Results

MRC Analysis of Protein Backbones

Figure 3. MRC Analysis of Calmodulin. White spheres represent the position of the calcium ions in the structure. The 143 residues (j=7.16) visible in the electron density map define the ribbon drawing on the far left. The 4 nearly identical "EF-hand" domains are given alternate colors. In the lower resolution versions (j=5,4,3,2) , the curve defined by the 32,16,8,4 segments is rendered as a thick tube.

Figure 3 shows the multiresolution curves of calmodulin [29]. Figure 4 shows the MRC analysis of neuraminidase [30]. Both figures show only the discrete integer resolution versions of the curves.

Figure 4. MRC Analysis of Neuraminidase. A ball-and-stick drawing marks the inhibitor DANA in the active site. The sheet residues are given different colors. The 389 residues (j=3D8.60) define the ribbon drawing in the upper left. In the lower resolution versions (j=3D8,7,...,0), the curve defined by the 256,128,...,1 segments is rendered as a tube.

Figure 5. MRC Visualization of Interferon. Left: The interferon-( dimer is rendered as a tube ribbon (j=3D6.93). Cylinders fit to the helical residues are shown. Right: The MRC version with j =3D 3.93. The tube is a larger diameter for helices.

Interferon-( [31] is shown in Figure 5. Each monomer has 123 residues, implying j=6.93. The B-spline curve defined by 32 segments (j=5) generally passes through the cylinders. The fractional level was lowered to visually assess the exact level where the topology is still recognizable. This subjective judgment is influenced by the style of the drawing presented. Thin tubes of a single color were contrasted with tubes colored by secondary structure. The tube radius was also varied as a function of secondary structure, affecting a 'tube-and-cylinder' style diagram.

Figure 6. MRC Visualization of Aldehyde Reductase. Left: Aldehyde reductase (j=3D8.32) is shown with transparent polygons connecting the hydrogen-bonding regions of this TIM barrel protein. The protein is rendered as a tube colored by secondary structure with the diameter of sheet ( helix ( other. Right: The MRC version with j =3D 5.63.

Aldehyde reductase [32] is shown in Figure 6. The chain has 320 residues, implying j=8.32. The B-spline curve defined by 64 segments (j=6) goes through the helices and hugs the central barrel. Lowering the level further caused degradation of the image.

About a half dozen coworkers examined the protein of their choice. One was disconcerted by the disappearance of the helices and thought reduction to 1/2 the number of points was the limit. Others feel reduction by 1/8 or greater to be possible, especially if there is additional color/texture information (e.g., fat helices). The deviation between the Ca and the Cr was monitored for database analysis on the low-resolution curves for the two types of ribbons described in Figure 2. Except at j=0, where the Ca ribbon is by construction Cr, the plots are virtually identical. This shows an equivalence of the various ribbon formulations at lower resolution. The data for the standard ribbon drawing is plotted in Figure 7. It is seen that helix and sheet residues are approximated slightly better than those in coils. The plots are similar, with slightly higher values for the fractional mode. The main difference is the point at which the deviation trends sharply upward.

Figure 7. Deviation from C( as a Function of Resolution. The y-axis is the distance between C( and Cr. The x-axis is the decomposition level. 0 is the original (padded) curve, at maximum j. For integer level plots, 1,2,3,... are the curves with 1/2, 1/4, 1/8,... the segments. The inset at the upper left gives the results, as does the thick gray line in the main graph. For the fractional level plots, 0 is also the original curve, but 1,2,... represent the initial fractional level minus 1,2,.... (For example, for neuraminidase 0,1,2,... is j =3D 8.6, 7.6, 6.6,....) The rms value over all residues in the database is marked with a tick at the center of error bars representing one standard deviation. Separate marks are shown for all the residues and the separate secondary structure classes: helix, sheet, other.

Topological Comparison

A visual comparison of proteins is shown in Figure 8. The fractional level of eight superimposed serine proteases [33] was adjusted to the lowest value at which the fold was still deemed clearly recognizable. This value is 2.5 less than the maximum value, implying about 1/6 the original information.

Figure 8. Topological Comparison with MRC representation. Left: Standard ribbon drawings of seven serine proteases aligned to model Factor D, labeled and colored by the PDB code. Factor D has 228 residues (j=7.85). Right: MRC representations with j=5.35.

Human complement Factor D is a serine protease with several flexible loops. The crystal structure [34] has two independent conformations, designated the A and B chains. Superposing the Ca's for all 228 residues gives an rms difference between Ca positions of 0.95A. For the 206 residues not in the flexible loops, the Ca's are fit to an rms difference of 0.35A. This latter transformation was taken as the best fit.

Least squares fits were determined based on a reduced number of points. The results of the transformation were assessed by measuring the rms difference of the 206 non-loop residues described above. Every 1/2, 1/4, 1/8, and 1/16 Ca positions were used, starting on every possible initial point. For example, using every fourth point, fitting 1,5,9,..., 2,6,..., 3,7,..., and 4,8,..., gave rms fits of 0.874, 0.727, 0.917, 1.176A, respectively.

Using the MRC representation and the maximum integer level j-2, or 1/4 the control points, and fitting based on the Cr points gave corresponding values of 0.633, 0.674, 0.715, 0.674 A, respectively. These latter results are slightly better and, more importantly, show less variation. However, little difference is seen between the Ca and MRC methods from visual assessment of the superpositions. Also, fits based on the control points of the MRCs, instead of the curve itself, gave poor results.

Multiresolution Editing

The user is presented with pickable control point spheres that define the B-spline ribbon. Dragging the picked point with the mouse changes the ribbon curve and demonstrates the local control property of B-splines: only the two curve segments (residues) on either side of the control point change.

Wavelets allow one to interactively choose the level of approximation. Each decrease in the multiresolution level doubles the number of residues that change when a control point is moved. Figure 9 is a representation. Picking a point in the center of a turn at the maximum level moves only the 4 residues in the turn. Changing the level to j=3 allowed a point in the low resolution version to be selected. Moving this point changed all the residues in a strand-loop-strand motif.

Figure 9. Multiresolution Editing. Thirty-two residues of ubiquitin (j=5) were manipulated. Spheres represent the pickable control points. Upon picking, a 'handle box' cube highlights the movable point. The thick curve is the high-resolution ribbon, the thinner curve the low-resolution version, and thin lines the atomic bonds. The dashed line/arrow sketches the motion of an edited control point in the panels on the right. The panels on the left show the picked point before editing. Upper left: Picking on the original curve. Upper-right: Only 4 local residues are altered. Lower left: Picking on the 1/4 resolution curve. Lower right: The entire 16 residue strand-loop-strand swings away from the helix.

A model of calmodulin was presented as an editable MRC. Also displayed was a model of calcium-free calmodulin, with a large conformational change in two helices of the calcium-binding domains [35]. An unsuccessful attempt was made to interconvert the structures by moving only one control point of a very low resolution curve. By raising the level and moving only two control points, the helix-loop-helix backbones could nearly be overlaid at the expense of considerable distortion of one of the helices.

Molecular Surfaces

Figure 10. Multiresolution Surfaces of DNA. From left to right: MSP surface colored by electrostatics. DNurbs surface j=5,4,3 textured by electrostatics, j=2 textured by curvature, MSP surface colored by curvature.

DNurbs representations were created with 32 x 32 surface patches modeling a 32 base pair model of DNA. MSP triangle surfaces are shown for comparison in Figure 10. The grooves disappeared by level j=3 (8 x 8 patches), although the texturing makes the groove pattern clear.

Figure 11. Multiresolution Surfaces of Neuraminidase. The inhibitor DANA is shown as a ball-and-stick model. From upper-left: Atomic spheres colored by temperature factor, NURBS globe j=5,4,3 textured by temperature factor.

The globular protein neuraminidase is shown as a space-filling model in Figure 11. The textured globe NURBS representations are shown at various resolutions. The method does a decent job at modeling the surface, but the active site progressively loses definition. For general surfaces, the progressive remeshing of the neuraminidase active site was taken as a test case. The MSP surface is approximated by 6,24,96,... triangles in Figure 12. Note this is just a remeshing; a precursor to actual wavelet analysis [27].

Figure 12. Multiresolution Surface Remeshing. The MSP triangular surface of the neuraminidase active site colored by electrostatics along with the inhibitor DANA is shown in the upper-left. In each successive panel moving clockwise, the original MSP is shown as a mesh along with the approximating surface. The lowest resolution level, M=0, has only 6 triangles. Each successive level of M has 4 times the number of triangles.