El enfoque constructivista en la alfabetización

We model lipid bilayers as graphene plates decorated with flexible phosphatidylcholine headgroups attached to them and study the behavior of the free energy of interaction between these plates immersed in water as a function of distance between plates. The interaction energy displays three regions similar to the regions observed in the experiments, and like in the experiments, these regions can be fitted to exponential curves. For the region when the fluid spacing between plates is large (in our case, this happens when the fluid thickness is above 0.9 nm) and when there is bulk-like water in the fluid space, the force is small. In the experiments, this force is mostly due to the membrane undulations. In our simulations, the bilayer undulations, although present due to the flexible nature of the headgroups, have very different character compared to the ones in experimental bilayers. Our simulations clearly show the presence of the force correlated to the removal of water structures, the so-called hydration force, when we remove two to three layers of interfacial water, when the fluid space thickness is changed from 0.7 to 0.1 nm. Once all of the interfacial water is removed, the steric factor due to the overlap of the headgroups is mostly contributing to the interaction force, although water is also still contributing, because some inner water remains in the system.

The main goal in this chapter is to show that even a simple model, as the one we chose, is able to display the same features in the behavior of the interfacial force, as observed in experiments measuring the force acting between lipid bilayers. Specifically, the force is not a simple exponential force, but it can be represented as a force where different regions are dominated by forces of different origin. We observed that the hydration force is correlated to the removal of structured water in the interfacial space, and also observed

the restructuring of the hydrogen bonding network, as plates move toward each other and therefore the contributions to the force will have energetic and entropic components. These can be estimated by performing simulations at different temperatures, although the results will be very sensitive to the numerical noise in the calculations. Direct interactions between our surfaces that include the electrostatic interactions between the headgroups and also van der Waals interactions between the opposing graphene plates and also headgroups contribute to the force in the intermediate region, and the value of their contributions to the total free energy can be calculated. We describe how to perform such calculations in the Appendix. In Figure 2-8, we display the decomposition of the PMF into the contributions from the direct interaction and from the water-mediated interaction. As we can see from this decomposition, the direct interaction which consists of the van der Waals and electrostatic interactions is attractive, and therefore, the water contribution is repulsive for all separation distances. This means that even the long-ranged repulsive character of the force, which is due to undulations in experiments, is due to water in our simulations. From the form of the curve for the water-mediated interaction, we also can conclude that water-mediated force is mostly active at the interval between 1.6 and 1.0 nm, in agreement with our previous conclusion obtained from the consideration of the PMF. One should also understand that water indirectly influences the direct force by changing the character of the fluctuations of the headgroups on the plates, and therefore, it makes sense to call the total force in the intermediate interval the hydration force.

Figure 2-8. Decomposition of the PMF (black) into the contributions from direct interaction (red) and from water-mediated interaction (green)

Our simulations also show that protrusions are not needed to obtain the hydration force, since the model excludes protrusions. Comparison of the results from this simulation and previous work of Lu and Berkowitz also shows that flexibility of the headgroups plays an important role. Due to this flexibility, proper boundary conditions can be created to establish a nice hydrogen bonding network in water, while rigid dipoles on the plates can produce frustrations for creation of the hydrogen bonding network which results in a small hydration force. As we can see, although our model is still relatively simple, it provides a useful insight into the phenomenon of the hydration force.

Appendix

The Gibbs free energy change between two adjacent states specified by r1 and r2 at a given temperature T was calculated on the basis of the following formula

assuming that the state of r2 is slightly perturbed from the state of r1.

Since the potential is a sum of pairwise additive interactions, we can separate U(r1) into two terms, that is, U(r1) = u(r1) + w(r1). Here, u(r1) represents the interaction between the atoms belonging to PC−headgroup plates, which does not explicitly depend on the coordinates of water molecules, whereas w(r1) is dependent on the coordinates of water

molecules. Using this separation, we get

where we denoted e−(u(r

1)−u(r2))/kBT 1 and e−(u(r

1)−u(r2))/kBT(e−(w(r1)−w(r2))/kBT − 1) 1 by A and B , respectively. The first term in the result above is the contribution from the direct interactions, and the second term is the contribution from the water-mediated interactions. Therefore, we can define the first and second terms as Δg1,d and Δg1,w, respectively, and get

This result shows that the contribution from the direct interactions can be calculated by using the same trajectories obtained from our calculations of the total PMF. Note that, in the absence of water, Δg1,w = 0, since w(r1) = w(r2) = 0. The water contribution can be

obtained as a difference between the total PMF and the direct contribution.