Restricted Diffusion of Methyl Groups in Proteins Revealed by Deuteron NMR:
Manifestation of Intra-well Dynamics
Liliya Vugmeyster1, Dmitry Ostrovsky2
1
Department of Chemistry, University of Alaska Anchorage, Anchorage, Alaska, 99508;
2
Department of Mathematics, University of Alaska Anchorage, Anchorage, Alaska,
99508
SI1. Comparison of the solution of the Smoluchowski equation obtained in this work
with the approach by Edholm and Blomberg.
Table SI1 presents several lowest eigenvalues obtained for the discrete form of the
Smoluchowski equation for the value of V0 / k BT fixed at 4.8 as well as the amplitudes
associated with them. The third column of Table SI1 shows the results from Edholm and
Blomberg 1. The comparison for the amplitudes can be done by consulting Table 1 of
Edholm and Blomberg. We omit the eigenvalues which do not contribute to the
relaxation process due to the 3-fold symmetry of the potential, which leads to zero
amplitudes for certain eigenvalues. Note also that the eigenvalue 22.7 is missing in Table
1 of Edholm and Blomberg, apparently due to a scribal mistake. The two approaches
are in a good agreement. The first non-zero eigenvalues calculated from our approach
agree with the calculations of Edholm and Bloomberg within 0.1%, and the next three
within 1%. The minimum number of sites required to achieve this agreement is 72,
corresponding to 5o steps for the angle .
Table SI1
eigenvalue
number
0
1
2
4
5
7
8
10
eigenvalues,
our results
0
0.075
13.5
22.7
31.1
53.0
66.5
98.0
eigenvalues,
Edholm&Blomberg
0
0.075
13.9
32.4
55.9
70.8
107
Amplitudes, our results
A00
1
0
0
0
0
0
0
0
A11
A22
0
0.945
0.039
0.016
2.5·10-4
3.2·10-4
4.1·10-6
4.1·10-6
0
0.772
0.183
0.011
3.3·10-2
9.0·10-6
1.0·10-3
1.1·10-7
A-21
0
0.854
-0.084
-0.013
-2.9·10-3
-5.4·10-5
-6.4·10-5
-6.7·10-7
Table SI2 presents the first non-zero eigenvalues for several values of V0 / k BT , obtained
by the two approaches. The differences are miniscule.
1
Table SI2
3/4
V0 / k BT
1
2
3
4
5
6
7
8
9
10
first
eigenvalue,
our results
7.104·10-1
3.400·10-1
1.366·10-1
4.940·10-2
1.666·10-2
5.360·10-3
1.668·10-3
5.069·10-4
1.513·10-4
4.454·10-5
First eigenvalue,
Edholm&Blomberg
7.10·10-1
3.40·10-1
1.37·10-1
4.94·10-2
1.66·10-2
5.36·10-3
1.67·10-3
5.07·10-4
1.51·10-4
4.46·10-5
SI2. Correlation function in the eigenvalue representation.
The eigenvalue representation for the Smoluchowski equation is well described in the
literature. 2,3 In the following we outline the derivation of Eqs. (12-14) of the main text.
The correlation function of the internal rotational diffusion of the methyl group, shown in
Eq.(15) of the main manuscript, is given by
C p ,q (t )  e ip (t ) e iq ( 0) 
2
2
0
0
ip iq
eq
 d  d 0 e e 0 P( , t |  0 ) P ( 0 ) ,
(S-1)
where P( , t |  0 ) is the conditional probability of transition from  0 to  during time t,
and P eq ( 0 )  e V ( ) / k BT is the equilibrium distribution. P( , t |  0 ) satisfies the
Smoluchowski equation with the initial condition P( ,0 |  0 )   (   0 ) . The
equilibrium distribution P eq ( 0 ) is the stationary solution of the Smoluchowski equation.
The operator on the right-hand side of the Smoluchowski equation is not self-adjoint, but
if one substitutes
P( , t |  0 )  P eq ( ) R( , t |  0 )
1
P eq ( 0 )
,
(S-2)
then R( , t |  0 ) satisfies the equation
2
 2
dR( , t |  0 )
1  d 2V 1  dV   

 R( , t |  0 )
D

 
  2 2k BT  d 2 2  d   
dt



2
(S-3)
with the initial condition R( ,0 |  0 )   (   0 ) . The operator on the right-hand side of
Eq. (S-3) is self-adjoint and, therefore, the set of its eigenfunctions  s ( ) can be chosen
as an orthonormal basis. The functions  s ( , t )  e  st s ( ) , in which ( s ) are the
eigenvalues for the eigenfunctions  s ( ) , are solutions of Eq. (S-3). Because of
orthonormality of the basis { s } , the eigenfunction decomposition for the initial
condition is
R( ,0 |  0 )   (   0 )   s ( ) s ( 0 ) . Then, the solution for the time-dependent
s
problem becomes
R( , t |  0 )   e  st s ( ) s ( 0 )
(S-4)
s
Thus, the correlation function can be written as
2
C p ,q (t ) 
2
 d  d e
0
0
0
ip
e iq0  e st s ( ) s ( 0 ) P eq ( ) P eq ( 0 ) .
(S-5)
s
As can be seen from Eq.(S-2), functions  s ( )   s ( ) P eq ( ) are eigenfunctions of the
original Smoluchowski equation, with the eigenvalues of ( s ) . Therefore, the
correlation function is
2
2
C p ,q (t )   e st  de ip  s ( )  d0 eiq0  s (0 ) ,
s
0
(S-6)
0
which coincides with Eqs. (12-14) of the main text with
Ap( s,q)   s , p s*,q and
2
 s , p   de ip  s ( ) .
0
We note that  0 is proportional to the equilibrium distribution and does not contribute
into the sum of Eq.(S-6).
SI3. Overview of previously published model
In order to explain the experimental results, shown in Figure 8 of the main text, we have
developed a model that involves a distribution of conformers distinguished by their
values of activation energy barriers Ea for three-site hops.4 The distribution is the source
of the non-exponentialy of the magnetization decay curves. We note that the samples
used in these studies had deuteron labels in single residues, so that the distribution
reflects heterogeneity of the energy landscape at each individual methyl group.
3
For the case of a continuous distribution of conformers, the overall observed
magnetization M(t) is defined by
M (t )   m( E a , t )dE a
(S-7)
in which m( E a , t ) is the magnetization density.
A simple Gaussian form of the
 (Ea  Ea ) 
 was used.
exp  
The longitudinal
2


2
2

2


relaxation rate depends on the activation energy through the three-site hop rate constant k.
The temperature dependence of the three-site hops rate constant in each conformer was
assumed to be Arrhenius. k ( E a , T )  k 0 e  Ea / T and the value of k 0 was assumed to be the
same for all conformers.
distribution
f (Ea ) 
2
1
Based on the data, we have also concluded that at high temperatures there is a presence of
conformational exchange between the conformers, which occurs on the time scales of
T1eff (~50 ms). The effect of exchange was only important for temperatures above about
250K, at which its most obvious contribution was to raise the modeled values of . The
quality of the fits for lower temperatures as well as the overall fitting parameters of the
distribution remained largely unaffected by the inclusion of the exchange. Therefore, in
the current analysis we neglect the effects of exchange and work in the static limit.
SI4. Analysis of various functional form of the distribution of barrier heights.
In this section we consider the effect of several functional forms of the distribution of
barrier heights on the relaxation behavior.
We analyse the following three cases, which include both symmetric and nonsymmetrical functional forms, such as
i) a flat (square) distribution of the activation energies
1

f ( E a )   2s

 0
for E a  E a  s
in which s defines the width of the distribution;
otherwise
ii) the Gamma distribution f ( E a ) 
b
E a 1 e bEa in which ( ) is the Euler gamma
( )
function. For this distribution the average value is given by
maximum probability density is achieved at E a  (  1) / b ;
4
E a   / b and the
E a  E a ,0
1
r
for E a  E a ,0  s

2
 2s
2
s
iii) a linear distribution f ( E a )  
where Ea , 0 is the


0
otherwise

central value of the ramp and r varies between -1 and 1.
Using these above functional forms of the distribution, we perform the fits that best
match the position of the T1eff minimum for L61 residue, as illustrated in Figure S1.
Figure S1. Fits of
T1eff and  for L61 residue using the minimization procedure in which the best-fit of the
T1eff minimum position is obtained. The results are demonstrated for several functional forms of the
distributions: Gaussian (black), square (green), Gamma (blue), and linear (red). For all cases we used Cq
=175 kHz and the nontetrahedral geometry around the methyl carbon axis.
The following parameters are obtained:
Gaussian lnk0=27.4, <Ea>= 11.8 kJ/mol,  = 1.7 kJ/mol.
Flat (square) distribution: lnk0=27.4, s = 2.4 kJ/mol, <Ea>= 11.8 kJ/mol.
5
Gamma distribution: lnk0=27.5, = 35.4, b= 51.1/ (kJ/mol), which lead to <Ea>= 12.0 kJ/mol and
the standard deviation  = 1.7 kJ/mol.
Linear distribution: lnk0=27.5, s =2.6 kJ/mol, Ea,0= 12.3 kJ/mol, r=0.32 .
For all of the cases in Figure S1, if the position of the T1eff minimum is matched, then a
relatively large discrepancy is seen between the modeled and experimental values of .
SI5. Analysis of Arrhenius temperature dependence of the diffusion coefficient.
Figure S2 shows the contour lines of constant T1 values for a range of lnD and V0 / k BT . The
choice of variables described in the figure legend was made for the best representation of the
analysis of Arrhenius temperature dependence of D. As obvious from this figure, in the selected
range of parameters the relaxation time for the diffusion in a three-fold potential depends very
weakly on lnD, provided that V0 / k BT is chosen in the combination V0 / k BT  a ln D . If
lnD depends linearly on 1/T such dependence would be almost completely masked by a different
choice of V0. The parameters of the model obtained through the fit to experimental data under the
assumption of constant D (Table 1 of the main text) fall within the range shown in Figure S2.
Figure S2. Contour plot of T1 calculated for the diffusion in the three-fold potential model for the Larmor
frequency of deuteron of 77 MHz. Two parameters of the model (D and V0/kBT) are expressed through
independent variables lnD and V0  V0 / k BT  a ln D , in which a = 1.09 is chosen as the best
approximation that makes T1 minimum position the same for any value of lnD; this position is given by
V0 = 20.5. The numbers on the contour lines indicate the values of T1 in milliseconds.
6
References
(1) O. Edholm and C. Blomberg, Chem Phys 42, 449 (1979).
(2) H. Risken The Fokker-Planck Equation: Methods of Solution and Applications,
(Springer, Berlin, 1984).
(3) R. B. Jones, J Chem Phys 119, 1517 (2003)
(4) L. Vugmeyster, D. Ostrovsky, K. Penland, G. L. Hoatson and R. L. Vold, J Phys
Chem B 117, 1051 (2013).
7