User:Membrains/sandbox

Helfrich energy with the Monge parametrization
Cell membranes are constantly subjected to thermal fluctuations, leading to deviations from a flat membrane configuration. The Monge gauge representation (or height representation) allows for the description of the shape of a fluctuating elastic membrane in terms of its height above a two-dimensional plane. Each point in the xy plane is ascribed a single height value, given by $$h(x,y)$$.

The Helfrich free energy for a membrane spanning an area $$ L^2 $$ is then given by

$$ H = \int_{L^2} d\mathbf{r}\Big[\frac{\kappa}{2}(\nabla^2 h(\mathbf{r}))^2 + \frac{\sigma}{2}(\nabla h(\mathbf{r}))^2\Big] + H_\text{int}$$

where $$ \kappa $$ represents the membrane's elastic bending modulus, $$ \sigma $$ the surface tension, $$ \nabla^2 h(\mathbf{r}) $$ the local curvature of the membrane at a given position $$ r = (x, y) $$, and $$ H_\text{int} $$ accounts for external interactions (such as due to proteins or applied forces).

Fourier convention
For calculational convenience, the membrane's shape can be represented as a set of Fourier modes. The height is then given by following Fourier transform convention.

$$ h_\mathbf{q} = \int_{L^2} d\mathbf{r} h(\mathbf{r})e^{-i\mathbf{q}\cdot \mathbf{r}} $$

$$ h(\mathbf{r}) = \frac{1}{L^2}\sum_\mathbf{q} h_\mathbf{q} e^{i \mathbf{q} \cdot \mathbf{r}} $$

Consequently, the Helfrich energy is

$$ H = \frac{1}{2L^2}\sum_q (\kappa q^4 + \sigma q^2)|h_q|^2 $$

The wave vector $$ q $$ must obey periodic boundary conditions such that $$ q = \frac{2 \pi}{L}(m, n) $$, where $$ (m, n) $$ are integers in the range $$ -N/2 < (m, n) < N/2 $$.

High $$ q $$ values represent shorter wavelengths, and vice-versa. Practically, $$ N $$ is chosen such that the minimum wavelength $$ a = L/N $$ is suitable for the desired simulation. Choosing a system with large $$ N $$ values entails greater computational expense due to finer discretization. For instance, $$ a $$ might be the size of a typical protein interacting with the membrane, or at an even finer resolution, the size of a single lipid. In the continuum limit ($$ a \rightarrow 0 $$), the sum over the wave-vectors can be converted to an integral as $$ \sum_\mathbf{q} = \Big(\frac{L}{2\pi}\Big)^d \int{d\mathbf{q}} $$, where $$ d = 2 $$ for a two-dimensional plane.

Accounting for thermal effects
In accordance with the equipartition theorem, each degree of freedom should correspond to $$ \frac{1}{2}k_B T $$ of energy, where $$ k_B $$ is the Boltzmann constant and $$ T $$ is the temperature. Since the membrane's height is fluctuating along a single vertical axis (a single degree of freedom) with respect to the flat plane, the mean-squared height in Fourier space can be derived as follows

$$ \begin{align} H &= \frac{1}{2L^2}\sum_q (\kappa q^4 + \sigma q^2)|h_q|^2\\ \frac{k_B T}{2} &= \frac{1}{2L^2}\sum_q (\kappa q^4 + \sigma q^2)|h_q|^2\\ \implies \langle |h_q|^2 \rangle &= \frac{L^2 k_B T}{\kappa q^4 + \sigma q^2} \end{align} $$

Since $$ \langle h_q \rangle = 0$$ for a flat membrane, $$ \langle |h_q|^2 \rangle $$ represents the height variance of the membrane in Fourier space.

The height variance in real space may then be derived as



\begin{align} \langle h(r)^2 \rangle &= \Big\langle \Big(\frac{1}{L^2}\sum_\mathbf{q} h_\mathbf{q} e^{i \mathbf{q} \cdot \mathbf{r}}\Big)\Big(\frac{1}{L^2}\sum_\mathbf{q} h_\mathbf{q} e^{i \mathbf{q} \cdot \mathbf{r}}\Big) \Big\rangle \end{align} $$