The structure of biological membranes, which are composed of amphiphilic lipid molecules, has been experimentally difficult to measure. However, nuclear magnetic resonance is an experimental technique that directly measures the dynamics of lipid molecules at an atomic level, via order parameters (which can be measured by dipole-dipole or quadrupolar interactions). The theoretical challenge has been to connect the observed dynamics to structure.

The first successful model is called the diamond-lattice (DL) model. This theory models the motion of methylene segments of the lipid chains as discrete tetrahedral orientations (akin to the diamond structure). However it is an over-parameterized model, therefore assumption has been that the acyl chains do not back fold (or upturn in the membrane) which reduces the parameter space. It has been discovered that this assumption is only appropriate for the chain segments near the head group of the lipid or for highly ordered membranes. This proves to be a problem as the model requires use of all of the order parameters from the head group down to the end of the chains to calculate a chain length. As a result it models the chains to be fluctuating too fast and estimates membranes to be too thin.

The second successful model is called the mean-torque (MT) model. As opposed to a discrete lattice, this theory treats the chain motions as a continuum under an orienting potential (equivalent to a mean torque). As a result this theory does not have to initially neglect back folding orientations and is shown to be more-accurate in modeling chain dynamics. However to avoid the over-parameterization problem it treats the orientating potential as a single parameter, and may not accurately model the chain dynamics under all circumstances (especially for highly mobile segments near the ends of the lipid chains).

As the presented models are more accurate for chain dynamics near lipid head groups, they are less accurate in handling the problem of directly calculating the chain length. Therefore the idea of methylene travel was presented. If we imagine a chain segment fluctuating in space, its fluctuations are only limited to the area they are given to move in. So if you can calculate the average travel of a methylene chain segment near the lipid head group, you can calculate the area per lipid molecule. And if you assume a constant chain volume, which experiments tend to agree on, then you can calculate the volumetric thickness of a chain segment. It should be noted that the area problem involves discontinuities present in the secant function, therefore a Taylor expansion is used which involves (again) neglecting chain segment upturns. Therefore structural calculations should only be used for the most-ordered lipid chain segments (or highest order parameters). In this regime, the mean-torque model has yielded more-accurate results when compared to experiment.

Below is a calculator which calculates lipid structure based on both the diamond-lattice (DL) and mean-torque (MT) models. Note that the order parameter already assumes fast axial-rotation of the methylene segments so it should only have a range from 0 to 1/2.

As with any JavaScript programming, the source code is embedded within this page. However it makes use of a table of values for the mean-torque model calculations. To generate this table I used the MATLAB software. You can download the MATLAB script below.

H. I. Petrache, S. W. Dodd, and M. F. Brown. 2000. Area per Lipid and Acyl Length Distributions in Fluid Phosphatidylcholines Determined by ^{2}H NMR Spectroscopy. *Biophys. J.* **79**:3172-3192.