Connecting NMR data to biomolecular structure anddynamics

David A. Case

Chem 538, Spring, 2014

Basics of NMR

All nuclei are characterized by a spin quantum number I, whichcan be 0, 1/2, 1, 3/2, 2 ....

Even atomic mass and number: I=0 (12C, 16O, 32S)

Even atomic mass, odd atomic number: I = integer (14N, 2H)

Odd atomic mass: I=half integer (1H, 15N, 13C, 31P)

Basics of NMR!"#$%&'()*+,(+&-

./,0($1*02(0&3',

Bo=0

Bo>0

!"# $"# %$&'

$()*(+#,(-./01/(2/,34,567

$" %$8(.594:./(;,:16)+7

%$ z

y

x

Energy gaps are very small!"#$%&'()*+,(+&-

./,0($1*02(0&3',

4/(5$3'6'3$5/,$/$7088'*'2)$&+&93/)0+2$:!;+$0,$*'3/)'7$

)+$)5'$'2'*?-$7088'*'2('$=-$)5'$.+3)@AA/2$70,)*0=9)0+2B

! "! #$#% &"'(

4$8+*$CD$/)$EFF$"D@$:.F$G$HIJ$K;$0,$LIM$N$CFOJ$P(/3QA+3

$$$$! "! #$CIFFFFRE

$K5'$,9*&39,$&+&93/)0+2$0,$,A/33$:',&'(0/33-$>5'2$(+A&/*'7$)+$ST$+*$U#;I

K5/)$*'27'*,$!"#$/$*/)5'*$02,'2,0)06'$)'(520V9'W

Comparison to other spectroscopies

!"#$%&'()*+,(+&-./'$'0'()*+1234')5($,&'()*61

1022

1020

1018

1016

1014

1012

1010

108

106

-rays

X-rays

Mossbauer

electronic ultraviolet

visible

infrared

microwave

radiofrequency

vibrational

rotational

NMR

600 500 400

300

200

100

1H 19F

31P

13C

10 8

6

4

2

0

aldehydic

aromatic

olefinic

acetylenic

aliphatic

/Hz

/MHz

/ppm

Chemical shift dispersion!"#$%&'()*+,(+&-

./'$0/'12(34$%/25)

678$"9:$;9$,&'()*

Expansion to two (or more) dimensions

NMR Spectroscopy

Protein NMR

2D NOESY

Chemical shifts: Environmental effects for protons

Nearby groups with magneticsusceptibilities

ring currentspeptide group contributionsparamagnetic metal sites

Electric fields

bond polarizability modelsimportant in hydrogen bonds

van der Waals (close contact)deshieldings

hydrogen bondsgeneral solvation effects

local (“through-bond”) contributions

−3 −2 −1 0 1

observed secondary shift

−3

−2

−1

0

1

pre

dic

ted

se

co

nd

ary

sh

ift

methyl shifts in proteins

Ösapay & Case, JACS113, 9436 (1991)

Buckingham, Schaefer, Schneider, JCP 32: 1227, 1960

Chemical shielding and magnetic susceptibilities

y

H

x

j

H

M

H0

effeff

P

M = χH

Spin transfer between nearby spins

NMR Spectroscopy

Protein NMR

2D NOESY

NMR 101

σij =(

∂ 2E∂ µi∂Bj

)

σ ∼ τcS2

r6

Fragment B3 of protein G

N–H correlation functions for GB3

effective decay time(6τ)−1 = eT ·D ·eBut beware: statisticaluncertainty in C(τ) isroughly(

2τT

)1/2[1−C(τ)]

Now consider intenal motions

0 2 4 6 8 10time, ns

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

<P

2[u(

0).u

(t)]

>

phe 30

ala 48

leu 12

gly 41

Hall & Fushman, JBNMR, 27, 261(2003)

Converting distances to structures

Metric Matrix Distance Geometry

To describe a molecule in terms of the distances between atoms,there are manyconstraints on the distances, since for N atoms there are N(N −1)/2distances butonly 3N coordinates. General considerations for the conditions required to "embed" aset of interatomic distances into a realizable three-dimensional object forms thesubject of distance geometry. The basic approach starts from the metric matrix thatcontains the scalar products of the vectors xi that give the positions of the atoms:

gij ≡ xi ·xj (1)

These matrix elements can be expressed in terms of the distances dij :

gij = 2(d2i0 +d

2j0 −d2ij ) (2)

If the origin ("0") is chosen at the centroid of the atoms, then it can be shown thatdistances from this point can be computed from the interatomic distances alone. Afundamental theorem of distance geometry states that a set of distances cancorrespond to a three-dimensional object only if the metric matrix g is rank three, i.e. ifit has three positive and N-3 zero eigenvalues. This may be made plausible bythinking of the eigenanalysis as a principal component analysis: all of the distanceproperties of the molecule should be describable in terms of three "components,"which would be the x , y and z coordinates.

Metric Matrix Distance Geometry (part 2)

If we denote the eigenvector matrix as w and the eigenvalues λ , the metric matrix canbe written in two ways:

gij =3

∑k=1

xik xjk =3

∑k=1

wik wjk λk (3)

The first equality follows from the definition of the metric tensor, Eq. (1); the upperlimit of three in the second summation reflects the fact that a rank three matrix hasonly three non-zero eigenvalues. Eq. (3) then provides an expression for thecoordinates xi in terms of the eigenvalues and eigenvectors of the metric matrix:

xik = λ1/2k wik (4)

Using imprecise distances

If the input distances are not exact, then in general the metric matrix will have more than threenon-zero eigenvalues, but an approximate scheme can be made by using Eq. (4) with the threelargest eigenvalues. Since information is lost by discarding the remaining eigenvectors, theresulting distances will not agree with the input distances, but will approximate them in a certainoptimal fashion. If one only knows a distance range, then some choice of distance to be usedmust be made.

Considerable attention has been paid recently to improving the performance of distancegeometry by examining the ways in which the bounds are "smoothed" and by which distancesare selected between the bounds. Triangle bound inequalities can improve consistency amongthe bounds, and NAB implements the "random pairwise metrization" algorithm developed by JayPonder. Methods like these are important especially for underconstrained problems, where agoal is to generate a reasonably random distribution of acceptable structures, and the differencebetween individual members of the ensemble may be quite large.

An alternative procedure, which we call "random embedding", implements the procedure ofdeGroot et al. for satisfying distance constraints. This does not use the embedding ideadiscussed above, but rather randomly corrects individual distances, ignoring all couplingsbetween distances.

Molecular dynamics-based structure refinement

Example: cyclophilin bound to cyclosporin A474

A

B 8 0 ~ 80

70 70 .%.

"~:. k i ~ , J "

G

Spitzfaden, Braun, Wider, Widmer & Wüthrich, J. Biomol. NMR 4, 463(1994)

NMR 102

Jij =

(∂ 2E

∂ µi∂ µj

)

Rex = pApB(∆ω)2τex

Three-bond backbone couplings in proteins

−180 −60 60 180

0

2

4

−180 −60 60 180

0

2

4

6

−180 −60 60 1800

2

4

6

8

10

12

−180 −60 60 1800

2

4

6

8

−180 −60 60 180

0

2

4

−180 −60 60 180

0

2

4

6

H−Hα H−Cβ

H−C’ C’−C’

C’−Cβ C’−Hα

Case, Scheurer,Brüschweiler, JACS122:10390, 2000

J-couplings across Watson-Crick hydrogen bonds

1.6 1.7 1.8 1.9 2.0 2.1

H...N distance, Angstroms

0

2

4

6

8

10

hJ(N

−N

) o

r hJ(H

−N

), H

z

solid: AT

dashed: GC

2hJ(N−N)

1hJ(H−N)

Getting dynamics from NMRNMR Spectroscopy

Protein dynamics

“Chemical exchange”

NMR Spectroscopy - Protein dynamics

Chemical exchange

A B k

k

N

CH3

CH3 O

N k

k N

CH3

CH3

O

N

Average time between jumps is !=1/k !=!/k

Effects of chemical exchange on NMR lineshapesNMR Spectroscopy - Protein dynamicsChemical exchange

Hydrogen/deuterium exchangeNMR Spectroscopy - Protein dynamics

Hydrogen/Deuterium (H/D) exchange

kop kch kcl

kobs=kop kch/(kop + kcl + kch)

EX2: kcl>>kch kobs=kop kch/(kcl) = Kop kch

Kop is referred to as the protection factor, P

!Gop = –RTlnKop

Protection factors illustrate slow protein dynamics

NMR Spectroscopy - Protein dynamics

Significant heterogeneity in the magnitude of the protection factors

Hydrogen/Deuterium (H/D) exchange

A large number of states define the native-state ensemble

There is another kinetic regime for H/D exchangeNMR Spectroscopy - Protein dynamics

Hydrogen/Deuterium (H/D) exchange

kop kch kcl

kobs=kop kch/(kop + kcl + kch)

EX1: kcl

Getting dynamics from NMR

NMR Spectroscopy - Protein dynamics

Hydrogen/Deuterium (H/D) exchange

kop

kch (s-1)

N-H (closed)

kcl

N-H (open) kch

D2O

[kobs=kop]

[kobs=(kop/kcl) kch)]