Intersecting polymers in lipid bilayers: cliques, static order parameters and lateral diffusion

We have modelled a macrolipid polymer composed of lipid molecules (monomers) embedded in a lipid bilayer or monolayer and polymerized via their polar groups. Because of fluctuations perpendicular to the plane of the bilayer, the polar region occupied by the polymer chain possesses sufficient space so that the polymer might exhibit 'self-intersection' if its conformational state is projected onto the plane of the bilayer/monolayer. We represent the plane of the bilayer/monolayer by a triangular lattice. Each site can be occupied by a monomer or be empty (and thus occupied by one of the unpolymerizable lipids which make up the bilayer/monolayer). A macrolipid is represented by a sequence of N monomers connected by N - 1 bonds. Bonds may be either short (connecting nearest neighbour monomers) or long (between second neighbour monomers), in accord with the average properties of the spacers between the polymerized lipids. We have carried out computer simulation of this system using the Carmesin-Kremer bond stretching algorithm. Although no two monomers can occupy the same site, bonds may cross each other. We analyzed the dependence of 〈R²〉 and 〈R_G²〉 ∼ N^2v_c and 〈N_sc〉 + 〈N_mc〉 ∼ N ^2σ_c, where N_sc and N_mc are the number of bond-crossings in the same macrolipid ('self-crossing') or in two different macrolipids ('mutual-crossing'). For single macrolipids, we confirmed that v_c 3 4 and have found that σ_c≈0.52, which we consider supports that σ_c = 1 2. For the dense case with monomer concentration, c 0.72, we found that v_c 1 2 and that σ_c ≈ 0.52 supports that σ_c 1 2. In the semi-dilute regime (c 0.2) we found crossover behaviour, although σ_c 1 2. The total number of bond crossings thus scale like N, independent of concentration. We studied the connectivity of the system by calculating the weight averaged cluster, or 'clique', size. Cliques are defined as being composed of all macrolipids which exhibit at least one crossing bond with one other member of the clique. We found that while the average clique contains about two macrolipids at low concentrations, the clique size approaches the maximum possible value at high concentrations if the macrolipids are sufficiently long. In the latter case a transition appears to occur as the macrolipid length increases. This transition occurs at length 40 when c 0.72. These observations should have experimental consequences for the viscoelastic properties of the system. We defined an order parameter, B, and showed that although B depended upon c, it appeared to be independent of N for both crossing and non-crossing macrolipids. We calculated the dependence of the lateral diffusion coefficient upon c for some values of N and compared it to the case of non-crossing macrolipids and single lipid molecules. We suggest how the existence of crossing bonds might be observed.