(untitled) (Communications in Computational Physics, 2009)
We study a continuum paradigm of the lipid bilayer based on minimizing the free energy of a mixture of water and lipid molecules. This paper extends previous work of Blom and Peletier [European J. Appl. Math., 15 (2004), pp. 487-508] in the following ways. (a) It formulates a more general model of the hydrophobic effect to facilitate connections with microscale simulations and first-principles analysis. (b) It clarifies the meaning and role of the model parameters. (c) It outlines a method for determining parameter values so that physically-realistic bilayer density profiles can be obtained, for example for use in macroscale simulations. Points (a)-(c) suggest that the model has potential to robustly connect some micro- and macroscale levels of multiscale blood flow simulations. The mathematical modelling in point (a) is based upon a consideration of the underlying physics of inter-molecular forces. The governing equations thus obtained are minimized by gradient flows via a novel numerical approach; this enables point (b). The numerical results are shown to behave physically in terms of the effect of background concentration, in contrast to the earlier model which is shown here to not display the expected behaviour. A “short-tail” approximation of the lipid molecules also gives an analytical tool which yields critical values of some parameters under certain conditions. Point (c) involves the first quantitative comparison of the numerical data with physical experimental results.