The 1AON – the structure of the object under consideration has been taken from the PDB (deposit 1AON) [15].

It is assumed that the presence of an external force field of hydrophobic character expressed by the three-dimensional Gauss function is able to direct the protein folding toward hydrophobic core generation, with the simultaneous exposure of hydrophilic residues toward the surface of the protein molecule.

The external force field is represented by the three-dimensional Gauss function. The value of the Gauss function (traditionally interpreted as probability density value) is assumed to represent the hydrophobic density in the protein body. The hydrophobicity density can be calculated for any point the space covering the protein molecule.

The three-dimensional Gauss function is given as follows: H t j = 1 H t s u m   exp ( − ( x j − x ¯ ) 2 2 σ x 2 )   exp ( − ( y j − y ¯ ) 2 2 σ y 2 )   exp ( − ( z j − z ¯ ) 2 2 σ z 2 )

The value of Ht_j is assumed to represent the hydrophobicity distribution at a particular point belonging to the protein body. The hydrophobicity maximum is localized in the center of the ellipsoid (x̄, ȳ, z̄) and decreases in a distance-dependent manner according to the Gauss function. The mean value at which the Gauss function reaches its maximum is localized at the (0.0.0) point in a coordinate system. The standard deviation values σ_x,σ_y,σ_z calculated for each dimension (axis) separately represent the size of the drop which depends on the length of the polypeptide under consideration [7].

The j-th grid point, for which the hydrophobicity is calculated, represents the effective atom position (averaged position of the side chain including Cα atoms) making possible attachment of a particular hydrophobicity density to a particular amino acid.

Before the external hydrophobic force field can be defined for any protein molecule, it shall be oriented in the space according to following procedure:

the geometric centre of the molecule shall be localized in the center of coordinate system.

This is how the geometric parameters of protein molecule can be interpreted according to the Gauss function.

Taking into account the high symmetry of the system under consideration, a user defined orientation of the coordinate system is necessary. Thus, the initial orientation determining the X-axis is defined by the position of the symmetrical units, so the averaged position of the top elements and averaged position of bottom elements (user-defined) determine the initial orientation of the molecule. The user-defined orientation of the molecule is available and necessary for any protein molecules or complexes before the “fuzzy oil drop” model can be applied. The X-axis defined this way is simultaneously the 7-fold symmetry axis.

The empirical (observed) distribution of hydrophobicity can be different than the idealized one. The empirical hydrophobicity distribution can be calculated according to Levitt function [16]: H ˜ o j = 1 H ˜ o s u m ∑ i = 1 N ( H i r + H j r ) { [ 1 − 1 2 ( 7 ( r i j c ) 2 − 9 ( r i j c ) 4 + 5 ( r i j c ) 6 − ( r i j c ) 8 ) ] 0   for   r i j > c   for   r i j ≤ cwhere Ho_j represents the empirical hydrophobicity value characteristic for the j-th grid point, H i r represents the hydrophobicity characteristic of the i-th amino acid, r_ij is the distance between the j-th grid point and i-th effective atom in the amino acid, and c expresses the cutoff distance, which has a fixed value of 9.0 Å following the original paper [16].

The continuity of the Gauss function allows calculation of the hydrophobicity density in any point in space (in the protein body). So any point can be treated as a grid point. It can also be the position of an effective atom in particular. This is why the index j may represent the position of j-th residue, as it is taken in this work. Ho_sum represents the sum of all the grid points hydrophobicity. Any hydrophobicity scale may be applied to calculate the observed density of hydrophobicity [16–21].

Since both values are standardized (the coefficient 1 H sum) the differences between theoretical and empirical values expressing hydrophobicity density in a particular point of space (position of the side chain of i-th amino acid) can be calculated according to: Δ H ˜ i = H t i − H o i

The profile of ΔH̃_i (expressing the value of difference for each amino acid) reveals some maxima, which are related to hydrophobicity deficiency. The hydrophobicity deficiency (ΔH̃_i > 0) seems to represent the potentially ligand binding site. The potential ligand may adhere in this area as the complementary element compensating the hydrophobicity deficiency producing in effect the regular smoothed hydrophobicity distribution. The values of negative ΔH̃_i represent the area of hydrophobicity higher than expected. The area of such characteristics, when localized on the surface of protein seems to represent the potential area responsible for protein-protein complex creation.

The profile of ΔH̃_i values can show the discrepancy between the idealized and empirical hydrophobicity density distribution revealing the fragments (or individual residues) representing the hydrophobicity excess (ΔH̃_i negative) and hydrophobicity deficiency (ΔH̃_i positive). It is expected that the hydrophobicity irregularity versus idealized one may express localization of biological function-related area in the protein body.

The 1AON is a quite large and complex protein molecule. This is why different approaches have been applied.

The complete molecule was treated as one uniform “drop” – the orientation of molecule was according to its 7-fold symmetry axis (the X-axis).

The partitioning shown above is aimed to define the folding unit and possible path leading to complex generation (Figure 1). 1AON represents the chaperonin additionally complexed with ADP and some Mg⁺² ions. The characteristics of localization of these ligands will be analyzed with respect to the construction of the “fuzzy-oil-drop”.

The cut-off of 3.9 Å was taken to identify the residues interacting with ions, ligand (ADP) and protein (chain). The cut-off value has been taken according to the criteria applied in PDBsum [22] data base to make possible the comparison of results.

All results have been obtained using our own program, written in Python [23] programming language. The program has been divided into multiple subroutines to ensure flexibility and diversity in dealing with PDB [24] files, which was required to work on protein partitions described above.

The first routine reads the input PDB file, removes all water residues and classifies non-empty chains into three groups: protein, nucleic and “hetero”. By “hetero” we mean neither protein nor nucleic acid chains, presumably containing only heteroatoms. All operations regarding PDB files are conducted using methods implemented in the Biopython [25] library.

The second routine is the core of the “fuzzy oil drop” evaluation: it performs an extraction of the residue subsets from the protein chains that the user is interested in and computes their effective atoms, which form the “drop”. The drop is placed in the origin and rotated to achieve a desired spatial orientation, as stated in the model description. After the drop size becomes known, theoretical and observed hydrophobicity are evaluated using optimized array operators from Numpy [26] (Numerical Python) library.

Identification of interactions is conducted using a fast k-d tree algorithm implemented in the Biopython library by checking position of each atom from selected residues against all atoms in file. Each neighbor atom placed within given radius of 3.9 Å is unfolded into parental residue and then classified as either protein, nucleic, ligand or ion contact. Contacts with protein residues from same chain are ignored.

The last routine sets output values for every residue (same for each atom) in PDB file: normalizes hydrophobicity discrepancy by overwriting the beta-factor column and contact type by overwriting the occupancy column. Plotting of graphical representation of results is made using the Matplotlib library [27].

The agglomerative hierarchical clustering “ahc” algorithm has been applied to analyze the agreement between subjective interpretation of the “fuzzy oil drop” model applicability and objective discrimination of elements representing the common characteristics [28]. Clustering is the assignment of objects (feature vectors) into groups called clusters so that objects from the same cluster are more similar to each other than objects from different clusters. The short description of “ahc” algorithm is as follows. Suppose we have a data set X = {x₁,..., x_N} of N objects to be clustered and a user-defined distance measure d(x_i, x_j) to state how similar the two objects x_i and x_j are, for any i and j. The “ahc” algorithm starts off clustering this data by putting each of the data objects x_i in a singleton cluster C_{i} = {x_i} and then keeps on joining the closest pair of clusters C_{i} ∩ C_{j} = C_{i,j} until there is only one large cluster C_{1,2,...,N}. Simultaneously, the clustering tree (dendrogram) is built from leaves towards root, where merging of clusters is depicted as a common parent for two sub-trees. The distance between clusters C_i = C_{i1,...,jp} and C_j = C_{j1,...,jr} can be measured in different ways. Three popular methods are single, complete and average linkage: (single) d min   ( C i , C j ) = min x ∈ C i , y ∈ C j d ( x , y ) (complete) d max   ( C i , C j ) = max x ∈ C i , y ∈ C j d ( x , y ) (average) d a v g ( C i , C j ) = 1 | C i | ⋅ | C j | ∑ x ∈ C i , y ∈ C j d ( x , y )where | C | denotes the number of objects in a cluster C. Thus, the distance between clusters is the minimum (or maximum or average) of the distances between one object from the first and another from the second cluster. Once the dendrogram is generated for assumed k clusters, the procedure cuts the k-1 longest links in a dendrogram. The described “ahc” algorithm was implemented as a function in the software package Matlab v.7.

First, the implemented “ahc” algorithm was used to cluster separately objects in each of the files according to the partition (section 2.3) into k=2 groups. Each object x_i in these files is represented as a four dimensional feature vector x_i = (x_i1, x_i2, x_i3, x_i4), where x_i1, x_i2, x_i3 are the spatial coordinates x,y,z of the object x_i while x_i4 is its estimated variable ΔH̃. For each object x_i its correct classification (being label ‘0’ or ‘1’ (i.e. “0” – means residue engaged in protein-protein interaction, ‘1’ denotes the residue not engaged in the protein-protein interaction) is known (according to PDBSum criteria). The clustering result for each partitioning form (as described in Section 2.3) compared with the correct classification allowed the calculation of the clustering performance defined as the number of objects correctly assigned to a groups divided by the total number of objects in a file.

The ΔH̃ profile of chain A taken to represent the chains A-G is shown in Figure 2. The residues engaged in ligand binding are distinguished as well as the residues involved in protein-protein complexation. According to “fuzzy oil drop” the ΔH̃ minima representing the hydrophobicity excess on the surface of protein is assumed to be potential protein-protein contact area. The ΔH̃ maxima (hydrophobicity deficiency) in domains distinguished in chain A are observed in accordance with the interpretation of “fuzzy oil drop” model being engaged in ligand (ADP) binding.

The residues of local ΔH̃ minima are engaged in protein-protein interaction, particularly well seen in the domain 192–371. This accordance disappears step-wise taking the parts larger than the domain as the unit for drop construction localizing the ligand molecule even in the area of negative ΔH̃ values. Both explanations for ligand binding are possible as the complementary accordance of target molecule and ligand. This observation is important for the interpretation of the “fuzzy oil drop” model.

Analysis of the ΔH̃ profiles for chain A suggests that the construction of domains present in the chain A structure appears highly accordant to the interpretation of “fuzzy oil drop” model. The local ΔH̃ minima are engaged in protein-protein complexation (particularly in the domain 192–371, while ligand molecule occupies the position of residues of local ΔH̃ maxima. The characteristics of ligand binding residues in the Gro-EL and GroEl-ES complex get changed.

The characteristics of residues engaged in P-P or ligand binding changes due to different relative localization of these residues versus in the “oil drop” construction.

The same analysis performed for chain H representing the 7-fold system of the chains H-N is presented in Figure 3. There is no difference between the amino acids sequence in chains A-G and H-N, although some small differences of ΔH̃ profiles are observed. It is the possible result of different complexation conditions (the chains H-N have no contact with the molecule Gro-ES) and no ligand is complexed to this ring.

The accordance of protein-protein interaction area with expectations based on “fuzzy oil drop” model is lower in chain H than in chain A.

According to expectations the residues involved in the protein-protein interaction represent the hydrophobicity excess (ΔH̃ <0). The accordance is also found for the protein-ligand interaction which engages the residues of ΔH̃>0 hydrophobicity deficiency, which may be seen in Figures 4 and 5. The frequency of residues engaged in protein-protein complexation decreases together with the ΔH̃ value (except the ring A-H which reaches the maximum for positive values of ΔH̃).

The ΔH̃ profile of the chain O as calculated for the Gro-ES fragment reveals rather large fragments of hydrophobicity deficiency in this part of the chaperonin (Figure 6). The residues engaged in the interaction with the chain A are of special importance. Representing the very low values of ΔH̃ (hydrophobicity excess) the residues of chain O fit well with the residues in chain A engaged in the interaction with this chain (residues 233-269), also representing the low values of ΔH̃. A high accordance can be seen in this case.

The residues of low ΔH̃ values at the N- and C-terminal fragments are engaged in the interaction with chains of the cis-ring. This interaction seems to be of the form of hydrophobic interaction although the fragments of hydrophobic deficiency character are also engaged in protein-protein interaction.

In conclusion one may say that the H and A chain domains (especially the domain containing the residues 192 – 371) are highly accordant with the model. The domains generated by N-terminal and C-terminal polypeptide fragments seem also to be accordant with the “fuzzy oil drop” model.

To make the interpretation of ΔH̃ profiles more objective the clustering analysis was applied (as described in 2.6.) was performed. The results are shown in Table 1.

The highest accordance between expected (according to the “fuzzy oil drop” model) and observed (according to the PDBSum criteria) was obtained for the domain 192–371 in chain A. This observation supports the assumption that this domain could be the first one spontaneously folded according to “fuzzy oil drop” model exposing on the surface the hydrophobic residues in contact with other polypeptide chains. The chain H, although representing the identical sequence, displays some differences versus the A chain, suggesting it to be folded under other conditions than the chain A. Lower accordance between expected and observed classification of all other fragments of the complex (partitioning) suggest that the proper unit to be applied for protein folding in the environment simulated by the “fuzzy oil drop” is the domain 192–371 of the chain A. The relatively high accordance observed for the entire complex (chaperonin molecule) may be interpreted as the reliability of the “fuzzy oil drop” model. It suggests that the protein-protein interaction in the complete molecule may be recognized by the minima of ΔH̃ in the profile. The results given in Table 1 seem to support the observations presented in Figures 4 and 5.

The “fuzzy-oil-drop” model may be also applied for biological function recognition of the protein under consideration [10–12]. The role of chaperonin molecule (complex) is to create the environment for folding proteins. Thus, the internal surface characteristics of the capsule seems to be of special importance in respect to the biological function of this molecule.

The ΔH̃ values of residues localized on the internal surface of the complex are presented in Figure 7 and the 3-D representation in Figure 8. These two figures show the residues ordered according to the X-axis localized on the internal surface of the capsule. The high excess of hydrophobicity in Gro-ES suggests the high participation of this type of interaction in substrate binding. The cis-ring (chains A-H), in contrast to the GroES part, presents highly differentiated characteristics expressing excess/deficiency hydrophobicity, although biased significantly toward the hydrophobicity deficiency. It may be interpreted as a specific distribution for stronger/weaker interactions with a substrate molecule. The trans-ring (H-N) presents rather low differentiation of hydrophobicity excess/deficiency with ΔH̃ values close to zero (particularly in the end area). This suggests high accordance of the hydrophobicity distribution with the expected one (accordant with “fuzzy oil drop” model).

Figures 7 and 8 present the hydrophobicity irregularity of the internal channel where the folding reaction takes place. These two pictures show the characteristics of the external force field of hydrophobic character. Its specific deformation (in the sense of irregularity versus the idealized Gauss function distribution) seems to represent the localization of possible anchorage for folding protein in the chaperonin capsule.

The discussion of hydrophobicity based structure–to-function relationships also concerns other parts of the complex presenting highly irregular hydrophobicity distributions. Particularly the excess hydrophobicity areas not engaged in protein-protein interactions (responsible for complex generation) seem to represent the regions potentially ready to interact under changed circumstances during the action of the chaperonin. The structural changes are reported to be a large deviation from the 7-fold symmetry of Gro-EL rings [14]. The possible protein-protein interaction can be simulated linking two hydrophobicity-excess areas of interacting chains. This can be seen in Figure 9.

The large–scale structural changes observed as accompanying the folding process engage residues potentially ready to interact. The residues localized on the surface (large distance versus the 7-fold symmetry axis) carrying highly negative ΔH̃ values seem to be ready for hydrophobic interaction (distinguished by red circle in Figure 9). The residues localized closely versus the 7-fold symmetry axis representing large positive ΔH̃ values are ready for non-bonding interaction (distinguished by green circle in Figure 9). The first possibility seems to be related to structural changes in the chaperonin molecule while the second one seems to be related to the interaction with the folding protein molecule (internal surface of the capsule) [29].