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The purpose of this research is to investigate physical properties of a lipid bilayer membrane using coarse-grained simulation methods. The goal is to determine a method to consistently and accurately calculate the bending modulus (κ.) A system of dipalmitoylphosphatdylcholine (DPPC) forms a membrane which is run using gromacs molecular dynamics software. The simulations apply pressure in the x dimension to the membrane, while holding the y and z pressures at a reference level. At high enough pressures, the membrane buckles to relieve the pressure. The membrane can be modeled as a sine wave. The properties of the sine wave, along with the pressure data from the simulation, are used to calculate the value of the bending modulus. Several approaches were attempted before derivation of the current model, which offers good agreement with the literature value. However, there is a linear trend in the calculated values for κ; which is inconsistent with our model – the value should be a constant physical property. Future work will be focused on determining the origin of this trend and evaluating the model by comparing systems of different size and composition.
The bending modulus (κ) is a physical property of elastic systems that determines the contribution to the free energy from bending. The bending energy per unit area is given by κ*R-2, where R is the radius of curvature for bending in one direction.1 The bending modulus is useful in determining whether a bilayer would be wrinkled or flat. The bending of bilayers is an integral part of the formation of vesicles, and by extension is of biological interest when considering membranes. The bending modulus plays an important part in determining the energetics of such cellular processes as exocytosis and endocytosis.
One of the primary advantages of a coarse grained (CG) simulation over a full atomistic simulation is that it is computationally less expensive and still retains a high degree of accuracy. As the name implies, an atomistic model treats every atom individually as it computes the interactions all over the system. In a CG model, groups of atoms are represented at single active sites, which can represent properties of the entire group, such as polarity, nonpolarity, lipophilicity, etc. For example, 4 water molecules, a total of 12 atoms, are treated as a single active site group. Goetz et al. have shown that these CG simulations are quite effective at determining the elastic properties of lipid bilayer systems.2
Initial Compression
- System: 256 DPPC molecules arranged into a bilayer
- Solvent: 6000 CG W (W =four water molecules)
- Constant Temperature = 323 K
- periodic boundary conditions on
- gromacs molecular dynamics software
- Linux cluster
First this system was compressed at a constant pressure in the x-dimension with the y-dimension and z-dimension held at 1.0 bar. The simulations were run for six to ten nanoseconds. The x and z-dimensions were allowed to adjust to cope with the pressure, while the y-dimension was held constant.
Different points from the initial compression were selected and allowed to relax.
Relaxation:
- Frames from the same bending simulation were used as starting points
- X and Y dimensions held constant, Z allowed to equilibrate for 40 ns
- Pressure data recorded from the equilibrated system
As can be seen in Figure 3, the buckled bilayer forms a simple sine wave with period equal to 2π/Lx. Initially, we focused on using a simple expression for determination of the bending modulus:
where Peχc is the excess pressure, Lz and Lx are the dimensions of the z and x-dimensions, respectively, and A is the amplitude of the best fit sine wave. A variety of methods were used to calculate the excess pressure from the individual components of the total pressure. Unfortunately, none of these methods yielded final values for the bending modulus that were near the literature value of 5 +/- 2 x 1020 J.3 Rederiving the formula with an additional term to account for compression of the system also failed to offer results better than the original model.
Our current model is based on a fresh derivation that treats the surface area and the curvature of the bilayer separately. The resultant formula is:
The values of κ for this system fall within the error range of the expected value. Recently we have attempted to repeat the experiment with a larger system, however, the new system (N = 512 Lx0 = 24.8096 nm) does not fit well when its curvature is low. We are currently investigating possible causes for this discrepancy.
In general, as Lx decreases, the curvature of the system increases
As the curvature increases, the pressure in the X and Y-dimensions decreases, indicating that buckling is able to relieve some of the applied pressure. In systems that did not successfully buckle, Pzz is much greater than 1.0 bar.
Since the system is roughly incompressible, the curve length remains approximately equal to Lx0, and the decrease in Lx correlates with an increase in amplitude.
The values of the bending modulus generally fall within the error bars of the literature value (the yellow area in Figure entitled “Apparent Bending Modulus vs Curvature”). However, the apparent bending modulus decreases as the system becomes more buckled. Since this should be a physical constant of the system, this leaves us with an open question – why does the apparent bending modulus decrease? Is this a property of the material, or an artifact of the analysis? Future work will address this concern.
This research is ongoing, with future plans directed towards working with larger bilayer systems and varying the applied pressure to bring about the buckling. Additional consideration will be given to the current model to attempt to understand what is causing the downward trend in calculated values. A system of dipalmitoylphosphatdylethanolamine (DPPE) has been equilibrated and prepared to repeat the procedures outlined in this presentation. Eventually we would like to move on to mixed systems of DPPC and DPPE, to see what the difference in headgroup size does to the distribution of the two lipids.
This material is based upon work supported by the Howard Hughes Medical Institute under Grant No. 52003727 and by the Office of Undergraduate Studies at Emory College through the Student Inquiry and Research at Emory award to Jeffrey Gaulding.
(1) Safran, Samuel. “Statistical Thermodynamics of Surfaces, Interfaces and Membranes.” Frontiers in Physics. 1994.
(2) Goetz, Rudiger, Gerhard Gompper, and Reinhard Lipowsky. “Mobility and Elasticity of Self-Assembled Membranes.” Physical Review Letters. 1999, 82.1, 221-224.
(3) Marrink, Siewert J., Alex H. de Vries, and Alan E. Mark. “Coarse Grained Model for Semiquantitative Lipid Simulations.” Journal of Physical Chemistry B. 2004, 108, 750-760.
As computers become more powerful, computer modeling and simulation is becoming a very useful research tool. One hindrance is that it remains time-consuming for even the fastest computers to stimulate any small system composed of many tiny atoms. One way around this problem is to use a "Coarse Grained" method, which analyzes small groups of atoms. These CG models are faster than the atomistic models but may not always behave realistically.
In order to make sure these models are valid, we want to try to measure properties of the computer model that can be compared to experimental data. This project was concerned with the elastic properties of lipid bilayer systems, particularly the bending modulus (a measure of the rigidity of the system.) By squeezing a bilayer system and forcing it to buckle, we can mathematically model the bilayer as a sine wave. This lets us derive a formula to calculate the bending modulus of the system from the pressure data and the sine properties of the bilayer. The calculated values of the bending modulus can then be compared to experimentally measured values to see how well this CG model represents physical properties of the system.
simulations using gromacs MD software, use of VMD visualization software, coarse grained simulation
molecular dynamics simulations, lipid bilayer modeling, bending modulus, elasticity of membranes, coarse grained computer modeling
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