A too simple model for protein folding

1
A too simple model forprotein folding

• Ethan Bolker
• Mathematics and Computer Science
• UMass Boston
• Clark University
• April 14, 2004

2
Preliminaries

• Problem source biology teaching need,
• Analysis mixes biology, cs, mathematics (
  applied mathematics)
• Ongoing help from Bogdan Calota
• See www.cs.umb.edu/eb/folding

3
How life works

• DNA (gene) makes RNA
• RNA makes polypeptide
• Polypeptide folds into protein
• Proteins interact (biochemistry)
• Cells organisms communities
• Natural selection makes gene mix evolve

4
Virtual teaching laboratories

• For Brian White (Biology, UMass Boston)
• Virtual Genetics Laboratory (VGL)
• Mendelian genetics
• http//intro.bio.umb.edu/VGL/index.htm
• Science, April 16, 2004
• GenExplorer
• the central dogma
• www.cs.umb.edu/genex/
• Watch this space

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Polypeptide ? protein

• Polypeptide sequence of amino acids
• chemical (biological) activity depends on
  three dimensional configuration
  (folding)
• Protein polypeptide folded into active shape
• Given the sequence, whats the shape?
• Wet lab
• lots of chemistry
• x-ray crystallography
• (newer tools)
• Virtual lab
• compute shape from chemical principles
• need supercomputer or grid

6
folding_at_home

• www.stanford.edu/group/pandegroup/folding/

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For beginning biologists

• Problem give students hands on experience
  showing how sequence determines shape
• Solution very simple model
• amino acid disk in the plane, hydrophobic index
  hi expresses wish to avoid wet environment
• fold polypeptide on hex grid to minimize energy
• energy S ( exposed edges) ? hi
  acids

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folding_at_umb
51882 possible configurations (5279 modulo
dihedral group symmetry) minimum energy
-131.17 minimum occurs once topology 0 2,
7 1 2 0, 7 3 4 5 6
7 0, 2
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folding_at_umb
51882 possible configurations (5279 modulo
dihedral group symmetry) minimum energy
-13.161 minimum occurs twice (second - obvious -
answer has same topology)
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Brute force search

• Try all nonintersecting walks of length n on
  plane grid of hexagons
• 1, 6, 30, 138, 618, 2730, 11946, 51882, 224130,
  964134, 4133166,
• Sequence A001334 in the
  Online Encyclopedia of Integer Sequences
  www.research.att.com/njas/sequences/
• No closed form expression
• Growth rate obviously O(5n), actual ? 4.25n
• To count foldings, divide by 12 (symmetry)

11
A (random) chain of length 17

• Five of the 11 minimum energy foldings
• All 11 show same 8 acid cool ring, hot core
• Essentially the same topology
• 12 hour computation

12
Open questions (statistical)

• How many minima?
• What is the energy distribution
• for one polypeptide, over all foldings?
• of minima, over all polypeptides of fixed length?
• Do all minima for a pp have same topology?
  (several possible definitions for topology)
• Do approximate minima have same topology?
  (several possible definitions for approximate)

13
Which amino acid universe?

• Random polypeptides acids chosen
• hi uniformly distributed in -1,1
• hi (1,-1) with probability (p, 1-p)
• from (Ala, Arg, , Tyr, Val) with
• measured hydrophobic indices
• measured probabilities of occurrence
• the natural universe

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Digression

• How do you interpolate visually between red and
  green?
• in RGB space, white is halfway
• in HSB space, yellow is halfway
• Application uses cubic interpolation to adjust
  contrast near the midpoint

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Cubic interpolation

• // Map a range of hydrophobic indices h to a
  continuum of
• // colors between RED and GREEN in HSB space.
• //
• // First map h linearly to x between 0.0 and
  1.1 so that we
• // can form convex combinations. To get
  better visual effect
• // replace x by
• // f(x) ax3 bx2
  cx
• // color(x) f(x)RED
  (1-f(x))GREEN
• // f(0) 0 means color(0) GREEN. Then find
  a, b and c so that
• // f(1) 1, f(1/2) 1/2 and f '(1/2) k
  (to be determined). Then
• // color(1) RED and color(1/2) 1/2
  (REDGREEN) YELLOW,
• //
• // When k 1, f(x) x is linear, not cubic
  (check the algebra).
• // That works well for the natural table. But
  for the virtual table it
• // provides too little contrast near the
  center. k ½ flattens out the
• // cubic at its inflection point there and
  seems to be just about right.

16
Open questions (biological)

• Nature isnt random naturally occurring
  polypeptides are not a random selection from the
  natural universe
• Which shapes can occur as the minimum energy
  configurations of polypeptides?
• which are beautiful? (polypeptide tangrams)
• which are interesting? (designer drugs)
• (I like cool rings, Brian White likes hot cores)

17
Folding algorithms

• Conjecture brute force is NP-complete
• Look for an approximate algorithm
• polynomial time
• close to true minimum with high probability
• not stochastic
• Conjecture no local algorithm will do

18
Incremental Folding

• int lookahead
• int step lookahead
• while there are acids to place
• explore all positions for the next
  lookahead acids that minimize the energy
  of configuration so far
• place the first step of those lookahead
  acids

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Incremental Folding

• lookahead step 1 is greedy
• lookahead step n is brute force
• time O( ? 4.xlookahead )
• linear in n, but exponential in lookahead

n step
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50 acids, randomly chosen from natural universe
seed 2255 minimum energy -352.38 lookahead 8,
step 1 time 139 seconds
21
50 acids, randomly chosen from natural universe
seed 2255 minimum energy -338.42 lookahead 8,
step 4 time 29 seconds
22
50 acids, randomly chosen from natural universe
seed 2255 minimum energy -351.54 lookahead 8,
step 5 time 27 seconds
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50 acids, randomly chosen from natural universe
seed 2255 minimum energy -343.98 lookahead 8,
step 7 time 15 seconds
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brute force folding for one random chain of
length 17
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incremental step sensitivity
brute force
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incremental lookahead sensitivity
13
14
11
12
10
9
5
6
8
7
brute force
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Incremental Folding

• Topology highly sensitive to step
• Energy not monotone with step or lookahead
• Can always be fooled
• May be realistic biologically
• Suffices for teaching goal

? ? ?
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More geometry

• Square grid folding is faster O(2.xlookahead)
  instead of O(4.xlookahead)
• But not nearly as pretty

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Folding in space

• Cubic grid has same folding complexity as hex
  grid in plane since each cell has six neighbors
• 3D analogue of hex grid is spherical close
  packing
• oranges at the market
• layers of hexagonally close packed planes
• cell is a rhombic dodecahedron
• each sphere has 12 neighbors
• folding complexity O(10.xn )

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Packing spheres
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H. SteinhausMathematical Snapshots
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Foldings in space
energy 37.8 time 18 seconds explored 752057
chains
energy 15.6 time 0 seconds explored 8185 chains
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Summary

• The customer is satisfied
• You can play with the applet
• The software needs work
• All the interesting questions are still open