Membrane fluidity and the probability of complement fixation

We develop a mathematical theory of the role of membrane fluidity in the initiation of the IgG mediated complement cascade. The basic assumption is that C1q must be at least doubly bound to activate C1r, but that once C1q is doubly bound, C1r still requires some mean finite time tau to become enzymatically active. If C1q dissociates during this time interval, C1r cannot be activated. We consider the consequences of the simplest model of fluidity--one in which the difference between "fluid phase" lipids and "non-fluid phase" lipids is to allow protein mobility, but not a change in protein conformation. We show that under these conditions fluidity will effect C1r activation only if the rate of formation of multiply bound C1q is limited by diffusion in the membrane. If diffusion in the membrane is not rate-limiting, then, within the framework of this model, fluidity has no effect whatsoever on C1r activation. Thus, an experimental determination that C1q binding is not rate-limited by diffusion in the surface, but that fluidity does effect activation, would suggest a protein conformational change resulting perhaps from altered lipid composition. If diffusion in the surface does rate limit multiple C1q binding, we predict the possibility of an optimum diffusion coefficient for activation. For suitably chosen and reasonable parameter values this optimum will occur in the range (10(-11) less than or equal to D less than or equal to 10(-8) cm2/sec. We predict further, under these circumstances, a precipitous drop in the probability of activation above the optimum. The abrupt switch from a high probability of activation to essentially no probability of activation suggests the possibility of a very sensitive control mechanism exploitable by relatively small changes in membrane lipid composition.