Connecting NMR data to biomolecular structure and dynamics

Transcription

1 Connecting NMR data to biomolecular structure and dynamics David A. Case Chem 538, Spring, 2014

2 Basics of NMR All nuclei are characterized by a spin quantum number I, which can be 0, 1/2, 1, 3/2, 2... Even atomic mass and number: I=0 ( 12 C, 16 O, 32 S) Even atomic mass, odd atomic number: I = integer ( 14 N, 2 H) Odd atomic mass: I=half integer ( 1 H, 15 N, 13 C, 31 P)

3 Basics of NMR./,0($1*02(0&3',!"# $"# %$&' B o =0 B o >0 $()*(+#,(-./01/(2/,34,567 $" %$8(.594:./(;,:16)+7 %$ z y x

4 Energy gaps are very small./,0($1*02(0&3', 4/(5$3'6'3$5/,$/$7088'*'2)$&+&93/)0+2$:!;<$/27$)5'$7088'*'2('$=')>''2$)5'$)>+$0,$*'3/)'7$ "! #$#% &"'( 4$8+*$ C D$/)$EFF$"D@$:.F$G$HIJ$K;$0,$LIM$N$CF OJ $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

5 ./'$'0'()*+1234')5($,&'()*61 Comparison to other spectroscopies -rays X-rays ultraviolet visible infrared microwave radiofrequency Mossbauer electronic vibrational rotational NMR H 19 F 31 P 13 C /MHz 10 aldehydic 8 aromatic 6 olefinic 4 acetylenic 2 aliphatic 0 /ppm /Hz

6 Chemical shift dispersion!"#$%&'()*+,(+&-./'$0/'12(34$%/25) 1')/-4 &*+)+=, 312>'$&*+)+=, 3*+13)2($*2=? $&*+)+=, 1')/-4'=' &*+)+=, 678$"9:$ ; 9$,&'()*<1$+5$3$,1344$&*+)'2=$ shielding frequency magnetic field

7 Expansion to two (or more) dimensions Protein NMR 2D NOESY

8 Chemical shifts: Environmental effects for protons Nearby groups with magnetic susceptibilities ring currents peptide group contributions paramagnetic metal sites Electric fields bond polarizability models important in hydrogen bonds predicted secondary shift methyl shifts in proteins van der Waals (close contact) deshieldings hydrogen bonds general solvation effects observed secondary shift Ösapay & Case, JACS 113, 9436 (1991) local ( through-bond ) contributions Buckingham, Schaefer, Schneider, JCP 32: 1227, 1960

9 Chemical shielding and magnetic susceptibilities P H 0 y Heff Heff M j x

10 Spin transfer between nearby 2D NOESY spins

11 NMR 101 ( ) 2 E σ ij = µ i B j σ τ cs 2 r 6

12 Fragment B3 of protein G

13 N H correlation functions for GB3 effective decay time (6τ) 1 = e T D e But beware: statistical uncertainty in C(τ) is roughly ( ) 2τ 1/2 [1 C(τ)] T

14 Now consider intenal motions phe 30 ala 48 <P 2 [u(0).u(t)]> leu gly time, ns Hall & Fushman, JBNMR, 27, 261 (2003)

15 Converting distances to structures

16 Metric Matrix Distance Geometry To describe a molecule in terms of the distances between atoms,there are many constraints on the distances, since for N atoms there are N(N 1)/2distances but only 3N coordinates. General considerations for the conditions required to "embed" a set of interatomic distances into a realizable three-dimensional object forms the subject of distance geometry. The basic approach starts from the metric matrix that contains the scalar products of the vectors x i that give the positions of the atoms: g ij x i x j (1) These matrix elements can be expressed in terms of the distances d ij : g ij = 2(d 2 i0 + d 2 j0 d 2 ij ) (2) If the origin ("0") is chosen at the centroid of the atoms, then it can be shown that distances from this point can be computed from the interatomic distances alone. A fundamental theorem of distance geometry states that a set of distances can correspond to a three-dimensional object only if the metric matrix g is rank three, i.e. if it has three positive and N-3 zero eigenvalues. This may be made plausible by thinking of the eigenanalysis as a principal component analysis: all of the distance properties of the molecule should be describable in terms of three "components," which would be the x, y and z coordinates.

17 Metric Matrix Distance Geometry (part 2) If we denote the eigenvector matrix as w and the eigenvalues λ, the metric matrix can be written in two ways: 3 3 g ij = x ik x jk = w ik w jk λ k (3) k=1 k=1 The first equality follows from the definition of the metric tensor, Eq. (1); the upper limit of three in the second summation reflects the fact that a rank three matrix has only three non-zero eigenvalues. Eq. (3) then provides an expression for the coordinates x i in terms of the eigenvalues and eigenvectors of the metric matrix: x ik = λ 1/2 k w ik (4)

18 Using imprecise distances If the input distances are not exact, then in general the metric matrix will have more than three non-zero eigenvalues, but an approximate scheme can be made by using Eq. (4) with the three largest eigenvalues. Since information is lost by discarding the remaining eigenvectors, the resulting distances will not agree with the input distances, but will approximate them in a certain optimal fashion. If one only knows a distance range, then some choice of distance to be used must be made. Considerable attention has been paid recently to improving the performance of distance geometry by examining the ways in which the bounds are "smoothed" and by which distances are selected between the bounds. Triangle bound inequalities can improve consistency among the bounds, and NAB implements the "random pairwise metrization" algorithm developed by Jay Ponder. Methods like these are important especially for underconstrained problems, where a goal is to generate a reasonably random distribution of acceptable structures, and the difference between individual members of the ensemble may be quite large. An alternative procedure, which we call "random embedding", implements the procedure of degroot et al. for satisfying distance constraints. This does not use the embedding idea discussed above, but rather randomly corrects individual distances, ignoring all couplings between distances.

19 Molecular dynamics-based structure refinement

20 474 Example: cyclophilin bound to cyclosporin A A Spitzfaden, Braun, Wider, Widmer & Wüthrich, J. Biomol. NMR 4, 463

21 NMR 102 ( ) 2 E J ij = µ i µ j R ex = p A p B ( ω) 2 τ ex

22 Three-bond backbone couplings in proteins H Hα H Cβ H C 4 2 C C Case, Scheurer, Brüschweiler, JACS 122:10390, C Cβ C Hα

23 J-couplings across Watson-Crick hydrogen bonds h J(N N) h J(N N) or h J(H N), Hz solid: AT dashed: GC 1h J(H N) H...N distance, Angstroms

24 Getting dynamics from NMR NMR Spectroscopy Protein dynamics

25 Chemical exchange A k k B CH 3 CH 3 N N O k k CH 3 CH 3 N N O

26 Effects of chemical exchange on NMR lineshapes NMR Spectroscopy - Protein dynamics Chemical exchange

27 NMR Spectroscopy - Protein dynamics Hydrogen/deuterium Hydrogen/Deuterium exchange (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

28 Protection factors Hydrogen/Deuterium illustrate slow protein (H/D) exchange dynamics Significant heterogeneity in the magnitude of the protection factors A large number of states define the native-state ensemble

29 NMR Spectroscopy - Protein dynamics There is anotherhydrogen/deuterium kinetic regime (H/D) for H/D exchange exchange kop kch kcl kobs=kop kch/(kop + kcl + kch) EX1: kcl<<kch kobs=kop

30 Hydrogen/Deuterium (H/D) exchange Getting dynamics from NMR N-H (closed) k cl N-H (open) k ch D 2 O k op [k obs =k op ] [k obs =(k op /k cl ) k ch )] k ch (s -1 )