^{1,a)}, Bernard R. Brooks

^{1}and Richard W. Pastor

^{1}

### Abstract

Molecular dynamics (MD) simulations of dipalmitoylphosphatidylcholine bilayers composed of 72 and 288 lipids are used to examine system size dependence on dynamical properties associated with the particle mesh Ewald (PME) treatment of electrostatic interactions. The lateral diffusion constant is and for 72 and 288 lipids, respectively. This dramatic finite size effect originates from the correlation length of lipiddiffusion, which extends to next-nearest neighbors in the 288 lipid system. Consequently, diffusional events in smaller systems can propagate across the boundaries of the periodic box. The internal dynamics of lipids calculated from the PME simulations are independent of the system size. Specifically, reorientational correlation functions for the slowly relaxing phosphorus-glycerol hydrogen, phosphorus-nitrogen vectors, and more rapidly relaxing CH vectors in the aliphatic chains are equivalent for the 72 and 288 lipid simulations. A third MD simulation of a bilayer with 72 lipids using spherical force-shift electrostatic cutoffs resulted in interdigitated chains, thereby rendering this cutoff method inappropriate.

This research was supported in part by the Intramural Research Program of the NIH, National Heart, Lung and Blood Institute. We thank Klaus Gawrisch and Scott Feller for helpful discussions.

I. INTRODUCTION

II. METHODOLOGY

A. Simulation details

B. Translational diffusion

III. RESULTS AND DISCUSSION

A. Translational diffusion

B. Finite size effects on bilayer structure and lipid reorientation

IV. SUMMARY

### Key Topics

- Lipids
- 91.0
- Diffusion
- 47.0
- Electrostatics
- 17.0
- Correlation functions
- 8.0
- Probability theory
- 8.0

## Figures

The lateral mean squared displacement of the lipids, , for 72 PME (an average of three independent simulations) and one 288 PME simulation for all (top) and for the interval of (bottom). Standard errors are denoted with vertical bars.

The lateral mean squared displacement of the lipids, , for 72 PME (an average of three independent simulations) and one 288 PME simulation for all (top) and for the interval of (bottom). Standard errors are denoted with vertical bars.

(Color online) Final snapshot of one 72 PME (left), 288 PME (middle), and 72 FSW (right) simulations. The ends of the aliphatic chains are shown in large yellow spheres.

(Color online) Final snapshot of one 72 PME (left), 288 PME (middle), and 72 FSW (right) simulations. The ends of the aliphatic chains are shown in large yellow spheres.

(Color online) The lateral trace of the center of mass (C.M.) of a lipid in one 72 PME simulation, with the five clusters with radius of .

(Color online) The lateral trace of the center of mass (C.M.) of a lipid in one 72 PME simulation, with the five clusters with radius of .

Jump times (defined by the clustering algorithm and denoted by vertical tick marks) of all of the lipids in 72 PME (top) and 288 PME (bottom).

Jump times (defined by the clustering algorithm and denoted by vertical tick marks) of all of the lipids in 72 PME (top) and 288 PME (bottom).

The average probability of the number of jumps in a block ignoring the first and last blocks for 72 PME (top) and 288 PME (bottom), and the Poisson distribution for these values of (solid lines in each panel).

The average probability of the number of jumps in a block ignoring the first and last blocks for 72 PME (top) and 288 PME (bottom), and the Poisson distribution for these values of (solid lines in each panel).

The two-dimensional radial distribution functions for one leaflet for P–P (top) and N–N (bottom).

The two-dimensional radial distribution functions for one leaflet for P–P (top) and N–N (bottom).

The probability of finding a neighbor with an angle defined by the angle formed between its P–N vector and those neighboring lipids less than of separation between the phosphorus atoms.

The probability of finding a neighbor with an angle defined by the angle formed between its P–N vector and those neighboring lipids less than of separation between the phosphorus atoms.

(Color online) Second rank reorientational correlation functions of the principal axis (PA), P–N, and the vector between the phosphate and adjacent glycerol hydrogen (P–H) (top), and C–H vectors of aliphatic carbons C2, C9, and C15 (bottom) for 72 PME and 288 PME.

(Color online) Second rank reorientational correlation functions of the principal axis (PA), P–N, and the vector between the phosphate and adjacent glycerol hydrogen (P–H) (top), and C–H vectors of aliphatic carbons C2, C9, and C15 (bottom) for 72 PME and 288 PME.

## Tables

Lateral diffusion constants for DPPC: short-time cage diffusion evaluated from the slope of the mean squared displacement [Eq. (1)] between 0 and , long-time lateral diffusion constant evaluated over the range of , lateral diffusion with the monolayer C.M. displacement removed , and for a jump model, from Eq. (2). All diffusion constants have units of and, unless otherwise noted, are calculated at .

Lateral diffusion constants for DPPC: short-time cage diffusion evaluated from the slope of the mean squared displacement [Eq. (1)] between 0 and , long-time lateral diffusion constant evaluated over the range of , lateral diffusion with the monolayer C.M. displacement removed , and for a jump model, from Eq. (2). All diffusion constants have units of and, unless otherwise noted, are calculated at .

Statistics for concerted motion: the number of diffusional jumps for each trajectory based on a cluster analysis, the number of jumps following another jump within , the number of jumps that would be expected from independent Poisson statistics [Eq. (6)], the probability of a concerted jump [Eq. (7)], and the value, which is the probability that would have been observed if the jumps were independent [Eq. (8)]. Lipids were divided into the first shell , second shell , and third shell , with the average number of lipids in each shell.

Statistics for concerted motion: the number of diffusional jumps for each trajectory based on a cluster analysis, the number of jumps following another jump within , the number of jumps that would be expected from independent Poisson statistics [Eq. (6)], the probability of a concerted jump [Eq. (7)], and the value, which is the probability that would have been observed if the jumps were independent [Eq. (8)]. Lipids were divided into the first shell , second shell , and third shell , with the average number of lipids in each shell.

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