Forward and inverse kinematics in RNA

backbone conformations

By Xueyi Wang and Jack Snoeyink

Department of Computer ScienceUNC-Chapel Hill

Outline RNA Structure &

Crystallography Ramachandran-like plots Measurements and Conformations Forward and Inverse Kinematics Future Work

RNA Structure

Large ribosome subunit

-- Chain0: 2914 bases

-- Chain9: 122 bases

RNA StructureResidue

Suite

α

βγ

δ

εζ

δγ β

αζ

δε

RNA Structure & Crystallography

Large RNA structures at 2.5 or 3Å resolution are considered good.

Electron Density Map--The Phosphates and Bases can be clearly

located.--Sugar puckers can be derived.--Other parts are ambiguous. Goal: Achieve correct RNA structures from

electron density maps.

Electron Density Map

Image Courtesy Richardsons’ Lab

All-Atom Contact Analysis

Image Courtesy Richardsons’ Lab

Complexity of RNA Backbone

α

β

γδε

ζ

Nucleic Acid:6 dihedrals

Amino Acid:2 dihedrals

φ

ψ

ψ

Cα

Cα

φ

φ

Complexity of RNA Backbone

RNA Backbone:Two ends and the base plane are fixed

Protein Side-chain:One end is fixed

α

β

γδε

ζ

Outline RNA Structure & Crystallography Ramachandran-like plots Measurements and Conformations Forward and Inverse Kinematics Future Work

Ramachandran Plot

φ

ψ

ψ

Cα

Cα

φ

φ

L. Murray, et al. PNAS:2003

99% backbone steric clashes are within suites

42 Conformations A-form RNA

accounts for 75% data

Observed Data

Space-filling Model forRNA Residue/Suite

Standard RNA structure parameters --From NDB (Nucleic Acid Database) Dihedrals are sampled at every 5°. Overlaps (distances of pairs of atoms

that are at least four bonds apart): --No Clash: > vdwi + vdwj - 0.2Å --Small Clash: < vdwi + vdwj - 0.2Å and > vdwi + vdwj - 0.5Å --Bad Clash: < vdwi + vdwj - 0.5Å

Valid Ranges of Dihedrals

Distribution of δ(Bimodal): Space-filling Model: -- C3’endo: [65°, 94°] -- C2’endo: [117°, 167°] Observed Data (L. Murray, et al. PNAS:2003) -- C3’endo: near 84°. -- C2’endo: near 147°.

δ

Valid Ranges of Dihedrals

Distribution of ε (Eclipsed): Space-filling Model: -- C3’endo: [-180°, -30°] [160°, 180°] whenδ=94° [-180°, -70°] [115°, 180°] whenδ=65° -- C2’endo: [-185°, -55°] whenδ=117° [-175°, -55°] whenδ=167° Observed Data (L. Murray, et al. PNAS:2003) -- C3’endo: mode=-150° -- C2’endo: mode=-100°.

δ

ε

Valid Ranges of Dihedrals

Distribution of ζ,α,β and γ: Space-filling Model: -- Peaks of ζ and α: p, m and t. -- Peaks of β: t. -- Peaks of γ: mode=t. Observed Data (L. Murray, et al. PNAS:2003) -- Peaks of ζ: p, m, t and -140 (only in C3’endo). -- Peaks of α: p, m, t and -110 (only in C3’endo). -- Peaks of β: t, 110, -135 and 135 and 80 (only in

C3’endo). -- Peaks of γ: p, m and t.

αβγδε

ζ

Demo δ-ε-ζ plots (and clash plots): -- C3’endo -- C2’endo α-β-γ plots: -- C3’endo -- C2’endo

Outline RNA Structure & Crystallography Ramachandran-like plots Measurements and

Conformations Forward and Inverse Kinematics Future Work

L. Murray, et al. PNAS:2003

99% backbone steric clashes are within suites

42 Conformations A-form RNA

accounts for 75% data

Observed Data

Measurements Known information in electron density

map: -- Phosphate positions -- base plane positions Goals: --Map the known positions to C3’endo and C2’endo puckers. --Map the known positions to 42 conformations.

Measurements

18 measurements -- distances: N1--N2, P--N1, etc. -- perpendicular distances: P -- C1-N1, P -- Sugar

Pucker -- angles: N1--P--N2, P--N1--N2, etc.

C1

N1 N2

C2

P

Criteria The measurement should well separate the C3’ endo

and C2’ endo puckers. The span of the measurement (SPANall) should be a

long range (>2Å or >60°). The ratio of the span of each conformation

measurement to the span of the whole value (SPANeach / SPANall or ΣSPANeach / SPANall) should be small.

The overlapping among different conformations should be small.

The overlapping of all SPANeach should cover the SPANall (i.e. no gaps).

Separate Sugar Puckers Space-filling Model: -- C3’endo: P -- N1-C1 > 2.537Å -- C2’endo: P -- N1-C1 < 2.313Å Proposed measurement from

Richardson’s lab: -- C3’endo: P -- First Base Plane > 2.9Å -- C2’endo: P -- First Base Plane < 2.9Å

Separate 42 Conformations

All 42 conformations -- (P--Sugar2, N1--N2 and P--N1--N2) and (P--Sugar2, C1--C2

and P--C1--C2). Conformations in the different sugar puckers -- C3’endo and C3’endo: (P--Sugar2, N1--N2 and P--N2--N1). -- C3’endo and C2’endo: (P--Sugar2, N1--N2 and P--N2--N1). -- C2’endo and C3’endo: (P--Sugar2, N1--N2 and P--N2). -- C2’endo and C2’endo: (P--Sugar2, N1--N2 and P--N2).

Outline RNA Structure & Crystallography Ramachandran-like plots Measurements and Conformations Forward and Inverse Kinematics Future Work

Electron Density Map

Image Courtesy Richardsons’ Lab

Forward and Inverse Kinematics

Forward Kinematics: -- One end is fixed. -- Fit some constraints. Inverse Kinematics: -- Both ends are fixed. -- At least 6 degrees of freedom.

Forward Kinematics

Start from phosphate. Fit bases.

Forward Kinematics

Start from base. Fit phosphates.

Inverse Kinematics

Start from two phosphates. Fit the sugar pucker.

Inverse Kinematics

Start from two bases. Fit the phosphate position.

Problems Too many degrees of freedoms. -- Use Ramachandran-like plots and the relations of

measurements and conformations to reduce the choices. Each phosphorus or sugar pucker will be used

two times. -- Keep several valid conformations calculated by forward

or inverse kinematics in each residue and suite. -- Merge the phosphorus or sugar pucker calculated from

adjacent residues or suites using the combination of the valid conformations.

Example: Solve Existing Bad Clashes

Forward Kinematics: Start from phosphorus and fits the bases.

Solve the bad clashes in the existing RNA structures.

-- Fix the atoms outside the suite and the base planes.

-- Do forward kinematics in two directions and meet all the constraints (bond lengths, angles, etc.).

-- Choose for no bad clash conformations. -- Do small adjustments if necessary.

Example: Solve Existing Bad Clashes

Suite 101 (residue 100 and 101) in ar0001.pdb Suite 50 (residue 59 and 60) in 1YFG.pdb

Improvements

Extend the forward kinematics to two residues. Solve slightly bad clashes (<vdwi+vdwj-0.4 and

>vdwi+vdwj-0.5) by wiggling atom positions.

Outline RNA Structure & Crystallography Ramachandran-like plots Measurements and Conformations Inverse and Forward Kinematics Future Work

Ramachandran-like plots

Find some good methods to project the 6D (in residue) and 7D (in suite) data into visible plots.

Analyze the collision boundaries between valid and invalid conformations.

Measurements and Conformations

Refine the relations of measurements and conformations.

Use the relations of measurements and conformations to accelerate the process of determining RNA structure.

Forward and Inverse Kinematics

Resolve bad clashes in existing RNA structures.

Build automatic tools to determine the RNA structures from electron density maps.

Acknowledgements:-- Prof. Jane Richardson, Prof. David

Richardson and Laura Murray.-- NSF grant 0076984.

The End