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Transcript
Protein Structure and Folding
NMR of folded and denatured proteins
There are many NMR parameters that can be measured from denatured proteins. Among the foremost are the relaxation
rate constants e.g. transverse relaxation rate R2 and relaxation rate for heteronuclear longitudinal magnetization i.e.
heteronuclear nOe (Nuclear Overhauser Enhancement). The relaxation rates report from dynamics, indeed the most
natural property of a molecule without a defined three-dimensional structure. However recent progress in NMR has
enable studies of the spatial properties using residual dipolar couplings (RDCs). Below is described the phenomenon for
structured i.e. folded proteins. Subsequently some guidelines are given how to treat the phenomenon for unfolded
proteins.
1.
Residual dipolar coupling
The dipolar coupling is the very same coupling that exists between two bar-magnets in a magnetic field. The stronger
the moments, the stronger is the interaction, i.e. 1 2. Furthermore the closer the magnets are each other the stronger
is the interaction, i.e. there is  1/rij3 distance dependence. Finally in the magnetic field there is also directional
dependence on angle  for the vector rij (secular approximation).


ij
ij
H dd  Dmax
I zi I zj 3 cos 2   1  Dmax
P2 cos
where P2(x)=1/2(3x2-1) (Legendre polynomial) and Dmaxij consumes the distance dependence and the magnitudes of the
moments. The distance dependence is of no major interest here because the length of a covalent bond remains the same
in the folded and denatured state. We focus our attention to the average of P2.
For the folded proteins with a single conformer P2 is time average due to the molecular tumbling in the solution.
For the denatured proteins P2 is also the time average due to the tumbling but also an ensemble average over the
conformations that exist in the solution. We work first the time average for a folded protein. The internuclear vector rij
is related to the molecular frame denoted by x, y, z by cosines of  (red) and the molecular frame is related to the
magnetic field direction by cosine of .
z
B
P2 (cos  ) 
o
3
1
( cos  x cos  x  cos  y cos  y  cos  z cos  z ) 2 
2
2
y
x
Using a short-hand notation for the cosines we have
P2 cos 

3
Cx
2
2
c x2  C y
2
c y2  C z
2

c z2  2 C x C y c x c y  2 C x C z c x c z  2 C y C z c y c z 
1
2
1
because the time average involves only the molecular rotation whereas the bond vector with respect to the molecular
frame remains invariant (for a folded single-conformer protein). We may further streamline the notation by introduction
what is known as Saupe order matrix
3
1
Ci C j   ij
2
2
S ij 
with the usual meaning of the Kronecker delta function to obtain
P2 cos 
S
c cj
ij i
ij x , y , z 
The 3 x 3 Saupe order matrix is traceless i.e. sum of diagonal elements vanish. The physical meaning is that the
resonance frequency does not depend on dipolar coupling (however it does depend on orientation for other reasons).
The matrix is also symmetric CiCj = CjCi because the dipolar coupling is symmetric with respect to the internuclear
vector.
Obviously a Saupe order matrix can be diagonalized so that the cross terms are zero then we have for the dipolar
coupling
P2 cos 

3
Cx
2
2
c x2  C y
2
c y2  C z
2

c z2  1
(make sure you understood how the –1 came about). The meaning of Ci2 is the probability of finding axis i along the
magnetic field. Clearly for each Cartesian direction the probability may depart from 1/3 i.e. Ci2 = 1/3 + Aij where we
have introduced a new matrix referred to as the molecular alignment tensor that describes how the molecule is oriented
in the solution.
Finally we make the transformation from the Cartesian system to the polar coordinates by setting  = z, cz = cos,
cx = sin cos,cy = sin sin, to obtain the main result (equation is drawn on right B vertical)

D ij  ,   
3 ij
Dmax sin 2  cos 2 Axx  sin 2  sin 2 Ayy  cos 2 Azz
2
3 ij  1
1

 Dmax
sin 2  cos 2 Axx  Ayy   3 cos 2   1 Azz 

2
2
2






3

ij  1
2
2
 Dmax
 2 Aa 3 cos   1  4 Ar sin  cos 2 
where the property of the traceless matrix Axx + Ayy = - Azz has been used and trigonometric identities 2sin2 =1 – cos2
and 2cos2 = 1+ cos2: The axial component Aa = 3/2Azz and the rhombic component i.e. departure from the cylindrical
symmetry Ar = Axx – Ayy.
Now we are in the position to calculate the time average
 2
  P  ,  sin dd
2
0 0
which is easy , at least in the isotropic solution by integration over the angles to give zero (make sure you get that). This
trivial but intuitively correct result means that the molecule will not prefer any particular orientation in the solution thus
the dipolar coupling will vanish.
2.
Alignment
However molecules may prefer a particular orientation in the magnetic field. For diamagnetic molecules the effect is
usually very small because the groups with magnetic susceptibility anisotropy in proteins (aromatic rings, peptide
bonds) have various orientations so that the resultant is small. For non-diamagnetic proteins e.g. those containing a
2
heme-group the effect is large enough to be measured quantitatively. Also double stranded DNA will give sizeable
effect because the bases are aligned axially to add up constructively.
The alignment of diamagnetic molecules may also be enhanced by steric obstruction i.e. by an anisotropic solution
that contains aligned particles. It is easy to imagine a solution containing large planes. In the vicinity of a plane an
elongated molecule will prefer certain orientations i.e. those that fit snugly next to the plane.
V
r
V
f
Consequently there is next to the plane a restricted volume Vr where concentrations for various orientations are unequal
and the dipolar coupling will not vanish. Away from the plane there is the free volume Vf where the average of the
dipolar coupling is 0 as usual. The effect seems very small and it would be tiny unless the dipolar coupling wasn’t so
large typically on the order of 10 kHz so that even one thousand of a fraction Vr/Vf =1/1000 will give few hertz
couplings.
The steric obstruction has been treated by the group of García de la Torre (Murcia Spain) by approximating the
molecular shape with an ellipsoid. Earlier the steric obstruction has been simulated by Zweckstetter and Bax (NIH).
Departures from the steric obstruction model result from interaction (Coulumbic) between the molecule and the plane,
the liquid crystalline particle). These can be handled to a degree by Debye-Hückel theory (Gerhard Hummel, Ad Bax,
Markus Zweckstetter).
3.
Denatured proteins
Now we are minimally equipped to tackle the problem of residual dipolar couplings from denatured proteins.
Apparently we face immediate problems in our attempts to calculate the time average because the internuclear vector is
no longer static and we do not have a common molecular frame to denote the orientation but an ensemble of
conformations. In other words we may think to have for each conformer instantaneous frame with angle  with respect
to the field and with respect to the instantaneous frame an internuclear vector direction defined by . Thus we need to
calculate both time and ensemble averages. In general this can be very complicated depending on a particular ensemble
in the solution.
We hope to simplify the problem by examining a fully denatured protein i.e. a coil. To make the problem tractable
we use a random coil distribution function
 3 
W ( x, y, z )dxdydz  
2 
  2nl 
3/ 2

e

3 x2  y 2  z 2
2 nl 2

dxdydz
which states the probability to find the chain with n elements each length l starting from the origin within an interval x ..
x + dx etc. Thus we should attempt to calculate
3
P2   P2 W ( x, y, z )dxdydzd sin d
which is easy for the case of isotropic solution because W is spherical so that dipolar couplings will vanish by the same
integrations as before. However, when there is an obstruction at hz from the origin the distribution itself will be
modified along the z-direction
 3 
W ( z )dz  
2 
  2nl 
3/ 2
3  2 hz  z 
  3 z 22

2
e 2 nl  e 2 nl

2

 dz

Importantly the residual dipolar coupling will be acquired from an internuclear pair whose orientation we must be
averaged over the distribution. RDC is not an average over the whole chain but gives information of the orientation of a
segment of interest. Thus we need to build a distribution function for any given segment in the chain by combining two
obstructed distributions, one for each half chain i.e. to the “left” and to the “right” of the segment of interest. The
averaging of P2 for the segment of interest over all orientation will involve integration over the probability. The
distribution is not spherically symmetric and is expected to give non-vanishing residual dipolar couplings. A longer
chain will have a more spherical distribution than a shorter chain thus RDCs for a long chain will be smaller than for a
short chain. The same argument applies qualitatively how RDCs will vary as function of the position in the chain.
This is the principle of calculation, however the calculation is somewhat tedious for the purposes of the course.
Thus we will not proceed any further.
Note:
a) Make sure you understand the physical basis of enhanced alignment.
b) Make sure you are able to calculate average of P2 in the free region.
c) Make sure you are able to calculate average of P2 for a needle-like object in a restricted region (in a given
position).
d) Make sure you understand why a random coil may give rise to residual dipolar couplings and how residual
dipolar couplings should vary along the chain.
4