Calcium-Binding Protein Protocols Calcium-Binding Protein Protocols

Shape and Dynamics of Calcium-Binding Proteins 295
G. Palmer, III, Columbia University). The statistical significance of changes in χ 2
are evaluated as outlined in Subheading 3.2., step10, where
χ 2 tot
= N
∑ i ( (T 1 /T 2 ) i exp – (T 1 /T 2 ) i th / σ(T1 /T 2 ) i exp ) 2 (9)
and (T th
1/T2) i is a function of the diffusion tensor via Eqs. 1–3 and 5, with S 2 = 1.
For the isotropic model, the isotropic diffusion constant D and for the axially
symmetric diffusion model, the isotropic diffusion constant D, the axial ratio of
the diffusion tensor, D ||/D⊥, and its orientation in the molecular frame θ and ϕ are
obtained by minimization of Eq. 9. Averages of these quantities for a family of
22 NMR structures of cbEGF32-33 (40) are shown in Table 1 and Fig. 3A.
4. The χ2 values of the isotropic fit χ2 iso and the axially symmetric fit χ2axial are used
to calculate the probability Q that the improvement in χ2 axial is obtained by chance
(see Subheading 3.2., step 10 and Table 1). The degrees of freedom are N–1 for
the isotropic model and N–4 for an axially symmetric tensor.
5. For an axially symmetric diffusion tensor the dependence of the (T1/T2) –1 ratios
R, as a function of the angle α of the N–H bond vector with the unique axis of the
diffusion tensor can be approximated by (41):
R(α) = R(0)(1 + ε sin2 α) (10)
with ε = (D ||/D⊥–1). Provided that the analysis indicates some anisotropic motion, a
plot of the (T1/T2) –1 as a function of sin2α, yields a straight line with y-intercept R(0)
and slope R(0)(D ||/D⊥–1) (see Fig. 3). Sampling of a wide range of angles is important
for obtaining a robust fit and a reliable estimate of the anisotropy.
3.4. Model-Free Analysis of T 1, T 2 and 1 H– 15 N NOE Data
1. Internal dynamics is characterized using the model-free approach of Lipari and
Szabo (22) as implemented in Modelfree4 (26,27). Parameters of internal
dynamics are obtained by χ 2 minimization (see Subheading 3.2., step 10):
χ n 2 = ∑i
[(T exp – Ti
th ) 2 /(σi exp ) 2 ] + [(NOEexp – NOEth )/(σnoe exp ) 2 (11)
i
where T 1 th , T2 th , and NOE th depend on S 2 , τe, and R ex via Eqs. 1–3 and 5 (see Fig. 4).
The uncertainties of the parameters are estimated using Monte Carlo simulations
(typically 200–300). For each residue, five models of increasing complexity are
tested, whereas the previously defined diffusion tensor is kept fixed (see Table 2).
For each residue, an appropriate model is selected according to the criteria outlined
in Subheading 3.2., step 10.
2. After an appropriate model has been found for each residue, all parameters
including the diffusion tensor and all parameters of internal dynamics are optimized
simultaneously in a final χ 2 minimization. The uncertainties of the free
parameters are estimated from a set of Monte Carlo simulations (typically 500).
4. Notes
1. For both samples, at this calcium concentration and in the absence of additional
salt, both Ca2+ -binding sites of the pair were deemed saturated, based on the