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Recent excitements in protein NMR: Large proteins and biologically relevant dynamics
Sai Chaitanya Chiliveri and Mandar V. Deshmukh
Equation SE1:
é I ( I +1) ùæ g ö2
ê H H
úç H ÷ ~ 16
I
I
+1
êë D ( D ) úûè g D ø
(SE1)
where, IH and ID are the magnetic moments of proton and deuterium, and H and D
are the gyromagnetic ratios of proton and deuterium.
Equation SE2:
The dipolar coupling between the internuclear vector P and Q is given by the
equation 2 (Tjandra and Bax 1997):
DPQ  ,   S
0 P Q 
3

Aa 3cos2   1  Ar sin 2  cos 2 
2 3

2
8 rPQ 



(SE2)
where, S is the generalized order parameter for internal motion of the vector PQ, 0 is
the magnetic permeability of vacuum, P and Q are the gyromagnetic ratios of P and Q, h is
Planck’s constant, rPQ is the distance between P and Q, and  and  are cylindrical
coordinates describing the orientation of the PQ vector in the principal axis system of A. A a
and Ar are the axial and rhombic components of the molecular alignment tensor.
Equation SE3-5:
2æ m ö
G1 = ç 0 ÷ g I2 ge2m B2 S ( S +1) J SB (w I )
5 è 4p ø
2
1  0  2 2 2

  I g e  B S S  14 J SB 0  3J SB I 
15  4 
(SE3)
2
2 
(SE4)
where, 0 is the permeability of the free space, I is the gyromagnetic ratio of the spin
I, e is the electronic g-factor, B is the Bohr magneton, S is the total electron spin quantum
number, and JSB() is the generalized spectral density function for the reduced correlation
function, given by:
J    r  6
c
2
1   c 
(SE5)
The correlation time is defined as, c-1=r-1 + s-1, where, r and s, are the rotational
correlation time of macromolecule and effective electron relaxation time, respectively.
Equation SE6 and SE7:
 PCS 


1 3 
3

r  ax 3 cos2   1   rh sin 2  cos 2 
12 
2

(SE6)
where, r is the distance between the paramagnetic center and the nucleus, and ax
and rh are the respective axial and rhombic components of the magnetic susceptibility
tensor, given as:
 ax   zz 

1
 xx   yy
2

 rh   xx   yy
(SE7)
Equation SE8:
DAB =
ü
hB0g Ag B -3 ì
3
rAB íDc ax (3cos2 q -1) + Dc rh sin 2 q cos2f ý
3
þ
240p kT î
2
(SE8)
where, rAB is the distance be between nucleus A and B, B0 is the external magnetic
field strength, A and B are the gyromagnetic ratios of nucleus A and B, respectively,  is the
angle between the A-B internuclear vector with the z-axis of the  tensor,  is the angle
between the projection of A-B internuclear vector on the x-y plane with the x-axis of the 
tensor, and ax and rh are described in Equation SE7.
Equation SE9-SE11:
d2 é
2
ù
ë J (w I - w S ) + 3J (w S ) + 6J (w I + w S ) + c J (w S )û
4
(SE9)
d2 é
c2
ë4J ( 0) + J (w I - w S ) + 3J (w S ) + 6J (w I ) + 6J (w I + w S )ùû + éë4J ( 0) + 3J (wS )ùû
8
6
(SE10)
R1 =
R2 =
NOE =1+
d2 gI é
ë6J (w I + wS ) - J (w I - w S )ùû
4R1 g S
(SE11)
where, J() is described in Equation 1.11, d = 0hIS/82, c = S/3, 0 is the
permeability of free space, h is Plank’s constant, I and S are the gyromagnetic ratios of the I
and S spins, respectively.
Equation SE12 and SE13:
The tilt angle (θ) of the effective field generated from the RF field in the rotating frame is given
by:
tan 
1
(SE12)

where,  is the resonance offset from the spin-lock career frequency.
R2 rates can be obtained from the following trigonometric equation (Equation 13), having prior
information on R1 rates.
R1  R1 cos2   R2 sin 2   sin 2 
p A pB  2kex sin 2 
kex  e
2
(SE13)
where, pA and pB are the fractional populations of A and B,  is the difference is
chemical shifts between the two sites A and B, and k ex is the exchange rate constant.