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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 (Denny et al., 1997; Wang and Stubbs, 1993). 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:

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

(15.1)

where $J_{n}$ is the

-th order Bessel function; $(r_j, \phi_j, z_j)$ are the cylindrical coordinates of atom

;

is the atomic scattering factor of atom

;

is the length of the axial repeat,

is the layer line number;

is the reciprocal space radius; and

is a function of the temperature factor and the occupancy factor of atom

. Because the diffracting units in fiber specimens are randomly oriented about the fiber axis, the observed intensity are cylindrical averaged:

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

(15.2)

For a fiber structure with simple helical symmetry,

is restricted by the selection rule

where

is the number of helical turns in an axial repeat unit,

is the number of subunits in the repeat unit, and

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 . In FX-PLOR, and 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:

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

(15.3)

where

is a scale factor and

is a weight which makes the gradient of the fiber effective energy comparable to the gradient of the empirical potential energies,

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:

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

(15.4)

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

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

(15.5)

where

$\begin{displaymath}\omega = 2\pi t / u \end{displaymath}$

(15.6)

$\begin{displaymath}\tau = \pm c / u \end{displaymath}$

(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):

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

(15.8)

and

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

(15.9)

where the multiplicity number

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.

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Xplor-NIH 2024-09-13