Random-walk model of the phase transition of hydrocarbon chains on a lattice

Fatty-acid side chains are simulated by the randon-walk path of a particle in three dimensions. Three independent Markov states are defined and made to correspond to trans and gauche states of real chains, by analogy to their packing properties. The stochastic matrix of transition probabilities is computed, and from it, the probability density for the occurrence of each Markov state is derived. We calculate the partition function for all configurations of a chain in the field of four nearest neighbors. The attractive interaction is of the form $\frac{1}{r^6}$ and depends upon the depth from the initial position. The phase transition experimentally observed in phospholipids is reasonably fitted by one free parameter in this model. The transition temperature, width, change in fraction of trans-to-total bonds, initial and final fractions—before and after the transition—, and enthalphy of transition, are calculated in good agreement with experimental results and models based on mean-field calculations. Applications of this approach to more complex lipid molecules are discussed.