# The Effect of Transmembrane Protein Shape on Surrounding Lipid Domain Formation by Wetting

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{i}(i = 1, 2, 3, ... m) parts of a lipid molecule of the i-th type, so that ${{\displaystyle \sum}}_{i=1}^{m}{\nu}_{i}=1$. The condition of phase equilibrium can be written as follows:

_{eq}is the chemical potential of the quasimolecule in the domain, and μ

_{i}is the chemical potential of the i-th component in the liquid-disordered phase. μ

_{eq}can be related to the grand potential of the wetting ordered phase, W. It is convenient to use the grand potential in the analysis of the wetting since its independent variables are temperature and chemical potential, which are homogenous throughout the membrane in equilibrium. We denote the total number of quasimolecules in the domain as n, the average area per quasimolecule as a and the total area of the domain as s. Then, according to the well-known thermodynamic relation [49], we obtain:

_{u}) and inner (R

_{d}) monolayers are determined by minimizing the free energy of the system. In the case of monolayer wetting, it is assumed that phase separation occurs in one (outer) monolayer of the membrane at a given temperature, i.e., this monolayer is saturated, but in the inner monolayer the global phase separation does not occur, i.e., the inner monolayer is subsaturated. In this case, the state of the system is determined by the radius of the ordered domain r, which is formed in the inner monolayer due to the wetting. Thus, in the case of bilayer subsaturation, we consider the dependence μ(W(s)), and in the case of monolayer subsaturation, we consider μ(W(r)).

_{i}can be written as follows:

_{i}is the activity of the i-th component in the surrounding membrane, ${c}_{i}^{eq}$ is the equilibrium activity of this component near the boundary of the surrounding membrane and the liquid-ordered domain, k

_{B}is the Boltzmann constant, T is the absolute temperature. Difference between activities c

_{i}and ${c}_{i}^{eq}$ is small near the phase transition, i.e., the subsaturation is small. This allows us to expand Equation (3) in a Taylor series. We keep the first non-zero term in the expansion and substitute the resulting expression into Equation (1). As a result, we get:

**n**, called directors, characterizing the average orientation of lipid molecules. This field is related to some surface passing inside the monolayer. The shape of the surface is determined by the field of unit normals

**N**to it; normals are considered to be directed towards the intermonolayer surface of the membrane. We take into account two deformation modes—tilt and bending. Deformations and elastic moduli are referred to the so-called neutral surface, on which bending and lateral stretching deformations are energetically independent. Bending deformation is quantitatively described by the divergence of the director along the neutral surface, and tilt deformation is described by the tilt vector

**t**=

**n**/(

**nN**) −

**N**≈

**n**−

**N**. We assume that membrane deformations are small. The energy of the deformed monolayer, measured from the state of a flat monolayer, can be represented as [51]:

_{0}is the neutral surface area in the initial undeformed state. The description of deformations within the framework of such a thermodynamic approach does not allow taking into account any large-scale fluctuations (for example, membrane shape fluctuations), but is focused on the consideration of the membrane deformations in the vicinity of the protein inclusion, at the distances of the order of several nanometers under the assumption of their rapid decay. There are several similar models that describe elastic properties of membranes by introducing deformation modes. They include, for instance, the model proposed by Fournier [52]. In this work, independent elastic moduli of bending, tilt, and tilt variation are introduced and estimated. At the same time, the Hamm-Kozlov model demonstrates that the variation of tilt and bending are additive modes and the contributions of both of them to the elastic energy are determined by the modulus known from measurements of the pure monolayer bending [51]. This allows us to use only two deformation modes, bending and tilt, with bending modulus measured in the experiment, and tilt modulus being estimated from the value of the interfacial tension between water and hydrocarbon tails of lipid molecules. However, it is noteworthy, that in particular cases the elastic models proposed by Fournier and by Hamm and Kozlov yield very similar nontrivial results. For example, the model by Fournier predicts decaying oscillations of the membrane thickness for a certain relation between the elastic moduli [52]. In the framework of the Hamm-Kozlov model the oscillations arises from bending deformation restricted by the condition of local volumetric incompressibility of lipids [31,32,33].

**n**→ n

_{r}= n,

**N**→ N

_{r}= N,

**t**→ t

_{r}= t. Within the required accuracy, the divergence of the director along the neutral surface can be written in the form: div(

**n**) → dn(r)/dr + n(r)/r. In addition, we take into account the condition of local volumetric incompressibility in the following form [51]:

_{0}is the thickness of the unperturbed monolayer. The thickness of the unperturbed ordered monolayer is denoted by h

_{r}, and the thickness of the unperturbed monolayer of the surrounding membrane is denoted by h

_{s}. Values related to the outer and inner monolayers are marked by the indices “a” and “b”, respectively. The shape of the neutral and intermonolayer surfaces H(r) and M(r) are described by the distance from the Or plane to the point with coordinate r on the surface, measured along a perpendicular to the plane Or. In such designations the thickness of the outer monolayer is written in the form h

_{a}(r) = H

_{a}(r) − M(r), the inner one is given by the equation h

_{b}(r) = M(r) − H

_{b}(r). Within the required accuracy, the radial projection of the normal vector to the neutral surface of the outer and inner monolayers equals to N

_{a}= dH

_{a}/dr and N

_{b}= −dH

_{b}/dr, respectively. One can express the functions H

_{a}(r) and H

_{b}(r) from the Equation (7) in terms of the shape of the intermonolayer surface M(r) and the radial projections of the directors of the outer and inner monolayers, a(r) and b(r). Using the simplified definitions of the tilt vectors t

_{a}= a − N

_{a}= a − dH

_{a}/dr, t

_{b}= b − N

_{b}= b + dH

_{b}/dr and the incompressibility condition (7), we express the tilt vectors in terms of the functions M(r), a(r), b(r) and their derivatives with respect to the coordinate r. In addition, we take into account that within the required accuracy:

_{a}(r) and H

_{b}(r) can be expressed through M(r), a(r), b(r) and their derivatives with respect to r, using Equation (7). We substitute the obtained relations into the elastic energy functional, Equation (6), for each monolayer. The functional of total elastic energy of the membrane equals to the sum of the elastic energies of its monolayers, which depends on three functions: a(r), b(r) and M(r). To find the extremals of this functional, we vary it with respect to the functions a(r), b(r), M(r). As a result, we obtain three Euler-Lagrange differential equations. We substitute the solutions of these equations into the functional of the total elastic energy of the membrane. The expressions for the functions a(r), b(r), M(r) obtained as a result of solving the system of Euler-Lagrange differential equations contain indefinite coefficients, which are determined from the boundary conditions and from the condition of minimum of the total elastic energy. The boundary conditions are determined by the geometry of the TMD of the protein, the requirement of continuity of directors and neutral surfaces, and the requirement of the finiteness of the deformations. The model is described in details in Appendix A.

_{B}T

_{0}(T

_{0}≈ 300 K is the room temperature, i.e., k

_{B}T

_{0}≈ 4.14⋅10

^{−21}J), K = 40 mN/m = 10 k

_{B}T

_{0}/nm

^{2}[51,53] for monolayer bending and tilt moduli, respectively. The equilibrium thickness of the ordered monolayer is taken to be equal to h

_{r}= 1.8 nm, the monolayer thickness of the surrounding membrane equals to h

_{s}= 1.3 nm. The lateral tension of the monolayer is taken equal to σ = 0.01 k

_{B}T

_{0}/nm

^{2}, which is of the same order of magnitude as it is determined for cell plasma membranes [54]. Subsaturation Δ both in the case of a bilayer and in the case of a monolayer wetting is taken to be equal to 1% = 0.01.

## 3. Results

_{p}, the radii of ordered domains in the upper and lower monolayers, R

_{u}and R

_{d}, respectively, as well as the radial projections of the boundary directors in the upper and lower monolayers, n

_{1}and n

_{2}, which can be different. Schematically, this model is presented in Figure 1.

_{p}, R

_{u}, R

_{d}, n

_{1}, n

_{2}. The obtained results are compared with the case of the epidermal growth factor receptor, for which it has been experimentally shown that a change in the conformation and, correspondingly, in the shape of TMDs of its dimer plays a key role in the activation of this receptor [42]. Using these data, we established system parameters that model the EGFR conformational transition from the closed state to the open one. Based on these parameters, we establish the effect of such a change in the shape of the protein on the formation of a liquid-ordered lipid domain in the membrane surrounding the protein. Upon transition from a closed to an open state, the radius R

_{p}increases from approximately 0.9 nm to 1.6 nm [42]. In addition, based on the available data, we can qualitatively assess the change in radial projections of the boundary directors n

_{1}and n

_{2}. We assume that in the closed state of the receptor n

_{1}< 0; n

_{2}≈ 0, and in the open state n

_{1}> 0; n

_{2}< 0 (see Figure 2). The adopted parameter values should be considered as semi-quantitative estimates. Thus, the variation of the radius of TMD by 0.1–0.2 nm should not significantly change the results; the major factor is only a substantial increase in the radius of the TMD when passing from one conformation to another.

_{p}, equals to twice the thickness of the monolayer of the ordered phase, i.e., H

_{p}= 2h

_{r}.

_{u}is formed in the outer monolayer due to the global phase separation. In the first case, a bilayer ordered domain is formed due to wetting of the protein. In the second case, wetting leads to the formation of the ordered domain in the inner monolayer, while the domain in the outer monolayer preexists as a result of the global phase separation.

_{1}and n

_{2}for the cases of closed and open conformations of EGFR. A typical dependence is shown in Figure 3.

_{1}and n

_{2}in the range from −0.7 to 0.7. This restriction was introduced for the reason of keeping the deformations small, so that the squared director projection at the TMD boundary was substantially less than unity. For each dependence E(s), we find the total domain area in the outer and inner monolayers s

_{eq}corresponding to the minimum of the free energy, i.e., equilibrium state of the system. Then we calculate the dependence of the elastic energy W on the radius of the domain R

_{d}in the lower monolayer for the fixed total area of the ordered phase around the protein s

_{eq}and find the value of the radius R

_{d}corresponding to the minimum of the energy W. The radius of the ordered domain in the upper monolayer R

_{u}is determined from the relation: s

^{eq}= π(R

_{d}

^{2}− R

_{p}

^{2}) + π(R

_{u}

^{2}− R

_{p}

^{2}). Thus, we determine the size of the domains of the liquid-ordered phase in both monolayers for each given shape of the protein TMD. In this case, a stable ordered domain can be either bilayer (R

_{d}, R

_{u}> R

_{p}) or monolayer (either R

_{d}= R

_{p}, R

_{u}> R

_{p}, or R

_{d}> R

_{p}, R

_{u}= R

_{p}).

_{p}= 0.9 nm, n

_{1}< 0; n

_{2}≈ 0), stable domains do not form. Moreover, at R

_{p}= 0.9 nm, stable domains are not formed for any physically reasonable values of the radial projections of the boundary directors n

_{1}, n

_{2}. At R

_{p}= 1.6 nm, a stable domain can form around the protein. The domain can be either bilayer or monolayer depending on the specific shape of TMD, i.e., on the value of the radial projections of the boundary directors. However, the range of values (n

_{1}, n

_{2}), at which stable ordered domains are formed, does not intersect with the region (n

_{1}> 0; n

_{2}< 0) corresponding to the open EGFR conformation, which is illustrated in Figure 4.

_{u}and R

_{d}of the domains in the upper and lower monolayers, respectively, and the radial projections of the boundary directors, n

_{1}and n

_{2}. We varied these parameters for TMD radii of 0.9 and 1.6 nm and determined the sets of parameters at which stable bilayer domains are formed. Calculations carried out for a large number of different sets of parameters R

_{u}, R

_{d}, n

_{1}, n

_{2}showed that the widths of stable domains in the outer and inner monolayers of the membrane, L

_{u}= R

_{u}− R

_{p}and L

_{d}= R

_{d}− R

_{p}, respectively, very weakly depend on the parameter values and are approximately equal to L

_{u}≈ 7 nm, L

_{d}≈ 4 nm. This allows us to fix the domain radii R

_{u}= R

_{p}+ 7 nm, R

_{d}= R

_{p}+ 4 nm, and analyze their stability at different radial projections of the boundary directors n

_{1}and n

_{2}. In more detail, the types of dependences of the free energy on the radius of an ordered domain in the inner monolayer, as well as the stability criteria for domains, are presented in Appendix B.

_{1}, n

_{2}), we determined the ranges of these parameters in which the formation of a stable ordered domain in the lower monolayer is possible for open and closed conformations of the TMD of EGFR. The radius of the ordered domain in the outer monolayer was considered equal to R

_{u}= R

_{p}+ 7 nm. The calculation results are presented in Figure 5.

_{1}< 0; n

_{2}≈ 0; for closed conformation, and n

_{1}> 0; n

_{2}< 0 for open conformation. Comparing these conditions with the range of parameters (n

_{1}, n

_{2}) under which the formation of a stable ordered domain in the inner monolayer is possible (violet areas in Figure 5), we concluded that the stable ordered bilayer domains do not form in the case of closed conformation of the protein (Figure 5a).

_{1}, n

_{2}) (the dark green region in Figure 5b). Namely, for the formation of stable domains, the radial projection of the boundary director n

_{1}in the outer monolayer should be greater than 0.4; in this case, the radial projection of the boundary director in the inner monolayer n

_{2}should be in the range from 0 to –0.3.

## 4. Discussion

_{1}and n

_{2}in two monolayers of the membrane. An increase in the radius of TMD itself should lead to an increase in wetting efficiency due to a decrease in Laplace pressure in the locally formed phase [48]. A comparison of the phase portraits for the open and closed conformations of the TMD of EGFR shown in Figure 5 demonstrate that the radius of the TMD affects, but does not completely determine the wetting process. Indeed, for an open conformation, characterized by a large radius of TMD, with large positive values of the boundary director projections (n

_{1}= n

_{2}= 0.6, for instance), a stable bilayer domain does not form (Figure 5b). However, it forms at the same values of n

_{1}and n

_{2}in the case of the closed conformation with a smaller radius of TMD (Figure 5a). This observation is reflected in Figure 6, showing the effectiveness of various geometries of protein TMD in terms of domain formation. This means that TMD involved into the signal transduction mediated by the lipid domains, should have similar structure and shape; moreover, small changes in the structure of TMD should lead to critical failures in signal transduction. This prediction of our model finds experimental evidence: it is known that mutation of the N-terminus of certain tyrosine kinase receptors leads to a change in the shape of the TMDs of the receptor dimer, which correlates with the appearance of various pathologies [63,64]. Especially interesting in the light of our results, is the work devoted to the study of the vascular endothelial growth factor receptor 2 (VEGFR-2) [64], which, like EGFR, belongs to the class of bitopic proteins. In this work, the authors build an activation model for the wild-type VEGFR-2. According to their model, upon binding to a ligand, the TMD of the receptor changes its conformation from hourglass to conical geometry. Mutations in TMD of the receptor induced spontaneous activation of the VEGFR-2 in the absence of interaction with the ligand. Thus, TMD of the wild-type VEGFR-2 in the inactive conformation has an hourglass geometry (see Figure 6), and the mutants have a conical or inverted conical shape, approximately corresponding to the shape of the active conformation of the wild type receptor. According to our calculations, this should lead to the preferred formation of wetting lipid domains around mutant TMD that is equivalent to spontaneous activation of the receptor.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{p}. We assume that both monolayers of the membrane are subsaturated; in this case, ordered domains formed in different monolayers can have different sizes. We assume that the lipid composition of a domain does not depend on its size. This condition equivalent to that the size of the domain changes due to the exchange with the liquid-disordered surrounding membrane by “quasimolecules”, each of which including the lipid components of the membrane in the same stoichiometry as the whole domain. Let the quasimolecule include the fraction ν

_{i}(i = 1, 2, 3, ... m) of the lipid molecule of each of the m lipid components, so that that ${{\displaystyle \sum}}_{i=1}^{m}{\nu}_{i}=1$. Then the change in the free energy of the system upon the addition of one quasimolecule to a domain reads as follows:

_{i}is the chemical potential of the i-th component in the liquid-disordered surrounding membrane. Under the equilibrium conditions Δμ = 0, i.e.:

_{e}is the equilibrium domain area. The chemical potential μ(s) of the quasimolecules in the domain can be associated with the large canonical potential of the membrane W. If n is the number of quasimolecules in the domain, then:

_{i}can be written as follows:

_{i}is the activity of the i-th component in the surrounding membrane, ${c}_{i}^{eq}$ is the equilibrium activity of this component in the presence of a liquid-ordered domain in the membrane. Near the phase transition, the activity c

_{i}and ${c}_{i}^{eq}$ do not differ much from each other, i.e., the subsaturation is small. This allows expanding the expression (A6) into a Taylor series by subsaturation; keeping the first nonzero term in the expansion, we get:

_{p}is the cross-sectional area of TMD of the protein. Substituting the expression dn = ds/a into Equation (A11), we obtain:

_{p}is the radius of the transmembrane domain of the protein. The expression (A10), which relates the chemical potential to the subsaturation, changes accordingly:

_{d}in the inner monolayer. It can be represented as follows:

_{a}(r), the inner monolayer H

_{b}(r), the intermonolayer surface M(r), the radial projection of the director in the outer monolayer a(r) and the radial projection of the director in the inner monolayer b(r). Using the conditions of local volume incompressibility, the shapes of the neutral surfaces of monolayers are expressed through the director projections a(r), b(r) and the shape of the intermonolayer surface M(r). Variation of the functional of elastic energy, Equation (6), with respect to these functions leads to three Euler-Lagrange differential equations. Their solutions a(r), b(r), M(r) contain indefinite coefficients, which are determined from the boundary conditions. These conditions are implied at the protein boundary (r = R

_{p}), at the boundary of the ordered domain with the surrounding membrane (r = R

_{u}, r = R

_{d}). The boundary conditions at the protein are determined by the TMD shape. The latter is determined by its length H

_{p}, as well as the radial projections of the boundary directors n

_{1}and n

_{2}in the upper and lower monolayers, respectively (see Figure 1); in this case, n

_{1}may be unequal to n

_{2}, which models the possible asymmetry of TMD of the protein. The boundary conditions have the following form:

_{p}, coincides with the thickness of the unperturbed ordered bilayer 2h

_{r}:

## Appendix B

_{u}in the outer monolayer. The membrane state depends on four variable parameters—the radii of the domains R

_{u}, R

_{d}and the radial projections of the boundary directors n

_{1}, n

_{2}in the outer and inner monolayers, respectively. Our goal is to determine the range of parameters at which a stable ordered domain can form in the inner subsaturated monolayer. For this, we calculate the dependence of the total free energy E on the radius of the domain r in the inner monolayer for various sets of three parameters: L

_{u}(L

_{u}= R

_{u}− R

_{p}), n

_{1}and n

_{2}. A typical resulting dependency is shown in Figure A1.

**Figure A1.**The dependence of the total free energy E on the radius r of the domain in the inner monolayer. (

**a**) For a fixed domain width in the outer monolayer L

_{u}= 3, 5, 7, 9 nm. The remaining parameters were considered equal to R

_{p}= 1 nm, n

_{1}= 0.5, n

_{2}= 0.5. (

**b**) For a fixed radial projection of the boundary director in the inner monolayer n

_{2}= −0.7, −0.3, 0.3, 0.7. The remaining parameters were considered equal to R

_{p}= 1.6 nm, L

_{u}= 7 nm, n

_{1}= 0.6.

_{min}in the minimum should be less than the energy in the initial state, at r = 0. As can be seen from Figure A1b, there are two types of dependence E(r) for which the formation of a stable domain is possible. In the first case, the free energy monotonically decreases down to the minimum point and then increases (the red curve corresponding to n

_{2}= −0.7). In the second case, there is an energy barrier between the global minimum point and the initial state r = 0. Our calculations show that the barrier-free dependences E(r) correspond to the width of the equilibrium domain in the inner monolayer of approximately 1 nm, i.e., one layer of lipids. Such cases are excluded from consideration since the boundary layer of lipids cannot be considered as a phase and cannot be considered within the framework of continuum models. Then we consider only the dependences E(r) exhibiting the energy barrier. We select those dependencies for which two conditions are satisfied: (i) the resulting domain is stable, i.e., E

_{min}< 0; (ii) the energy barrier E

_{B}of the formation of the equilibrium domain is relatively small (less than 5 k

_{B}T).

_{1}and n

_{2}, stable domains in the inner monolayer correspond to a very narrow range of domain widths L

_{u}in the outer monolayer. Therefore, according to Figure A1a, the minimum free energy with a relatively small initial energy barrier is achieved at L

_{u}= 7 nm. This value weakly depends on the values of the remaining parameters. Therefore, in further calculations, we fixed L

_{u}= 7 nm.

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**Figure 1.**Schematic representation of a lipid-protein domain formed around transmembrane domain (TMD) of the protein by the wetting mechanism. TMD (shown in blue) has the length H

_{p}and the radius in the plane of the membrane R

_{p}. The boundary directors in the upper and lower monolayers are designated as

**n**

_{1}and

**n**

_{2}, respectively. Liquid-ordered lipid domains in two monolayers of the membrane are highlighted in pink.

**Figure 2.**Schematic representation of the conformational transition of TMD of the epidermal growth factor receptor (EGFR) protein from the closed (left) to the open (right) state upon ligand binding. A change in the state of a protein is accompanied by a change in the radius of its TMD, R

_{p}, and radial projections of the boundary directors n

_{1}and n

_{2}.

**Figure 3.**The dependence of the total free energy E on the total area of the ordered phase s for cases of different shapes of the TMD. The radius of the TMD of the protein is 1.6 nm, which corresponds to the open conformation of the EGFR receptor. For each case, a schematic representation of the corresponding geometry of TMD is given.

**Figure 4.**Phase portrait of the system in coordinates (n

_{1}, n

_{2}) in the case of bilayer wetting at R

_{p}= 1.6 nm. The area corresponding to the shape of the TMD of EGFR in the open conformation is highlighted in light green shading. The violet color shades the region of parameters at which the formation of a stable monolayer domain is possible; the green color shades the region of parameters at which the formation of a stable bilayer domain is possible.

**Figure 5.**Phase portraits of the system with closed, R

_{p}= 0.9 nm (

**a**), and open, R

_{p}= 1.6 nm (

**b**), EGFR conformations in (n

_{1}, n

_{2}) plane in the case of monolayer wetting. Light green shading marks the parameter areas (n

_{1}, n

_{2}) that are characteristic of the corresponding conformation. Violet regions indicate the range of parameters at which the formation of the stable domain in the inner monolayer of the membrane is possible. The dark green color indicates the parameter region corresponding to the open conformation of TMD of EGFR, in which the formation of a stable ordered domain in the inner monolayer is possible. The radius of the ordered domain in the outer monolayer was considered equal to R

_{u}= R

_{p}+ 7 nm.

**Figure 6.**Comparison of the wetting efficiency of TMD of various shapes. Wetting efficiency decreases from the left to the right. The barrel-type TMDs (n

_{1}> 0, n

_{2}> 0) are most effectively wetted, the hourglass-type TMDs are least effective (n

_{1}< 0, n

_{2}< 0). Conical (n

_{1}> 0, n

_{2}< 0) and inverted conical (n

_{1}< 0, n

_{2}> 0) TMD are wetted with intermediate efficiency.

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**MDPI and ACS Style**

Molotkovsky, R.J.; Galimzyanov, T.R.; Batishchev, O.V.; Akimov, S.A.
The Effect of Transmembrane Protein Shape on Surrounding Lipid Domain Formation by Wetting. *Biomolecules* **2019**, *9*, 729.
https://doi.org/10.3390/biom9110729

**AMA Style**

Molotkovsky RJ, Galimzyanov TR, Batishchev OV, Akimov SA.
The Effect of Transmembrane Protein Shape on Surrounding Lipid Domain Formation by Wetting. *Biomolecules*. 2019; 9(11):729.
https://doi.org/10.3390/biom9110729

**Chicago/Turabian Style**

Molotkovsky, Rodion J., Timur R. Galimzyanov, Oleg V. Batishchev, and Sergey A. Akimov.
2019. "The Effect of Transmembrane Protein Shape on Surrounding Lipid Domain Formation by Wetting" *Biomolecules* 9, no. 11: 729.
https://doi.org/10.3390/biom9110729