Biological membranes are generally formed by lipids and proteins. Often, the membrane properties are studied through model membranes formed by phospholipids only. They are molecules composed by a hydrophilic head group and hydrophobic tails, which can present a panoply of various motions, including small localized movements of a few atoms up to the diffusion of the whole lipid or collective motions of many of them. In the past, efforts were made to measure these motions experimentally by incoherent neutron scattering and to quantify them, but with upcoming modern neutron sources and instruments, such models can now be improved. In the present work, we expose a quantitative and exhaustive study of lipid dynamics on DMPC and DMPG membranes, using the Matryoshka model recently developed by our group. The model is confronted here to experimental data collected on two different membrane samples, at three temperatures and two instruments. Despite such complexity, the model describes reliably the data and permits to extract a series of parameters. The results compare also very well to other values found in the literature.

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Fluid lipid bilayers are the building blocks of biological membranes. Although there is a large amount of experimental data using inconsistent quasi-elastic neutron scattering (QENS) techniques to study membranes, very little theoretical works have been developed to study the local dynamics of membranes. The main objective of this work is to build a theoretical framework to study and describe the local dynamics of lipids and derive analytical expressions of inconsistent diffusion functions (ISF) for QENS. As results, we developed the dynamical Matryoshka model which describes the local dynamics of lipid molecules in membrane layers as a nested hierarchical convolution of three motional processes: (i) individual motions described by the vibrational motions of H-atoms; (ii) internal motions including movements of the lipid backbone, head groups and tails, and (iii) molecule movements of the lipid molecule as a whole. The analytical expressions of the ISF associated with these movements are all derived. For use in analyzing the QENS experimental data, we also derived an analytical expression for the aggregate ISF of the Matryoshka model which involves an elastic term plus three inelastic terms of well-separated time scales and whose amplitudes and rates are functions of the lipid motions. And as an illustrative application, we used the aggregated ISF to analyze the experimental QENS data on a lipid sample of multilamellar bilayers of DMPC (1,2-dimyristoyl-\textit{sn}-glycero-3-phosphocholine). It is clear from this analysis that the dynamical Matryoshka model describes very well the experimental data and allow extracting the dynamical parameters of the studied system.