Computer practical room

Nov. 10 & Nov. 24: Physics Cip pool (ground floor)

Starting Dec. 1: SR1 (A.01.101)

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 structureTertiary structure = 3D fold of one polypeptide chain

Mainly alpha-helical

Protein structures: tertiary structureTertiary structure = 3D fold of one polypeptide chain

Mainly beta sheets

Protein structures: tertiary structureTertiary structure = 3D fold of one polypeptide chain

OmpX (pdb 2M06)

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

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?

box

or

droplet?

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






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

-8000P

ote

ntia

l (kJ

/mol)

Numerical instability: Accumulation of kinetic energy in to one degree of freedom.

~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

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)