Optimization algorithms for proteinprotein interactions

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Optimization algorithms for protein-protein
interactions

• Rahul S. Sampath
• George Biros
• University of Pennsylvania

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Overview

• Electrostatic protein-protein interactions
• VCP SPICE
• Poisson-Boltzmann Equation (PBE)
• Sensitivity analysis
• Fast Algorithms
• Preliminary results

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Electrostatics protein-protein interactions

• Need to study protein-protein interactions
• Vital for many biological processes
• Immune system pathogen mechanisms
• Application in drug design
• Why is electrostatics important?
• Many bio-molecules are charged
• Electrostatic force is long-ranged
• Initiate binding
• Sometimes the only dominant force

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Apps for Protein-Protein Simulations

• Protein Assembly
• Will 2 proteins bind?
• Binding Sites?
• Shape of complexes?
• Describe thermodynamics of binding
• Protein Design
• Design mutants (slightly different sequence)
• Complementary proteins Facilitate binding
• Example VCP/SPICE

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VCP SPICE

• Variola smallpox causing virus
• Vaccinia used in smallpox vaccine
• VCP Vaccinia complement control protein
• SPICE Smallpox inhibitor of complement enzymes
  (genetically engineered Variola protein)
• Complement proteins Immune system

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VCP SPICE

• SPICE gt 90 similar to VCP
• SPICE gt 1000 times as powerful as VCP in
  controlling complement proteins
• Small proteins electrostatics likely dominant

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Electrostatic Potential around VCP,SPICE their
Mutants (Lambris et al. )

• Mutants selected randomly
• Potential distribution using Coulomb's law (Swiss
  PDB Viewer)
• Good correlation with experiment

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Shortcomings of the current approach

• All mutations done experimentally
• No deterministic method to select mutants
• Could we predict sites for mutations numerically?
• Modeling errors
• Atomistic vs continuous
• Coulomb's vs Poisson-Boltzmann
• Solvent Implicit vs explicit

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Atomistic Vs Continuum approx.

• Atomistic model Molecular Dynamics
• Expensive
• Small sized problems
• Continuum Model Poisson-Boltzmann Equation

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Treating Solvent as a Continuum

• Explicit Solvent simple
• Solvent treated separately
• Implicit Solvent more realistic, complex

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Implicit Solvent Model Solvent Accessibility
Free Region (?f) Set of centers of the solvent
probes which do not intersect any of the
atoms. Solvent Accessible Region (?s) Union of
volumes of the solvent probes with centers in
?f.
2-D view of Protein in solvent Courtesy Michael
Levitt Britt H Park (1993)
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Building the material properties from the geometry

• PDB file gives coordinates of centers of atoms
• PDB2PQR adds additional data charges radii
• Build a domain ¼ 3 times protein size. Generate
  grids.
• Protein is partitioned among different processors
• ?r , ? in each element are average of nodal
  values.
• Every node in the mesh is either in ?f or ?s or
  ?p.

VCP
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The Poisson-Boltzmann Equation

• Poisson-Boltzmann Equation (PBE)

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The Linearized Poisson-Boltzmann Equation

• Linearized Poisson-Boltzmann Equation (LPBE)

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Difficulties with PBE

• Nonlinearity
• Dirac-delta functions as sources
• Complicated geometry
• Material discontinuity

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Estimation of mutation sites
Introduce auxiliary objective function
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Adjoint-Sensitivity Operator for PBE
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More on the Adjoint Formulation

• Linear
• Independent of the perturbation term
• Solve the adjoint problem once Vs solving for
  sensitivity many times
• Essentially combinatorial
• q not arbitrary
• 20 amino-acids to choose from

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Solving the PBE

• Nonlinear solver
• Inexact Newton iterative linear solvers
• PBE is an elliptic equation
• Multigrid (MG) works well with elliptic equations
• A Multigrid Tutorial, Briggs et al.
• BlackBox Multigrid, Dendy et.al.
• Baker et.al., APBS. IBM J. R D (2001)
• Baker et.al., Electrostatics of nano-systems
  Application to microtubules and the ribosome.
  PNAS (2001)

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Full Multi-grid Algorithm
Fig. CourtesyPaul Heckbert
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What is new in our approach?

• Matrix-Free implementation
• Developed on top of PETSc
• Gives a lot of handles to change the smoothers
• Extend PETSc restriction/prolongation using
  matrix-free operators to reduce memory usage
• Application to sensitivity analysis

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Matrix Free Methods

• Problem of storage for large problems
• Creating and destroying matrices is expensive
• Iterative solvers only need the result of a
  Matrix-Vector multiplication, not the matrix
  itself

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Implementing a Matrix Free Multi-grid

• Matrix-free Restriction and Interpolation
• Coarse grid operator in Multi-grid
• A2h R Ah P (Galerkin Coarsening)
• No Explicit Galerkin Coarsening for Matrix-Free

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Components in the algorithm

• Use PDB2PQR to add missing data from PDB.
• Partition the protein.
• Assign material properties.
• Solve Main problem MG (LPB) or Newton-MG (NLPB)
• Solve Adjoint problem MG (LPBy) or Newton-MG
  (NLPBy)
• Generate VTK files display using Paraview.
• Solve optimization problem (To be implemented)

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Approximation of Jacobians in each Element

• J1 and J2 LPBE
• J3 NLPBE Adjoint equations.
• J3 expensive, poor MG performance. J4 ¼ J3
• The residual is still computed exactly.

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Performance of the approx Jacobian

• The exact jacobian does not converge
• Failure to satisfy the Galerkin condition
• Must use Matrix-based explicit Galerkin approach
• Approx jacobian converges
• Performace does not deteriorate with mesh size
• 643, 73 iterations, 13 min on 1 processor
• 1283, 70 iterations, 120 min on 1 processor

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Matrix based DD Vs Matrix free MG

• MFMG outperforms DD for large problems
• MFMG is asymptotically cheaper
• Performance of DD deteriorates, not for MG

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Scalability for more processors
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Potential Iso-contours
Generated Using Paraview LPB (Blue , Red
-) NLPB (Green , Yellow -)
Using Swiss-PDB viewer Coulomb's law
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Acknowledgements

• John D. Lambris, Department of Pathology
  Laboratory Medicine, University of Pennsylvania.
• PETSc Team
• The computations were performed in part on the
  National Science Foundation HP GS1280 system at
  the Pittsburgh Supercomputing Center.
• The other members of our group Santi, Shravan,
  Cosmina, Hari and Bharath.

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• Questions?