81 1 3d structure calculation. structure calculation in general some form of restrained molecular...

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3D Structure calculation

Structure Calculation In general some form of restrained Molecular

Dynamics (MD) simulation is used to obtain a set of low energy structures that satisfy the NMR restraints.

Procedure:• Create a starting structure from sequence• Optimization of the structure• MD calculation with restraints from NMR• Repeat this several times• Selection of 'final' structures

Starting structure and optimization

• The amino acid sequence of the protein is used by the user- interface (Builder) of the modelling program to create an 'extended' starting structure.

• Optimization is then done by Energy Minimization (Molecular Mechanics).

Optimized starting structureStarting structure: extended chain, often in a box of water molecules

Molecular Dynamics Simulation

A Molecular Dynamics Simulation is a computer calculation of the movement of the atoms in a molecule by solving Newton's equation of motion for all atoms i:

(mi mass, ri position, Fi force)

ii

i dt

dm F

r

2

2

The force Fi is calculated from tabulated potential energy terms V (the force field) and the current position ri:

The empirical potential energy function V contains terms like:

The Force Field

ticelectrostaWaalsdervandihedralanglebondlengthbond VVVVVV

ii

V

rF

Potential Energy Function

Potential Energy Function for Bond Length

l

l0 l

E

Bond stretching(vibrational motion)

The NMR Restraints

In addition to potentials of the force field:

• Non-physical restraints for distances

and dihedral angles (and others) from NMR

• These are extra terms in the potential

energy function V Restrained MD

NOE distance restraints

Restraints for upper (uij) and lower (lij) bounds for the distance rij:

ijijijij

ijijij

ijijijijNOE

lrrlk

url

ururkV

if)(

if0

if)(

2

2

Molecular Dynamics

Typical time-scales for molecular motions

Time scale Amplitude Description

short femto to pico 10-15 - 10-12s

0.001 - 0.1 Å - bond stretching, angle bending - dihedral motion

medium pico to nano 10-12 - 10-9s

0.1 - 10 Å - unhindered surface side chain motion - loop motion, collective motion

long nano to micro second 10-9 - 10-6s

1 - 100 Å - folding in small peptides - helix coil transition

micro to seconds 10-6 - 10-1s

10 - 100 Å - protein folding

Local or Global Energy minimum

Structural landscape contains peaks and valleys.

Energy Minimization protocol always moves “down hill”. Difficult to cross over local maxima to get to global minimum.

Therefore: Simulated Annealing

• Often used with Restrained MD• Potentials are 'down-scaled' in the beginning• Higher degree of freedom ('sampling a bigger conformational space')• In later steps the potentials are slowly brought to their final values. This is like first heating up the molecule and thencooling it down in small steps.

Family of Structures

Usually a large number of calculations is done in parallel resulting in a family of structures, from which an average structure can be calulated or the one with the minimum energy selected.

Family of structures of the protein crambinFamily of structures of the protein crambin

Final 3D structures of biomolecules

Ribbon presentation

Summary: Restrained Molecular Dynamics

• Choose a force-field • Add constraints from NMR data• Starting coordinates of all atoms (starting structure)• Starting velocities of all atoms ('random seed numbers', Maxwell Boltzmann distribution)• Solve Newton's classical equation of motion for very small steps (few femto-seconds)• Calculate new coordinates, forces and velocities• Repeat the last two steps to find structure with lowest energy

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Increasing the NMR size limit

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Advances in hardware & techniques• Higher magnetic field strength

- Increased resolution & sensitivity

- Maximum now 1000 MHz (1 GHz)

• Cryoprobes

- Cooling of probe coil with He gas (~20 K)

- Reduces thermal noise generated by electric circuits

- Increase of sensitivity by factor of 3-4

• Dynamic nuclear polarization (DNP)

- Transferring spin polarization from electrons to nuclei

- Requires saturation of electron spins by Gyrotron irradiation

- So far only for solid-state NMR

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Protein deuteration• Reduce 1H-1H dipolar interactions

- γH/γD ~ 1/6.5

- Longer T2 → sharper lines 30 kDa

15N 15N, 90% 2H

Garret DS et al. Biochemistry (1997)

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TROSY• Transverse optimized

spectroscopy

- Lines from 1H-15N multiplet have differential relaxation

➡ Interference between dipole-dipole and CSA relaxation

- TROSY only selects the narrow, slowly relaxing line

- TROSY effect more pronounced at high magnetic field-strength

➡ CSA is field-dependent

40 kDa @ 750 MHz

Pervushin K et al. PNAS (1997)

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Relaxation & Dynamics

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NMR time scales

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Local fluctuating magnetic fields• Bloc(t) = Bloc[iso] + Bloc(t)[aniso]

- Isotropic part is not time dependent

➡ chemical shift

➡ J-coupling

- Only the anisotropic part is time dependent

➡ chemical shift anisotropy (CSA)

➡ dipolar interaction (DD)

r

B0

anisotropic

interactions

13C

CSA dipole-dipole

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Components of the local field• Bloc(t) xy components

- Transverse fluctuating fields

- Non-adiabatic: exchange of energy between the spin-system and the lattice [environment]

α

βnon-adiabatic

transitions

T1 relaxation

transitions between states restore Boltzman equilibrium

α

β

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Components of the local field• Bloc(t) z component

- Longitudinal fluctuating fields

- Adiabatic: no exchange of energy between the spin-system and the lattice

- Effective field along z-axis varies

➡ frequency ω0 varies

adiabatic variations of ω0

B0

Bloc(t)•ez

z-component: frequency ω0 varies due to local changes in B0

xy-component: transitions between states reduce phase coherence

T2 relaxation

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Spectral density function Frequencies of the random fluctuating fields

- Spectral density function J(ω) is the Fourier transform of the correlation function C(τ). It gives the probability of finding a component of the fluctuation at frequency ω.

- The component of J(ω) at the Larmor frequency ω0 can induce T1 relaxation transitions. J(0) is important for T2.J(ω)

ω

5 ns10 ns20 ns

)1(

2)(

22c

cJ

τc

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Molecular tumbling and relaxation

fast tumblingfast tumblingsmall moleculesmall molecule

slow tumblingslow tumblinglarge proteinlarge protein

Since the integral of J(ω) over all frequencies is constant, slow tumbling (large molecule) gives more contributions at low frequencies, fast tumbling (small molecule) more at higher frequencies.

J(ω)

)1(

2)(

22c

cJ

Logarithmic scale

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Molecular tumbling and relaxation

slow tumblinglarge protein

Inverse line widthT2 ~ 1/Δ

fast tumblingsmall molecule

c [s] correlation timethgiewralucelom

Proteins >10kDa

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Effects of relaxation on protein NMR spectraslower tumbling in solution fast decay of NMR signal broad lines

larger number of signals more signal overlap

c 4 nsMW 8 kDa

8 ns16 kDa

12 ns24 kDa

25 ns50 kDa

linewidth Δν1/2 = 1/πT2

78910 ppm78910 ppm78910 ppm78910 ppm

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Protein backbone dynamics• 15N relaxation to describe ps-ns dynamics

- R1: longitudinal relaxation rate

- R2: transversal relaxation rate

- hetero-nuclear NOE: {1H}-15N

• Measured as a 2D 1H-15N spectrum

- R1,R2: Repeat experiment several times with increasing relaxation-delay

- Fit the signal intensity as a function of the relaxation delay

➡ I0. exp(-Rt)

- {1H}-15N NOE: Intensity ratio between saturated and non-saturated experiment

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15N relaxation ratesR2

2HzNy

Nx

15N chemical shift evolution

CPMG

relaxation

delay

-Nz15N chemical shift evolution

Nx

Relaxation delay

R1

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Relaxation rates

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NMR time scales

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Conformational exchange

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Conformational exchange

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Measuring kex with CPMG• Carr-Purcell-Meiboom-Gill

- Refocussing the 15N chemical shift when measuring the 15N R2 relaxation rate

• Relaxation dispersion

- Determine the R2,eff as a function of CPMG frequency (i.e. frequency of 180° pulses)

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Relaxation dispersionCan provide information about “invisible” state

- Fitting of dispersion curves at more than one magnetic field

➡ Time-scale of the interconversion (kex=kA+kB)

➡ Populations of the two states (pA, pB)

➡ Chemical shift difference (Δω = ωA-ωB)

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• Wide range of time scales

• Fluctuating magnetic fields

• Correlation function, spectral density function

• rotational correlation time (ns)

• fast time scale (ps-ns): flexibility (fast backbone motions) from 15N relaxation and 1H-15N NOE

• slow time scale (μs-ms): conformational exchange from relaxation dispersion CPMG

Key concepts relaxation