User:IndyWP/sandbox

= Fluctuations in elastic cell membranes = Fluctuations are an ever-present feature of elastic cell membranes. Membranes are constantly buffeted by molecules from the surrounding medium (water or cytoplasm) due to thermal effects. Although these thermal collisions occur locally, the elastic properties of the membrane result in the coupling of membrane undulations at different locations on the cell.

Experimentally, these fluctuations are often tracked to determine the membrane’s elastic properties, such as its bending modulus. From a theoretical point of view, fluctuating membranes serve as an insightful model system for statistical physics.

Membrane fluctuations are also of great biological relevance. Fluctuations facilitate the adhesion of cells to surfaces of different topographies by extending the range of contact, whilst steric repulsions between fluctuating membranes can prevent the aggregation of cells such as RBCs. Membrane fluctuations also serve as an indicator of the condition of the cell: older cells showed less complex fluctuation spectra as compared to younger cells, and infected or diseased cells (such as by malaria or sickle cell disease) displayed distinct fluctuation amplitudes as compared to healthy cells. Subcellular processes depending on ATP in red blood cells can induce active fluctuations that do not obey a Gaussian distribution (otherwise observed in ATP-depleted RBCs). . In nucleated cells such as human macrophages, stimulation by inflammatory cytokines substantially increased the cell membrane's fluctuation amplitudes.

Helfrich energy with the Monge parametrization
Cell membranes are constantly subjected to thermal fluctuations, leading to deviations from their flat 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
The membrane's shape can be represented as a set of Fourier modes. The membrane 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 two dimensions.

Accounting for thermal fluctuations
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_{q} h_{q} e^{i {q} \cdot {r}}\Big)\Big(\frac{1}{L^2}\sum_{p} h_{p} e^{i p \cdot r}\Big) \Big\rangle\\ &= \Big\langle \frac{1}{L^4}\sum_q \sum_p h_q h_p e^{i q \cdot r} e^{i p \cdot r}\delta_{-q,p}\Big\rangle\\ &= \frac{1}{L^4}\sum_{q}\Big \langle h_{q} h_{-q} \Big \rangle\\ \end{align} $$

where $$ \delta_{-q,p} $$ is the Kronecker delta. Since $$ h(r) $$ must always be real, it follows that $$ \text{Re}[h_{q}] = \text{Re}[h_{-q}] $$, and $$ \text{Im}[h_{q}] = -\text{Im}[h_{-q}] $$.

Therefore, $$ h_{-q} = h_q^* $$, and consequently



\begin{align} \langle h(r)^2 \rangle &= \frac{1}{L^4}\sum_{q}\langle |h_{q}|^2 \rangle = \frac{1}{L^2}\sum_{q}\frac{k_B T}{\kappa q^4 + \sigma q^2} \end{align} $$

Time-evolution of fluctuating membranes
The dynamical equation for evolving $$ h_\mathbf{q} $$ modes as a function of time is given by



\begin{align} \frac{\partial h_\mathbf{q}(t)}{\partial t} = \Lambda_\mathbf{q}\{F_\mathbf{q}[h(\mathbf{r},t)] + \zeta_\mathbf{q}(t)\} \end{align} $$ where $$\Lambda_\mathbf{q} = \frac{1}{4\eta \mathbf{q}}$$ is the Oseen tensor, and $$\zeta_\mathbf{q}(t)$$ is a white noise term with $$\langle \zeta_\mathbf{q}(t) \rangle = 0$$ and the autocorrelation $$\langle \zeta_\mathbf{q}(t)\zeta_\mathbf{q'}(t') \rangle = 2k_B T L^2 \Lambda_\mathbf{q}^{-1}\delta_{\mathbf{q},-\mathbf{q}}\delta(t - t')$$.

$$F_\mathbf{q}[h(\mathbf{r},t)]$$ is the force as a functional of the height field and is constituted by the sum of the bending and interaction forces, as $$ F_\mathbf{q}^\text{bend} = -(\kappa q^4 + \sigma q^2)h_\mathbf{q} $$ and $$ F_\mathbf{q}^\text{int} $$ is obtained as the Fourier transform of the functional derivative $$ \frac{-\delta H_\text{int}}{dh(\mathbf{r})} $$.



This dynamical equation can also be simplified to

\begin{align} \frac{\partial h_\mathbf{q}(t)}{\partial t} = -\omega_\mathbf{q} h_\mathbf{q} + \Lambda_\mathbf{q}\zeta_\mathbf{q} \end{align} $$

which allows the wave-vector relaxation frequency to be expressed as $$ \omega_\mathbf{q} = \frac{\kappa q^4 + \sigma q^2}{4 \eta q} $$.

Simulation algorithm
An algorithm to simulate such fluctuating membranes (titled Fourier Space Brownian Dynamics, or FSBD) was proposed by Lin and Brown.



\begin{align} h_\mathbf{q}(t + \Delta t) = h_\mathbf{q}(t) + \Lambda_\mathbf{q}F_\mathbf{q}(t)\Delta{t} + R_\mathbf{q}(\Delta{t})\\ R_\mathbf{q}(\Delta t) \equiv \Lambda_\mathbf{q}\int_t^{t + \Delta t} dt' \zeta_\mathbf{q}(t') \end{align} $$

Computationally, the noise $$ R_\mathbf{q}(\Delta t) $$ is introduced by sampling from a Gaussian distribution with $$ \mu = 0$$ and $$ \sigma^2 = 2L^2k_B T \Lambda_\mathbf{q}\Delta{t}$$ for the modes that are explicitly real. For the other independent modes, both the real and imaginary components are sampled with $$ \mu = 0$$ and $$ \sigma^2 = L^2k_B T \Lambda_\mathbf{q}\Delta{t}$$. The $$ h_\mathbf{-q} $$ modes are not evolved independently since $$ h_\mathbf{-q} = h_\mathbf{q}^* $$. It is also computationally convenient to choose an odd number of modes (odd $$ N $$) such that the only explicitly real mode is the $$ 0^{\text{th}} $$ mode, which is generally not evolved. For numerical feasibility, the timestep must be small, such that $$ \Delta t $$ ≪ $$ \frac{1}{\omega_\mathbf{q}}$$.

Experimental techniques to measure membrane fluctuations
Measuring membrane fluctuations experimentally can be challenging for several reasons. Membranes fluctuate over very small length scales in terms of lateral displacement, which requires fine spatiotemporal resolution (often only a few nanometers), but they also span over surface areas of several microns. As a result, the techniques used to probe these fluctuations must be large enough to map the desired membrane area, but still offer respectable spatiotemporal resolution. Also, the instrument used to observe membrane fluctuations must not itself physically perturb the membrane and introduce unintended biases to the measurement. This makes it undesirable to use techniques such as atomic force microscopy, where the AFM tip might alter the membrane's dynamics substantially. Therefore, the most widely used techniques for studying membrane fluctuations are light-based, emphasizing minimal direct contact with the membrane.

Flicker spectroscopy


Flicker spectroscopy (FS) is a convenient technique to measure membrane fluctuations. It involves the use of camera imaging to study the changes in the shape and fluctuations of large membranes, such as those on the periphery of giant unilamellar vesicles (GUVs) or cells. The fluctuation spectra is measured by tracking the lateral fluctuations amplitudes at four points $$ \delta n(j) (\text{for } j = 0, 1, 2, 3 $$) spaced regularly around a phase-contrast video recording of the cell, allowing for the generation of time series data. The fluctuation spectra are then Fourier-transformed such that $$ \delta n_m = \frac{1}{4}\sum_{j=0}^{3}\delta n(j)e^{i2\pi n m/4} $$. The azimuthal modes $$ m = 0, \pm 1, \pm 2 $$ correspond to cell radius fluctuations, whole cell translations, and elliptical deformations, respectively. $$ \delta n_m $$ is used to compute the mean-square amplitude $$ \langle \delta n_m^2 \rangle $$, which can then be used to extract the membrane's bending modulus $$ \kappa $$ and surface tension $$ \sigma $$ as fit parameters.

Since flicker spectroscopy relies on the reflectivity of the membrane being tracked, poor light contrast can be a limitation that is compensated for by using contrasting agents such as fluorophores. However, using too many fluorophores can potentially alter the membrane's dynamics. Additionally, the camera limits the spatiotemporal resolution, which is generally about 80 nm and 10 ms.

Localized displacement detection


DODS (Dynamical Optical Displacement Spectroscopy) is a technique based on FCS, and can be used to track a fluorescently labeled membrane at a precise location. It offers high spatiotemporal resolutions (about 20 nm and 10 $$\mu$$s). In addition to being able to track fluctuations on GUVs, DODS also allows for measuring fluctuations in optically inhomogeneous membranes, such as those in nucleated cells.

The membrane region being tracked is included in the confocal detection volume (CDV), scanned axially. This is subject to an illuminating light beam that obeys a Gaussian intensity profile. As the membrane fluctuates, intensity measurements can be made along the z-axis. In FCS, the membrane is generally labeled with a low fluorescent dye concentration (about 0.001%) to track lateral molecular diffusion. With DODS, the membrane is enriched with a 1% fluorescent dye concentration, such that for a given detection volume there is very little fluctuation in the number of dye molecules diffusing laterally, effectively ignored as noise. Instead, the primary source of intensity variation in the detection volume is due to fluctuations of the membrane along the $$ z $$-axis, and these intensity variations are used to infer the membrane's height fluctuations using the following linear relationship

$$ \delta I(t) = -I_\text{max} \exp{\Bigg[-2\frac{h_0^2}{z_0^2}\Bigg]}\Bigg\{\frac{4 h_0 \delta h(t)}{z_0^2}\Bigg\} \equiv m \cdot \delta h(t) $$

where $$ \delta I(t) $$ is the instantaneous intensity change, $$ I_\text{max} $$ is the maximum intensity at the center of the Gaussian intensity peak, $$ z_0 $$ is the axial radius of the CDV, $$ h_0 $$ is the mean membrane height along the z-axis, and $$ \delta h(t) $$ is the instantaneous change in membrane height. By measuring changes in intensity, the slope m can be used to extrapolate the corresponding change in the membrane's height. Fluctuations in intensity inform fluctuations in membrane height most sensitively when the mean membrane height is positioned along the CDV at the inflection point of the light-beam's Gaussian intensity profile.

Interference microscopy
RICM (Reflection Interference Contrast Microscopy) is an interference-based technique that is particularly useful for measuring membrane fluctuations in the vicinity of a substrate, or when adhered to it. RICM does not require the use of fluorophores and offers a spatiotemporal resolution of 5 nm and 10 ms, but is restricted to a distance of about 1 $$ \mu $$m from the substrate surface. Typically, RICM entails a vesicle as pictured in the figure on the right, where the substrate, the medium, the membrane and the vesicle interior have different thicknesses and refractive indices. Rays of light are reflected at the substrate-medium interface, the medium-membrane interface, and at the membrane-interior interface. These reflected rays interfere with each other constructively or destructively depending on the height $$ h_m $$ of the membrane above the substrate. The intensity of these reflected rays is captured using a camera over the region of interest and represented with an intensity interferogram.

To extrapolate the height of the membrane $$ h_m = h_m(\mathbf{x},t) $$ at each position, the experimentally measured RICM intensities are compared to the following theoretical relation (based on Fresnel equations)

$$ I(h_m) = \frac{S}{2} - D\frac{\sin(y)}{2y}\cos\Bigg(2kn\Bigg[h_m \cos^2\Big(\frac{\alpha}{2}\Big) - h_\text{offset}\Bigg]\Bigg) $$

where $$ y = 2kh_m \sin^2\Big(\frac{\alpha}{2}\Big) $$, $$ \alpha $$ is the angle of illumination, $$ S $$ and $$ D $$ are the sum and difference of the minimal and maximal intensities, $$ k = 2\pi/\lambda $$ is the wave-vector, and $$ n $$ is the refractive index of the medium.

Originally, RICM (single wavelength of illumination) was limited in its ability to resolve $$ h_m $$ unambiguously except when the membrane was very close to the substrate, since a decrease in the intensity could be either due to an increase or decrease in $$ h_m $$ due to the periodic nature of the intensity-height curve. This resulted in multiple height values corresponding to the same intensity value. To remedy the degeneracy, dual-wavelength RICM was developed. DW-RICM entails the use of two different wavelengths of illumination to generate the interferogram such that each $$ h_m $$ corresponds to a pair of intensities (as pictured on the right). This enables the membrane height to be determined uniquely.

Neutron spin echo
NSE is a widely used scattering technique to determine the change in the polarization of an incident neutron beam due to a sample. It is suitable for studying membrane fluctuations on very small length and timescales (as small as 1 to 100 angstroms, from picoseconds to nanoseconds), and is frequently used to investigate the influence of lipid composition on the physical properties of the membrane. When the incident beam scatters on a membrane, it results in a dephasing of neutron spins, and the spins are subsequently rephased and detected to determine the momentum that was transferred to the membrane. When repeated for various scattering angles, the collective motion of the membrane is revealed by the structure factor $$ S(q, t) \propto e^{-(\Gamma(q)\cdot t)^{2/3}} $$, where the relaxation $$ \Gamma(q) \sim \frac{q^3}{\eta(T)\kappa^{1/2}} $$ depends on the membrane viscosity and bending modulus, and can be used to test the $$ q $$-dependence of the membrane relaxation for various lipid compositions and additives.