Optimisation of the diffusion tensor parameters

The parameters of the Brownian rotational diffusion tensor belong to the set $\mathfrak{D}$. This set is the union of the geometric parameters $\mathfrak{G}= \{\mathfrak{D}_{iso}, \mathfrak{D}_a, \mathfrak{D}_r\}$ and the orientational parameters $\mathfrak{O}$,

$$\mathfrak{D}= \mathfrak{G}\cup \mathfrak{O}.$$ (15.4)

When diffusion is spherical solely the geometric parameter $\mathfrak{D}_{iso}$ is optimised. When the molecule diffuses as a spheroid the geometric parameters $\mathfrak{D}_{iso}$ and $\mathfrak{D}_a$ and the orientational parameters θ (the polar angle) and φ (the azimuthal angle) are optimised. If the molecule diffuses as an ellipsoid the geometric parameters $\mathfrak{D}_{iso}$, $\mathfrak{D}_a$, and $\mathfrak{D}_r$ are optimised together with the Euler angles α, β, and γ.

This category is defined as the optimisation of solely the parameters of $\mathfrak{D}$. The model-free parameters of $\mathfrak{F}$ are held constant. As all selected residues of the macromolecule are involved in the optimisation, this category is global and can be more complex than the optimisation of $\mathfrak{F}_i$ or $\mathfrak{T}_i$. The dimensionality of the problem nevertheless low with

$$\mathfrak{D}= 1, \qquad \dim \mathfrak{D}= 4, \qquad \dim \mathfrak{D}= 6,$$ (15.5)

for the diffusion as a sphere, spheroid, and ellipsoid respectively.