The object of analysis is a de novo designed protein dimer with the structure available in PDB ID-2GJH. ²⁷ The analysis of structural changes related to dynamics presented in ²⁸ is concerned with the structural changes and stability of this dimer under different external conditions.

Our analysis presents an assessment of the structures in question from the point of view of hydrophobicity distribution in monomers as well as in dimers, taking into account the environment, using a fuzzy oil drop model (FOD-M) to assess the structure. ¹⁶

Table 1. presents a list of proteins and their complexes discussed herein.

The phenomenon of the spontaneous process of formation of structures referred to as micelles relies on the variable interaction of parts of bi-polar molecules in an aqueous environment. The polar parts of the molecules locate on the surface providing an entropically favourable arrangement with the surrounding water. The hydrophobic parts locate in the centre of the system, creating a concentration of hydrophobicity – hydrophobic core - isolated from water contact.

By interpreting the structure of amino acids as bi-polar molecules with a varying ratio between the polar and non-polar parts, the process of protein folding can be compared to the process of micellization.

This trend was identified by the oil drop model introduced by Kauzmann. ¹⁵ The current version of the oil drop model changes the discrete two-layer system – polar shell and hydrophobic core – into a continuous form. The level of hydrophobicity varies from zero at the surface to the highest concentration in the centre of the system. This type of distribution is expressed by a 3D Gaussian function spanning the body of the protein expressed by the function: H i T = 1 H sum T exp ( − ( x i − x ¯ ) 2 2 σ x 2 ) exp ( − ( y i − y ¯ ) 2 2 σ y 2 ) exp ( − ( z i − z ¯ ) 2 2 σ z 2 ) eq.1. where x, y, z are the coordinates of the geometric centre of the protein, x _i, y _i, z _i are the coordinates expressing the positions of the effective atoms of the amino acid (effective atom – the averaged position of all the atoms comprising a given amino acid). The σ _x, σ _y and σ _z are parameters expressing the size and shape of the encapsulated protein under consideration.

However the actual hydrophobicity distribution is the result of an inter-residual hydrophobic interaction expressed by a function introduced by M. Levitt ³⁸ : H i O = 1 H sum O ∑ j { ( H i r + H j r ) ( 1 − 1 2 ( 7 ( r ij c ) 2 − 9 ( r ij c ) 4 + 5 ( r ij c ) 6 − ( r ij c ) 8 ) ) for r ij ≤ c 0 , for r ij > c eq. 2. where H ^r denotes the intrinsic hydrophobicity of the amino acid in question (any scale can be used ³⁹ ). The “ c” stands for cutoff distance – after ³⁸ c = 9 Ǻ. The r _ij are the distances between the positions of the i-th and j-th effective atoms (the averaged position of the co-ordinates of the atoms comprising the amino acid in question).

Both of these normalised distributions (the first expressions in eq. 1 and eq.2) can be compared using the Kullback-Leibler divergence entropy ⁴⁰ : D KL ( P | Q ) = ∑ i = 1 N P i log 2 P i Q i eq 3

D KL ( P | Q ) = ∑ i = 1 N P i log 2 P i Q i where distribution P is the distribution under consideration – in the FOD-M model it is distribution O ( eq. 2), distribution Q – is the reference distribution – in the FOD-M model, it is distribution T.

Distribution T is an idealised, perfect distribution consistent with an ideal micelle with a centrally located hydrophobic core and a polar surface. Hydrophobicity levels in the protein body decrease continuously from a maximum in the centre to zero at the surface.

However, the value of D _KL (O|T) cannot be interpreted directly (entropy). Therefore, a second reference distribution R is introduced, where each R _i value assigned to successive effective atoms is equal and expressed as R _i = 1/N where N is the number of amino acids in the protein. This reference distribution expresses a constant, non-variable distribution in a protein lacking a hydrophobic core.

The relationship D _KL(O|T) < D _KL(O|R) indicates an approximation of the O distribution to the T distribution, which is interpreted as the presence of a hydrophobic core in the protein under consideration. To avoid describing one object with two parameters, the RD (Relative Distance) parameter was introduced, defined as: RD = D KL ( O | T ) D KL ( O | T ) + D KL ( O | R ) eq. 4.

An RD value <0.5 indicates the presence of a hydrophobic core.

Protein analysis confirms the validity of the adopted model. Indeed, groups of proteins with very low RD values (<0.3) have been identified. This group includes proteins: fast-folding. Ultra-fast-folding, down-hill and antifreeze type II proteins. ⁴¹ The results of the simulation performed, taking into account the environment, indicate the need for a suitable module in the programmes to express the contribution of the environment to the procedures simulating the protein folding process. ⁴² Other groups of proteins, e.g. membrane proteins, show RD values >0.5.

The water environment directing the folding process towards the generation of a micellar system is not the only one in which proteins show their activity. Membrane proteins, for stability in a hydrophobic membrane environment during folding, expose hydrophobic residues by concentrating hydrophilic residues in the centre (especially in the case of proteins acting as an ion channel).

In this situation, the expected hydrophobicity distribution takes the inverse form: T i = ( T MAX − T i ) n eq.5

Index n denotes normalisation.

After analysis of numerous proteins, it became apparent that the distribution of hydrophobicity in the protein body mapped a distribution that was the sum of both functions: M i = [ T i + K ( T MAX − T i ) n ] n eq.6

The contribution of the aquatic environment is corrected by the presence of other components that reduce the polar character of the environment. The contribution of the modifying factor varies. This is expressed by the value of the K parameter, which for the proteins mentioned above (down-hill) is equal to K = 0.0 or very low K < 0.4. This means that the folding of a given protein takes place with the active participation of only the aqueous environment directing the folding process towards the generation of a micelle-like system.

The value of the K parameter is determined by an iterative procedure identifying the minimum value of D _KL(O|M) ( Fig. 1.C). The principle of the model described is illustrated in Fig.1.

Interpretation of the FOD-M model parameters: 1.

The RD value indicates the degree to which the hydrophobicity distribution in the protein body is aligned with the idealised micelle-like distribution (maximum hydrophobicity in the centre with zero hydrophobicity on the surface). Low values of RD < 0.5 indicate the presence of a hydrophobic core

The program to calculate the RD and K parameters is available for free at https:\hphob.sano.science.

The experimental investigation (according to ²⁹ ) of subdomain of the chicken villin headpiece indicates that the substitution of two lysines by norleucine increased the folding rate 6-fold at 300 K and amounts to 0.7 micros. \.

The structure of this domain is available in the PDB resources – ID 2F4K. ²⁹ When analysed based on the hydrophobicity distribution (using the FOD-M model), it shows an alignment of the O distribution with the T distribution. This means that a centrally located hydrophobic core with a polar surface is present in the structure of this domain. A value of RD = 0.295 and K = 0.0 indicates an almost perfect reproduction of the hydrophobicity distribution that results from the folding process being directed by the water environment ( Fig. 2).

The determined residues responsible for reconstructing the local maxima forming the hydrophobic core are: 47F, 51F, 53 M, 58F ( Fig. 2 – red squares). These residues were visualised on the 3D presentation ( Fig. 2) as red squares on the horizontal axis of the hydrophobicity distributions and red space filling on the 3D presentation.

From the point of view of an interpretation based on the FOD-M model, the speed and ease of the folding process of this domain in an aqueous environment is not surprising.

Such results were also obtained for a number of other fast-folding, ultra-fast-folding, down-hill and antifreeze type II proteins. ⁴¹

Complexation-dimerisation analysis, based on in silico methods, determines the conditions for the dimerisation process determined based on molecular dynamics simulations ²⁸ for both monomers and dimers of the Top7-CFr protein (PDB ID – 2GJH ²⁷ ).

Experimental studies in ²⁸ have shown the conditions leading to partial unfolding and refolding by operating the force field of the CHARM22 program for a temperature of 400 K – a temperature close to the experimentally observed process referred to as melting. In the monomer-containing box, The 6574 water molecules were located, while in the dimer-containing box – 12,497 water molecules. The presence of 0.1 M NaCl corresponding to 0.1 M of this salt. As a result, the protein concentration is expressed as a concentration of ∼8 mM. ²⁸

In the cited simulation, the presence of water is recorded by interaction with amino acid atoms located on the surface of the protein coming in contact with the atoms of water molecules. A description of monomer-folding phenomenon, starting from the extended form and corresponding to the experimentally available form ²⁸ (PDB ID - 2GJH), was obtained. Forms of intermediates leading to a structural form consistent with the experiment were achieved. In the analysis, some differences to the experimental observations were observed. It also concludes that “overall folding may depend on external conditions.” This observation is important for assessing the type of stabilisation of both monomers and dimers.

The results of the monomer and dimer structure analysis of the Top7-CFr protein in the FOD-M model assessment are given in Table. 2.

The RD and K parameter values given in Table. 2 reveal a very high ordering of hydrophobicity in the protein body consistent with the idealised distribution. This means obtaining a structure that is the result of the influence of the aqueous environment on the structuring process, consistent with the orientation of this type of external force field for both monomer and dimer structuring ( Fig.3. and Fig. 4.). This high ordering applies both to chains treated as individual structural units (3D Gaussian functions generated for each unit individually) and to chains treated as components of a complex (assessment of the status of the T, O and M (for K = 0.2) profile fragments for chains, respectively) ( Fig. 4.).

The assessment of the structural stability of the Top7-CFr monomers as well as the dimer is due to the clearly high ordering of the hydrophobicity distribution consistent with a micelle-like arrangement. The presence of a hydrophobic core is identified as a factor in the stabilisation of the protein’s tertiary structure (in addition to the presence of SS-bonds absent here). In the case of the protein in question, stabilisation is definitely based on the presence of a clearly defined hydrophobic core for both the monomers and the dimer.

The status of the monomers is much closer to micelle-like structuring in the FOD-M model classification, which is assumed to be due to the influence of the aqueous environment as a factor directing the folding process towards generating a hydrophobic core with a polar outer shell. P-P status – interface status – in the dimer, it represents the effect of a joint involvement in building dimer stabilisation – a very low RD value for the interface. This means that the interacting residues form a common core for the dimer.

Examples of other proteins with a structure associated with the presence of a micelle-like hydrophobicity distribution include the SH3 domain. Table. 3. presents a set of examples demonstrating a high degree of micelle-like ordering of hydrophobicity.

This domain represents a conservative (from prokaryotes through viruses and eukaryotes) protein module. It plays a role in cellular signalling processes by controlling inter-protein interactions. This is of particular relevance to cytoskeletal structure or kinase activity.

In the light of the assessment of structuring based on the FOD-M model of the SH3 domain, the structure of this domain represents another example of a hydrophobicity distribution adapted to the conditions imposed by the aqueous environment leading to an ordering of hydrophobicity according to the micelle-like model. A centrally located hydrophobic core with a polar surface promotes the stabilisation of the structure achieved by this domain and ensures the functioning of this domain.

This term indicates the need for one of the monomers to be present for the folding of the other chain. One of the chains is a target for the folding and for the adjustment of the partner in the complex.

Signalling protein 2LCS ³⁷ – a protein that binds multiple targets – peptides that show similar strong affinity. In the example discussed here, the SH3 domain interacts with a 16-amino-acid peptide with serine/threonine-protein kinase STE20 from Saccharomyces cerevisiae.

From the point of view of the applications of the FOD-M model, this example forms the basis for the analysis of induced folding.

The SH3 domain is a target for the peptide, which in the interpretation of the FOD-M model means that the SH3 domain provides an external force field imposing the structuring of the peptide. The peptide in question, due to its small number of amino acids, is unable to produce its own tertiary structure and adapts relatively easily to the target, which in this case is the SH3 domain. Nevertheless, using the in silico procedure for protein and peptide folding, the introduction of an external force field in the form of the parameter K = 1.5 would arguably provide the structure represented in this case ( Fig. 5.).

The alignment of the structure with the external force field is clearly visible in the form of a very low K value for the complex in question. This value is minimally different from K describing the SH3 status of the domain. This implies an alignment of the peptide also from the point of view of the hydrophobicity distribution of the dimer. Examples are given in Table. 4.

Analysis of the contribution of individual chains to the 2LCS dimer structure shows chain A status as a dimer component described with the values RD = 0.363 and K = 0.1 and for chain B with the values RD = 0.594 and K = 0.4. This set of parameter values expresses the relationship between the chains and the contribution to the construction of the common dimer structure. Dominant chain A shows a status comparable to that of the complex, while chain B expresses its role in determining the status of the complex with values significantly lower compared to its status as an individual structural unit. The value K = 0.4 can also be interpreted as determining the external force field provided by chain A, to which chain B has adapted. This example can be regarded as induced folding, which can also be expressed as folding under the influence of a local external force field. Chain A also acts as a chaperone imposing the folding of the chain according to the field generated by chain A.

In the case of the SH3 domain complex with the peptide from pbs2-kinase (PDB ID - 2VKN), the status of the peptide is determined by the value K = 0.4. This means that the SH3 domain provides an external force field directing folding to a lower degree than is the case in the example discussed above.

Another example of inducing folding is the folding of chain Alpha of haemoglobin assisted by AHSP, which essentially acts as a chaperone directing the folding of this chain. The role of AHSP is to prevent independent folding to the final form that the chain would probably have achieved without the directing factor. ⁴³ This chain treated as an individual structural unit (as available in PDB: 1W0B, 1W0A, 1W09) shows structuring with hydrophobicity distribution consistent with a micelle-like arrangement ( Fig. 6.) ( Table. 5). This means that the protein obtains its structure spontaneously in the aqueous environment.

The protein with the code (PDB ID – 1Y01 ³⁶ ) is the complex of AHSP with human Fe (II)-alphaHb. The low RD and K values for AHSP suggest the acquisition of structure by this protein exclusively under the influence of the aqueous environment ( Fig. 7). The status of AHSP in complex with haemoglobin Alpha-chain is comparable to that shown as an individual structural unit.

The values of the RD parameters ( Table. 6, Fig. 8) reveal the status of the residues involved in the interaction (P-P) and the part lacking the amino acids remaining in contact with the other chain. (No P-P) In the case of the complex, the status of the interface is favourable from the point of view of hydrophobicity distribution, which implies the participation of the interface in the construction of the common hydrophobic core. The remaining part (No P-P) devoid of the interface shows an increased RD value, indicating a significant role for the interface in the structure of the dimer. In contrast, the status of the residues involved in the interaction determined within the individual chains show elevated RD values for the residues comprising the interface. This entails preparing an exposure of hydrophobicity on the protein surface, and the exposure is used to form the complex.