The Extended Eighth-Shell method for periodic boundary conditions with rotational symmetry

Simulating asymmetric membranes using P21 periodic boundary conditions

Molecular dynamics (MD) simulations of symmetric lipid bilayers are now well established, while those of asymmetric ones are considerably less developed. This disjunction arises in part because the surface tensions of leaflets in asymmetric bilayers can differ (unlike those of symmetric ones), and there is no simple way to determine them without assumptions. This chapter describes the use of P21 periodic boundary conditions (PBC), which allow lipids to switch leaflets, to generate asymmetric bilayers under the assumption of equal chemical potentials of lipids in opposing leaflets. A series of examples, ranging from bilayers with one lipid type to those with peptides and proteins, provides a guide for the use of P21 PBC. Critical properties of asymmetric membranes, such as spontaneous curvature, are highly sensitive to differences in the leaflet surface tensions (or differential stress), and equilibration with P21 PBC substantially reduces differential stress of asymmetric bilayers assembled with surface area-based methods. Limitations of the method are discussed. Technically, the nonstandard unit cell is difficult to parallelize and to incorporate restraints. Inherently, the assumption of equal chemical potentials, and therefore the method itself, is not applicable to all target systems. Despite these limitations, it is argued that P21 simulations should be considered when designing equilibration protocols for MD studies of most asymmetric membranes.

Spontaneous curvature generation by peptides in asymmetric bilayers

Peptides and proteins play crucial roles in membrane remodeling by inducing spontaneous curvature. However, extracting spontaneous curvatures from simulations of asymmetric bilayers is challenging because differential stress (i.e., the difference of the leaflet surface tensions) arising from leaflet area strains can vary substantially among initial conditions. This study investigates peptide-induced spontaneous curvatureδc 0 p in asymmetric bilayers consisting of a single lipid type and a peptide confined to one leaflet;δc 0 p is calculated from the Helfrich equation using the first moment of the lateral pressure tensor and an alternative expression using the differential stress. It is shown that differential stress introduced during initial system generation is effectively relaxed by equilibrating using P21 periodic boundary conditions, which allows lipids to switch leaflets across cell boundaries and equalize their chemical potentials across leaflets. This procedure leads to robust estimates ofδc 0 p for the systems simulated, and is recommended when equality of chemical potentials between the leaflets is a primary consideration.

Developing initial conditions for simulations of asymmetric membranes: a practical recommendation

It has been proposed that the surface tension difference between leaflets (or differential stress) in asymmetric bilayers is generally nonvanishing. This implies that there is no unique approach to generate initial conditions for simulations of asymmetric bilayers in the absence of experimentally derived constraints. Current generation methods include individual area per lipid (APL) based, leaflet surface area (SA) matching, and zero leaflet tension based (0-DS). This work adds a bilayer-based approach that aims for achieving partial chemical equilibrium by interleaflet switching of selected lipids via P2₁ periodic boundary conditions. Based on a recently proposed theoretical framework, we obtained expressions for tensions in asymmetric bilayers from both the bending and area strains. We also developed a quantitative measure for the energetic penalty from the differential stress. The impacts of APL-, SA-, and 0-DS-based approaches on mechanical properties are assessed for two different asymmetric bilayers. The lateral pressure profile and its moments differ significantly for each method, whereas the area compressibility modulus is relatively insensitive. Application of P2₁ periodic boundary conditions (APL/P2₁, SA/P2₁, and 0-DS/P2₁) results in better agreement in mechanical properties between asymmetric bilayers generated by APL-, SA-, and 0-DS-based approaches, in which changes are the smallest for bilayers from the SA-based method. The estimated differential stress from the theory shows good agreement with that from the simulations. These simulation results and the good agreement between the predicted and observed differential stress further support the theoretical framework in which bilayer mechanical properties are outcomes of the interplay between intrinsic bending and asymmetric lipid packing. Based on the simulation results and theoretical predictions, the SA/P2₁-based, or at least the SA-based (when the differential stress is small), approach is recommended as a practical method for developing initial conditions for asymmetric bilayer simulations.