Bioisis

NORMALIZED (DIMENSIONLESS) KRATKY ANALYSIS ... by Robert P. Rambo, Ph.D.

The dimensionless (normalized) Kratky plot provides a semi-quantitative approach to assessing the state of your biopolymer in solution. For applications regarding intrinsically disordered proteins (IDPs) see Receveur-Brechot and Durand or general considerations see Rambo and Tainer (Biopolymers). The plot is made dimensionless by multiplying the q-vector (x-axis) by the particle's Rg and multiplying I(q) by (q·Rg)2 instead of q2 (see Durand D, Vives C, et al, 2010 ). To normalize for particle mass and concentration, the modified I(q) is divided by the experiment's I(0). For particles obeying Guinier's approximation, there will be a peak at √3 with a magnitude of 3·e-1 (1.104).

Starting with Guinier approximation:

Guinier_equation

Multiply by (q·Rg)2 and divide by I(0):

Kratky_guinier_equation

Do a change of variables letting u = q·Rg:

U_substituted

Find the first maxima by taking the derivative and solving for f'(u) = 0:

Derivative_u Solve_for_zero

Take the square root of both sides thus solving for u or (q·Rg)=√3:

Final_u_3

This type of dimensionless Kratky plot should show for most globular compact particles, obeying Guinier's approximation, a peak maxima at √3 regardless of particle size, composition and concentration. Deviations from this suggest either particle flexibility or asymmetry (see Receveur-Brechot and Durand (2012) or Durand et al. (2010)).

Figure 1

Normal_kratky_figure_1

In Figure 1, a 21 kDa globular protein, xylanase (cyan), shows a peak maxima at √3. For asymmetric particles, see Figure 1: P4-P6 RNA domain (orange) and SAM-1 riboswitch (purple), the peak shifts right with a maxima > 1.104. For an intrinsically disordered protein (RAD51-AP1), we get the classic hyperbolic plateau (magenta). Attaching RAD51-AP1 to the well-folded maltose binding protein (blue), the peak shifts right (q·Rg = 3.5) and reaches a new maxima. The Rg-based dimensionless Kratky plot is based on I(0) and Rg derived from either a Guinier analysis or real space transform. Scatter creates both of these plots automatically if the real space parameters have been previously determined.

Another dimensionless plot can be made using Vc (Figure 2). Here, we are using q2·Vc instead of q·Rg2 to make the plot dimensionless. We still normalize to I(0) but peak height is now directly proportional to Vc/Rg2, a ratio inversely related to the particle's surface-to-volume ratio. The maximum theoretical height will be that for a sphere which is 0.82.

Figure 2

Normal_kratky_figure_2

This form of a dimensionless Kratky plot can be more informative for inferring changes in a particle conformation. For instance, compare SAM riboswitch with (+, purple) and without (-, green) ligand in Figures 1 and 2. In the Rg-based plot, it is clear there is a conformational difference as the SAM(-) curve (green) is more hyperbolic suggesting a flexible ensemble. However, in the Vc-based plot (Figure 2), the hyperbolic feature is preserved for SAM(-) curve but shifts downward suggesting an increase in the surface-to-volume ratio of the particle in the absence of ligand.

SCÅTTER can make both types of dimensionless Kratky plots simply by clicking on the button "Normal Kratky" (Figure 3, red arrow). It will make four plots, two plots (real and reciprocal space) using only I(0) and Rg parameters and two plots (real and reciprocal space) using I(0) and Vc.

Figure 3

Normal_kratky_figure_3