4 nm

Molecular dynamics simulation of Aquaporin-1

i~@t (r, R) = H (r, R)

He e(r;R) = Ee(R) e(r;R)

Molecular Dynamics Simulations

Schrödinger equation

Born-Oppenheimer approximation

Nucleic motion described classically

Empirical Force field

1

Molecular Dynamics Simulations

Interatomic interactions

„Force-Field“

Molecular Dynamics SimulationMolecule: (classical) N-particle system


Newtonian equations of motion:

with

Integrate numerically via the „leapfrog“ scheme:

(equivalent to the Verlet algorithm)

with

Δt ≈ 1fs!

“Aquaporin” water channel

Human hemoglobin

Lipid membranes

Today’s lecture

• Protein structures • Notes on force calculations • Setup of a simulation • Organize force field parameters • Algorithms used during simulation • Energy minimization and equilibration of

initial structure

• Analysis of a simulation

Protein structures: primary structure

• 20 different amino acids encoded in the DNA

• 3-letter and 1-letter codes

www2.chemistry.msu.edu

Primary structure = amino acid sequence

KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

Lysozyme

• From N- to C-terminus

Protein structures: secondary structure

Secondary structure = 3D fold of local AA segments

Lysozyme:

alpha-helices, beta sheets, connected by loops

• alpha helix

• beta sheet

• Turns, 310-helix,…

Protein structures: tertiary structure

Tertiary structure = 3D fold of one polypeptide chain

Mainly alpha-helical

Protein structures: tertiary structure

Tertiary structure = 3D fold of one polypeptide chain

Mainly beta sheets

Protein structures: tertiary structure

Tertiary structure = 3D fold of one polypeptide chain

OmpX (pdb 2M06)

Protein  structures:  ter-ary  structure

Alpha helices and beta sheets

Protein  structures:  ter-ary  structure

Alpha helices and beta sheets

Protein structures: quaternary structure

Arrangement of multiple folded polypeptides

Example: Haemoglobin• four subunits

Interesting:
Cooperative oxygen binding 


through quaternary transitions

Multiple Time Stepping

H. Grubmüller, H. Heller, A. Windemuth, K. Schulten; Mol. Sim. 6 (1991) 121

1. Taylor expansion

Multipole Methods

Exact for infinite multipole series

O(N2)

i

i j

j

Fast Multipole Method (FMM)

+ arbitrary accuracy

- high order expansions required 
 to achieve moderate accuracy

à O(N)

L. Greengard and V. Rokhlin, J. Comp. Phys. 73 (1987) 325

Fast structure-adapted multipole methods: O(N)

M. Eichinger, H. Grubmüller, H. Heller, P. Tavan, J. Comp. Chem. 18 (1997) 1729

Ewald summationAnother very popular method to efficiently compute Coulomb forces of without simple cutoffs


(applicable for periodic systems)

q

x x x

q q

Idea: Rewrite the charge density as a sum of two terms:

• Quickly varying density: Potential can be computed accurately with cut-offs (“direct space calculation”)

• Slowly varying density: potential can be efficiently computed in reciprocal space using the Fast Fourier Transform (FFT); O(N log(N))

Charge density:

Point charges

Fourier transform of charge densityEwald, Ann. Phys. 64:253-287 (1921)

Simulation system setup 1

• Get PDB structure and check for ‣ missing atoms/groups ‣ inaccuracies (flipped histidine ring) ‣ missing ligands ‣ chemical plausibility ‣ mutations (e.g., to facilitate crystallization) ‣ read the paper!!

• Choose force field ‣ “all-atom” or “united-atom”, e.g. CH2, CH3 as one atom ‣ implicit or explicit hydrogen atoms ‣ polarizable force field required? ‣ QM methods required (chemistry?)

• Add hydrogen atoms to protonable (“titratable”) groups (Histidine!)

Simulation system setup 2

• Choose periodic boundary conditions or not

Role of environment - solvent

explicit or

implicit solvation?

box or droplet? 




Typical: box with periodic boundary conditions,

avoid surface artefacts

periodic boundary conditions and the minimum image convention

Surface (tension) effects?

~xi(t = 0) done!

Simulation system setup 2

• Choose periodic boundary conditions or not • if membrane protein: add lipid membrane atoms • add water molecules • add ions as counter ions (if possible, according to Debye-

Hückel)

b(i)0 ,K(i)b for all bonds

�(j)0 ,K(j)� for all angles

Simulation system setup 3

• Define V(x1,...xN) via force field

‣ bond parameters

‣ angle parameters

‣ dihedrals, extraplanars 


‣ partial charges

‣ Van-der-Waals parameters
(Lennard-Jones potential)




VLJ = 4✏

⇣�r

⌘12�⇣�r

⌘6�

qi for all atoms

�i, ✏i for all atoms

Simulation system setup 4• For frequently reoccurring chemical motifs 


define atom types, e.g.: ‣ hydrogen HC ‣ carbon CH2

• parameter file: list properties of atom 
types and their bonds, angles, ... 


HC q=+0.2 m=1.0 # charge, massCH2 q=-0.4 m=12.0

HC -CH2 K=200 b=1.1 # bondsCH2-CH2 K=500 b=1.5

HC-CH2-HC K=20 118° # anglesHC-CH2-CH2 ...

Simulation system setup 5

‣ Topology file: defines • atoms • bonds • angles • dihedrals etc. of the simulation system 


[ atoms ]; nr type name … 1 HC HA1 
 2 HC HA2 
 3 HC HB1 
 4 HC HB2 
 5 CH2 CA 
 6 CH2 CB

[ bonds ] 1 5 HC-CH2 2 5 HC-CH2 3 6 HC-CH2 4 6 HC-CH2 5 6 CH2-CH2

[ angles ] 1 5 2 HC-CH2-HC 1 5 6 HC-CH2-CH2...

1

25

3

46

Simulation phase - algorithms

‣ Integration of Newton’s equations of motion 


Integrate numerically via the „leapfrog“ scheme:

(equivalent to the Verlet algorithm)

with

Δt ≈ 1fs!

where

~P =N

atomsX

i=0

~pi

~pi0 = ~pi �

miM

~P

Simulation phase - algorithms

‣ Integration of Newton’s equations of motion ‣ Constrain bond lengths (LINCS, SHAKE) 


idea: eliminate fastest vibrations (C-H) to 
 increase the integration time step from 1fs to 2fs 
side-effect: better descriptions of QM vibrations

‣ Remove overall translation (and rotation): 
Avoid drift of the molecule: remove translation (and rotation) of the entire simulation system: 


Remove overall
momentum:

Remove angular 
momentum analogously

Simulation phase - algorithms

‣ Remove overall translation (and rotation): 
Avoid drift of the molecule: remove translation (and rotation) of the entire simulation system:

0 1000 2000 3000 4000 5000

Time (ps)

0

500

1000

1500

2000

Coord

inate

(nm

)

Center of mass

0 1000 2000 3000 4000 5000

Time (ps)

-10000

-8000

Pote

ntia

l (kJ

/mol)

Numerical instability: Accumulation of kinetic energy in to one degree of freedom.
 (Flying ice cube problem)

~vi ~vi

s

1� �t⌧

✓T

T0� 1

◆

T =2

3

1

NkB

NX

i=1

m

2v2i

Simulation phase - algorithms

‣ Choose thermodynamic ensemble 

NVE (microcanonical ensemble)
NVT (canonical ensemble, isochoric): T-coupling 
NPT (canonical ensemble, isobaric): T-coupling and 
 P-coupling

‣ T-coupling, e.g. Berendsen thermostat 
After each step Δt:

‣ P-coupling: analogous, by scaling volume ‣Write out coordinates at some frequency 


𝝉 = coupling time constant


T0 = target temperature

Mimimization/equilibration: 1) Energy minimization

☞ Reduce the steric strain by a moving along the 
steepest descent in

V (~x1, . . . , ~xN )

☞ Notes:

• Protein moves in to local minimum



• Attention: proteins don’t tend towards the local minimum in V(x), but towards the global minimum in the free energy!

☞ Entropy/ensembles are important!

BPTI: Minimization

Mimimization/equilibration: 2) Thermalization

☞ Heat the system to, e.g. 300K by assigning Maxwell-distributed velocities

p(vx

) / e�mv

2x

2kB

T , p(vy

) / · · ·

Trick to avoid distortion of the protein: • assign velocities to to the system• keep protein backbone restrained• equilibrate for ~100ps

Mimimization/equilibration: 3) Equilibration

How long? → Multiple checks:

• Convergence of energy contributions (particularly Coulomb and 
Lennard-Jones) and box dimensions

• Room-mean square deviation (RMSD) from the crystal/NMR structure

RMSD(t) =

✓1

N

XNi=1

[~xi(t)� ~xi(0)]2◆1/2

Typically:

0 1 2 3 4 5 6 7 8 9

Time (ns)

0.00

0.05

0.10

0.15

RM

SD

(n

m)

picosecond jumpconformationalsampling

?

Mimimization/equilibration: 3) Equilibration

Reasons for RMSD increase/drift:

• Fast fluctuations → picosecond jump ☞ OK• slow conformational motions 


→ nanosecond drift ☞ OK 



• Conformational transitions → stairs ☞ OK




• Structural drift due to ☞ NOT OK - bad X-ray structure- inaccurate force field- software bug- …

Mimimization/equilibration: 3) Equilibration

Judgement of RMSD:

• RMSD does not converge ⟹ simulation is not OK.• But: RMSD converges ⇏ simulation is OK.

Better check, e.g., PCA projections (see later lecture)

Flow chart of MD simulation

Get initial positions of atoms
(e.g., from the PDB)

Compute forces using your force field

Update atom positions & velocities
(“integration step”)

Update time step 
t ! t+�t

Repeat up to requested
simulation time

Take care of pressure and temperature

e.g.

109

times

Prepare simulation system
(add hydrogen atoms, water, ions)

Choose force field

Specify simulation parameters
(time step, temperature, …)

Preparation Simulation

Energy minimisation

Set initial velocities

Simulation analysis

Available after simulation:

• Positions:


e.g., T = 10ns, N = 100.000, Δt = 2fs 


☞ 5·106 × 105 × 3 × 4 Byte = 6 TByte !

• Velocities

• Temperature

• Potential energies:

• Anything you can program…

~x1(ti), . . . , ~xN (ti), ti = 0,�t, 2�t, . . . , T

~v1(ti), . . . ,~vN (ti)

T (ti) =1

(3N � 6)kB

NX

i=1

miv2i (ti)

Vbond

(ti), Vangle(ti), Vdih(ti), VCoul(ti), VLJ(ti),

Simulation analysis

Observables that may be interesting: everything that can be 
measured

• Size of atomic fluctuations








Note: ensemble average ⟨⋯⟩ ≠ time average

• Anything that helps to understand the protein function:

- Movie (!), motion of groups

- interaction energies, hydrogen bonds, radial distribution 


functions, transition rates, change in secondary structure 


x̄j = M�1

MX

i=1

~xj(ti)

h(~xj � h~xij)2i ⇡1

M

MX

i=1

⇥~xj(ti)� x̄j

⇤2

BPTI: Molecular Dynamics (300K)

Opening transition of the enzyme ATCase

4 nm

Molecular dynamics simulation of Aquaporin-1