Elastic models

Canham and Helfrich Hamiltonians

Back in 1970, Peter Canham proposed one of the first models to duly account for the peculiar biconcave shape of human red blood cells, without considering any other organelle, but the cellular membrane.¹¹⁴ The key idea behind this model was to identify bending (rather than stretching or torsion) as the relevant quantity that characterizes the free energy of the lipid bilayer. Relying on elastic theory, he proposed a generalization of the beam-bending energy to describe the elastic energy of the membrane:

U=

where the integral extends over the whole surface of the bilayer,κis the bending modulus andc₁andc₂are the local, principal curvatures of the membrane. Few years later, in 1973, Wolfgang Helfrich found a generalization of this expression that became the cornerstone for most mesoscopic, mean-field models describ-ing the thermodynamics of lipid bilayers. Instead of directly relydescrib-ing on elastic theory, Helfrich started from a general quadratic expansion for the membrane elastic energy in terms of the spatial derivatives of the local bilayer normal.

Then, appealing to physical grounds and symmetry arguments, he justified the omission of irrelevant terms in the expansion.¹¹⁵ The reasonings followed at that stage were similar to those used by Oseen¹¹⁶and Frank¹¹⁷ when deriving the free energy of liquid crystals. Beside the curvature-independent term, γ, which accounts for energy contributions due to membrane stretching, the Hel-frich generalization for the bilayer free energy is:

U=

Compared to the Canham’s Hamiltonian, Eq.3.1, there are two additional coefficients in this expression. The first one is the spontaneous curvature of the bilayer,C₀. Due to symmetry, this quantity should vanish in completely homo-geneous membranes. However, the presence of heterogeneities may lead to a preferred curvature, thereby settingC₀ 6= 0. For example, this may be the case in membranes whose two apposing monolayers are composed of lipids with different chemical identities or packing geometries. The second coefficient, ¯κ, is the so-called saddle-splay or Gaussian modulus. It accounts for the membrane

preference to adopt concave/convex or saddle shapes. This contribution, how-ever, may be omitted when studying membranes with closed geometry (such as closed vesicles) or idealizations where some of the lateral dimensions of the membrane extend infinitely (such as infinite planar membranes or cylinders).

The reason for this, relies in the celebratedGauss-Bonnet theorem, that relates the surface integral of the local Gaussian curvature,K=c₁c₂, of a manifoldS, to the line integral of the geodesic curvature,kg, along the contour ofSand the Euler characteristic, χ(S): a scalar invariant that describes the shape of a topological space regardless of how it’s bent.

Z

The immediate consequence of this result is that, for membranes without boundaries, the second term in the Helfrich Hamiltonian, Eq. 3.2, may be ne-glected, since it will integrate into a constant with no influence on the physical properties of the system. Taking this into account, the Helfrich Hamiltonian can be rewritten as:

were we have defined the mean curvature,H = ^c1+c₂ ². The practical use of this expression will still depend on the way curvatures are computed, which in turn will depend on the parametrization of the membrane.

The Monge gauge

One of the most appealing ways to describe membranes, thinking of them as extended objects, is the so-called Monge gauge. In this description, the local height, h, is measured with respect to a reference plane located at z = 0, as shown in Fig. 3.1. This way, the coordinates of any point on the membrane are given byR= R(x,y,h(x,y)). Of course, this assumes thath(x,y)is single-valued and, thus it’s restricted to the study of quasi-planar surfaces.

Within the Monge gauge, the mean curvature can be evaluated via the di-vergence of the membrane normal, 2H = −∇ ·n. However, the resulting ex-ˆ pressions allow for analytic ways to find equilibrium configurations (those that minimize the resulting Helfrich Hamiltonian) only in a very limited number of cases. For this reason, it is usually assumed that h(x,y) is a smooth-enough function (i.e., |∇h| 1), so that a small gradient expansion results in an ap-propriate description of membrane fluctuations. These simplifications yield the most common approximation to the Helfrich Hamiltonian:

U= 1

x

y

h ( x , y )

Figure 3.1: Monge parametrization of a planar membrane patch.

Fluctuation spectra

The quadratic dependence of the Helfrich Hamiltonian on curvature, Eq.3.5, is enormously convenient when computing statistical averages and, in partic-ular, it will provide an analytical path for evaluating equilibrium properties.

Another key aspect of this harmonic model is that shape fluctuations are com-pletely determined by two uncoupled contributions: out-of-plane deformations (bending) and in-plane compressions. By measuring the spatial correlation of these undulations, we can determine the conditions favoring the appearance of one kind of mode or another. To carry out this study, it is convenient to evaluate the Fourier expansion of height fluctuations:

h(r) =X

q

h_qe^iq·r with q= 2π

L (nx,ny), nx,ny ∈N, (3.6) where, for simplicity, we have assumed a membrane patch with equal lateral dimensions, L, replicating infinitely via periodic boundary conditions. With this definition the Helfrich Hamiltonian, Eq. 3.5, can be rewritten as follows:

U_q = L² 2

X

q

h_q

2h

κq⁴+γq²i

. (3.7)

Then, invocation of the equipartition theorem, which states that every quadratic term in the Hamiltonian contributes exactlyk_BT/2 to the average energy of the

system, yields the desired expression for the membrane fluctuation spectra:

D|h_q|²E

= k_BT

L²[κq⁴+γq²]. (3.8) Now it is clear that compression modes will become dominant whenever q < p

γ/κ. Another remarkable feature of this expression is the simple depen-dence on the phenomenological parameters of the model. This is one of the reasons why measuring the fluctuation spectra has become one of the preferred techniques to determine the bending modulus of lipid bilayers.