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Geometry for Biomolecules

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Geometry for Biomolecules Powered By Docstoc
					Geometry for Biomolecules


           Patrice Koehl
  Computer Science and Genome Center
        http://www.cs.ucdavis.edu/~koehl/
                         The Importance of Shape

 Sequence
KKAVINGEQIRSISDLHQTLKK
WELALPEYYGENLDALWDCLTG
VEYPLVLEWRQFEQSKQLTENG
AESVLQVFREAKAEGCDITI
                              Structure



                                             Function




                                                    ligand
Enzyme – Substrate Binding

  Substrate           Enzyme
              +
  (ligand)            (receptor)




        Induced Fit
Co-factors may induce the fit: allostery

                                         Ligand


                                         Receptor




       Co-factors bind                            Ligand binds




                     Co-factors
                     induce
                     conformational
                     Change: allostery
   Geometry for Biomolecules
• Geometric measures of molecules
  – Space filling diagrams and cavities
  – Applications:
     • Accessibility
     • Drug binding
     • Meshing
• Geometry for Interactions
  – Docking
   Geometry for Biomolecules
• Geometric measures of molecules
  – Space filling diagrams and cavities
  – Applications:
     • Accessibility
     • Drug binding
     • Meshing
• Geometry for Interactions
  – Docking
  Representations of Biomolecules
Cartoon             Space-filling Model
 Computing the Surface Area
and Volume of a Union of Balls
         Computing the Surface Area
        and Volume of a Union of Balls

Power Diagram:
              Computing the Surface Area
             and Volume of a Union of Balls

Decomposition of the
Space-filling diagram
                 Computing the Surface Area
                and Volume of a Union of Balls



                      i

                           i
                 i




Surface Area                                     Volume

                                            4     N

                                                  
          N
A  4    i   i
                  2                      V         i3  i
         i 1
                                             3     i 1
              Computing the Surface Area
             and Volume of a Union of Balls




The weighted Delaunay triangulation is the dual of the power diagram
                 Computing the Surface Area
                and Volume of a Union of Balls




The dual complex K is the dual of the decomposition of the space-filling diagram
          Computing the Surface Area
           and Volume of a Protein
                                                    K complex




                                                    Pocket
Protein      Delaunay Complex



                        http://www.cs.ucdavis.edu/koehl/ProShape/
                 Computing the Surface Area
                    and Volume of RNA         K complex




P4-P6 domain         Delaunay Complex
Group I intron

                                                Pocket
   Geometry for Biomolecules
• Geometric measures of molecules
  – Space filling diagrams and cavities
  – Applications:
     • Accessibility
     • Drug binding
     • Meshing
• Geometry for Interactions
  – Docking
Experimental measures of accessibilities

                   Hydroxyl radical footprinting:




                            H5’’
                              H3’

                                           H1’
                                     H2’
                      H5’
                               H4’
                                       HO2’
                 Footprinting count / Ribose H accessibility




Residue number
   Geometry for Biomolecules
• Geometric measures of molecules
  – Space filling diagrams and cavities
  – Applications:
     • Accessibility
     • Drug binding
     • Meshing
• Geometry for Interactions
  – Docking
      BINDING POCKETS IN 16S RIBOSOMAL RNA



                      Hygromycin B




PDB structure: 1HZN
BINDING POCKETS IN 16S RIBOSOMAL RNA




                                       8Å

                        Probe Size
              1.4 Å
BINDING POCKETS IN 16S RIBOSOMAL RNA
   Geometry for Biomolecules
• Geometric measures of molecules
  – Space filling diagrams and cavities
  – Applications:
     • Accessibility
     • Drug binding
     • Meshing
• Geometry for Interactions
  – Docking
                      Meshes




•   Unstructured mesh have advantages over structured mesh
    on boundary conformity and adaptivity
•   Smooth surface models for molecules are necessary for
    unstructured mesh generation
        Molecular Surface

                     Disadvantages
                         • Lack of smoothness
                         • Cannot be meshed with good quality




An example of the self-intersection of molecular surface
                     Molecular Skin
     • The molecular skin is similar to the molecular
       surface but uses hyperboloids blend between
       the spheres representing the atoms
     • It is a smooth surface, free of intersection




Comparison between the molecular surface model and the skin model for a protein
Mesh Quality
            Example




Skin mesh




                      Volumetric mesh
   Geometry for Biomolecules
• Geometric measures of molecules
  – Space filling diagrams and cavities
  – Applications:
     • Accessibility
     • Drug binding
     • Meshing
• Geometry for Interactions
  – Docking
                      What is docking?
Docking is finding the binding geometry of two interacting molecules
with known structures

The two molecules (“Receptor” and “Ligand”) can be:

        - two proteins
        - a protein and a drug
        - a nucleic acid and a drug

Two types of docking:

        - local docking: the binding site in the receptor is known,
                         and docking refers to finding the position
                         of the ligand in that binding site

        - global docking: the binding site is unknown. The search
                          for the binding site and the position of the
                          ligand in the binding site can then
                          be performed sequentially or simulaneously
                What is docking?

Water


                              L
            R                      R
                L
                          L
                      R
        L                      L
            R
                               R
                        What is docking?

Two types of docking:

        - bound docking: the goal is to reproduce a known complex,
                        where the starting structures for the receptor
                        and ligand are taken from the structure of
                        the complex
                        (testing docking method)

        - unbound docking: the structures of the receptor and ligand are
                         taken from data on the unbound molecules
                       (actual docking)
                                  “Docking” scenarios
                                                             “Historical”
                                                             techniques:
                        Rigid Receptor, Rigid Ligand         Lock and Key
Increasing difficulty




                        Rigid Receptor, Flexible Ligand

                        Flexible Receptor, Rigid Ligand
                                                             Current
                        Flexible Receptor, Flexible Ligand
                                                             Research area


                                           Hopefully soon…
             Why is docking hard?

• The main problem is the dimension of the
  conformational space to be explored:
            - rigid structure alignment: 3D (hard)

            - rigid docking: 6D (hard)

            - flexible docking: 6D + Nfb (impossible!)
              Docking Scoring Criteria

• Geometric match:
   –   Prevent overlap between atoms of the receptor and ligand
   –   Maximum shape compatibility
   –   Large surface burial
   –   No large cavity at interface
• Energetic Match (Force-field + Statistical potential)
   – Good hydrogen bonding
   – Good charge complementarity
   – Polar/polar contacts favored, polar / non polar contacts
     disfavoured
   – Low “free energy”
         Docking Search Strategies

• Full search
  – Grid approaches (FFT…)
• Directed search
  – Spherical harmonics surface triangle
  – Geometric hashing
• Pseudo Random
  – Simulated annealing / Monte Carlo
  – Genetic algorithms
DOCK: the first Docking Program (Kuntz, 1982)




                    http://dock.compbio.ucsf.edu/
    Global Rigid Docking: a FFT approach
1. Representation:
Receptor:
                              Assign value to each cell:

            R                         Exterior: a(i,j) = 0
                     R
                                      Surface: a(i,j) = +1

                                      Interior: a(i,j) = -15


Ligand:

                                      Exterior: b(i,j) = 0

            L        L                Surface: b(i,j) = +1

                                      Interior: b(i,j) = -15
                Global Rigid Docking: a FFT approach
   2. Scoring:
                                                                     L

                      R                 R                        R
                                          L
Translation Y




                     L


                                         Translation X


                     Score   a(i, j )b' (i, j )       where b’ is the grid for the
                               i   j                     ligand after rotation and
                                                         translation
   Global Rigid Docking: a FFT approach
2. Scoring:
 Test all possible positions of ligand on receptor:
            - Test all rotations of ligand
            - For each rotation, test all translations of ligand
              grid over receptor grid
                        Score(i,j) = Receptor(i,j)*Ligand(i,j)

 Rotation R; Translation: T = Tx + Ty + Tz:



    S R, T    a (i, j, k )b' i  Tx , j  Ty , k  Tk 
                      N   N    N


                     i 1 j 1 k 1


   For each R, this requires N6 operations…


But, for a given rotation, this is a correlation product, that can be computed
in Fourier Space!
        Global Rigid Docking: a FFT approach

                                               Fourier
    R          Discretize                      transform
                               R
                                                           A=DFT(a)




                                                            C=A*B     S=iDFT(C)

                                             Fourier
    Rotation                                 transform
L                             L                          B=DFT(b)

               Discretize



                            Computing cost: N3log(N3)!
             Biogeometry
• Many, many more applications!!
  – Visualization
  – Shape recognition, shape matching
  – ….

				
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posted:6/30/2012
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