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A too simple model for protein folding. Ethan Bolker Mathematics and Computer Science UMass Boston Clark University April 14, 2004. Preliminaries. Problem source: biology teaching need, Analysis mixes biology, cs, mathematics (= applied mathematics) Ongoing help from Bogdan Calota

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a too simple model for protein folding

A too simple model forprotein folding

Ethan Bolker

Mathematics and Computer Science

UMass Boston

Clark University

April 14, 2004

preliminaries
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
how life works
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
virtual teaching laboratories
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 …
polypeptide protein
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, what’s the shape?
    • Wet lab
      • lots of chemistry
      • x-ray crystallography
      • (newer tools)
    • Virtual lab
      • compute shape from chemical principles
      • need supercomputer or grid
folding@home
folding@home

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

for beginning biologists
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 = Σ(# exposed edges)  hi acids

folding@umb
folding@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]

folding@umb9
folding@umb

51882 possible configurations (5279 modulo dihedral group symmetry)

minimum energy -13.161

minimum occurs twice (second - obvious - answer has same topology)

brute force search
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 Sequenceswww.research.att.com/~njas/sequences/
  • No closed form expression
  • Growth rate obviously O(5n), actual  4.25n
  • To count foldings, divide by 12 (symmetry)
a random chain of length 17
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
open questions statistical
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)
which amino acid universe
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

digression
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
cubic interpolation
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) = ax^3 + bx^2 + 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 (RED+GREEN) = 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.

open questions biological
Open questions (biological)
  • Nature isn’t 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)

folding algorithms
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
incremental folding
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

incremental folding19
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

slide20
50 acids, randomly chosen from natural universe

seed 2255

minimum energy -352.38

lookahead 8, step 1

time 139 seconds

slide21
50 acids, randomly chosen from natural universe

seed 2255

minimum energy -338.42

lookahead 8, step 4

time 29 seconds

slide22
50 acids, randomly chosen from natural universe

seed 2255

minimum energy -351.54

lookahead 8, step 5

time 27 seconds

slide23
50 acids, randomly chosen from natural universe

seed 2255

minimum energy -343.98

lookahead 8,

step 7

time 15 seconds

incremental lookahead sensitivity
incremental: lookahead sensitivity

13

14

11

12

10

9

5

6

7

8

brute force

incremental folding27
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

● ● ●

more geometry
More geometry
  • Square grid folding is faster: O(2.xlookahead) instead of O(4.xlookahead)
  • But not nearly as pretty
folding in space
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 )
foldings in space
Foldings in space

energy 15.6

time 0 seconds

explored 8185 chains

energy 37.8

time 18 seconds

explored 752057 chains

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