Lecture 9: 2D spectroscopy, N O ESY
HOMEWORK
p(a) = \jt + + ^2z)
HOMEWORK:
iti\ Tm
p(a) = \jt + \^iz + ^2z)
PÍP) =       + - ^2y)
HOMEWORK
p(a) = \jt + ?k(siz + s2z)
-j ^
P(C) = 2^ + 2« (-cll^ly + Sll^lar ~ C2l^2y + S21^2x)
en -+ e-^.!*! cosCQ^i) Sll e-^iti sinCQ^i) c2i -» e-^i cos(Q2*i)      «21    e-^i sin(Q2*i)
HOMEWORK:
p(a) = \jt + \*(S\z + ^2z)
p(b) =       + \k{-J\v - J?2y)
p(c) =       + ^ (-CllAy + SH^lx - C21^2y + S21^2x)
en -> e-^2-1*1 cos(Qiti) an -> e"^1*1 sin(Qiti) c       e-«2,2*i cos(Q2*i)       «21 -> e"^2*1 sin(Q2*i)
p(<j) = \jt +      {-cn^lz + sn^ix - C2l^2z + «21^2x)
HOMEWORK
1 1
Mz relaxes with Ri, Mx,My relax with R2'. Tm = 0.2 s, i?i = 1 s-1, and i?2 = 20 s-1
e-i?2Tm = e-20x0.2 _ e-4 ^ q.02
=^1x5 ^ly: ^2x1 ^2y contributions reduced to 2% « 0
^     g-i^irm = e-lx0.2 _ e-0.2 ^ o.82
^\z^2z contributions survive (82% « 1)
HOMEWORK
HOMEWORK:
A\ = -e_jRiTmcn      A2 = -e~RlTmc2\
HOMEWORK:
A\ = -e_jRiTmcn      A2 = -e~RlTmc2\
p(f) =        + ^!^ly + A2^2y
p(t2) = \*t
+^i(cos(Qit2)^iy - sinCQi^)^)
+^2(C0S(Q2*2)^2y - Sin(Q2*2)^2a;)
MODULATION IN 2D EXPERIMENT
M+=M (7l(/lx + \Tly) + 72(T2x + \T2y))
(M+) = Tr{p(í2)M+}
= JV7M1 (e_jR2.lť2 cos(nií2) - ie-jR2,i*2SJn(Q1Í2)
+ JV7M2 (V^2.2^ COS(fi2*2) - Íe-jR2,2Í2sjn(Q2Í2)
/
y (o;) = A/7/1
^1^2.1
^2^2,2
v
- iA/7^
^2,1 +      - AO5     ä£2 + 0" - ^2)
/    ^i(cu-Qi) ,42(cj-Q2) N
+ (w - Q1)2    ħ 2 + (w - ^2)2
v
MODULATION IN 2D EXPERIMENT
Tm
e-i?i,irme-fi2,i*i cos(Qiti) (e^1*2 cos(Qit2)
e-fil,2rme-ii2,2«l COS(Q2*l) (e^2'2*2 COS(Q2t2)
ie--R2,i*2 sin(^2it2)) + ^-^2,2*2 Sin(Q2i2))
)
8/cBT
■i2i,i-nne-i22,i*i cos(Qiti)
#2,1
8/cBT
e-Ri,2Tme-R2,2ti cos(Q2^i)
#2,1 + (w - ^l)2 #2,2
#2,2 + (w - ^2)2
MODULATION IN 2D EXPERIMENT
Tm
rl2h2B0 .
e-R!,irme-R2,iti cos(Qiti)(e-fi2'lt2 cos(Qit2) e-fli,2Tme-«2,2*i Cos(Q2ti) (e^2-2*2 COS(Q2t2)
je-^2>li2sjn(Qlt2)^ _|_
^-#2,2*2 Sin(Q2£2))
)
Sty = Mf^0e
SkBT l2h2B0
■i2l,l-nne-i22,l*l cOS(fiiti)
«2.1
8kBT
e-Ri,2Tme-R2,2ti cos(Q2^i)
«2,1 + (W - ^l)2 «2,2
«2,2 + (W - ^2)2
MODULATION IN 2D EXPERIMENT
Tm
e-Ri,irme-fi2,i*i cos(Qiti) (e^1*2 cos(Qit2) e-fii,2Tme-fi2,2*i cos(Q2ti) (e^2-2*2 cos(Q2i2)
ie--R2,i*2Sjn(n1t2)) + ie-fl2,2*2 sin(Q2*2))
)
SkBT
i2h2B0
■i2l,l-nne-i22,l*l cOS(fiiti)
«2.1
8/cBT
e-i?i,2Tme-#2,2ti Cos(Q2*i)
«2,1 + (w - ^l)2 «2,2
«2,2 + (w - ^2)2
Transmitter on
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ti
7"m
t2
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ti
tm
*2
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ti
tm
*2
*2
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ti
tm
*2
*2
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ti
tm
*2
*2
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ti
tm
*2
*2
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ti
7"m
t2
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t2
ti
7"m
t2
f:
f:
07 ( H7) 3 04 06
C5—C4 N7 /(
// _ \ H8_f ^05^
H6-C6    I      N3H3 9°       w NNim
\ / N3 \
02 N2
H22
1 2  3 4  5 6 C T G A A T H H H
H H H
G A C T T A
6' 5' 4' 3' 2' 1'
G6' G3       T3' T2 T2' T6
f2
Relaxation due to dipolar coupling
Bloch-Wangsness-Redfield theory applicable dipolar coupling: different Hamiltonian, large effect
dipolar b =
47rr3
dA(!f1") = -^(6J(w0)i)+2J(wo,i - w0,2) + 127(^0,1 + ^o,2))A(Ml2)
dt 8
+ — (2J(w0,l - w0)2) - 12J(w0,l + W0,2))A(M2^> = -ÄalA<Miz) - ßxA(M2z)
dA(jf2") = -^(6J(o;0,2)+2J(w0)l - "0,2) + 12J(w0,i + ^0;2))A(M2,> dt 8
b2
+ — (2J(w0,i - wo,2) - 12J(w0,l + w0,2))A(Mu) = -fia2A(M2z) - ßxA(Mlz)
= -^(4J(0) + 3J(u;o,l)+6J(cuo,2) dt 8
+ J(wq,1 -^0,2) + 6J(w0,l + "0,2))(^1+)
= - (#0,1 + |Äal) <Mi+> = -Ä2,i(M1+)
(M+> = a/VK
Ai (e-^1*2 cos(Qií2) - ie~-R2,ií2sin(Qií2)) + X2    (e-ß2,2i2 Cos(Q2í2) - ie-^2'2*2 sin(Q2í2))
)
sfty = A/7 Ä2,1
+A/7/1 ^2
#2,2
R\ 2 + (w - Q2)2
NOESY
ÖA{^lz) = RalA(Mlz) + RxMM2z) dA<d^) = Ra2MM2z) + ÄxA(Mis)
NOESY
ÖA{^lz) = RaA(Mlz) + #xA(M2z) ót
NOESY
ÖA{^2z) = RaA{M2z) + ÄxA(Mu>
HOMEWORK
Sections 9.4, 9.5.1, 9.5.2
NOESY
dA(HM2z) = R^{M2z) + RxMMiz) ót
Ä! = -((1 - C)e_i?2.líl cos(Qiíi) + Ce_i?2>2Í1 cos(Q2íi))e"(i?a+i?x)Tm
A2 = -((1 - C)e_i?2.2Í1 cos(Q2íi) + Ce"^2'1*1 cos(Qiíi))e-(i?a+i?x)Tm
1 2 3 4 5 6 C T G A A T
H H-H G A C T T A 6' 5' 4' 3' 2' 1'
G6' G3       T3' T2 T2' T6
f2
10 kDa
protein:
0.5H
1-oH
E 1.5H
CL CL
2.0H
2.5H
3.0J
9.0
8.5 8.0 _i_i_._i_i_j_i—_i_
o íj   ó K>
7.5
7.0
J_I_I_I_I_I_I_I_L
-0 •
)0 §
o •
0
0 o
0
9.0
6.5
J_I_I_L.
6.5
h0.5
1.0
1.5
h2.0
h2.5
h3.0
C02 - 1H (ppm)
NOESY CROSS-PEAK HEIGHT Yt
If rm < 1/Äx :
Vmax oc -- (eRxTm - e~RxTmj e~RaTm ss -R
^o\2i4^(Jio)_6J(2uJo))Tm
xt m
\8tt/
r
J(0) = -rr       J(2w0) = c
Slow motions, long tq.
2u0tc > 1 => J(0) = §rc > 6 J(2^0) « 0     ymax > 0
Fast motions, short tq.
2ujQTC < 1     J(0) = |rc < 6 J(2w0) « 6 x |rc     ymax < 0
NOESY CROSS-PEAK HEIGHT Yt
If rm < 1/Äx :
Vmax oc -- (eRxTm - e_jRxTm) e~RaTm « -R
r ^-( J(0) - 6 J(2cü0))rm
xT m
\8tt/
r
•J(O) = -rC        ^(2cu0) = C
5 5 1 + (2w0rC)2
Slow motions:
2üJ0tc > 1 => J(0) = |rc > 6 J(2w0) ~ 0 =>■ Fmax > 0
Fast motions:
2cj0tc < 1     j(0) = |rc < 6 j(2cu0) « 6 x |rc =j> Fmax < 0
NOESY CROSS-PEAK HEIGHT Yt
Yr
max
Mmax,ref
v r /
r
rref
-Mnax.ref
max
Reference protons distance
geminal in methylene H-C-H       0.17 nm
vicinal in aromatic ring H-C=C-H 0.25 nm meta in aromatic ring   H-C=CH-C-H 0.42 nm