Coarse Representations
      for the Essential Features of the Protein Surface

                    Mark Gerstein
             (mbg@cb-iris.stanford.edu)

The protein surface is usually represented (and viewed) in terms of
thousands of intersecting atoms (spheres). The great amount of detail
in such a representation obscures the overall shape of the surface and
creates computational problems for many popular docking and surface
matching schemes. Two approaches are presented for removing
unnecessary detail and representing the essential features of the
protein surface.

(1) Reverse-crystallography: The protein surface is represented in
terms of a resolution-dependent Fourier series. This approach allows
for hierarchical, resolution-sensitive shape matching and very
efficient docking.

(2) The hydration surface: The protein surface is defined by the
second shell of water molecules surrounding it. The hydration surface
is similar to the commonly used molecular surface, but the "probe"
water positions are determined in the course of a molecular
simulation, rather than just purely geometrically, so it is argued
that this surface is more chemically meaningful.

MMBJC seminar 
Beckman 402
3:30 PM Wednesday 8 June 1994
        
The following references are summarize this work:

1	M Gerstein (1992). “A Resolution-Sensitive Procedure for Comparing Protein Surfaces and its Application to the Comparison of Antigen-Combining Sites,” Acta Crystallographica A48: 271-276.
2	M Gerstein & R Lynden-Bell (1993). “Simulation of Water around a Model Protein Helix. 1. Two-dimensional Projections of Solvent Structure,” Journal of Physical Chemistry  97: 2982-2990.
3	M Gerstein & R Lynden-Bell (1993). “What is the Natural Boundary for a Protein in Solution?” Journal of Molecular Biology  230: 641-650.
4	M Gerstein & R Lynden-Bell (1993). “Simulation of Water around a Model Protein Helix. 2. The Relative Contributions of Packing, Hydrophobicity, and Hydrogen-Bonding,” Journal of Physical Chemistry  97: 2991-2999.

1
Coarse Representations
for the Essential Features
of the Protein Surface

Mark Gerstein
R Lynden-Bell (Cambridge Chemistry)
M Levitt (Stanford Struc. Biol.)

8 June 1994

Overheads Available Electronically with URL
ftp://cb-iris.stanford.edu/pub/mbg/CoarseSurf/
CoarseSurf.94Jun8.talk.
{ps, word.rtf, word.hqx, txt, abstract}
2
Coarse Representations
for the Essential Features
of the Protein Surface
•	Surface representations are important for: 
°	docking & rational drug design
°	understanding molecular recognition
•	Customary representations: 
°	VDW surface: all the nooks & crannies
°	Richards’ accessible surface & molecular surface smooth away some of the detail with a probe sphere
dot surface	Slide 01 8.VI.94
vdw surface	Slide 02 8.VI.94
Connolly surface	Slide 03 8.VI.94
•	My new representations: 
1	Reverse-Crystallography: 
represent the protein surface in terms of a Fourier Series 
2	A Hydration Surface:
represent the protein surface in terms of the position of water molecules in a molecular simulation
3
Reverse-Crystallography
•	Procedure
°	Threshold VDW potential and fill in to create an envelope (1 inside, 0 outside)
°	Fourier transform envelope
	F(s) = FFT[ f(x) ]
x = 3D position in Cartesian space
	s = 3D position in frequency space
•	Resolution-Sensitive Shape Comparisons
°	Compare transforms vs resolution S 
°	Compare to reference R(s) = FFT[ r(x) ]
°	Comparison metrics :
1.	straight difference 
D(S) = BBC·((F - R) (F - R)*)
2.	correlation
C(S) = F( BBC·(dFdR* + dF*dR),2sFsR) 
 where dF = F-BBC·(F)
4
Implementation
•	Immunoglobulin CDR regions
°	Surface diversity 
with common structural framework
°	McPC603, 17/9, D1.3, HyHEL-10
REI is reference
°	Cut out loops, threshold, put in P1 cell, and transform

5
Implementation (Cont.)

6
Results
D & C vs S	Slide 05 8.VI.94
•	Want to detect similarities in overall features, such as similar clefts
•	Implicit scaling in the correlation, so:
°	D best for very dissimilar shapes 
( compare bird’s face with person’s face )
°	C best for similar shapes
( compare two peoples’ faces)

D(0) = (V-VR)2 

7
Discussion of
Resolution-Sensitive Procedure
•	In the future can use other than rectangular basis sets
°	Spherical harmonics YS(l,m)
(slow, but no choice of origin or reference necessary)
°	Basis functions designed for discrete data: i.e., the Hadamard transforms from signal compression
•	Hierarchial Representation
°	A few numbers sum up overall low-res shape 
ﬁ Automatic shape classification
°	Other terms describe high-res “texture”
(cf. fractal measures of surface roughness)
•	Very Efficient for Docking....
8
Docking
•	Want to compute a score for many different relative orientations of two bodies
°	Score = correlation = c(t) 
°	Orientations in terms of  a 
3D translation t & a 3D rotation R
°	This involves a 6D search.
•	Can use FFT to reduce docking to a 3D search by eliminating the translation search
°	Katchalski-Katzir et al.  (1992). 
PNAS  89: 2195
°	Real-space correlation is a
reciprocal-space multiplication
°	Like to molecular replacement in crystallography
Katchalski	Insert 06 8.VI.94

N   = number of grid points on an axis
9
Hydration surface
•	Protein surface should be defined in terms of the environment in which it resides — i.e. in water.
•	Where is the protein surface chemically? 
°	Protein has “region of influence” through its Hbonds & hydrophobic hydration.
°	“What is the natural boundary of a protein in solution?”
•	Richards tried to take into solvent with
Molecular & Accessible Surfaces 
°	Roll simple probe sphere (radius 1.4 Å) on surface of protein. Locus of probe sphere centers is accessible surface (1971). 
°	Molecular surface = 
Contact Surface + Re-entrant Surface (1977)
°	Implemented by Connolly
(1983)
Connolly surface	Slide 06A 8.VI.94

10
The Plan
•	Propose to use the 2nd shell of water molecules in a simulation to define a hydration surface
°	Use real water as “probe sphere”
°	Effectively a smoothed surface
1	Characterize water around single helix.
°	Density, orientation, and energy.
°	Represent results structurally
2	Competition between packing and Hbonding
°	Compare water distribution to that of
simple solvents, such as Ar (l) 
3	Use 2nd shell	to define natural boundary 
(hydration surface)
°	When are 2 helices together & apart?
4	Think about constructing & comparing hydration surfaces of real proteins

11
Simple Model Systems
•	1 or 2 polyalanine a-helices 
°	14 residues of :
ACO1(—N—, ,(Me))ACO1(Ca,|,Cb)ACO1(—, , )ACO1(C,||,O)ACO1(—, , )
°	in 22.219 ¥ 22.219 ¥ 20.93 Å cell
containing 321 waters
•	Very hydrophobic for a protein surface
but our interest is in hydrophobic hydration
•	Exploit symmetry for averaging
S2	Single Helix Slide
S11	TwoSys
12
Monte-Carlo Method
•	Want to calculate average densities, energies, dipoles, quadrupoles, &c
	· µ(x,y) Ò = F(1,Z)  I(A(phase,space),, µ(x,y) eSUP10(-bH(qN))) dqN
Sample states randomly: 
•	from a uniform distribution &
weight them by Boltzmann factor
(Forever integration).
•	from a Boltzmann distribution &
weight uniformly
(Monte-Carlo integration). 
°	Metropolis Rule tells how to sample according to Boltzmann distribution.

•	Constructed Highly Parallel
MC program for water in NVT ensemble
13
Simple Pair Potential
				  Coulomb	+	Lennard-Jones	
Eij = ISU(i,, )ISU(j,, ) B( F(qiqj,rij)  +  F(Aij,rS(12,ij))  –  F(Bij,rS(6,ij)) )
Parameters
•	CHARMM on 14 residues of alanine
•	321 TIP3 waters
°	3-site model consistent with protein
°	H: q= +0.4e
O: q= -0.8e	s=3.2Å	 e=0.6kJ/mol
°	Other non-polarizable potentials (TIP4P) make little difference. 
14
Projections 
•	Straight Projection	down axis of helix
	g(Rf) =	· g(r) Òz
	g(r) oxygen density in 3d 
•	Helical Projection 
along path of polypeptide chain
°	Or “unwind helix” then project straight
	· g(R,f,z) ÒHLX = 	BBC·(g(L(R, f + F(2πz,p), z)))z	
					F(2π,p) = F(360°,5.4Å) = F(100°,1.5Å)

°	Apply atoms in helix & get 4 quadrants:
B	= Backbone	(—N—Ca—C—)
C	= Carbonyl	(=O)
M	= Methyl		(—Cb)
G	= Gap			(    )
•	Show results in ‘gray-level’ graphs 
(black is high) rather than featureless 1d plots 
15
Oxygen Density
•	Straight
3 shells: 6.3 Å, 1.3; 4.6 Å, 1.1; 9Å 
•	Helical
°	3 peaks in Methyl & Backbone
°	Inner 2 peaks merge in Carbonyl to 1 big peak (3.5 vs 1.4)
•	Water not compressed
•	Compare to Richards-Connolly
Molecular Surface 
°	1.4 Å sphere rolling on surface
°	‘Argon’ hydration surface 
•	Use to tell when 2 helices touch.
S3	Ost
S4	ZO
S5	Z0-3d
lj321-Ost	lj272-Ost
S12	Oseq
16
Hydration Surface  
=	2nd shell = boundary relative to bulk 
°	Differing orientation of waters & energetic effects die off after 1st shell
•	Structural Aspects of Hydrophobicity
°	Density in gap is low (0.23)
°	Are narrow clefts more hydrophobic by virtue of their shape? 
•	Qualifications?
°	Electrostatic Interactions Longer-ranged 
than 2nd shell (DNA 20Å)
°	Our model system does not have strongly charged groups.
17
Long Calculation
but Parallel Program
•		2 ¥ 106 MC cycles for >900 waters
ª	4 ns MD (2fs MD = MC cycle)
ª	100 days CPU on 1 i860  (4.5 s/cycle)
ª	200 days on an iris (R3000)
ª	10 days on 8-16 i860s (Alliant = 24 i860s)
•	Coarse-grained Parallelism (COVI)
°	concurrent MC runs & then sum results
°	heat to 800K, slow cool, then do averages
°	ideal for networks of workstations (Linda)
 
	for(R=0; R< num_runs ; R++)
		if  (fork() == IS_CHILD) {
	for(S=0; S<num_steps ; S++, T=heat_cool_run(S) ){ 
		 for(W=0; W< num_waters ; W++){ 
	for(P= nbr_list[P] ; *P != END_LIST; P++){
	} /* P */
		} /* W */
		analysis(results[P],config,S);
	} /* S */
		} /* R */
	wait(); 
	for(R=0; R<num_runs; R++)
		totals = accumulate(results[R],totals);

18
Coarse Representations
for the Essential Features
of the Protein Surface
•	Reverse-Crystallography
°	Represent the protein surface in terms of a Fourier series
°	Resolution-sensitive comparisons
°	Efficient docking by eliminating translation search
•	Hydration Surface
°	Use 2nd shell of water molecules in a simulation to define a surface
°	That is, use a more realistic probe sphere than in Richards’ molecular surface