Analysis of Protein Geometry, Particularly Related to Packingat the Protein Surface

Copyright 1997 Mark Gerstein

All Rights Reserved

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Analysis of Protein Geometry, Particularly Related to Packingat the Protein Surface

CSB Seminar, 1 May 1997

Mark Gerstein

What is Protein Geometry?

• Coordinates (X, Y, Z’s)

• Derivative Concepts
  • Distance, Surface Area, Volume, Cavity, Groove, Axes, Angle, &c

• Relation to
  • Function, Energies (E(x)), Dynamics (dx/dt)

Packing at Interfaces

• Voronoi volumes (and D. triangulation) to measure packing

• Tight core packing v. Loose surface packing

• Grooves & ridges: close-packing v. H-bonding

• How packing defines a surface (hydration surface)

• Implications for Motions

Classic Papers

• Lee, B. & Richards, F. M. (1971). “The Interpretation of Protein Structures: Estimation of Static Accessibility,” J. Mol. Biol. 55, 379-400.

• Richards, F. M. (1974). “The Interpretation of Protein Structures: Total Volume, Group Volume Distributions and Packing Density,” J. Mol. Biol. 82, 1-14.

• Richards, F. M. (1977). “Areas, Volumes, Packing, and Protein Structure,” Ann. Rev. Biophys. Bioeng. 6, 151-76.

Lego

Packing ~ VDW force

• Longer-range isotropic attractive tail provides general cohesion

• Shorter-ranged repulsion determines detailed geometry of interaction

• Billiard Ball model, WCA Theory

Close-packing is Default

• No tight packing when highly directional interactions (such as H-bonds) need to be satisfied

• Packing spheres (.74), hexagonal

• Water (~.35), “Open” tetrahedral, H-bonds

Water v. Argon

Voronoi Volumes

• Each atom surrounded by a single convex polyhedron and allocated space within it
  • Allocation of all space (large V implies cavities)

• 2 methods of determination
  • Find planes separating atoms, intersection of these is polyhedron
  • Locate vertices, which are equidistant from 4 atoms

Voronoi Volumes, the Natural Way to Measure Packing

Packing Efficiency

= Volume-of-Object-----------------Space-it-occupies

= V(VDW) / V(Voronoi)

• Absolute v relative eff.
  V1 / V2
  

• Other methods
  • Measure Cavity Volume (grids, constructions, &c)

Delauney Triangulation, the Natural Way to Define Packing Neighbors

• Related to Voronoi polyhedra (dual)

• What “coordination number” does an atom have? Doesn’t depend on distance

• alpha shape

• threading

Standard Residue Volumes

• Database of many hi-res structures (~100, 2 Å)

• Volumes statistics for buried residues (various selections, resample, &c)

• Standard atomic volumes harder… parameter set development...

Clustering into a set of Atom Types

• Which atoms are equivalent? How many types valid?

• 18 types, [CNOS][34]H[123][bsu]

Atoms have different sizes

• Difficulty with Voronoi Meth.Not all atoms created equal

• Solutions
  • Bisection -- plane midway between atoms
  • Method B (Richards)Positions the dividing plane according to ratio
  • Radical Plane

• VDW Radii Set

Set of VDW Radii

• Great differences in a sensitive parameter (Radii for carbon 1.87 vs 2.00)

• Complex calculation: minimizing SD, iterative procedure, from protein structures

• Look for common distances in CCD

• Preliminary Solution

Atom Bondi New

C4__ 1.87 1.88

C3H1 1.76 1.76

C3H0 1.76 1.61

O1HO 1.40 1.42

O2H1 1.40 1.46

N___ 1.65 1.64

S___ 1.85 1.77

Standard Core Volumes (Prelim.)

PPT Slide

PPT Slide

Small Packing Changes Significant

• Exponential dependence

• Bounded within a range of 0.5 (.8 and .3)

• Many observations in standard volumes gives small error about the mean (SD/sqrt(N))

ALREADY MADE:Apply Voronoi Polyhedra to Protein Surface

• Helps to understand why ligands bind
  • Elements of molecular recognition
  • Docking & rational drug design

• Problem of missing atoms (waters) in crystal structures

• Simulation to put in waters

• Ultra-high resolution structures (1.5 Å or better)

ALREADY MADE:Surface Volume Increase in 20 High Resolution Structures

• 20 ultra hi-res structures (ə.5 Å),crystal symmetry

• Volume (Å3) of -CH2-
  • Protein core 23.5
  • In solution 26.5
  • In organic solvent 29.0
  • On protein surface 24.7

• Similar results for all atom types
  • 55% non-polar 45% polar
  • Missing charged atoms

ALREADY MADE:Simulation

• ENCAD NVE MD, F3C waters, ~1 ns
  • PTI 2300 H2O + 454 protein atoms
  • RNase 3732 H2O + 951 protein atoms

• Sim. & xtal. struc. similar overall results
  • Vol. incr. -CH2- on surface ~7%
  • Vol. incr. all atoms on surface ~6%

• However, the few charged atoms shrink, due to electroconstriction
  • -O (D,E) Surface 9 atoms 13.2 Å3
  • -O (D,E) Core 1 atom 13.7 Å3
  • -NH2 (R) Surface 14 atoms 28.0 Å3
  • -NH2 (R) Core 2 atoms 29.7 Å3

• Water volume can be calculated and classified on basis of protein atoms water shares most polyhedra faces with (“near”)
  • Near charged 28.8 Å3
  • Near polar 29.2 Å3
  • ? Bulk 29.7 Å3
  • Near apolar 29.9 Å3

ALREADY MADE:picture of PTI simulation

Ways of Rationalizing Packing

ALREADY MADE:Compressibility

• Fluctuations in polyhedra volume over simulation related to compressibility
  • ° Same way amplitude of a spring is related to spring constant
  • ° =
  • ° Rigorous for NPT only, approximately true for part of NVE

• Simulation Results (avg. fluctuations as %SD and compressibility)
  • Protein core 9.7 % .14
  • Protein surface 11.7 % .29
  • Water near protein 13.2 % .50
  • Bulk water 11.9 % .41
  • ° Consistent with more variable packing at protein surface

• Results verified by doing high-pressure simulation (5000 atm, 10000 atm)
  • ° Allows calculation of compressibility from definition C17

ALREADY MADE PICTURE:[C19: 7rsa.surfcv.cont.rgb.gz]

• ° Fractional Accessibility (Us, Richards)
• ° SurFractal (Tainer, Getzoff)

• Expansion of Grooves
  • Avg. ?V of a Groove atom 3.5 Å3
  • Avg. ?V of a Ridge atom 3.2 Å3
  • Avg. ASA of a Groove atom 5.8 Å2
  • Avg. ASA of a Ridge atom 19.2 Å2
  • Groove ?V per unit surf. area 0.60 Å
  • Ridge ?V per unit surf. area 0.16 Å
  • Atoms in grooves also appear to be more compressible.
  ALREADY MADE:Structural Hydrophobicity
  
  • Sharp, Nicholls, & Honig
    • Hydrophobicity related to surface structure (curvature) not chemical types of atoms
    • Concave surfaces are more hydrophobic
  • Can Structural Hydrophobicity be Observed?
    • ° Physically explained by difficulty of a water to fit into narrow surface crevice
    • \ Narrow grooves might not be packed that tightly
    • \ Look at relation between packing efficiency and surface curvature
  ALREADY MADE:Expansion of Surface Grooves
  
  • Various measures of curvature to divide surface into ridges and grooves
    • ° Occluded solid angle (Sharp, Honig)
  Packing defines the “Correct Definition” of the Protein Surface
  
  • Voronoi polyhedra are the Natural way to study packing!
  • How reasonable is a geometric definition of the surface in light of what we know about packing
  • The relationship between
    • accessible surface
    • molecular surface
    • Delauney Triangulation (Convex Hull)
    • polyhedra faces
    • hydration surface
  Defining Surfaces from Packing: Convex Hull and Layers of Waters
  
  Defining a Surface from the Faces of Voronoi Polyhedra
  
  Accessible Surface as a Time-averaged Water Layer
  
  The Hydration Surface: Trying to Model Real Water
  
  ALREADY MADE:Toy System
  
  • Number density
    • g = Normal water,straight & helical projections
    • For usual RDF “volume elements” are concentric spherical shells
    • Here, they are tiny vertical columns and helices perpendicular to page
    • More intuition about groove expansion
  • Compare water packing with that of simple liquid (“re-scaled Ar”)
  • Also, look at F3C v. TIP4 and MC v. MD
  Second Solvent Shell:Water v LJ Liquid
  
  ALREADY MADE:helical projections
  
  • [C23: ZO.ps.Z] & [C25: lj272-ZO.ps.Z]
    • 2nd shell forms a “boundary” around protein
    • Higher water density near partially charged atoms (O and N) than around uncharged atoms (CA and CB)
    • “Uncharged” water in st. & hel. proj.
  Hydration Surface
  
  • Bring together two helices
    • Unusually low water density in grooves and crevices — especially, as compared to uncharged water
    • Fit line through second shell
  Packing at Interfaces
  
  • Voronoi volumes (and D. triangulation) to measure packing
  • Tight core packing v. Loose surface packing
  • Grooves & ridges: close-packing v. H-bonding
  • How packing defines a surface (hydration surface)
  • Implications for Motions
  Credits
  
  Cyrus Chothia
  
  Jerry Tsai
  Mike Levitt
  Ruth Lynden-Bell
  Yehouda Harpaz
  Robin Taylor
  http://bioinfo.csb.yale.edu/Geometry/csb-seminar
  References
  
  • Gerstein, M. & Chothia, C. (1996). Packing at the Protein-Water Interface. Proc. Natl. Acad. Sci. USA 93, 10167-10172.
  • Gerstein, M. & Lynden-Bell, R. M. (1993). Simulation of Water around a Model Protein Helix. 2. The Relative Contributions of Packing, Hydrophobicity, and Hydrogen Bonding. J. Phys. Chem. 97, 2991-2999.
  • Gerstein, M. & Lynden-Bell, R. M. (1993). What is the natural boundary for a protein in solution? J. Mol. Biol. 230, 641-650.
  • Gerstein, M., Tsai, J. & Levitt, M. (1995). The volume of atoms on the protein surface: Calculated from simulation, using Voronoi polyhedra. J. Mol. Biol. 249, 955-966.
  • Harpaz, Y., Gerstein, M. & Chothia, C. (1994). Volume Changes on Protein Folding. Structure 2, 641-649.
  18 (or 11) different atom types
  
  • [CONS] [34] H [0123] [bsu]
  • 18 11 5 Comment
  • C3H0b C3H0_ C3___ carboxyl and carbonyl Cs without branching and aromatic C without H
  • C3H0s "" "" carbonyl carbons with branching (mainchain carbonyls Cs with C in CB)
  • C3H1b C3H1_ "" aromatic carbons with one hydrogen, larger for facing away from chain
  • C3H1s "" "" aromatic carbons with one hydrogen, small for usw. facing towards chain
  • C43Hu C4H3_ C4___ aliphatic carbons with three hydrogens, methyls
  • C4H2b C4H2_ "" aliphatic carbons with two hydrogens, big
  • C4H2s "" "" aliphatic carbons with two hydrogens, small
  • C4H1b C4H1_ "" aliphatic carbons with one hydrogen and no branching
  • C4H1s "" "" aliphatic carbons with one H and branching from all 3 heavy atom bonds
  • N3H0u N3___ N____ amide nitrogens with no hydrogens (proline's N)
  • N3H1b "" "" amide nitrogens with one hydrogen (on sidechains)
  • N3H1s "" "" amide nitrogens with one hydrogen (all other mainchain N's)
  • N3H2u "" "" all amide nitrogens with 2 hydrogens (all at the end of side chains)
  • N4H3u N4___ "" amide nitrogen charged, with 3 hydrogens (lysine's NZ)
  Compare Volumes: Core v. Amino Acid Crystals
  
  • Example residue volume: Leu (Å3)
    • Residue in the protein core 165
    • VDW envelope 128
    • Absolute packing efficiency 78 %
    • Sidechain in the protein core 101
    • Sidechain in a.a. crystal 110
    • Sidechain in solution 107
  • Example atomic volume: -CH2- (Å3)
    • Protein core 23.5
    • In solution 26.5
    • In organic solvent 29.0
  ALREADY MADE: Compare standard volumes with amino acids SOLUTION volumes
  
  • Use already made overhead
  • Example the volume of a Leu (Å3)
    • Residue in the protein core 165
    • VDW envelope 128
    • Sidechain in the protein core 101
    • Sidechain in a.a. crystal 110
    • Sidechain in solution 107
  • ? Solution volumes originally determined by Cohn et al. ; recently, refined by others
    • ° Cohn et al. (1934). JACS 56: 784
    • ° Mishra & Ahluwalia (1984). JPC 88: 86
    • ° Rao et al. (1984). JPC 88: 3129
  • ? How does sidechain volume change upon protein folding ? [?SC]
    • -4 aliphatic residues pack more tightly in core than soln.
    • 0 aromatic, hydroxyl, sulfur-containing same volume in core as in soln.
    • +7 charged and amide residues do not pack as well in core as in soln.
  SKIP:Compare standard residue volumes with those amino acids occupy in solution (Detail)
  
  • Use already made overhead
  • Residue &Side Chainin protein interior aliphatic residues –4
  • aromatic, hydroxyl, or sulfur-containing 0
  • charged and amide residues +7
  SKIP:Paradoxabout protein folding
  
  • 1 Solution-transfer models of the hydrophobic effect predict an increase in volume of hydrophobic residues upon folding
    • ° ?V(Ala from water to cyclohexane) > 0(Kauzmann, 1959, Adv in Protein Chem.. 14: 1 )
  • Volume (Å3) of -CH2-
    • ° in Protein Interior 23.5
    • ° in Water 26.5
    • ° in organic solvent 29.0
  • 2 Experiment shows : no net change in protein volume (Brandts et al. ,1970, Biochemistry 9: 1038 )
  • Both core residue volumes and amino-acid solution volumes can be used to estimate accurately the partial specific volume of proteins (within 1%).
  SKIP:Our ResultsResolve this Paradox
  
  • 3 We find :a decrease in volume of hydrophobic residues on folding, which is counterbalanced by an increase in volume of charged & amide residues, giving no net change.
  • This cancellation provides a new and consistent explanation for the experimental fact.