Simulation of Self-Assembly of Ampiphiles Using Molecular Dynamics

Simulation of Self-Assembly of Ampiphiles Using Molecular Dynamics Haneesh Kesari, Reza Banki, and Misty Davies Project Final Paper ME346 Stanford University December 15, 2005 1 Introduction Understanding the physics behind the self-assembly of ampiphiles, particularly lipids in water, has broad implications and potentially high benefit for technological advances in fields as diverse as biology and nanotechnology. For example, biological membranes such as cell walls are composed of lipid bilayers in water, and a more complete understanding of the parameters that affect self-assembly may lead to advances in drug delivery [8] and to an understanding of protobiological evolution [6]. As another of many examples, it is possible that lipid bilayers could be used to synthesize the nanometer-scale structures that are needed as components for nanoelectronic devices. [10] Experimental techniques have provided heuristic characterizations of the macroscopic behavior of the membranes, but it is difficult to measure the positions and motions of molecules in a way that would help us to understand the structure and dynamics at the molecular level. There are currently a variety of techniques to study self-assembly at the atomic or molecular level; including both theoretical studies [9, 5] and molecular dynamics (MD) simulations. Theoretical analyses are often advanced by assuming continuum models that neglect the bilayer structure. At the other extreme, the MD simulations at the atomic scale are computationally expensive limiting the simulations to between 50 and 200 molecules. [2] Ampiphilic molecules consist of a hydrophilic large head group of atoms and a hydrophobic tail made up largely of hydrocarbons. [4] See Figure 1 for a representative example of an ampiphilic molecule with one head and two tails. This molecular structure suggests a model in which, instead of the interactions between each atom, you instead model groups of atoms within the heads and tails as single entities with interactions that roughly describe the bond and bond angle interactions within the molecule, and interactions that describe the relationships between the molecules and the water ions. The groups of atoms that form the head of the molecule become one kind of bead, and the tails are modeled as a series of another kind of bead with springs between each bead to capture some of the dynamics of the bonds between the groups of atoms. This bead 1

Figure 1: An Ampiphilic Molecule With One Head and Two Tails Fig. 2.2 in Nielsen and Klein [7] and spring model allows for a coarse grained simulation technique that still provides knowledge about the structure without the severely limiting computational cost. [7] Using the techniques of periodic boundary conditions, neighbor lists, and cutoff distances, we studied ampiphile self-assembly with a model similar to the one proposed by Goetz and Lipowsky. [2] This method allows for a small number of characterizing parameters, and a tradeoff between structural detail and computational time that allowed the previous authors to simulate the self-assembly of 1440 particles on a 150 MHz machine in approximately one day. We eventually hope to use Goetz and Lipowsky s method to compare with a method that replaces the water molecules with two terms from the dissipative particle dynamics (DPD) simulation method as proposed by Groot and Warren. [3] This should allow us to more quickly drive the system to thermodynamic equilibrium while retaining many of the ampiphile dynamics including the formation of lipid bilayers created by the hydrodynamic forces. 2 Methodology Our MD model is built of three different types of particles a hydrophilic head (h), a hydrophobic tail (t), and water particles (w). We modeled cases where all ampiphilic molecules have one head and four tails. Unbonded interactions between two heads (h h), between two tails (t t), between two water particles (w w), or between a head and a water particle (h w) are modeled by a Truncated Lennard-Jones (LJ) potential as follows: ( ( r ) 12 ( r ) ) 6 U 6 12 (r) 4ɛ 0 + B r + A (1) σ σ where B 4ɛ 0 r c ( 12 ( rc σ ) 12 ( rc ) ) 6 6 σ (2) 2

( (rc ) 12 ( rc ) ) 6 A 4ɛ 0 B r c (3) σ σ The interaction between a head and a tail (h t), or between a tail and a water particle (t w) is purely repulsive and is modeled as: ( ) 9 ( ) 9 r rc U 9 (r) 4ɛ 0 4ɛ 0 + 36ɛ ( ) 9 0 rc (r r c ) (4) σ rep σ rep r c σ rep For our model we chose ɛ 0 = 3.321 10 24 kj, σ = 1/3nm, all of the cutoff radii to be r c = 2.5σ and σ rep = 1.05σ, in order to match the parameters given in Goetz and Lipowsky. [2] The potentials and forces are shown as a function of distance in Figures 2 and 3. In Figure 3 you should note that a central-differenced numerical approximation to the LJ-9 force is indistinguishable from the analytical expression of the LJ-9 force, validating our derivation of the analytical expression for the LJ-9 force. Since we are modeling molecules, it is imperative to account for the energy in the bonds between the head and tail particles, and more accurate models also consider the energy in bending. Goetz and Lipowsky (See Figure 4) obtained self-assembly behavior for both completely flexible models and for models that had some rigidity with respect to bending. Those models are defined in Figure 5 and as follows: U bond (r) k bond ( r i+1 r i σ) 2 (5) U bend (r) k bend (1 cos(φ i )) (6) where k bond = 5000 ɛ 0 σ and k bend = 50 ɛ 0 when the bending energy is used. Our preferred bond angle is zero degrees as measured using the method shown in Figure 5. The bond stretching and bond bending potentials are orders of magnitude larger than the LJ potentials. Therefore, the LJ interactions between bonded particles are neglected. We will adopt here the common notation of lower case letters to denote molecules with flexible tails and upper case letters to denote molecules with some rigidity to their tails. For example, HT 4 will be used to represent a molecule with 4 tails, one head, and bending energy, and ht 4 will be used to represent the same molecule with no bending energy. We should note here that our current implementation only allows for rigidity between tail particles. There needs to be further implementation to allow simulation for molecules with more than one head. The total potential energy of the system is then defined as U total (r) h t,t w U 6 12 + h t,t w U 9 + molecules U bond + i molecules U bend (7) To perform our MD Simulations, we implemented a new function in Professor Wei Cai s MD++ software [1]. The shape and size of the molecules are prescribed using an input script, along with the concentration of lipid molecules and the density and size of the simulation box. The number of lipid molecules is then calculated using the formula c s = (N t + N h )/N (8) 3 i

Figure 2: The Potential Energy in Electron Volts as a Function of Distance in Angstroms Between Particles for Both the Standard Lennard-Jones 6-12 Potential Function and the Purely Repulsive Lennard-Jones 9 Potential Function Figure 3: The Forces in Electron Volts per Angstrom Derived from the Potential Energy Functions 4

Figure 4: Structures Formed by ht 4 Molecules in Order of Increasing Concentration Fig. 5 in Goetz and Lipowsky [2] Figure 5: Geometry Used for Calculating Potential Energies in Bond Stretching and Molecular Bending Fig. 2 in Goetz and Lipowsky [2] 5

where c s is the lipid molecule concentration, and N t, N h, and N are the number of tail particles, the number of head particles, and the number of total particles. The lipid molecules and the water particles are placed randomly within the simulation box, relaxed using a conjugate gradient algorithm, and then a molecular dynamics simulation is run using periodic boundary conditions and a Verlet neighbor list. The simulation box size for all of our results was L x = L y = 12σ = 40 angstroms and L z = 15σ = 50 angstroms. The density for all of the simulations was 0.006 particles per cubic angstrom. It is important to note here that this is a third of the density of the Groetz and Lipowsky simulations. We found heuristically that this was the largest density that would allow relaxation using the conjugate gradient algorithm and random atom placement within MD++. We were also not able to complete a simulation using the timestep (1.4 ps) used by Groetz and Lipowsky both t = 1.4 ps and t = 0.1 ps were too large. Our simulations were run at a timestep of t = 0.001 ps. There was no attempt made to find the largest possible timestep. 3 Results Even with the crude bead and spring models for the lipid molecules and with the low densities of our simulations, we did see self-assembly of the ampiphiles. Self-assembly of most cases occurred within 100 ps, which is faster than the times noted by Groetz and Lipowsky at a higher density. The wall clock time for the simulations was on the order of 30 minutes on a 2 GHz Linux machine. Groetz and Lipowsky do not mention any sort of relaxation before they begin their simulation, so it may be possible that the conjugate gradient relaxation creates an initial position of the molecules that leads to a quicker selfassembly. This possibility is suggested by the locations of the molecules after conjugate gradient relaxation as shown in Figures 6, 8, 10, 12, 14, 16, 18, and 20. For all of the following figures, the red beads are groups of head molecules, the yellow are tail molecules, and the blue represent water molecules. We chose concentrations of 0.069, 0.208, 0.347, and 0.417 to match some of the results shown in Groetz and Lipowsky s paper, although as previously mentioned, the density of these simulations was one-third of the density of the simulations run in that paper. The ht4 molecules were quicker to relax using the conjugate gradient algorithm, likely due to the less complicated energy surface. While the ht4 molecules aggregrated, the structures that they formed did not seem as cohesive as those formed by the HT4 molecules. At a concentration of 0.069, as shown in Figure 7, the lipid molecules seemed to cluster and then break apart. It was difficult to say for sure that there was ever any aggregation. In Figure 9, at a concentration of 0.208, you can see what looks to be two spherical micelles. These micelles did not seem to be moving towards further aggregration at the completion of the simulation. When the concentration is increased to 0.347, as in Figure 11, a more obvious aggregate is formed. There seems to be a large spherical micelle spanning the top and bottom of the simulation cell, with a few lipids still remaining. At the largest concentration, 0.417, the lipids begin to form a bilayer, as shown in Figure 13. The HT4 molecules took much longer to relax, and the conjugate gradient did not always converge to within our strict tolerances. For example, we would allow MD sim- 6

ulation of the HT4 molecules to begin with a conjugate gradient tolerance of 1e-3, even though the ht4 molecules would converge in much less time to within a tolerance of 1e- 6. As you can see in Figures 14 and 15, even at a concentration of 0.069, the molecules formed a micelle within 100 ps of MD simulation after conjugate gradient relaxation. At a concentration of 0.208, it looked as if the molecules were trying to form two spherical micelles (Figures 16 and 17). At a concentration of 0.374 the lipids formed a cylindrical micelle. Note that, due to the periodic boundary condition the molecules at the top of the picture shown (Figure 19) are actually part of the bottom layer of the aggregation. At a concentration of 0.417, the ampiphiles form what appears to be a bilayer unfortunately the bilayer was formed at the edge of the simulation box and the results take a little effort to visualize (Figure 21). 4 Recommendations for Future Work Although it is currently possible to form lipids with more than one head using our implementation, it is not currently possible to model the bending energies of the bonds between those heads. It is also not currently possible to create a simulation using more than one kind of ampiphile. We also recommend implementing a mechanism to determine the moment of assembly. This would allow comparison of time to assembly with other data sets. This mechanism might also help with visualization, by placing the completed aggregate in the center of the simulation cell. As previously mentioned, the conjugate gradient algorithm had difficulty relaxing the simulation for densities larger than approximately 0.006 particles per cubic angstrom. Larger densities might be simulated if the initial placement of particles was not random or if there was a way to replace the actual particle interactions between the lipids and the water molecules with representative forces. We recommend comparing the results obtained with the purely conservative MD model with the results obtained by replacing the water molecules with two nonconservative forces inspired by the dissipative particle dynamics (DPD) model in Groot and Warren [3] a drag force and a random force given by: F D ij γw D ( r ij )( rˆ ij v ij ) rˆ ij (9) F R ij Ωw R ( r ij )θ ij rˆ ij (10) where rˆ ij = ( r i r j )/ r ij, Ω 2 = 2γk B T, θ ij (t) = ζ ij t 1/2, ζ is a random number chosen for each pair and timestep with zero mean and unit variance, and w D (r) [ w R (r) ] 2 { (rc r) 2 : r < r c 0 : r r c (11) Since the force in this method is a function of the velocity, it is impossible to use the same version of the velocity-verlet algorithm that was possible with the strictly conservative system. Instead, we recommend the predictor-corrector version of the velocity-verlet algorithm proposed by Groot and Warren. [3] 7

Figure 6: ht 4 Molecules at a Concentration of 0.069 After Conjugate Gradient Relaxation Figure 7: ht 4 Molecules at a Concentration of 0.069 After 100 ps 8

Figure 8: ht 4 Molecules at a Concentration of 0.208 After Conjugate Gradient Relaxation Figure 9: ht 4 Molecules at a Concentration of 0.208 After Formation of Two Spherical Micelles 9

Figure 10: ht 4 Molecules at a Concentration of 0.347 After Conjugate Gradient Relaxation Figure 11: ht 4 Molecules at a Concentration of 0.347 After Formation of a Spherical Micelle 10

Figure 12: ht 4 Molecules at a Concentration of 0.417 After Conjugate Gradient Relaxation Figure 13: ht 4 Molecules at a Concentration of 0.417 After Formation of a Bilayer 11

Figure 14: HT 4 Molecules at a Concentration of 0.069 After Conjugate Gradient Relaxation Figure 15: HT 4 Molecules at a Concentration of 0.069 After Formation of a Spherical Micelle 12

Figure 16: HT 4 Molecules at a Concentration of 0.208 After Conjugate Gradient Relaxation Figure 17: HT 4 Molecules at a Concentration of 0.208 After Aggregation 13

Figure 18: HT 4 Molecules at a Concentration of 0.347 After Conjugate Gradient Relaxation Figure 19: HT 4 Molecules at a Concentration of 0.347 After Formation of a Cylindrical Micelle 14

Figure 20: HT 4 Molecules at a Concentration of 0.417 After Conjugate Gradient Relaxation Figure 21: HT 4 Molecules at a Concentration of 0.417 After Formation of a Bilayer 15

We believe that using this method would allow reproduction of the results for selfassembly of surfactants by concentration as demonstrated in Goetz and Lipowsky, with the eventual attainment of a lipid bilayer, as shown in Figure 4. The two non-conservative forces, F D ij and F R ij, should lead to assembly in a shorter number of time steps than reported for the previous work. References [1] W Cai. MD++ Software. http://www.stanford.edu/ caiwei. [2] R Goetz and R Lipowsky. Computer Simulations of Bilayer Membranes: Self- Assembly and Interfacial Tension. Journal of Chemical Physics, 108(17):7397 7409, May 1998. [3] RD Groot and PB Warren. Dissipative Particle Dynamics: Bridging the Gap Between Atomistic and Mesoscopic Simulation. J. Chem. Phys., 107(11):4423, September 1997. [4] AR Leach. Molecular Modelling: Principles and Applications. Prentice Hall, Edinburgh, 2001. [5] R Lipowsky and E Sackman. Handbook of Biological Physics, volume 1. Elsevier, Amsterdam, 1995. [6] HJ Morowitz. Beginnings of Cellular Life. Yale University Press, New Haven, 1992. [7] SO Nielsen and ML Klein. A Coarse Grain Model for Lipid Monolayer and Bilayer Studies. Lecture Notes in Physics, 605:27 63, 2002. [8] RO Potts and RH Guy. A Predictive Algorithm for Skin Permeability:The Effects of Molecular Size and Hydrogen Bond Activity. Pharmaceutical Research, 12(11):1628 1633, November 1995. [9] C Tanford. The Hydrophobic Effect: Formation of Micelles and Biological Membranes. Wiley, New York, 1980. [10] MH Lamm Z Zhang, MA Horsch and SC Glotzer. Tethered Nano Building Blocks: Toward a Conceptual Framework for Nanoparticle Self-Assembly. Nano Letters, 3:1341 1347, 2003. 16