Having established a mathematical description of surface bending and free energy in Lecture 5, we now turn our attention to a statistical treatment of surface fluctuations.


6.1 Monge parameterisation

To begin, we need a way to describe the shape of a deformed surface. For this, we will follow Boal (2002) and adopt an approach using the work of French mathematician Gaspard Monge (1746–1818), known as Monge parameterisation. Slide 2 shows how a deformed surface is represented.


On a surface embedded in three dimensions, a point, P, has a position vector s,

s = (x, y, h(x, y)),

Equation 1


where h represents the "height" away from the x, y surface. For complex surfaces, h may not be single-valued (e.g. if there are overhangs), but in the Monge representation it is constrained to be single-valued.

We can construct tangent vectors along x and y, represented by rx and ry, by taking a unit step in the x (or y) direction, and a step in the z direction,

Equation 2
Equation 2


This leads to

Equation 3
Equation 3


These are not unit vectors and are not generally orthogonal. They define the plane tangent to the surface at point P,

Equation 4
Equation 4


The position vector sx of another point rPx a distance dx from P in the x-direction is

sx = (x + dx, y, h(x + dx, y)) ≅ (x + dx, y, h(x, y) + hx(x, y)dx).

Equation 5


The vector from P to Px is

sxs ≈ (1, 0, hx)dx.

Equation 6a


Similarly, for a point Py, a distance dy from P in the y direction,

sys ≈ (1, 0, hy)dy.

Equation 6b


The vectors defined in Equation 6 span a parallelogram (Slide 3), whose area, dA, is equal to the modulus of the cross product between these vectors,

Equation 7
Equation 7



For small displacements, h, relative to the average membrane plane, z = 0, the area of a surface element can be obtained by expanding Equation 7,

Equation 8
Equation 8


The actual increase in the infinitesimal area element arising from the deformation (relative to the plane) is dA – dxdy. This approximation (Equation 8) is of a form that can be treated easily analytically (but larger deformations require much more complex differential geometry).

The metric, g, of the surface is defined to be

Equation 9
Equation 9


so that

dA = √g dx dy.

Equation 10


6.2 Deformations cause curvature

Here we will follow the approach used by Safran. A surface can also be defined by its curvature, as set out in Equations 1–4 of Lecture 5 (Slide 4). We can usefully add

Equation 11
Equation 11


where s is the arc length.


Relating our definitions of curvature to the case of the two-dimensional surface described in the Monge representation (Slide 5) is not at all trivial. It is much simpler to see (Question 1) that in one dimension the curvature is related to the function that describes the curve by

Equation 12
Equation 12


where only the final step is an approximation, valid for small gradient,

|ƒ'| << 1


A similar formula applies in two dimensions,

Equation 13
Equation 13


If the membrane spans a square frame of size L x L (Slide 6), the total increase in energy due to the work, δEσ, against membrane tension, σ (stretching the membrane), and bending the membrane (δEB), can be found using Equation 14, which is Equation 7 from Lecture 5,

Equation 14
Equation 14


To first order in small deformations, the total work done is given by

Equation 15
Equation 15



In general, membranes will try to minimise the free energy of Equation 14, subject to any constraints or boundary conditions. For example, there may be a constraint that the overall membrane has a closed topology, like the vesicles described in Lecture 5, in which case the best shape for the vesicle may be spherical. In considering instead an isolated bilayer, an example of a boundary condition may be the position of the membrane edges, and any force applied on those edges.

6.3 Spectrum of fluctuations

An important problem is to consider the shape of a membrane described by the free energy introduced above, when it is subject to thermal noise. To proceed further in this problem, and in many others with a similar mathematical structure, it is useful to consider Fourier-expanding the displacement field, in this case the membrane shape, h(r). We assume for convenience periodic boundary conditions, so that the shape can be expanded in Fourier integral (Slide 7),

Equation 16
Equation 16



Note that this is a two-dimensional transform, and that the area A in the numerator is

A = L x L.

Equation 17


The inverse transform is

Equation 18
Equation 18


Substituting this decomposition of h (x, y) into the expression for the fluctuation energy (Equation 15), we get

Equation 19
Equation 19


Note that for a finite area, we should have used a Fourier series. What has been done is to consider the continuum limit, where the discrete nature of the mode numbers becomes irrelevant. Then the Fourier series becomes an integral, taking care to put the correct mode density as the integrating measure.

Knowing that the membrane surface, h (r), is a real function, the complex Fourier modes must satisfy the condition

h(q) = h*(-q)

Equation 20


The square of the integrals in Equation 19, which need to be written in terms of separate variables q1 and q2, are simplified because integrating over dx and dy generates a delta-function in each spatial component, so Equation 19 can be simplified (Slide 8) to

Equation 21
Equation 21



From equipartition of this energy into modes q, we expect

Equation 22
Equation 22


where we have simplified the notation by using

q2 = q2x + q2y.

Equation 23


Therefore (finally),

Equation 24
Equation 24


Equation 24 is called the fluctuation spectrum or static structure factor of a membrane. It tells us the mean square amplitude of membrane modes. Since they are thermally excited, they are also proportional to temperature. Importantly, the fluctuation spectrum depends on the bending modulus, κ, and on the applied tension, σ. Measuring the fluctuation spectrum is thus a viable method to extract the bending modulus in an experiment. The method is called flicker spectroscopy.

Note that the factors in front of the fraction on the right of Equation 24 depend on the Fourier transform convention that is chosen (Equations 16 and 18). Giving a structure factor result, as the one here, without specifying the Fourier transform would have little meaning.

Note that ½ kBT and not kBT is given to each mode, even though there are two dimensions, to avoid double counting, given that the modes depend only on the modulus of q. Note, too, the particularly strong dependence on q in the term relating to the bending modulus.

Whether a particular undulation mode costs predominantly bending energy or tension energy is a question of the wave vector. For wave vectors smaller than

Equation 25
Equation 25


that is on large length scales, tension is the dominant energy contributing to the fluctuation amplitude. Conversely, for wave vectors bigger than qcrossover, that is on small length scales, bending dominates.

6.3.1 Average amplitude of fluctuations

What average fluctuation amplitude do we expect for the entire membrane, and not just for a single mode? The full membrane amplitude (Slide 9) is the sum over all individual modes, and we can calculate the mean amplitude (or mean roughness),

Equation 26
Equation 26


where we have introduced a large wavelength cutoff

Equation 27
Equation 27


and a small wavelength cutoff

Equation 28
Equation 28


where d is comparable to the bilayer thickness.


The final approximate relation gives rise to a nice rule of thumb. Since a very typical value for the bending stiffness is

κ ≅ 20kBT,

Equation 29


inserting it we find

Equation 30
Equation 30


that is, the root mean square amplitude of the membrane fluctuations under vanishing tension are typically about 1% of the lateral extension of the membrane. If the membrane is under tension, this value is reduced.