the converted fragment. Table 1. The Kullback-Leibler divergence entropy expressing the distance between observed (O) versus idealized (T) and observed (O) versus random (R) distributions for the two proteins under consideration in their native structural forms (left column). The DKL was calculated also for the segments of proteins which differ in terms of secondary structure (central part of the table). The right column shows the D KL values for the protein deprived of the C-terminal fragments of different secondary structure. The values in bold represent the cases of accordance. -to- converted fragments (39-48aa) O/T O/R 0.3323 0.0598 0.0399 0.1067 Rest of molecule (1-38aa) O/T 0.0760 0.0608 O/R 0.1211 0.0802 PROTEIN 2PIJ 3BD1 Complete protein O/T 0.1183 0.0652 O/R 0.1232 0.0854 Discussion The results shown in Figure 2 and Table 1 reveal the structure of hydrophobic core accordant with the idealized one particularly for complete protein molecule. The high accordance of the protein fragment of common structure in two homological proteins (the structurally different fragment removed) seems to make the molecule stable assuming the hydrophobic interaction as responsible for tertiary structure stabilization. The structurally differentiated fragment appeared accordant in 3BD1 and not accordant in 2PIJ. This suggests the better fit for helical structural form although both molecules appear stable as the whole. The hydrophobic core, which according to Kauzman  model was assumed to be responsible for tertiary structure stabilization was extended to the form of "fuzzy oil drop" model. Kauzman's model originally of discrete character was extended to the form of the "fuzzy" form in "fuzzy oil drop" model. The recognition of proteins molecules of the structure accordant with the assumed "fuzzy oil drop" model supports the applicability of this model as the general mechanism for protein folding process. It makes possible simulation of this process in the presence of external force field of hydrophobic character. The folding polypeptide directs the hydrophobic residues toward the central of the "drop" with simultaneous exposure of the hydrophilic residues on the surface. This optimisation is planned to follow the traditional optimization of the non-bonding interactions inside protein molecule. The groups of proteins have been found to represent the structure accordant with the presented model. They are: fast folding proteins (downhill proteins) and antifreeze proteins. The hydrophobicity profile in these proteins is of good accordance with the model. Although some proteins have been found where the irregularity of the hydrophobic core construction appeared to be aim-oriented. The positions of residues representing hydrophobicity deficiency appeared to generate the ligand binding cavity, while the residues representing hydrophobicity excess appeared to be responsible for protein-protein complexation. The "intentional" character of hydrophobicity irregularity additionally supports the reliability of the assumed model. The requirement of highly specific character of the ligand binding cavity (or active center in enzymes) supported the assumption of the necessary presence of the specific ligand in the folding surrounding. The ligand (or substrate) molecule occupies the appropriate position (and orientation) in the gradually differentiated environment influencing the folding process of the polypeptide and ensuring the generation of specific binding cavity. This is why the folding of haemoglobin  and ribonuclease  was performed in the presence of heme and nucleotide inhibitor. The irregularities in hydrophobicity density distribution appeared to be aim-oriented creating the specific ligand binding cavity . The "fuzzy oil drop" model was positively verified also in case of transmembrane proteins . The Kullback-Leibler divergence entropy shows that the converted fragment does not fit well to the assumed model, suggesting a rather low significance of this fragment in the folding process. The hydrophobic core in these proteins seems to be sufficiently strong to stabilize the final structure The detailed analysis based on the presented model applied to the quite large set of structures of mutants of proteins reveals the influence of particular mutations on the structure.
Bio-Algorithms and Med-Systems – de Gruyter
Published: Jan 1, 2012
Access the full text.
Sign up today, get DeepDyve free for 14 days.