ps ns s ms

nm

m

mm

Ab-initio methods

Statistical and continuum methods

Atomistic methods

•Simulations where atoms or groups of atoms are treated as classical particles.Cannot simulate chemical reactions.Electrostatic properties are approximated (electronic polarizability).

What are atomistic simulations?

Other Approximations•The interaction potential is effective pair potential. •Boundary conditions.

• Molecular Dynamics (MD).• Monte Carlo (MC).• Brownian Dynamics (BD) etc.Choice of the technique depends on the problem and the

availability of computing resources.

Simulation techniques

• Initialize positions, velocities and potential parameters

• For each time step

Molecular dynamics algorithm :

o Compute energy and forces

o Update configuration

a) Non bonded forces

b) Bonded forcesc) External forcesd) Pressure tensor

etc.a) Compute new configuration by solving F = m ab) Apply constraints and correct velocitiesc) Scale simulation cell according to the ensemble

V = Vbond + Vangle + Vdihedral + VLJ + VElecrostatics

F =−∇V

VLJ =Cij

(12)

rij12 −

Cij(6)

rij6

i< j∑ VElectrostatics = f

qiqj

εrriji< j∑

Vbond =1

2kij (rij - bij )

2

Vangle =1

2kijkθ (θijk -θijk

0 )2

Vdihedral = kijklφ (1+ cos(nφijkl -φijkl

n0 ))n∑

Force parameters

• Determination of partial charges on the atoms. • Get bond and angle parameters from

spectroscopic data.• Work on smaller fragments of the molecule

and determine LJ parameters by fitting the density and heat of vaporization. E.g. Linear chain alkanes were used to determine the LJ parameters of hydrocarbon chains in lipids.

• Determination of dihedral parameters.

Again tune the parameters to match with known experimental details of the system.

-0.84

0.42

0.42

• Usually static lists are enough and are easy to implement.

• Parameter values depend on 1-4 non-bonded interactions.

Bonded forces :

• Lennard-Jones (LJ) potential• short range potential• can be truncated with a cutoff rc

(errors??, dynamic neighbor list)• Electrostatics

• long range potential• cutoffs usually produce erroneous

results• Particle Mesh Ewald (PME)

technique with appropriate boundary corrections are used. (poor scalability)

Non bonded forces :

Algorithms part here

Membrane as a barrier

Thanks to David Bostick

Fluid mosaic model

Experiments are performed on model liposome vesicles

What is a lipid bilayer ?

Hydrophilic

Hydrophobic

lipid bilayer

Depending on the cross sectional area and the chain length the molecules can form vesicles, micelles, hexagonal phase

Fatty acid

Fatty acidPhosphate

Glycerophospholipid

Key physical properties of lipid bilayers

• Thickness.•increases with the length of the hydrocarbon chains •decreases with the number of unsaturated bond in chains •decreases with temperature

• Area per molecule•increases with the length of the hydrocarbon chains•increases with the number of unsaturated bond in chains•increases with temperature •also depends on the properties of polar region

Key physical properties of lipid bilayers

•Order parameter

€

SCD = P2(cos(β ) ≡1

23cos2(β ) −1

Key physical properties of lipid bilayers: Phase transition

•Bilayer exhibits first order phase transition with respect to temperature• The transition temperature Tm depends on

• length of hydrocarbon chains• number of unsaturated bonds in the chain• properties of polar headgroup

• Effect of phase transitiono order parameterso thicknesso repeat distance in multilamellar vesicleso area per lipido bending and compressibility modulus.o lateral and transverse diffusion coefficients.

How real are the simulations ?

Area per lipid (Å2)

experimentssimulationsLipid

~ 38*~ 27Cholesterol

~ 53~ 53Sphingomyelin (18:0)

~ 54~ 53.6Dipalmitoylphosphatidylserine(DPPS with Na+ counter ions)

~ 72~ 71Dioleylphosphatidylcholine

(DOPC)

~ 64~ 62Dipalmitoylphosphatidylcholine

(DPPC)

How real are the simulations ?

Pandit, Bostick and Berkowitz, Biophys. J. (2003), 84, 3743-3750

Simulation of 128 DPPC bilayer in SPC water

How real are the simulations ?Sometimes the comparison is not obvious

e.g. Binding constants of various ions

Binding constant of an ion is given by Langmuir isotherms

where C is the concentration of ions at the membrane surface, is the ratio of number of bound ions to the number of lipids on the

surface.

Experimental values of binding constants of Na+ and Cl- to DPPC surface areKNa = 0.15 0.10 M-1 and KCl = 0.2 0.10 M-1

K =

(1−)C

How real are the simulations ?

KNa ~ 1.4 M-1 ?? KNa ~ 0.15 M-1

Pandit, Bostick and Berkowitz, Biophys. J. (2003), 84, 3743-3750

Predictability of atomistic simulations

Pandit, Bostick and Berkowitz, J. Chem. Phys. (2003), 119(4), 2199-2205

• Water shells were observed using the surface to point correlation function.

Heterogeneous Mixures in bilayers

• Lipids with other membrane components.• Lipid mixtures with different headgroups and polar region (charged and zwitterionic lipids).• Lipid mixtures with different chain lengths and saturation. Mixture with cholesterol

face face

Hydrophilic

• Cholesterol increases mechanical strength of the bilayer• Ordering of chains and phase behavior.

Phase diagram with cholesterol

Reproduced from Vist and Davis (1990), Biochemistry, 29:451

• Lo or phase

Liquid ordered phase where chain ordering is like gel phase but the lateral diffusion coefficients are like fluid phase.

Rafts

Rafts are liquid ordered domains in cell plasma membranes rich in sphingomyelin (or other sphingolipids), cholesterol and certain anchor proteins.

Rafts are important membrane structural components in:• signal transduction• protein transport • sorting of membrane components• process of fusion of bacteria and viruses into host cells

How do rafts function and maintain their integrity?

How does Cholesterol control the fluidity of lipid membranes?

A model for raft

Heterogeneous model membranes• Non-homogeneous domain formation is also observed in model membrane systems.

(a) 2:1 mixture of POPC:cholesterol

(b) 2:1:1 mixture of POPC:cholesterol:SM

1:1:1 mixture of DOPC:SM:chol.

Dietrich, C. et al, Biophys. J (2001), 80, 1417

• These domains are rich in sphingomyelin, cholesterol.•They are usually in liquid ordered or gel phase.

Simulation studies of rafts

Study of liquid ordered phaseStudy of ternary mixtures which form raft like domains.

• Mixture of cholesterol and saturated lipid (DPPC and DLPC).• Mixture of cholesterol and sphingomyelin.

Preformed domain

Spontaneous domain formation

Ternary mixture of DOPC, SM, Chol with a preformed domain.

Ternary mixture of DOPC, SM, Chol with random initial distribution

simulated systems:120 PC molecules (DPPC/DLPC)80 cholesterol molecules (~40 mol %)5000 water moleculesduration: 35 ns / 18 nsconditions: NPT ensemble (P = 1 atm, T = 323 / 279 => )

Study of Lo phase

Pandit, Bostick and Berkowitz, Biophys. J. (2004), 86, 1345-1356.

€

Treduced ≡T −TmTm

= 0.029

DPPC+Cholesterol DLPC+Cholesterol

Pandit, Bostick and Berkowitz, Biophys. J. (2004), 86, 1345-1356.

Pandit, Bostick and Berkowitz, Biophys. J. (2004), 86, 1345-1356.

Conclusions from these simulations

• Cholesterol increases order parameter of the lipids. The order parameters are close to liquid ordered phase order parameters.• Cholesterol forms complexes with two of its neighboring lipids. Life time of such a complex is ~10ns.• Complex formation process is dependent on the chain length of the lipids. Shorter chain lipids do not form large complexes.

Simulation of domain

•1424 DOPC, 266 SM, 122 cholesterols and 62561 waters (284,633 atoms)• Duration: 12 ns (with electrostatic cutoff) and 8 ns (with PME), 3 fs steps• Ensemble: NPT (P = 1 atm (isotropic), T = 293 K)• Configuration Biased Monte Carlo (CBMC) is used to thermalise chains.

Pandit et al., Biophys. J. (2004),87,1092-1100.

Area per sphingomyelin: 49.5 Å2 /mol Area per DOPC: 61.0 Å2 /mol (~72 Å2 /mol)

Area per Cholesterol: 29.6 Å2 /mol

Pandit et al., Biophys. J. (2004),87,1092-1100.

Area/molecule of DOPC vs distance from SM-cholesterol domain

DOPC molecular area affected by the ordered domain out to 8 nm

Pandit et al., Biophys. J. (2004),87,1092-1100.

Early stages of nanoscopic domain formation?

For the lipids in bilayer : D= 5 10 -12 m2/s

< r2 > = 4 D t

t = 250 ns r ~ 22 Å

In a system with 50 DOPC and 50 SM molecules< rSM-SM > ~ 9 Å

If one can simulate a system for 100s of ns then one can probably see some of the processes involved in the early stages of domain formation

Pandit, Jakobsson and Scott., Biophys. J. (2004), 87, 3312-3322.

Binary mixture: 100 DOPC, 100

18:0 SM, 7000 water (31600 atoms)

0 ns

Pandit, Jakobsson and Scott., Biophys. J. (2004), 87, 3312-3322.

250 ns

Ternary mixture: 100 DOPC, 100 18:0 SM, 100 CHOL, 10000 water

(43500 atoms)0 ns

Pandit, Jakobsson and Scott., Biophys. J. (2004), 87, 3312-3322.

250 ns

Orientation of cholesterol

Body-centered axes for cholesterol: (x, y, z)

€

g(r,ϕ ) =N(r,ϕ )

2πρrδrδϕ

Pandit, Jakobsson and Scott., Biophys. J. (2004), 87, 3312-3322.

- face of cholesterol

SM packs well around the - face of cholesterolPandit, Jakobsson and Scott., Biophys. J. (2004), 87, 3312-3322.

Speculation about structure of domain

• Based on these simulations we speculate the structure of a ordered domain

• To validate this speculation we need simulations that involve 103 to 105 lipids and simulation time scale of 10 s to 1 ms. This length and time scale is beyond the scope of current atomistic simulations.

Hybrid simulations with MD and MC

• Combine MD with swapping moves of molecules.1. MD simulation2. Choose two molecules randomly and

exchange their positions.a) Place similar atoms of the molecules.b) Place remaining atoms using CBMC.

3. Repeat step 2. for several pairs of molecules.4. Go to step 1.

Coarse grain mapping

3D lipid bilayer 2D lattice field

lipid chain Lattice field si

Cholesterol Hard rod

Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.

Binary mixture of DPPC and cholesterol

Order parameter fields after 20 s of simulation with 10000 chains and 7%, 11%, 18%, 25% and 33% cholesterols.

Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.