The model-free equations

In the original model-free analysis of Lipari and Szabo (1982a) the correlation function C(τ) of the XH bond vector is approximated by decoupling the internal fluctuations of the bond vector C_I(τ) from the correlation function of the overall Brownian rotational diffusion C_O(τ) by the equation

C(τ) = C_O(τ)⋅C_I(τ).

(15.57)

The overall correlation functions of the diffusion of a sphere, spheroid, and ellipsoid are presented respectively in section 15.9.1 on page

, section 15.10.1 on page

, and section 15.11.1 on page

. These three different equations can be combined into one generic correlation function which is independent of the type of diffusion. This generic correlation function is

C_O(τ) = $\displaystyle {\frac{{1}}{{5}}}$ $\displaystyle \sum_{{i=-k}}^{k}$ c_i⋅e^-τ/τ_i,

(15.58)

where c_i are the weights and τ_i are correlation times of the exponential terms. In the original model-free analysis of Lipari and Szabo (1982a,b) the internal motions are modelled by the correlation function

C_I(τ) = S² + (1 - S²)e^-τ/τ_e,

(15.59)

where S² is the generalised Lipari and Szabo order parameter which is related to the amplitude of the motion and τ_e is the effective correlation time which is an indicator of the timescale of the motion, albeit being dependent on the value of the order parameter. The order parameter ranges from one for complete rigidity to zero for unrestricted motions. Model-free theory was extended by Clore et al. (1990) to include motions on two timescales by the correlation function

C_I(τ) = S² + (1 - S²_f)e^-τ/τ_f + (S²_f - S²)e^-τ/τ_s,

(15.60)

where the faster of the motions is defined by the order parameter S²_f and the correlation time τ_f, the slower by the parameters S²_s and τ_s, and the two order parameter are related by the equation S² = S²_f⋅S²_s.

The relaxation equations of Abragam (1961) are composed of a sum of power spectral density functions J(ω) at five frequencies. The spectral density function is related to the correlation function as the two are a Fourier pair. Applying the Fourier transform to the correlation function composed of the generic diffusion equation and the original model-free correlation function results in the equation

J(ω) = $\displaystyle {\frac{{2}}{{5}}}$ $\displaystyle \sum_{{i=-k}}^{k}$ c_i⋅τ_i $\displaystyle \Bigg($ $\displaystyle {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2)(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}}$ $\displaystyle \Bigg)$ .

(15.61)

J(ω) = $\displaystyle {\frac{{2}}{{5}}}$ $\displaystyle \sum_{{i=-k}}^{k}$ c_i⋅τ_i $\displaystyle \Bigg($ $\displaystyle {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}}$ + $\displaystyle {\frac{{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$ $\displaystyle \Bigg)$ .

(15.62)