Forward and inverse kinematics in RNA backbone conformations

1
Forward and inverse kinematics in RNA backbone
conformations

• By
• Xueyi Wang and Jack Snoeyink
• Department of Computer Science
• UNC-Chapel Hill

2
Outline

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

3
RNA Structure

• Large ribosome subunit
• -- Chain0 2914 bases
• -- Chain9 122 bases

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RNA Structure

• Residue

Suite
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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.

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Electron Density Map

• Image Courtesy Richardsons Lab

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All-Atom Contact Analysis

• Image Courtesy Richardsons Lab

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Complexity of RNA Backbone
Nucleic Acid 6 dihedrals
Amino Acid 2 dihedrals
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Complexity of RNA Backbone
RNA Backbone Two ends and the base plane are
fixed
Protein Side-chain One end is fixed
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Outline

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

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Ramachandran Plot
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Observed Data

• L. Murray, et al. PNAS2003
• 99 backbone steric clashes are within suites
• 42 Conformations
• A-form RNA accounts for 75 data

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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 gt vdwi vdwj - 0.2Å
• --Small Clash lt vdwi vdwj - 0.2Å and
• gt vdwi vdwj - 0.5Å
• --Bad Clash lt vdwi vdwj - 0.5Å

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Valid Ranges of Dihedrals

• Distribution of d(Bimodal)
• Space-filling Model
• -- C3endo 65, 94
• -- C2endo 117, 167
• Observed Data (L. Murray, et al. PNAS2003)
• -- C3endo near 84.
• -- C2endo near 147.

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Valid Ranges of Dihedrals

• Distribution of e (Eclipsed)
• Space-filling Model
• -- C3endo -180, -30 160, 180
  whend94
• -180, -70 115, 180
  whend65
• -- C2endo -185, -55 whend117
• -175, -55 whend167
• Observed Data (L. Murray, et al. PNAS2003)
• -- C3endo mode-150
• -- C2endo mode-100.

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Valid Ranges of Dihedrals

• Distribution of ?,a,ß and ?
• Space-filling Model
• -- Peaks of ? and a p, m and t.
• -- Peaks of ß t.
• -- Peaks of ? modet.
• Observed Data (L. Murray, et al. PNAS2003)
• -- Peaks of ? p, m, t and -140 (only in
  C3endo).
• -- Peaks of a p, m, t and -110 (only in
  C3endo).
• -- Peaks of ß t, 110, -135 and 135 and 80
  (only in C3endo).
• -- Peaks of ? p, m and t.

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Demo

• d-e-? plots (and clash plots)
• -- C3endo
• -- C2endo
• a-ß-? plots
• -- C3endo
• -- C2endo

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Outline

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

19
Observed Data

• L. Murray, et al. PNAS2003
• 99 backbone steric clashes are within suites
• 42 Conformations
• A-form RNA accounts for 75 data

20
Measurements

• Known information in electron density map
• -- Phosphate positions
• -- base plane positions
• Goals
• --Map the known positions to C3endo and
• C2endo puckers.
• --Map the known positions to 42 conformations.

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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.

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Criteria

• The measurement should well separate the C3 endo
  and C2 endo puckers.
• The span of the measurement (SPANall) should be a
  long range (gt2Å or gt60).
• The ratio of the span of each conformation
  measurement to the span of the whole value
  (SPANeach / SPANall or SSPANeach / 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).

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Separate Sugar Puckers

• Space-filling Model
• -- C3endo P -- N1-C1 gt 2.537Å
• -- C2endo P -- N1-C1 lt 2.313Å
• Proposed measurement from Richardsons lab
• -- C3endo P -- First Base Plane gt 2.9Å
• -- C2endo P -- First Base Plane lt 2.9Å

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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
• -- C3endo and C3endo (P--Sugar2, N1--N2 and
  P--N2--N1).
• -- C3endo and C2endo (P--Sugar2, N1--N2 and
  P--N2--N1).
• -- C2endo and C3endo (P--Sugar2, N1--N2 and
  P--N2).
• -- C2endo and C2endo (P--Sugar2, N1--N2 and
  P--N2).

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Outline

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

26
Electron Density Map

• Image Courtesy Richardsons Lab

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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.

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Forward Kinematics

• Start from phosphate.
• Fit bases.

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Forward Kinematics

• Start from base.
• Fit phosphates.

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Inverse Kinematics

• Start from two phosphates.
• Fit the sugar pucker.

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Inverse Kinematics

• Start from two bases.
• Fit the phosphate position.

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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.

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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.

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Example Solve Existing Bad Clashes

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

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Improvements

• Extend the forward kinematics to two residues.
• Solve slightly bad clashes (ltvdwivdwj-0.4 and
  gtvdwivdwj-0.5) by wiggling atom positions.

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Outline

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

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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.

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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.

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Forward and Inverse Kinematics

• Resolve bad clashes in existing RNA structures.
• Build automatic tools to determine the RNA
  structures from electron density maps.

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The End

• Acknowledgements
• -- Prof. Jane Richardson, Prof. David Richardson
  and Laura Murray.
• -- NSF grant 0076984.