Mobile Lipids

We return to the boundary value problem, eq. (5.10), seeking to determine again the zero pressure line in the parameter space, but now for the case of mobile lipids. From now on, we focus on oppositely charged membranes, because the case of like charged membranes leads always to repulsion, be the lipids mobile or not [Pars 72]. Therefore, we demand thatq₁/q2 <0 andσ₁^∞/σ₂^∞ = Φ^∞₁ /Φ₂^∞ < 0. In addition, 0< η₁ ≤1 and 0 < η₂ ≤ 1. Note that bothq₁Φ₁^∞ andq₂Φ₂^∞ must be positive quantities. Again we start from the expression for the pressure, eq. (5.15), which is zero if either AorBis zero. AssumingAto be zero, the first boundary condition in eq. (5.10) results in

−κB=−4πλ_Bσ₁^∞ h

1−q₁(1−η₁)(B−Φ₁^∞)ⁱ. (5.20) This equation is solved by

B= Φ₁^∞. (5.21)

leading us to the potentialΦ(x) =Φ₁^∞e^−κxwhich is again the familiarGouy-Chapman potential. So, like in the case of immobile lipids, the potential (and thus the ion dis-tribution) in the slab 0 < x < l_min between the membranes is identical to the one that one would find if membrane 1 were alone, provided of course that the two mem-branes are separated just by the zero-pressure distancelmin. We note that due to this property two of the parameters specifying the surface properties of membrane one, namelyq₁andη₁, are now immaterial for the following considerations.

Inserting the potential Φ(x) = Φ₁^∞e^−κx into the second boundary condition of eq. (5.10) and using eq. (5.11), yields

−Φ₁^∞e^−κlmin =Φ₂^∞ h

1−q₂(1−η₂)Φ^∞₂ (Φ₁^∞e^−κlmin/Φ^∞₂ −1)ⁱ, (5.22)

0 0.5 1 1.5

Figure 5.6: Regions of attraction and repulsion between two oppositely charged membranes, as a function of the intermembrane distance and the ratio of surface charge densities. Similar to the plot for immobile lipids (fig. 5.5) we allow this time lipid mobility. In both plots, we show the line of zero net force for four different values of the surface fraction of charged lipidsη2(η2=1.00,0.75,0.50,0.25).

The sign of the effective intermembrane force in the brown-background region of the parameter-space can be inferred from a corresponding point in the grey-background region.

and hence eq. (5.18). It is clear from our previous discussion that the caseB=0 can not provide new information, one can check this by redoing the calculation with B = 0, which leads to a potential Φ(x) = Φ^∞₂ e^κx and again to eq. (5.23) but with the label 1 re-placing the label 2. We show in fig. 5.6 the zero-pressure line in the parameter-space calculated from eq. (5.23) for four different values ofη2, in fig. 5.6a) forΦ₂^∞q2 = 0.05 and in fig. 5.6b) forΦ₂^∞q₂ = 0.5. For completeness, we have also added the curves for the case B = 0 (|σ₁^∞| < |σ₂^∞|, brown-background region). Via eq. (5.11), q₂Φ^∞₂ is related to the two quantities characterising the electrolyte solution, λ_B andκ, and can thus be experimentally changed through variation of temperature, salt concentra-tion, or the choice of the solvent (e). In particular,q₂Φ₂^∞increases (withσ₂^∞remaining constant) on increasingλ_B, that is, if the significance of electrostatic forces relative to thermal forces increases.

We now discuss the findings of fig. 5.6. We have seen that the surface properties of the membrane having the higher surface density of charged lipids remain completely

unaffected by the presence of the other membrane, that is, if, for example, |σ₁^∞| >

|σ₂^∞|, thenΦ(0) = Φ^∞₁ andσ₁ = σ₁^∞ so that there is no difference between the lipid distribution on those parts of the membrane belonging to the interaction region and those belonging to the reservoir. However, on the other membrane having the lower density of charged lipids (membrane 2 if|σ₁^∞|>|σ₂^∞|), the lipid surface density in the interaction region changes in response to the presence of membrane 1. Assuming that membrane 1 is positively charged (q₁ >0,q₂ <0) and recalling thatΦ^∞₁ /Φ₂^∞ <0, we see that∆Φ₂ = Φ₁^∞e^−κlmin−Φ₂^∞ must be positive. Consequently, the surface charge density in the interaction zone of membrane 2,

σ₂ =σ₂^∞ h

1−q₂(1−η₂)∆Φ₂ i

, (5.24)

must be larger in magnitude than its reservoir valueσ₂^∞. In other words, atl = l_min additional lipids must have flowed from the reservoir into the interaction zone of membrane two. One may say that the membrane with the higher density of charged lipids, while remaining itself inert, attracts charged lipids on the other membrane out of the reservoir into the interaction zone. The other conceivable mechanism—

that one membrane expels charged lipids from the interaction region of the other membrane—is not observed. That, of two interacting membranes, only one mem-brane shows a major rearrangement of lipids, is a surprising result. Lipid mobility is thus completely irrelevant for the membrane with the higher surface density of charged lipids; it suffices to model this membrane just by a planar wall with a homo-geneous surface charge density.

To understand the implications which the observed lipid redistribution has on the effective interaction, let us return for a moment to the case of immobile lipids (η₂ =1) and assume that the system of two membranes are at a state-point in parameter-space where there is zero pressure (solid thick curves in the |σ₁^∞| > |σ₂^∞| regions of fig. 5.6. If now the lipid mobility is switched on (η₂ < 1), additional charged lipids are allowed to flow into the interaction zone of membrane 2. This results in a change from zero pressure to negative pressure at this particular point in parameter-space; the effective interaction between the membranes will now be attractive. In other words, lipid mobility is responsible for an additional attractive contribution to the effective intermembrane interaction potential, the latter being nothing but the total free energy of the system as a function ofκl. We see from fig. 5.6 that as a result of this additional attractive component in the free energy, the region in parameter-space where the two membranes attract each other increases with decreasing η₂ and/or increasingq₂Φ₂^∞. This statement, of course, applies to the part of the parameter-space where|σ₁^∞| > |σ₂^∞|; in the|σ₁^∞| < |σ₂^∞|part (brown-background region) the relevant quantities to look at areη₁andq₁Φ₁^∞.

0

Figure 5.7: Regions in parameter-space where touching contact (κl = 0) between two oppositely charged membranes corresponds to the equilibrium configuration, as a function of σ₁^∞/σ₂^∞ and the surface fraction η2of mobile lipids on the membrane with the lower surface densities of lipids. The dashed line is the function−1/η2. See text for details.