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Permeation of Organometallic Compounds through Phospholipid Membranes

von Dr. Raycho Yonchev

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[This also means that the surface tension is equal to the derivative of the free energy with respect to the area at] constant temperature and volume: $\gamma=\left ( \frac{\partial F}{\partial A} \right )_{T,V}$ . Unstressed lipid bilayers are free to adjust their surface area to attain an equilibrium with the surroundings. In terms of energy, the bilayer tends to adjust its surface area so as to minimize its free energy. Owing to the hydrophobic effect, an increase of the area leads to an increase in free energy. If this were the only parameter to consider, a membrane would minimize its area, going into an ordered gel phase whatever the temperature. The entropic contribution to the free energy has also to be considered: the entropy of the system grows upon increasing the area, which reduces the free energy. A small surface area forces the chains into a more ordered state, reducing the entropy of the system and thus increasing the free energy. To reach its thermodynamic equilibrium, the membrane has thus to find a balance between the enthalpic and entropic terms. When the minimum in free energy is attained, the surface tension is then, per definition, equal to zero. In 1995, Chiu et al. introduced a surface tension into the simulation of a DPPC membrane, arguing that the surface tension of a bilayer is twice that of the corresponding monolayer [51]. This assumption was however doubtful since the main contribution to the surface tension in an air-monolayer-water system comes from the interface between the air and the hydrocarbon chains and not between the headgroups and water. Feller and Pastor also adopted the explicit inclusion of a surface tension in a series of DPPC bilayer simulations [52]. They recognized that the surface tension of a bilayer patch is surely zero on a large length scale (microns), but put forward that a bilayer on the typical length scale of a MD simulation (nanometers) may exhibit a finite surface tension at the surface area that minimizes its free energy. They claimed that the application of a nonzero surface tension may be appropriate to compensate for the absence of long wavelength undulations in the small systems used in simulations, the applied surface tension being then considered as a correction for a finite-size effect. As the suppression of undulations is even reinforced by the confining effect of PBC, a nonzero surface tension would be only the expression of the suppressed undulations. Or, in other words, a tension would be required to remove the undulations, which are normally present in real membranes, but absent in simulated ones. Marrink and Mark however showed in an extensive series of glycerolmonoolein (GMO) bilayer simulations that, at stress free conditions, the equilibrium area per lipid does not strongly [depend on the system size and concluded that the application of an external surface tension to compensate for suppressed fluctuations is not necessary [54].]

51. Chiu, S. W.; Clark, M. M.; Balaji, V.; Subramaniam, S.; Scott, H. L.; Jakobsson, E. Biophys. J. 1995, 69, 1230.

52. Feller, S. E.; Pastor, R. W. J. Chem. Phys. 1999, 111, 1281.

54. Marrink, S. J.; Mark, A. E. J. Phys. Chem. B 2001, 105, 6122.

[page 66]

This also means that the surface tension is equal to the derivative of the free energy with respect to the area at constant temperature and volume: $\gamma=\left ( \frac{\partial F}{\partial A} \right )_{T,V}$. Unstressed lipid bilayers are free to adjust their surface area to attain an equilibrium with the surroundings. In terms of energy, the bilayer tends to adjust its surface area so as to minimize its free energy. Owing to the hydrophobic effect, an increase of the area leads to an increase in free energy. If this were the only parameter to consider, a membrane would minimize its area, going into an ordered gel phase whatever the temperature. The entropic contribution to the free energy has also to be considered: the entropy of the system grows upon increasing the area, which reduces the free energy. A small surface area forces the chains into a more ordered state, reducing the entropy of the system and thus increasing the free energy. To reach its thermodynamic equilibrium, the membrane has thus to find a balance between the enthalpic and entropic terms. When the minimum in free energy is attained, the surface tension is then, per definition, equal to zero [102]. In 1995, Chiu et al. introduced a surface tension into the simulation of a DPPC membrane, arguing that the surface tension of a bilayer is twice that of the corresponding monolayer [97]. This assumption was however doubtful since the main contribution to the surface tension in an air-monolayer-water system comes from the interface between the air and the hydrocarbon chains and not between the headgroups and water. The explicit inclusion of a surface tension was also adopted by Feller and Pastor in a series of DPPC bilayer simulations [99].

[page 67]

They recognized that the surface tension of a bilayer patch is surely zero on a large length scale (microns), but put forward that a bilayer on the typical length scale of a MD simulation (nanometers) may exhibit a finite surface tension at the surface area that minimizes its free energy. They claimed that the application of a nonzero surface tension may be appropriate to compensate for the absence of long wavelength undulations in the small systems used in simulations, the applied surface tension being then considered as a correction for a finite-size effect. As the suppression of undulations is even reinforced by the confining effect of PBC, a nonzero surface tension would be only the expression of the suppressed undulations. Or, in other words, a tension would be required to remove the undulations which are normally present in real membranes, but absent in simulated ones. Marrink and Mark however showed in an extensive series of glycerolmonoolein (GMO) bilayer simulations that, at stress free conditions, the equilibrium area per lipid does not strongly depend on the system size and concluded that the application of an external surface tension to compensate for suppressed fluctuations is not necessary [103].

[97] S.-W. Chiu, M. M. Clark, V. Balaji, S. Subramaniam, H. L. Scott, and E. Jakobsson. Biophys. J., 69:1230–1245, 1995.

[99] S. E. Feller and R. W. Pastor. J. Chem. Phys., 111:1281–1287, 1999.

[102] F. Jähnig. Biophys. J., 71:1348–1349, 1996.

[103] S. J. Marrink and A. E. Mark. J. Phys. Chem. B, 105:6122–6127, 2001.

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