Structural organization of the N and C terminal domains in one surface and interactions between the two IF surfaces will be presented and discussed below. The results considered include volume fraction profiles, ϕ(z), of N and C terminals for both models, corresponding profiles of the basic residues, ϕ+(z), and free energy of interactions between the two IF surfaces with the attached terminal domains, V(D). We start with the case of balanced charge densities for the surface and chains, |σs|=σNC, at low ionic strength, c
=10−5 M. Then we consider the effect of added salt and discuss the options when the surface charge in absolute value is higher or lower than the charge on the grafted chains. At the end, in order to obtain better insights into the properties of each type of terminal domains, we consider the interactions of the surfaces with only one type of the chains (N or C) grafted.
IF surfaces and tails at equal absolute charge: volume fraction profiles
The volume fraction profiles, ϕ(z), show the monomer density of the grafted N and C domains at distance z from the surface. These distributions provide an estimate of how far the grafted chains extend from the surface and what is the most probable location of any specified monomers. The volume fraction profiles for the whole N and C chains are given in Figure 3 while Figure 4 shows the distributions for only positively charged monomers of N and C domains, ϕ+(z). In order to obtain the spatial distribution for an unaffected N and C chains, the profiles were obtained at large surfaces separation, so that the grafted chains do not interact (this corresponds to the limit of an isolated IF in solution).
Distribution of N and C tails
In Figure 3(A) we present the volume fraction profiles for the PG model of N and C tails. The profiles for the two tails are quite different: the monomer distribution of the more hydrophobic N tails is more narrow compared with the profile for C tails, with most of the monomers located in the first 10 layers from the surface and the maximum density at z=4 a0. The extension of the N tails does not exceed z=13 a0. More polar C tails have lower density near the surface and more extended profiles. The maximum density is slightly shifted away from the surface, z=6 a0, and the profiles extend up to z=20 a0. With the simple block-copolymer model for terminal domains we obtained the two distinct populations of the chains: (i) more hydrophobic N tails are collapsed near the surface and (ii) more polar C tails are projected farther into the solution. However, we should notice that both types of chains are actually quite compact near the surface. With contour lengths of 161 a0 and 138 a0 the chains do not spread out more than 13 a0 and 20 a0 respectively. We attribute this behaviour not only to the hydrophobic nature of both tails, major component of which is glycine, but also to the attraction of the positively charged end-monomers to the negatively charged surface, causing formation of loops.
The profiles for more realistic (AA) model for terminal domains, presented in Figure 3(B), show much smaller difference between the distributions for N and C tails. In general, the behaviour of the all four chains is similar: the distributions are quite narrow; most of the monomers are located within the first 10 layers from the surface, with the maximum density at z=2– 3 a0. The heights of the density maxima reflect chain lengths, with the highest maximum for the longest N1 tail and lowest one for the shortest C10. Having the contour length of 124– 180 a0 all the chains are in collapsed state and do not protrude far into the solution due to their hydrophobicity (∼50% of glycine) and the electrostatic attraction to the surface. The highest value of volume fraction is obtained for N1, the longest domain (180 a0). With the maximum in layer 2, the chains do not extend more than z=14 a0.
The high monomer density near the surface reflects the strong hydrophobic properties of N1 tail—with 24% of non-polar and 40% of glycine residues the chains prefer to be in compact conformation, reducing contacts with the polar solvent. Similar tail extension is observed also for N10 except that the maximum density value is lower than that for N1, because N10 chains are shorter and slightly less hydrophobic (17% of H monomers and 47% of G). The volume fraction profiles for C tails are slightly more extended than those for N tails. In particular, the distribution for C10 tail extends farthest, up to z=20 a0, and the monomer density near the surface is reduced. Even though C tails also consist of about 50% G residues, the fraction of non-polar H monomers is much smaller, 9% and 2% for C1 and C10, respectively. Being more polar than N tails, C tails extend a little farther into the solution.
In Figure 3(C) and (D) we compare the monomer distributions for the two models. Figure 3(C) shows the profiles separately for N and C tails and Figure 3(D) compares the total profiles for N+C tails together. The simplified PG model of the N tails gives the density distribution quite similar to the combined profile for N1+N10 tails, see Figure 3(C). Even though in the more detailed AA model the maximum is slightly closer to the surface and the extension of the profile is slightly larger (dashed line), these differences are comparatively small. As for the C tails, the difference between the profile for the PG model and the combined C1+C10 profile for the AA model is more pronounced. The general shape of the profiles is similar, so is their extension (to z=20 a0), but the density maximum for the PG model is lower and shifted away from the surface. That gives the impression that the hydrophilicity of the C tails in the PG model is somewhat overestimated; the more accurate AA model predicts that the C tails are more hydrophobic. Nevertheless, the relatively narrow profiles for the terminal domains coincide with the prediction of the compact structure of the tails due to formation of the glycine loops . The glycine loops hypothesis predict that quasi-repetitive, glycine-rich terminal domains of epithelial keratins comprise flexible and compact glycine loops, where sequences of glycine make loops between the stacked non-polar residues. Even though SCF method does not allow obtaining such structural loops, it predicts compact conformation of the terminal domains near the surface. Therefore, despite some discrepancies in individual profiles for N and C tails, in the two models, the combined profiles for all (N+C) tails are fully consistent with each other, see Figure 3(D). The simple glycine multi-block model for N and C terminal domains reasonably well reflects the density distributions of terminal domains for K1/K10 IF.
Distribution of the basic residues of the tails
The volume fraction profiles for N and C domains provide the information about spatial distribution of the chains as a whole, while the location of the ends of the chains can be obtained from the distribution of the positive residues, ϕ+(z). The basic residues in the PG model are located only at the end of the tails and in the AA models they are scattered along the chains with higher concentration near the ends. The volume fraction profiles for basic residues in N and C tails, ϕ+(z), are presented in Figure 4.
For the PG model the distributions of positively charged monomers for N and C tails are practically the same. The positive monomers for both tails are located near the surface, with maximum at the first layer followed by abrupt decrease in the monomer density with the distance from the surface. For the distances z>5 a0 the fraction of basic monomers becomes very small. That result allows us to conclude that the basic residues, and, therefore, the end of the chains are located at the surfaces, so the tails form either loops back to the grafting surface or bridges with the opposite one.
As for the AA model, the distributions of positive residues for N and C tails differ both from those in the PG model and between each other. First, both distributions for the AA model are wider, especially for the C tails, and second, the difference between the ϕ+(z) profiles for N and C tails is more noticeable. For the N tails, ϕ+(z) is similar to that for the PG model, with the maximum at the first layer and subsequent decrease of the density with distance. At distances z>10 a0 very small fraction of basic monomers can be found. The total volume fraction is higher than that for the PG model because the amount of the positively charged monomers is higher. In the PG model there are only 5 basic monomers in each tail, while for N1 and N10 the numbers of basic monomers are 15 and 8, respectively. Taking into account that the grafting density of N tails for the PG model is the same as the sum of the grafting densities for N1 and N10, the calculated total amount of the positive charges for both N tails in the AA model is more than twice higher than that for the PG model. That results in about double the value of volume fraction of basic monomers for the AA model. Positively charged monomers for C tails distribute much wider and spreading gradually over ∼17 layers from the surface. The maximal density is again in the first layer but its value is more than half than that for the N tails, even though the number of positive charges for the C tails is not much smaller, 9 and 8 for C1 and C10, respectively.
The density profiles for all the tails show that the maximum density for the basic monomers is always at the first layer. The fact that the highest concentration of those residues is at the surface confirms our hypothesis that the charged monomers adsorb onto the surface, so the chains form loops and/or bridges between the surfaces. Broader volume fraction profiles of basic monomers for the AA model possibly result from the different distribution of the charged monomers along the chains. In the more detailed AA model, the basic monomers are not located exactly at the end of the chains, but somehow distributed along the whole length of the chains, with higher concentration at the ends. Thereby, the more uniformly distributed charges in the AA model give a thicker adsorbed layer while the clustered charges in the PG model adsorb flat on the surface, producing a very thin layer, similar to that of highly charged polyelectrolytes.
IF surfaces and grafted tails at equal absolute charge: interaction potential profiles
The interactions between the two surfaces (IF cores) covered by grafted N and C terminal domains can be evaluated by calculating free energy of interactions between the surfaces at each separation D. The free energy of interactions A(D) is calculated from the partition function under conditions of restricted equilibrium, described by Evers et al. . Under such conditions some components of the system are free to diffuse from the gap between the two surfaces to the bulk solution (e.g. water molecules, ions, free amino acids) and the others are restricted to stay within the gap (e.g. grafted N and C domains). The net interaction potential, V(D), is the difference between the free energy value at separation D and its value when the surfaces are far apart, V(D)=A(D)−A(D
) and it is measured in units of . The “far apart” separation, D
, is such that the two surfaces do not interact; in our calculations D
ranges between 150 a0 and 1000 a0, depending on the model and salt concentration. When the interaction potential is negative, V(D)<0, the two surfaces attract each other, while the positive potential, V(D)>0, implies the repulsive interactions between the surfaces. It can be shown that the interaction force between the two polymer-covered surfaces can be evaluated from the obtained interaction potential [37, 62].
Low ionic strength
The interaction potential for surfaces with attached N and C terminals (for both models) in conditions of charge balance between surface and chains, |σs|=σNC, is presented in Figures 5 and 6. There is no need for additional counterions to satisfy the charge neutrality condition. Ideally we would run the SCF calculation in the absence of added ions, but we are forced to introduce an extremely small concentration of added salt to maintain convergence. Still, the extremely low salt concentration case captures the experimental set up of two IF surfaces immersed in deionised water, where a small salt concentration cannot be avoided. Such is the case of the Jokura experiment  where the water extractable materials (NMF) from the SC sample were first released and then deionised water was added. Other extreme cases, where the surface charge is higher or lower than the charge on the terminal chains and addition of certain amount of salt (counterions) is required to obtain the charge neutrality, will be presented and discussed further below.
Figure 5 compares the interaction potential for both AA and PG models at low salt content, c
=10−5 M. For both models the interaction potential has a well pronounced minimum at D=17 a0 (6.8 nm) for the PG model and D=18 a0 (7.2 nm) for the AA model, corresponding to net attractive interactions in the system. It is interesting that the separations at which the attractive minimum occurs (D≈7 nm) are in agreement with the experimental values for the distance between the two IFs, D≈8.2 nm . The attraction between the surfaces at separations D≈15– 35 a0 (where V(D)<0) occurs due to the well known polyelectrolyte bridging effect [34, 37, 63, 64] and favorable electrostatic conditions (ionic strength). In the limit of low surface coverage, we believe that positively charged end-monomers are attracted to the opposite surface forming bridges across them. The possibility to be simultaneously attracted to more than one surface is more entropically favorable. The volume fraction profiles of the charged monomers discussed above support this picture as the positively charged residues are mostly located near the surface, which indicates the possibility of formation either loops or bridges (if the surfaces are close enough). At larger surface separations, D>35 a0 the interaction potential approaches zero, indicating that the grafted chains do not interact. However, at short separations, D<15 a0, the potential is positive due to strong steric repulsion between the chains.
We should draw attention to the fact that our simple PG model for N and C tails, based on the repetitive motif of glycine blocks, very well reproduces the result of the more complex AA model based on the amino acid sequence. Both characteristics of the system—the volume fraction profiles and the interaction potential between the IF surfaces—are in a good agreement between the two models. We believe that the PG model can be slightly improved, for example, by introducing some H residues into the C tail model and/or by distributing the charge less blockwise along the chain. Despite its simplicity, the PG model reflects well the properties of the N and C domains and, therefore, it probably can be used as a starting point for more refined (and computational intensive) modeling techniques, such as MC, MD or DPD.
High ionic strength and NMF
The interaction potential for the two models at different ionic strength is given in Figure 6. As we already discussed, at low salt content, c
<0.1 mM, the interaction potential develops an attractive minimum at short separations between the surfaces and levels to zero at longer separations. At higher ionic strength, c
≈1– 10 mM, the minimum becomes shallower and a repulsion apprears at larger separations. Our two models for terminal domains give qualitatively similar results but in the PG model more added salt is required to destroy the attraction, i.e. the PG model still shows a small attraction at c
=10 mM, while for the AA model the interactions are already repulsive at all separations at c
=5 mM. That occurs because the attraction to the surface of the charged block at the end of the PG chains is stronger than that of the AA chains (more uniform charge distribution along the chains), so more salt is needed to affect the attraction. In every case, at salt concentration near physiological conditions, c
=0.1 M, the strong repulsion between the surfaces with grafted chains is obtained for both models. High salt content leads to electrostatic screening, so the grafted chains mediate essentially a steric repulsion between the two opposing surfaces, much like polymer brushes. Obviously, at even higher ionic strength (c
>0.1 M) the repulsion between the surfaces becomes stronger and more short-ranged.
Calculations at varying levels of salt concentration were aimed at mimicking the Jokura experiments with normal (healthy) and reduced amounts of natural moisturizing factors (NMF) in SC. The amount of NMF is directly responsible for hydration level and elasticity of the skin [23–27]. NMF is made mostly of free amino acids derived from the enzymatic degradation of filaggrin, as well as organic and inorganic salts [23–30]. As charged amino acids and ions are important components of NMF, we have used high ionic strength as the first approximation. Increasing amounts of added salt tips the IFs interaction from attractive to repulsive. Experiments show that treatment with potassium lactate, could restore the SC hydration .
The next level of increasing complexity in our model was to account for the complex mixture of amino acids in the suspending matrix between the IFs. In order to create our coarse model of NMF we adopted the amino acid composition form Jacobson et al.  and then divided all amino acids into the same groups (H, P, G, +, −) used in the model for N and C tails. The water content was set to 30% as reported in the literature [23, 27, 65, 66] and addition of neutraliser was necessary to ensure charge neutrality in the bulk. The detailed composition of this NMF + water model is given in Table 2.
The interaction potentials for two IF surfaces with grafted terminals immersed into NMF solution are presented in Figure 6 by dashed lines. The graphs clearly illustrate that the more complex NMF-water mixture leads to an even stronger repulsion. We have observed that the mixture of free amino acids has stronger effect on the interactions between IF surfaces than just adding salt to the solvent; the NMF not only provides a strong repulsion between the approaching surfaces but also “pushes” the surfaces further away from each other. We believe the reason for such a strong repulsion between the surfaces rests in the high amount of free charged species (ions and amino acids), but what makes this forces more long ranged is the presence of free neutral amino acids. Solution of only neutral amino acids only slightly decreases the attraction between the surfaces and shifts of the attraction minimum to larger separations.
This result does not support the Jokura et al. finding that neutral amino acids improve mobility of keratin fibers, as well as basic amino acids, but not acidic ones . Our results showed that only charged species in solution can affect the attractive intermolecular forces between negatively charged IF cores with grafted positively charged terminal chains. We should also mention, that in our coarse-grained model we could not reveal the specific effect of basic amino acids, as the properties of positive and negative free amino acids are the same except of the charge and the charge neutrality is required in the bulk. In order to examine the effect of specific ions a more sophisticated model and/or method is required.
Surface charge higher or lower than the charge of the terminals
Previously we have described the case when charge on the surface is fully balanced by charge on the grafted chains only. As we already mention, some parts of the protofilaments and, therefore, some of the charged amino acids could be hidden inside the IF core and do not contribute to the charge density. That assumption may leads to the situation when the charges of the grafted tails and surface are not balanced. In order to account for such possibility, in this section we consider cases when the charge of the surface is higher or lower than the charge on the grafted tails. In order to obtain charge neutrality in under these conditions certain amount of counterions is needed. We do so by setting the concentration of salt c
in the bulk, which is in equilibrium with the gap between the decorated IF surfaces.
In Figure 7 we present the interaction potential, V(D), for the surface charge densities of σs=−0.0655 e, −0.0660 e, −0.0664 e, −0.0670 e, −0.0675 e, and −0.0710 e for AA model and σs=−0.0405 e, −0.0410 e, −0.0415 e, −0.0420 e, and −0.0425 e for the PG model. The charge on the grafted chains is kept constant at the values of σNC=0.0664 e for the AA model and σNC=0.0415 e for the PG model. Thus, for the AA model, surfaces with charge density |σs|<0.0664 e are “undercharged” (in the specific sense that the surface charge density is smaller in absolute value than that needed to balance the charge on the grafted chains) and, correspondingly, with |σs|>0.0664 e they are “overcharged”. For the PG model the threshold values of surface charge density for undercharged and overcharged surfaces would be, respectively, |σs|<0.0415 e and |σs|>0.0415 e.
For each surface charge we found the values of the salt concentration which provide charge balance. The surface charge densities with the balancing salt concentrations are given in Table 3.
In the cases of under- or overcharged surface at low salt concentrations, repulsive electrostatic forces dominate, so the bridging attraction between the covered surfaces could not be seen. At high ionic strength, the repulsion decreases due to screening. When the surface is overcharged, i.e. when the charge on the surface is higher in absolute value than the charge of the chains, it is possible to find balancing salt concentration, under which the interaction potential between the surfaces would be the same as for the case when the surface charge is fully balanced by the charge of the chains only. The graphs in Figure 7 show that at certain amount of added salt the interaction potential profiles for |σs|≥0.0664 e for the AA model and for |σs|≥0.0415 e for the PG model completely overlap. Table 3 also shows that the stronger the charge imbalance (difference between surface and chains charge), the higher the amount of salt is required to neutralize the charge in the system. However, when the surface is undercharged, the attractive part is reduced and the potential always displays long-ranged repulsion, which increases with increasing charge imbalance. This phenomenology is the consequence of charge screening, as the following simplified model calculation shows. Consider a plane surface with a negative surface charge density σs, surmounted by a charge cloud at charge density σNC uniformly distributed over a thickness H. We solve the linearised Poisson-Boltzmann equation for this problem,
where φ is the electrostatic potential (in units of kBT/e), κ
is the inverse Debye screening length (), and lB is the Bjerrum length (lB≈0.72 nm). The boundary conditions are d φ/d z=−4π lB|σs| at the wall and φ→0 as z→∞, and φ should be continuous at z=H with a continuous first derivative. This problem can be solved analytically. The behaviour of the potential at distances z>H from the surface is the relevant piece of information,
The prefactor indicates there is a special balance point where the potential vanishes completely for z>H. This point occurs when |σs|/σNC= sinh(κ
H). The right hand side is an increasing function of κ
H, and only approaches unity for κ
H→0. Thus we see the surface has to be overcharged in order to reach the balance point and a higher degree of overcharging requires a larger value of κ
H to compensate, corresponding to higher salt, exactly as found above. The reason for this is that for z≥H the surface charge density is screened by an factor relative to the diffuse oppositely-charged cloud.
The interactions between negatively charged surfaces covered by positively charged polyelectrolytes were investigated experimentally [67, 68] by Monte-Carlo simulations , and theoretically . The results of these studies have shown that the attractive bridging can dominate only when the charges of the polymers and ions balance the charge of the surface. Claesson and Ninham  demonstrated that attractive forces between mica surfaces covered by adsorbed chitosan were observed only when electrostatic double layer disappeared, i.e. when surface charges are exactly balanced by the charge of adsorbed polysaccharide. When charge of chitosan, controlled via variation of solution pH, was higher or lower than the charge of the mica surfaces, the electrostatic double layer repulsion forces dominate. Dahlgren et al.  measured the force acting between two mica surfaces covered by MAPTAC polyelectrolyte and also carried out MC simulations for two surfaces covered by oppositely charged polyelectrolytes. When PE adsorption was such that the surface charge was balanced by the polyelectrolyte, a strong attractive force was observed at short surface separations. Addition of salt to the MAPTAC solution facilitates the increased adsorption of polyelectrolyte, that leads to a reduced attraction and the appearance of a repulsive double-layer force. The authors concluded that the attractive bridging mechanism will only dominate when the polyelectrolyte adsorption approximately neutralizes the surface charge density. Borukhov et al.  proposed a theoretical approach to explain the behaviour of polyelectrolytes between charged surfaces. Their calculations show that at low ionic strength the attractive interactions between the surfaces take place when polymer adsorption balances surface charge. At high ionic strength the surface charge is balanced both by polymers and ions and the stronger the polymer charge, the more salt is needed to achieve the charge neutrality. The authors also considered values of adsorbed polymer higher or lower than the equilibrium adsorbed amount. When the adsorbed amount was lower than the equilibrium one, the attraction was weaker. However, when the adsorbed amount was higher than the equilibrium one, the results show stronger attraction between the walls and also appearance of strong long-ranged repulsion, similar to those shown in Figure 7.
In the experiments described by Jokura  loss of elasticity was observed for SC samples with extracted NMF and further hydrated by addition of deionised water. The authors suggested that loss of elasticity happens due to attractive intermolecular forces between keratin fibers. NMF, mainly free amino acids, reduces intermolecular forces through nonhelical regions of keratins (N and C terminal domains), so the keratin filaments acquire their elasticity. Our modelling results, theoretical consideration and literature analysis [67–69] show that the attractive interactions between IF at low salt content occur only when |σs|=σNC or the IF surfaces are slightly overcharged. We conclude that the Jokura experiments could take place only at condition that surface charge is equal or slightly higher than the charge on the nonhelical chains. Thus, the charge of IF cores could not be much higher or lower than the charge on the unstructured terminal domains.
Role of each type of terminal domains
Why has Nature used two types of unstructured terminal domains of similar length scale for each keratin protein? Does each domain type has a specific function and, if so, what is it? Is it necessary to capture the specific differences in a model? Different authors have taken different approaches. In modelling neurofilament projection domains [43–47], the much shorter globular N domains were not included in the study; only the C projection domains were considered. On the contrary, in 3RS tau protein research  the authors focused on the 196 amino acid long unstructured N domains. Thus, as the role of each domain in keratins is yet unknown, in order to generate insights, we decided to take advantage of fast computer models and examine the interactions mediated by each type of terminal domains separately.
Figure 8 shows the interaction potentials for the IF cores grafted only with N domains or only C domains. The graphs from Figure 8 summarize the results and compare them against the full model calculations. For the calculation of only one type of domains, the grafting density of the chains was kept the same as before, σ=0.00415; this is half the total grafting density for both chain types together (σ=0.0083). The charge on the surface was then adjusted to neutralize the charge from the chains, σs=−0.0332 e for the AA model and σs=−0.02075 e for the PG model.
When only N chains are present, the minimum becomes much deeper and is shifted closer to the surface. Even with the addition of 0.1M of salt this attraction minimum is still quite deep (data not shown). Apparently, the more hydrophobic N tails behave as a “glue”, holding together the two surfaces. In contrast, the C tails behave in the opposite way. The interaction potentials for a similar model including C-tails only result in much smaller attraction minimum, pushed away from the surface. The more polar C-tails contribute much less to the attraction between the IF surfaces.
It is tempting to propose that both N and C domains play important roles in the structure and interactions of skin keratin IFs. The more hydrophobic N chains bring about a strong attraction between the IF surfaces while the more polar C tails push the surfaces away from each other, so that the two types of domains work together to keep IFs at the optimal separation. Therefore, we believe that it is the combination of both types of the domains balances the interactions between the intermediate filaments.