Forward and inverse kinematics in RNA backbone conformations By Xueyi Wang and Jack Snoeyink Department of Computer Science UNC-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 Structure Residue Suite α β γ δ ε ζ δ γ β α ζ δ ε

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 γ δ ε ζ Nucleic Acid: 6 dihedrals Amino Acid: 2 dihedrals φ ψ ψ CαCα CαCα 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 γ δ ε ζ

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

Ramachandran Plot φ ψ ψ CαCα CαCα φ φ

L. Murray, et al. PNAS: % backbone steric clashes are within suites 42 Conformations A-form RNA accounts for 75% data Observed Data

Space-filling Model for RNA 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: > vdw i + vdw j - 0.2Å --Small Clash: < vdw i + vdw j - 0.2Å and > vdw i + vdw j - 0.5Å --Bad Clash: < vdw i + vdw j - 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°] when δ =94° [-180°, -70°] [115°, 180°] when δ =65° -- C2’endo: [-185°, -55°] when δ =117° [-175°, -55°] when δ =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. α β γ δ ε 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: % 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 N1N2 C2P

Criteria The measurement should well separate the C3’ endo and C2’ endo puckers. The span of the measurement (SPAN all ) should be a long range (>2Å or >60°). The ratio of the span of each conformation measurement to the span of the whole value (SPAN each / SPAN all or ΣSPAN each / SPAN all ) 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-0.5) 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 The End