Abstract
The lateral distribution of proteins and lipids in plasma membranes of mammalian cells is not uniform. Lipid-protein domains enriched with sphingomyelin and cholesterol are called rafts. The size of cell membrane rafts is extremely small, and this significantly complicates their experimental study. In model membranes, the lipid composition of which is close to the composition of the plasma membrane, the formation of ordered domains is possible as a result of phase separation induced by a decrease in temperature. An effective method for studying model rafts is atomic force microscopy, which makes it possible to register domains whose size is less than the diffraction limit of visible light. However, this method involves the deposition of the membrane on a solid support. Due to electrostatic, van der Waals, and hydration interactions with the support, the closest membrane monolayer appears in physical conditions that differ from the conditions of the monolayer distant from the support. The asymmetry of physical conditions leads to alteration of the structure and energy of the raft boundary. As a result, of the two possible states of the boundary of equal energy in a free membrane, only one is practically realized on the support. This can lead to a change in the efficiency of specific line-active lipids that accumulate at the boundary of domains and regulate their size by decreasing the boundary energy, similar to the action of surfactants in three-dimensional systems.
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ACKNOWLEDGMENTS
The work was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation.
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APPENDIX
APPENDIX
For simplicity, we consider the case of a large radius raft, such that the curvature of the boundaries of its constituent domains can be locally neglected and the boundaries can be considered as approximately straight lines. In this case, the system has translational symmetry along the boundaries, i.e., all variables depend only on the coordinate perpendicular to the boundary. We introduce a Cartesian coordinate system, the origin of which is located at the boundary of one of the domains (for definiteness, the lower one), the Ox axis is directed perpendicular to the boundaries, the Oy axis is directed along the boundaries, and the Oz axis is directed perpendicular to the membrane plane. In the considered case of translational symmetry along the Oy axis, all variables depend only on the x coordinate. In this case, vector quantities can be replaced by their projections onto the Ox axis, i.e., n → nx = n, N → Nx = N. For small deformations, div n ∼ dn/dx, N ≈ dH(x)/dx = H ′(x), where H(x) is a function characterizing the shape of the monolayer surface, the prime here and below denotes the derivative with respect to the x coordinate.
We assume that the hydrophobic part of the lipid monolayer is locally volumetric incompressible, i.e., the volume of any element does not change upon deformations. The condition for local volumetric incompressibility can be written in the form [29]:
where h(x) is the current thickness of hydrophobic part of the monolayer; h0 is the thickness of the hydrophobic part of the monolayer in undeformed state. For simplicity, we call the thickness of the hydrophobic part of the monolayer as just monolayer thickness.
The shapes of the surfaces of the upper and lower monolayers, as well as of the monolayer interface of the membrane, are characterized by three functions: Hu(x), Hd(x), and M(x), respectively. In such notation, the volumetric incompressibility conditions for the upper and lower monolayers can be written as:
where nu and nd are director projection onto Ox axis in the upper and lower monolayers, respectively; hu and hd are thicknesses of the upper and lower monolayers, respectively, in undeformed state; the argument x is omitted. Taking into account Eqs. (5) and (A1), the surface density of the elastic energy of the monolayer region can be written as:
for the upper and lower monolayer, respectively. The functional of the total energy of the bilayer membrane with account for Eqs. (3) and (A2) is given by the expression:
Variation of the functional over the functions nu(x), nd(x) and M(x) leads to the following system of differential Euler–Lagrange equations:
Let’s denote the equations in (A5) as E1, E2 and E3. Combination \(\frac{{{{E}_{1}}}}{{h_{u}^{2}}} - \frac{{{{E}_{2}}}}{{h_{d}^{2}}} - \frac{{d{{E}_{3}}}}{{dx}}\) allows expressing M ′(x) via director projections nu and nd and their second derivatives. Taking the derivative with respect to x of the expression, and substituting M ″ into E3, we obtain the equation, from which M(x) can be expressed via director projections nu and nd, their first and third derivatives. This allows excluding the function M(x) from Eq. (A5). Analogously, by excluding nd from two resulting differential equations, we obtain an isolated equation for the director projection nu:
Roots λj ( j = 1, 2, …, 8) of the characteristic polynomial can be found analytically, but they are very bulky. All roots are complex and can be written as follows:
where α, β, φ, ϕ are real coefficients depending on the system parameters; i is the imaginary unit. Spatial distribution of the director projection of the upper monolayer can be written as:
where cj ( j = 1, 2, …, 8) are complex coefficients, which should be determined from the boundary conditions. Eight complex coefficients are equivalent to sixteen real coefficients. However, the function nu(x) should be real at any real x. This condition reduces the number of the independent real coefficients to eight. From the distribution of the projection of the director of the upper monolayer nu(x), the distribution of the projection of the director of the lower monolayer nd(x) and the shape of the monolayer interface M(x) are found. Then, from the condition of local volumetric incompressibility (Eq. (A2)), the shape of the surface of the upper, Hu(x), and the lower, Hd(x), monolayers can be found. Substituting the functions nu(x), nd(x), M(x) into equation (A4) and integrating over x, we find the energy of the deformed region of the membrane deposited on a solid support.
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Galimzyanov, T.R., Akimov, S.A. Configurations of Ordered Domain Boundary in Lipid Membrane on Solid Support. Biochem. Moscow Suppl. Ser. A 15, 239–248 (2021). https://doi.org/10.1134/S1990747821040048
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DOI: https://doi.org/10.1134/S1990747821040048