Supporting Information:
Measurement of the signs of methyl 13C chemical shift differences between
interconverting ground and excited protein states by R: an application to
B-crystallin
Andrew J. Baldwin and Lewis E. Kay
Numerical evaluation of the evolution of magnetization during an off-resonance spinlock
kGE


 E , where the populations of the ground
Consider a two-site exchanging system, G 
kEG
(G) and excited (E) states are given by pG and pE, as described in the text. The time evolution
of magnetization that is ‘locked’ along an off-resonance spin-lock field can be described by




d
dt 




I XG   R2G  kGE
 
IYG  


G 
IZ  
0

E 

IX
kGE
 
E 

IY
0


E

0
I Z  

0
kEG
0
0
R2G  kGE
1
0
kEG
0
1
R1  kGE
0
0
kEG
0
R  kEG
  
0
kGE
0
  
R  kEG
1
0
kGE
0
1
R1  kEG
0
E
2
E
2









 

I XG 

IYG 

I ZG 
I XE 

IYE 

I ZE 
[S1]
where I Gj , I jE are the j  {X,Y,Z} magnetization components for G and E, respectively, R2G
and R2E are the intrinsic transverse relaxation rates, 1 (rad/sec) is the strength of the spinlock field applied along the X axis, =G-SL, with G, SL the positions of the ground state
resonance and the applied spin-lock, respectively and  = E – G. The longitudinal
1
relaxation rates for corresponding nuclei in G and E are assumed to be equal. Extra terms that
reflect equilibrium Z-magnetization that would normally be included (Allard et al., 1998)
have been set to 0 in these expressions since experiments are recorded in a manner whereby
the spin-locked magnetization decays to zero (Sklenar et al., 1987). Immediately prior to the
application of the spin-lock field, before the first ‘red’ pulse of Figure 1c, it is assumed that
only Z-magnetization for both G and E is present, in the fractional amounts given by pG and
pE, respectively. Note that the delay of duration eq following point b in Figure 1c allows
equilibration of the populations of both exchanging states so that
I  (0, 0,1  pE , 0, 0, pE ) +
[S2]
where the superscript ‘+’ denotes transpose so that I is a column vector (corresponding to the
vector in Eq. [S1]). Application of a  pulse (first red pulse of Figure 1c), tan = 1/, along
the Y axis rotates the vector given in Eq [S2]
I (0)  ((1  pE )sin , 0,(1  pE )cos , pE sin , 0, pE cos ) +
[S3]
so that the magnetization is ‘placed’ along the spin-lock axis. Subsequently, the effects of
relaxation can be accounted for by solving Eq. [S1] numerically
I (Trelax )  exp(MTrelax ) I (0)
[S4]
where M is the 6x6 matrix given in Eq. [S1]. Following the spin-lock period magnetization is
rotated by a  pulse along –Y and only the Z-component of the ground state magnetization is
retained. Thus,
I G  R( ) I G (Trelax )
 cos
and R( )   0

 sin
[S5]
0  sin
1
0
0 cos




[S6]
2
where I G is the magnetization vector for G (i.e., a 3x1 vector, containing the first 3 elements
of I (Trelax ) ) and R() is a rotation matrix that rotates magnetization back to the Z-axis after
the spin-lock.
Eq. [S1] can be used to evaluate the optimal experimental values of |  | and 1 for
each methyl group for which the sign of C is sought. This is achieved by maximizing the
difference




 | exp R1  (|  |)Trelax  exp R1  ( |  |)Trelax |
[S7]
as a function of |  | and 1 where the Z-component of the vector IG,
 (Trelax ) is proportional
to exp(R1  (|  |)Trelax ) . This is calculated by setting  |  |  |  ex | ,  = || in Eq.
[S1], with tan = 1/|  | and    O   SL . The value of |ex| can be calculated from Eq.
[3] of the text or as described below using {|C|,kex,pE,R2} obtained from
13
C methyl RD
CPMG studies. Note that exp(R1 ( |  |)Trelax ) is calculated as the Z-component of
IG,
 (Trelax ) obtained by substitution of    |  |  |  ex | and  = || in Eq. [S1], with
tan = -1/|  | (i.e.,  in Eqs. [S3],[S5],[S6] replaced by –). Values of R2G were obtained
from the plateau values in CPMG dispersion curves, while R1 was set to 2.5 s-1. Typically R2E
is set to R2G unless estimates of R2E are obtained from spin-state selective experiments
(Hansen et al., 2009). For |C| > |Cmin| optimal values are relatively insensitive to R2.
All other values used in the calculations are those fitted from dispersion profiles.
Numerical evaluation of ex=2ex
The equations describing the evolution of transverse magnetization can be written compactly
as
3
G
d  I
 E
dt  I 
  k  RG
GE
2
 
 
kGE
kEG
i  R2E  kEG
  IG

 E
  I 



[S8]
where I+ = IX + iIY. In Eq. [S8] it has been assumed (without loss in generality) that the spin
in the ground state is on resonance. The imaginary part of the eigenvalue for G gives the
exchange induced shift of the ground state resonance position, ex.
4
Table S1 C values measured directly from samples of apo- and ligand-bound Abp1p SH3,
5oC (Baldwin et al., 2009) along with optimized |  / (2 ) | and 1 calculated as described in
the text (kex=160 s-1, pE =7.5%). The signs obtained from the R1 and the H(S/M)QC
measurements are indicated, together with the predicted value of  (Eq. [7] of the text, Trelax
= 40 ms).
 (%)
|  / (2 ) |
Leu491
Val321
Val322
Leu382
Val551
Leu182
Leu381
Leu492
Leu181
Val552
Ile261
Leu571
Leu411
Val212
c (direct) (ppm)
+2.33
+1.44
-1.44
-0.34
-0.41
-0.36
-0.34
+0.28
+0.27
-0.27
-0.10
-0.06
+0.05
-0.04
(Hz)
446
267
267
80
62
53
48
39
37
37
16
16
4
4
1 (Hz)
99.5
79.5
79.5
42.9
37.6
34.3
32.2
28.6
28.0
27.6
17.6
18.9
6.7
5.3
Sign (R1)
+
+
+
+
-
Sign (H(S/M)QC)
+
+
+
+
-
33.8
31.9
31.9
23.9
23.9
19.7
18.7
16.3
15.7
15.4
3.1
0.8
0.1
0.1
References
Allard, P., Helgstrand, M., & Hard, T. (1998) The complete homogeneous master equation
for a heteronuclear two-spin system in the basis of cartesian product operators. J Magn
Reson 134, 7-16.
5
Baldwin, A. J., Hansen, D. F., Vallurupalli, P., & Kay, L. E. (2009) Measurement of methyl
axis orientations in invisible, excited states of proteins by relaxation dispersion NMR
spectroscopy. J Am Chem Soc 131, 11939-11948.
Hansen, D. F., Vallurupalli, P., & Kay, L. E. (2009) Measurement of methyl group motional
parameters of invisible, excited protein states by NMR spectroscopy. J Am Chem Soc
131, 12745-12754.
Sklenar, V., Torchia, D., & Bax, A. (1987) Measurement of carbon-13 longitudinal relaxation
using 1H detection. J. Magn. Reson. 73, 375-379.
6