Fiber Structure Refinement

Xplor-NIH now incorporates the functionality of the FX-PLOR package of computer programs for macromolecular refinements against X-ray fiber diffraction data (Wang and Stubbs, 1993; Denny et al., 1997). Most features in X-PLOR suitable for crystal structure refinement and structure analysis can be utilized for fiber structure refinement and analysis.

In X-ray fiber diffraction, the Fourier-Bessel structure factor is defined as:

$$\mbox{\boldmath$G$}_{nl}(R) = \sum_j f_j Q_j J_n(2\pi Rr_j) \exp (-n\phi_j+2\pi lz_j/c)i$$ (15.1)

where $J_{n}$ is the $n$-th order Bessel function; $(r_j, \phi_j, z_j)$ are the cylindrical coordinates of atom $j$ ; $f_j$ is the atomic scattering factor of atom $j$; $c$ is the length of the axial repeat, $l$ is the layer line number; $R$ is the reciprocal space radius; and $Q_j$ is a function of the temperature factor and the occupancy factor of atom $j$. Because the diffracting units in fiber specimens are randomly oriented about the fiber axis, the observed intensity are cylindrical averaged:

$$I(R,l) = {\cal G} (R,l)^2 = \sum_n \mbox{\boldmath$G$}_{n,l} (R) \mbox{\boldmath$G$}_{n,l}^* (R)$$ (15.2)

For a fiber structure with simple helical symmetry, $n$ is restricted by the selection rule $l = tn + um$ where $t$ is the number of helical turns in an axial repeat unit, $u$ is the number of subunits in the repeat unit, and $m$ is an integer.

Fiber diffraction intensities are usually measured at small sampling unit $\sigma_s$ along the layer line, where $\sigma_s$ is a constant in reciprocal space. Because the sampling unit is usually small, it is convenient to index the intensity data with reciprocal space index $R_s = R/\sigma_s$ and layer line number $l$. In FX-PLOR, $R_s$ and $l$ are stored in H array and L array, respectively. Moreover, because structure factor ${\cal G}$ in fiber diffraction is a multidimensional vector rather than a two-dimensional vector, only amplitude of ${\cal G}$ is stored in the real components of FCALC array and FOBS array.

In order to include X-ray fiber diffraction information in a molecular dynamics refinement or energy minimization, the effective energy for fiber diffraction is defined as a function of the discrepancy between observed and calculated intensities:

$$E_{FIBER} = S_f \frac {\sum_{R_s,l} w ({\cal G}_{obs} - k {\cal G}_{calc})^2 } {\sum_{R_s,l} w {\cal G}_{obs}^2 }$$ (15.3)

where $k$ is a scale factor and $S_f$ is a weight which makes the gradient of the fiber effective energy comparable to the gradient of the empirical potential energies, $w$ is the individual weight for each observed intensity. The effective energy term for fiber diffraction can be turned on or off by including or excluding XREF in flag-statement (see X-PLOR section 4.5), respectively.

To reduce computational time, look-up tables for $\sin$, $\cos$ and Bessel function may be set up for structure factor calculation, The size of the Bessel look-up table depends on the following factors: resolution limit of the diffraction data, maximum number of Bessel terms on a layer line (BLMAX) and grid size of the table for Bessel function (TGRID).

The basic symmetry relation of a helical structure is a screw rotation $({\cal R, T})$. In FXPLOR, these symmetry relations can be utilized in potential energy calculation for non-bonded interactions between atoms in different subunits. The rotation matrix $\cal R$ of the screw rotation is:

$$\left ( \begin{array}{rrr} \cos\omega & -\sin\omega & 0 \sin\omega & \cos\omega & 0 0 & 0 & 1 \end{array} \right )$$ (15.4)

and translation vector along the rotation axis $\cal T$ is:

$$\left ( \begin{array}{c} 0 0 \tau \end{array} \right )$$ (15.5)

where

$$\omega = 2\pi t / u$$ (15.6)

$$\tau = \pm c / u$$ (15.7)

The sign in formula (15.7) is determined by the handedness of the helix. A right handed helix has a positive value while left-handed helix has a negative value. Other symmetry relations can be derived from (15.4) and (15.5):

$$\left ( \begin{array}{rrr} \cos m\omega & -\sin m\omega & 0 ... ...n m\omega & \cos m\omega & 0 0 & 0 & 1 \end{array} \right )$$ (15.8)

and

$$\left ( \begin{array}{c} 0 0 m\tau \end{array} \right )$$ (15.9)

where the multiplicity number $m$ is an integer. The helical symmetry relations may be defined by NCS-stric-statement (see X-PLOR section 18) and/or fiber_refin-helical_symmetry-statement.