Eur. Phys. J. E (2023) 46:62 https://doi.org/10.1140/epje/s10189-023-00320-8 THE EUROPEAN PHYSICAL JOURNAL E Regular Article - Living Systems How hydrophobicity shapes the architecture of protein assemblies Juan A. Cedano , Enrique Querol , and Angel Mozo-Vilları́asa Institut de Biotecnologia i Biomedicina and Departament de Bioquı́mica i Biologia Molecular, Universitat Autónoma de Barcelona, Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain Received 11 May 2023 / Accepted 7 July 2023 / Published online 27 July 2023 © The Author(s) 2023 Abstract The interactions that give rise to protein self-assembly are basically electrical and hydrophobic in origin. The electrical interactions are approached in this study as the interaction between electrostatic dipoles originated by the asymmetric distribution of their charged amino acids. However, hydrophobicity is not easily derivable from basic physicochemical principles. Its treatment is carried out here considering a hydrophobic force field originated by “hydrophobic charges”. These charges are indices obtained experi- mentally from the free energies of transferring amino acids from polar to hydrophobic media. Hydrophobic dipole moments are used here in a manner analogous to electric dipole moments, and an empirical expression of interaction energy between hydrophobic dipoles is derived. This methodology is used with two examples of self-assembly systems of different complexity. It was found that the hydrophobic dipole moments of proteins tend to interact in such a way that they align parallel to each other in a completely analogous way to how phospholipids are oriented in biological membranes to form the well-known double layer. In this biological membrane model (BM model), proteins tend to interact in a similar way, although in this case this alignment is modulated by the tendency of the corresponding electrostatic dipoles to counter-align. 1 Introduction of the most active and fascinating fields of study. The forces and interactions that give rise to these assem- One of the most interesting problems in the life sci- blies by establishing the proper hydrogen bonds, are the ences, both from a theoretical and experimental point same ones that give these macromolecules their ability of view, is the elucidation of the capacity for assembly to fold and thus acquire their three-dimensional (3D) –and self-assembly of proteins to form large functional conformations. structures, essential for both health and disease. This Hydrophobicity is a merging property in complex area of interest and importance also includes the abil- molecular systems that does not manifest itself in each ity of proteins to interact with nucleic acids and with of the individual components. Hydrophobicity is an ligands to form essential structural and functional com- entropic effect (the “hydrophobic effect” [2]) and rises plexes for the maintenance of life. Examples include the as a consequence of the relative high affinity that water dynamic process of polymerization of actin or tubulin or polar molecules have for each other as compared monomers to form filaments and microtubules, essen- with the affinities that other non-polar molecules (i.e. tial for cell function. The stacking of nucleosomes in hydrocarbons) exert on other non-polar molecules or the chromatin of cells is another conspicuous —and even the affinities that these molecules exert on water. more complex— example. Or the pathological associ- Consequently, the hydrophobic force is considered the ation of small amyloid peptides in large and symmet- overwhelming short-range force (up to 100 Å), related ric agglomerates in cells that cause degenerative dis- to water structuring effects [3]. And as hydrophobic eases such as Alzheimer’s, Parkinson’s and others. It energy, together with the electrostatic energy, is a fun- is known that the 3D conformations are the result of a damental part of the whole energy content of both pro- nuanced balance between hydrophobic and electrostatic tein–protein and protein-DNA complexes, it is essential forces [1] that the amino acids exert on each other or to have a useful theoretical tool to describe it quanti- on nucleotides and/or with the external environment. tatively. However hydrophobicity is not easily derivable In short, the ability of these macromolecules to asso- from basic physicochemical principles. Many attempts ciate and the mechanisms used to do so, constitute one have been directed towards the generation of models of hydrophobic interactions [4–8] that are alternative a e-mail: angel.mozo@bioinf.uab.cat (corresponding descriptions to classical phenomenological surface-area author) models. 123 62 Page 2 of 10 Eur. Phys. J. E (2023) 46 :62 Several authors [9–11] have studied the hydropho- 2 The hydrophobic potential bic character of many molecules, in particular the con- stituent amino acids of proteins and nucleotides of The present work proposes a phenomenological for- nucleic acids. By measuring the free energy needed to mulation for the energy of hydrophobic interaction transfer a given amino acid from an aqueous or polar between proteins in their macromolecular constructs. medium to a hydrophobic medium, these authors have This energy employs the hydrophobicity indices of the generated scales in which a value or hydrophobicity individual amino acids described above as ”hydropho- index is attributed to each amino acid and nucleotide. bic charges” in a hydrophobic force field. The pecu- Most of them are based on empirical approximations liarity of the hydrophobic interaction is that these being the computation of logPo/w one of the most hydrophobic charges attract each other if they are of common, where Po/w is a partition function for 1- the same sign, while they repel each other if they are octanol/water [8] and is determined empirically for each of opposite sign. system [12–15]. These scales have been refined over The fact that both electrical and hydrophobic charges time and the most updated hydrophobicity values are are distributed inhomogeneously provides a protein used in this work [16, 17]. Hydrophobic amino acids with a dipolar character. This allows for the definition and nucleotides are assigned a minus sign, whereas of both electrical and hydrophobic dipole moment vec- hydrophilic amino acids have a positive sign. These val- tors. However, since in most proteins the total electro- ues and signs of intrinsic hydrophobicities (“hydropho- static charge and the total hydrophobic charge are not bic charges”) for each amino acid and nucleotide allow zero, we must substitute the exact dipole moment vec- establishing, for a given 3D distribution of amino acids tor definition, D = ΣQ i . r i , for a pseudo dipole moment (e.g. a protein), a hydrophobic force field in a way anal- vector defined as: ogous to an electric field created by a distribution of electric charges [18].   Since hydrophobic forces are ultimately the result D = Q+ . c+ −c− (1) of many electrical interactions between many atoms that give rise to their own interaction laws, they may where Q + is the total positive charge (either electro- enter in competition with other simpler and indepen- static or hydrophobic) and c + and c − are the posi- dent electrostatic interactions also present in molecu- tive and negative centroids of the charges (either elec- lar complexes. In the end, it is the balance between trostatic or hydrophobic). Only in the case of neutral both electric and hydrophobic interactions that gives proteins, both definitions are coincident. See a detailed rise to the myriad of native macromolecular structures description of these magnitudes in ref. [22]. Results and functions found in Nature. Hydrophobic forces are obtained by using the alternative definition of dipole thus, paramount in the interactions of proteins and moments D = Q − . (c + −c – ), or even with the exact nucleic acids and in the formation of higher structures definition, differed from those obtained by the use of and activities [19]. There is an excellent and classi- Eq. 1, as expected. Nevertheless, the behaviour of the cal description of hydrophobicity and the hydropho- variations of the moments does not depend on the spe- bic effect by Tanford [2] and more recently by Kron- cific definition, as long as definitions are not mixed. berg [20]. Another consideration on hydrophobicity is Since we are specifically interested in hydrophobicity, that the hydrophobic effect occurs at relatively small Eq. 1 is being used in this work. scales (1–10 Å), as well as relatively larger scales up We define the Biological Membrane (BM) effect as to 100 nm [see for example refs. 4, 5, 14 and literature the tendency of the hydrophobic dipoles of phospho- cited therein], where entropic and enthalpic forces may lipids that constitute a biological membrane to join in enter into significant competition. There does not seem parallel, forming one of the layers of the membrane to be a convincing theory that encompasses all scales. (Fig. 1). The model presented here is meant to be applied at These definitions allow us to consider the interaction small scale hydrophobicity, where distances and dimen- between monomers in a protein assembly as the interac- sions are of the same order of magnitude as biological tion between their electrostatic or hydrophobic dipoles. membranes. In the electrostatic case [23], the familiar expression Although protein–protein interactions will be dealt with explicitly in what follows, the conclusions hold for protein-DNA interactions as well. An example worth 3(ur · D 1 )(ur · D 2 ) − (D 1 · D 2 ) enD = − (2) mentioning here is that of genetic regulatory proteins 4πεr3 (small motifs or domains involved in the interaction with the DNA) and the short stretches of nucleotides provides the energy enD stored in a system formed by targeted by them [21]. two electric dipoles D 1 and D 2 , separated by a dis- The present work is a coarse-grain study and we do tance r and unit vector u r . The dot (scalar) products not consider interactions at the atomic level. Peptides of (u r . D 1 ), (u r . D 2 ) and (D 1 . D 2 ) represent the mutual any size, proteins and even groups of proteins are made projections of the three vectors involved. to interact as independent units. The results obtained In the same way that a distribution of electrical do not depend on the size of these molecules. charges whose centroids do not coincide, gives an elec- tric dipole character to a protein, a distribution of 123 Eur. Phys. J. E (2023) 46 :62 Page 3 of 10 62 entropic tendency of molecules to come together due to the exclusion effect of water molecules. But since a force is implied we can treat it formally in analogy with the electrical case. The hydrophobic potential gives rise to a hydrophobic field that is derived from the spatial vari- ations of the hydrophobic potential (see figure S1 and full derivation in Apendix1 of Supplementary Informa- tion). The hydrophobic potential due to the two charges of a dipole can be expressed in polar coordinates as V2 − V1 = κ.q.d. cos θ/r0 · exp(−r/r0 ) where θ is the angle formed by the dipole and vector Fig. 1 Biological Membrane (BM) effect. A Schematic position r . The field H created by dipole 1 is then: representation of two single phospholipids. The pseudo- hydrophobic moment H (green arrow) is defined as the     vector obtained from multiplying the hydrophobicity of the H = e−r/r0 κ · 1/rr0 H 1 + 1/r02 − 1/rr0 · (H 1 · ur ) · ur phospholipid h+ , times the distance vector d (black arrow) joining the hydrophobic centroid somewhere in the tail, and the hydrophilic centroid in the polar head. B Representa- The hydrophobic energy stored in a hydrophobic tion of a lipid bilayer showing the total hydrophobic vectors dipole formed by dipoles H 1 and H 2 is then given by of each layer (green arrows). The biological membrane joins enH = −H1 · H 2 , and thus: two lipid layers with their H vectors in opposition   1 1 r − r0 (ur · H 1 )(ur · H 2 ) − 1r (H 1 · H 2 ) hydrophobic charges can show a hydrophobic polariza- enH = κ r0 .er/r0 tion. Based on their force field, we derive a hydropho- (4) bic pseudo-energy between two hydrophobic dipoles in a phenomenological analogy with the electrical case described in Eq. 2. Two protein hydrophobic dipoles This equation is virtually of the same type as Eq. 2 can interact with each other in such a way that they since it depends on the same dot products (u r . H 1 ), orient one with respect to the other. (u r . H 2 ) and (H 1 . H 2 ), which describe the mutual To derive the hydrophobic energy of interaction we action of two dipolar entities. However, the exponential must start by considering the potential created by an dependence marks the difference with Eq. 2, both in the idealized hydrophobic charge in space by simple anal- numerator and in the denominator, when the decay con- ogy with the electrostatic potential. Many authors have stant and the exponential function appear. This expo- been faced with a very difficult problem to deal with in nential dependence in the denominator comes from the trying to find ways to quantitatively describe hydropho- experimental results on the dependence of hydrophobic bicity at all levels of sizes, ranges and situations. In forces on distance [4, 5, 25, 26]. In both Eq. 2 and Eq. 4, spite of these difficulties, some researchers experimen- constants ε and κ are strongly dependent on the par- tally found ways to define a potential in a hydrophobic ticular microenvironments of the interacting proteins force field. One of the first works to propose a decreas- and their relative orientations. Consequently, enD and ing exponential dependence with distance was that of enH in the present treatment cannot be quantitatively Marcelja et al. [24], for distance ranges between 0 and compared and we take constants in both formulas to be 1. Parameter r 0 cannot have the same value for all 100 Å. These studies later served as a basis for other cases studied, as is to be expected, since this parame- groups [4, 5, 25, 26 (and papers cited therein)] where ter depends on the environment, shapes, sizes, etc., of it was experimentally established that the dependence the proteins considered. For the choice of r 0 , limit val- of the hydrophobic action with distance, at least for ues were sought for which the expression derived and distances less than 100 Å, is of an exponential decay. used (Eq. 4), would start yielding absurd or unfeasible In the present work we use this experimental result and values of the energy enH (for example, when a clearly employ a simple definition of the hydrophobic potential attractive association would result in values of enH > created by a hydrophobic charge q as: 0). These limit values found are above 5–10 Å for all the cases studied. Given the scarce information available in V (r) = κ.q. exp(−r/r0 ) (3) the literature, for pragmatical purposes, a value of r 0 = 3 Å was adopted as a safe value for all cases. It is the where r 0 is the decay length and κ a phenomenological most frequent value found in the literature applied to constant that depends on the media where this charge is these ranges of distances and sizes [3, 4, 6, 7, 15, 24–26]. immersed. This concept corresponds to an idealization The most important point of using these two equa- of the distance effect exerted by a hydrophobic charge. tions is to determine whether the interaction energies, As noted in the Introduction, it is known that there both hydrophobic and electrostatic, are negative (and is not really a hydrophobic interaction, but rather the therefore attractive) for each case studied. They will 123 62 Page 4 of 10 Eur. Phys. J. E (2023) 46 :62 (Eq. 2). It should be noted that rotations performed for y the H2 vector over H 1 are independent of those per- H1 z formed for the D 2 vector over D 1 . Figure 3A shows two consecutive monomers of the Ebola virus matrix protein ur VP40 N-terminal domain (PDBid: 1H2C. Figure 3B renders the angular distribution of energies enH and enD in a rotation simulation of their respective H and H2 D vectors. The alignment of the H vectors clearly influences the alignment of the D vectors and, as a consequence, in most cases of protein assembly, the H vectors cannot x align perfectly parallel. This fact is reflected in the sym- metry of the angular distribution of enH and of enD. To semi-quantitatively characterize the degree of par- allel coupling of the dipole moments, the asymmetry Fig. 2 Two proteins shaped like a cylinder and a cone (H 1 index AS is used. This index is defined as the average and H 2 vectors) interact in order to form a dimer. The or the deviations of the three maxima: AS = (d1 + interaction energy enH is computed by means of Eq. 4. d2 + d3 )/3 of the angular distribution (see Fig. 3B) H 2 is then rotated from its initial native position in three from 180°. According to this definition perfect symme- orthogonal directions, with respect to H 1 in steps of 10°. try corresponds to AS = 0° and for total asymmetry AS These directions of rotation are: 1. rotation around the = 180°. In Fig. 3(B), AS = 69.1° for enH ; AS = 64.2° x–axis defined as the direction of the joining distance vec- for enD. Thus, a degree of asymmetry of 0° implies an tor between H 1 and H 2 , (u r , blue axis); 2. rotation around optimal mutual orientation between the two dipoles. the y–axis defined as the direction perpendicular to the Any value that deviates from 0° gives us an idea of how plane formed by the x–axis and vector H 2 , (green axis); 3. rotation around the z–axis as the direction perpendic- far away a given structure is from stability. On the con- ular to both x–axis and y–axis (red axis). For each simu- trary, values of this asymmetry index for distributions lated rotation angle, enH is computed and plotted against of repulsive energy, range around 180°. There is a cer- the rotated angle. The same procedure is applied to electric tain degree of counter action between both interactions dipole moments D 1 and D 2 , and enD is computed (Eq. 2) since a good alignment of H 2 with H 1 also implies the alignment between D 2 and D 1 , which is unfavourable to the union. The native configuration is an energetic even allow us to determine whether an eventual attrac- compromise between the simultaneous actions of both tion between the dipoles is energetically optimal. This forces. point is discussed in the next section. 3 Rotation simulations 4 Applications for the verification of the Biological Membrane model Energies enD and enH are highly dependent of the rel- ative orientation of the interacting dipoles. To verify The fast development of the Protein Data Bank (PDB) whether the energy enH obtained with Eq. 4 is the has allowed easy access to the coordinates of thou- optimal hydrophobic interaction energy between two sands of crystallized protein systems. The methodology proteins (and their dipoles), a simulation of rotation described above can thus be applied to the assembly of one of the dipoles (H 2 ) over the other (H 1 ) was of proteins. In order to verify our Biological Membrane carried out. The interaction energy enH is computed effect model, the interaction energies computed with by means of Eq. 4, and gives the energy of the native Eq. 2 and Eq. 4 were tested on a large set of protein conformation. H 2 is then rotated from its initial native systems obtained from the PDB (see graphic table in position in three orthogonal directions, with respect to Appendix 2 of Supplementary Information). We com- H 1 , in steps of 10º. Figure 2 shows schematically how puted the electrical and hydrophobic interaction ener- rotations are performed. gies between protein dipoles using algorithms created These directions of rotation are: 1. rotation around with Wolfram Mathematica. According to this model, the x–axis defined as the direction of the joining dis- monomers of a protein assembly species are attracted tance vector between H 1 and H 2 , (u r ); 2. rotation to each other due to their hydrophobic character, with around the y–axis defined as the direction perpendicu- their individual H vectors aligning like phospholipids lar to the plane formed by the x–axis and vector H 2 ; 3. do in a membrane. In their encounters, the individual rotation around the z–axis as the direction perpendicu- monomers explore the entire space of union possibili- lar to both x–axis and y–axis. For each simulated rota- ties with other monomers, remaining the longest time tion angle, enH is computed and plotted against the in those unions in which the strongest and most numer- rotated angle. The same procedure is applied to elec- ous hydrogen bonds can be formed. However, in many tric dipole moments D 1 and D 2 , and enD is computed instances these interactions have to coexist with the 123 Eur. Phys. J. E (2023) 46 :62 Page 5 of 10 62 Fig. 3 A PDBid:1H2C is an example of system in which enD (in red). Both variations, enH and enD versus rotated both hydrophobic and electrical interactions are attractive angle have been superimposed in the same plot in order to (enH < 0, enD < 0). A Two consecutive monomers (c and simplify all plots. It is important to keep in mind that values d) of the assembly showing their respective H vectors (blue of both enH and enD at 0° are those of the native species. arrows) and D vectors (red arrows). The abscissa on the left The relative degree of symmetry of the angular distributions in B represents the variation of enH (in green) on the three of enH and enD, serves as a qualitative assessment of the directions as defined above. Circles: variation on the x-axis; degree of alignment of the interacting vectors. This is done squares: variation on the y-axis; triangles: variation on the by computing the AS index which is the average deviation z-axis. The abscissa on the right corresponds to variations of of the curves maxima from 180° (see text) tendency of electric dipoles to orient themselves in an from total symmetry (AS = 77°) reveals some misalign- anti-parallel configuration. ment shown by the resultant vector of the two stacks. In order to illustrate the applicability of our model, The lack of total symmetry becomes apparent when the two examples of very different complexity are shown. total moment of the entire system, represented by the vertical vector in Fig. 4B, is calculated. The appear- ance of this vector suggests that this set associates (in 4.1 A simple system opposition) with an identical set following the BM effect principle. This is precisely what happens as reported by One of the simplest systems corresponds to the assem- Fitzpatrick et al. [27]. They also show associations of bly of cross-ß amyloid fibrils of the type involved in two and three groups in total agreement with the BM degenerative diseases such as Alzheimer’s or type II effect (Fig. 4 E, F, AS = 7.7°). diabetes, PDBid: 2M5N [27]. The Tyr-Thr-Ile-Ala-Ala- Leu-Leu-Ser-Pro-Tyr-Ser peptide, under the right con- ditions, is the monomer that self-assembles generating 4.2 A complex system long and compact fibres of variable morphology that accumulate in cells, impairing their normal function. Proteins can also be considered as electrostatic dipoles. Figure 4 shows a fragment of 8 + 8 of these peptides. We have already seen that since in most proteins, the The hydrophobic moment of each individual peptide alignment of the hydrophobic dipoles also implies the has been computed and drawn (Fig. 4A). The H vec- alignment of the associated electric dipoles, this electric tors are strongly associated by lining up in parallel. alignment can distort the hydrophobic alignment. The The resultant vector is also represented. The final fibres final 3D configuration adopted by the monomers will are formed by joining two of these stacks so that their be an energetic compromise between the two types of hydrophobic vectors oppose each other, as would hap- forces. This gives rise to all the variations of native pen with a biological membrane (Fig. 1) [18, 21, 28]. A protein assembly conformations found in nature. simulation of rotation (Fig. 4B) of any monomer around As stated earlier, in the present model it is not pos- its neighbour reveals the perfect symmetry of the union sible to make a direct quantitative comparison between occurring with maximum energy at 0° of the angular energies enD and enH . And for this same reason, at distribution (AS = 0°). this point, comparisons cannot be made between enH The association of the two stacks is also due to values obtained for different proteins either, since these hydrophobic attraction. However, in this case the asso- energies critically depend on specific 3D configurations. ciation is not as strong or symmetric as revealed in a There are many examples of self-assembly of com- simulation of rotations (Fig. 4 C, D). This deviation plex systems, ranging from filament formation in cells 123 62 Page 6 of 10 Eur. Phys. J. E (2023) 46 :62 Fig. 4 The simplest system, PDBid: 2M5N [27]. This sys- totally counter-aligned, yielding a resultant vector (dark tem consists of compact clusters of the linear peptide cross-β blue) perpendicular to the fibres. This misalignment of the amyloid YTIAALLSPYS, responsible for diseases related H vectors appears as an asymmetry in a rotation simula- to age dementia. The assembly does not have appreciable tion (D). This resultant vector serves to associate this set electric dipole vectors. A Four of these peptides correspond of columns to other sets. E Multiple associations of these to two facing columns. The respective H vectors in each columns as they appear in the images obtained by Fitz- column are perfectly aligned as is also reflected in the per- patrick et al. [27], assemble following the BM effect model. fect symmetry of the angular distributions of the rotation The simulation of rotations F of column 2 with respect to simulations (B). Note that these β-sheets are parallel to column 1 shows a fairly good symmetry. Note that the red each other due to the “biological membrane effect” and the circles in all the rotation simulations (angle 0°) correspond proper hydrogen bonds between them are then formed. C to values of enH of the native structures Interacting H vectors (green) of opposing columns are not 123 Eur. Phys. J. E (2023) 46 :62 Page 7 of 10 62 Fig. 5 A complex system, PDBid: 3J4F [29]. A One sin- components in the plane vanishes. The hydrophobic interac- gle turn of the helicoidal tubular capsid of the HIV virus tion between the normal components is attractive, showing formed by assembled hexamers. In the assembling of each a quite symmetric angular distribution of enH in a rotation hexamer (B, C), the consecutive H and D vectors of the simulation of two consecutive monomers (D). The normal monomers (blue and red arrows) are quasi-aligned result- components of the electric dipoles are also aligned, resulting ing in an attractive hydrophobic interaction and electro- in a repulsive force not strong enough to prevent the forma- static repulsion. The H and D vectors of each monomer tion of the hexamer. Green and yellow arrows in B are the have two components: one in the plane of the hexamer (C) resultant H and D vectors respectively (not to scale) and one perpendicular to it (B). The resultant vector of the to virus capsids (see Appendix 2 in Suppl. Info.). in each monomer, the D vectors must be also partially All of them have hydrophobic and electrostatic dipole aligned causing mutual repulsion and forcing misalign- moments and most of them show assemblies following ment until a conformational equilibrium between both the BM effect. We study in more detail the capsid struc- forces is reached. Secondly, unlike the simple system ture of the mature HIV-1 virus, PDBid: 3J4F [29]. This seen above, these monomers are relatively large globu- system consists of the assembly of homohexamers to lar proteins and perfect alignment of their H vectors form a tubular helicoid (Fig. 5A). Since this assembly would cause steric clashes of the amino acids in what is hierarchical, we study the system at three levels. The is known as “geometric frustration” [30, 31]. Figure 5D first level is the formation of a hexamer. The second shows a simulation of mutual rotation of two contigu- level is the interaction between hexamers that builds ous monomers of the hexamer, where these facts are the tubule. The third is the local interaction between reflected. The hydrophobic attractive energy enH is the facing surfaces of two consecutive hexamers. quasi-optimal, although the angular distribution in a Each hexamer is composed of six globular proteins rotation simulation does not render perfect symme- (Fig. 5B, C). The interaction of any two consecutive try (AS = 15.5°), reflecting that the two consecutive monomers of the hexamer relies on hydrophobic attrac- hydrophobic dipole moments are not perfectly parallel. tion through the quasi-alignment of their respective The energy enD is repulsive but obviously not strong hydrophobic moments. However, these H vectors do enough to counter the hydrophobic attraction. Atten- not align perfectly mostly due to two reasons. First, tion must be drawn to the fact that when this inter- due to the mutual orientation of the H and D vectors action is repeated with six monomers, the assembly, 123 62 Page 8 of 10 Eur. Phys. J. E (2023) 46 :62 Fig. 6 PDBid: 3J4F. A Two adjacent hexamers showing hydrophobic bond energy is high as compared to the other their H (blue) and D (red) vectors. B A simulation of interactions due to the close proximity of the interacting rotations shows an attractive hydrophobic interaction with- elements. D The angular distribution in the rotation simu- out perfect alignment. C The bond between the adjacent lation shows relatively high symmetry of the angular distri- hexamers is realized through the surface elements (grey bution, reinforcing the fact that this is the permanent bond and blue respectively) of the two facing hexamers. This of any hexamer with the other surrounding six hexamers instead of being a linear array of monomers, ends up geometric frustration. Contrary to the case of the build- closing in on itself, fitting six elements. Figure 5D shows ing up of each individual hexamer, the combination of the distribution of vectors H of the monomers of a hex- these three effects force the hexamers to form a chain amer. The individual H and D vectors of the monomers that cannot close in on itself but ends up adopting a in the hexamer can be decomposed into two compo- helicoid formation (Fig. 4A). Actually, and by the same nents, one in the plane of the hexamer and the other token, any hexamer interacts laterally with six other normal to this plane. The resultants of both H and hexamers, and instead of a planar surface of hexam- D vectors in the plane of the hexamer vanish. Con- ers, the interaction generates a curved surface due to versely, the resultant in the direction normal to the the balance of all of the three mentioned interaction plane provides the hexamer with a resultant net vec- factors. tor H . This “vertical” component makes the hexamer The energy enH found in interacting hexamers is capable of interacting with other hexamers according to weak compared to that of the monomers within a hex- the principle of the BM effect (Fig. 6A). A simulation amer but it orients the biomolecules in their exploration of mutual rotations between two consecutive hexamers of their configurational landscapes. As a consequence, illustrates these interactions. the union of two consecutive hexamers is realized by the Figure 6A shows two consecutive hexamers of the strong hydrophobic attraction that takes place at the first turn of the assembly (AS = 6.7°). The result is local surfaces of the interacting hexamers, as depicted again, the sum of the three actions, hydrophobic attrac- in Fig. 6C. It can be seen that both the H and D tion (dominant), electrical repulsion (in this case) and vectors of the interacting surfaces are almost perfectly counter-aligned, as revealed by a slight asymmetry in 123 Eur. Phys. J. E (2023) 46 :62 Page 9 of 10 62 the angular distributions in a rotations simulation (AS has been reported elsewhere, so at the present state of = 4.2°). Nevertheless, in this case the bond energy enH the matter, simplified approaches, as ours are valuable is relatively high as compared to the other levels of since they help to predict to a certain degree, protein interaction. interaction and complexes. It can be observed that at the three levels of interac- tion, the unions are carried out by hydrophobic attrac- tion, in which some electrostatic repulsion plus steric hindrances also intervene to modulate the final confor- 6 Conclusions mation of the system, locally as well as globally. It is experimentally verified by means of the Protein Data Bank, that the monomers of a protein assem- 5 Discussion bly adopt a quasi-parallel configuration, giving rise to the great variety of morphologies of protein assemblies The fact that both electrical and hydrophobic charges found in Nature. It can be concluded that this morpho- are distributed inhomogeneously provides a protein logical variety is the result of the combination of elec- with a dipolar character. The dipole character helps trical and hydrophobic forces plus the steric hindrance to orient a dipole protein relative to other dipole pro- effect (geometrical frustration). teins towards the most favourable relative orienta- We have postulated a definition of hydrophobic tion for a stable conformation. Our model rationalizes potential due to hydrophobic charges, which in turn the evidence that proteins presenting a hydrophobic leads us to a definition of a hydrophobic dipole. Follow- dipole moment tend to unite in such a way that their ing the electrical analogy, an expression of the energy hydrophobic dipole moments align in parallel until they exerted by two hydrophobic dipoles has been reached, are close enough to establish more permanent hydrogen which explains the affinity and relative orientation of bonds of the native conformation. the monomers in an assembly, in compliance with the Although this mechanism of hydrophobic attraction so-called Biological Membrane effect. is general for all self-assembled systems, there are some After studying other systems, either protein–protein systems in which not only the hydrophobic interaction or protein–nucleic acid, ranging from the simplest to is attractive but also there is attraction between the the most complex, we find that the mechanism of the electric dipole vectors when they tend to counter-align. BM effect is general to all assemblies [18, 21, 28]. No Consequently, whether attractive or repulsive, the elec- exceptions to this effect have been found so far, so it tric dipole interaction interferes with the hydropho- can be considered a global effect probably extendable bic interaction modulating the native configuration of to any type of particles and biological corpuscles since assembled systems. There are also some systems in hydrophobic attraction is present in most of the molec- which the hydrophobic interaction is repulsive, in which ular levels, being a fundamental pillar for the support case the assembling is done exclusively by attraction of of life and health. the electric dipoles. In all of the cases and scales stud- Supplementary Information The online version con- ied, it is observed that the BM effect occurs, regardless tains supplementary material available at https://doi.org/ of the size of the species studied, as shown by the exam- 10.1140/epje/s10189-023-00320-8.. ple of the case of the PDBid: 3J4F system and in any other studied (see, for example, the cases shown in the Acknowledgements This work was supported by the Supplementary Information). We thus consider that the Ministerio de Economı́a y Competitividad of Spain biological membrane effect is a general principle appli- [BIO2017-84166R and PID2021-125632OB-C22] and by the cable to all kinds of macromolecular associations. Centre de Referència de R+D de Biotecnologia de la Gen- The interaction between electric dipoles is weaker eralitat de Catalunya. The authors thank Lynn Strother for than that between electric charges. It might appear revising the English text that the interaction of proteins as dipoles is not signifi- cant. In fact, the unions between monomers to form an assembly are due to relatively strong hydrogen bonds. However, it is in the search for interacting partners where the interactions between dipoles are important in defining the proper relative orientation of the different Author contribution monomers in order to adopt the native configuration, after having explored other possible configurations in A.M-V contributed to conceptualization (lead); writ- space. ing—original draft; writing—review & editing. E.Q con- Resorting to exact methods developed in classical tributed to conceptualization (supporting); writing—review electrodynamics and statistical mechanics, would make & editing. J.C contributed to investigation (supporting); this task overwhelming. The method developed here resources (supporting). takes advantage of the simplicity of the experimentally verified hydrophobic potential formulation. As far as Funding Open Access Funding provided by Universitat we know, no theoretical physics analysis of this kind Autonoma de Barcelona. 123 62 Page 10 of 10 Eur. Phys. J. E (2023) 46 :62 Data availability The authors declare that all the crystal- 13. R.M. Jackson, M.J.E. Sternberg, Prot. Eng. 7, 371 lographic data supporting the findings of this study (both in (1994). https://doi.org/10.1093/protein/7.3.371 main text and its supplementary information) are available 14. D. Chandler, Nature 437, 640 (2005). https://doi.org/ from the Protein Data Bank (https://www.rcsb.org/). 10.1038/nature04162 15. A.J. Patel, P. Varilly, D. Chandler, J. Phys. Chem. B 114, 1632 (2010). https://doi.org/10.1021/jp909048f Open Access This article is licensed under a Creative 16. C. Peters, A. Elofsson, Proteins 82, 2190 (2014). 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