Goutaland Quentin, Fournier Jean-Baptiste
Laboratoire "Matière et Systèmes Complexes" (MSC), UMR 7057 CNRS, Université de Paris, 75205, Paris Cedex 13, France.
Eur Phys J E Soft Matter. 2019 Dec 17;42(12):156. doi: 10.1140/epje/i2019-11922-8.
We show that a very good analytical approximation of Saffman-Delbrück's (SD) law (mobility of a bio-membrane inclusion) can be obtained easily from the velocity field produced by a pointlike force in a 2D fluid embedded in a solvent, by using a small wavelength cutoff of the order of the particle's radius a . With this method, we obtain analytical generalizations of the SD law that take into account the bilayer nature of the membrane and the intermonolayer friction b . We also derive, in a calculation that consistently couples the quasi-planar two-dimensional (2D) membrane flow with the 3D solvent flow, the correction to the SD law arising when the inclusion creates a local spontaneous curvature. For an inclusion spanning a flat bilayer, the SD law is found to hold simply upon replacing the 2D viscosity [Formula: see text] of the membrane by the sum of the monolayer viscosities, without influence of b as long as b is above a threshold in practice well below known experimental values. For an inclusion located in only one of the two monolayers (or adhering to one monolayer), the SD law is influenced by b when b < [Formula: see text]/(4a) . In this case, the mobility can be increased by up to a factor of two, as the opposite monolayer is not fully dragged by the inclusion. For an inclusion creating a local spontaneous curvature, we show that the total friction is the sum of the SD friction and that due to the pull-back created by the membrane deformation, a point that was assumed without demonstration in the literature.
我们表明，通过使用与粒子半径(a)量级相同的小波长截止，从嵌入溶剂中的二维流体中一个点状力产生的速度场可以轻松获得萨夫曼 - 德尔布吕克（SD）定律（生物膜内含物的迁移率）的一个非常好的解析近似。通过这种方法，我们得到了考虑膜的双层性质和层间摩擦(b)的SD定律的解析推广。我们还在一个将准平面二维（2D）膜流与三维溶剂流一致耦合的计算中，推导了当内含物产生局部自发曲率时对SD定律的修正。对于跨越平坦双层的内含物，发现只要(b)高于一个实际上远低于已知实验值的阈值，SD定律只需将膜的二维粘度([公式：见文本])替换为单层粘度之和即可成立，且不受(b)影响。对于仅位于两个单层之一（或附着于一个单层）中的内含物，当(b < [公式：见文本]/(4a))时，SD定律受(b)影响。在这种情况下，迁移率最多可增加两倍，因为相对的单层不会被内含物完全拖动。对于产生局部自发曲率的内含物，我们表明总摩擦力是SD摩擦力与由于膜变形产生的回拉摩擦力之和，这一点在文献中是未经证明而假设的。
