Protein Folding

1
Modeling and Simulation of Protein Folding
Zhijun Wu Department of Mathematics Program on
Bio-informatics and Computational Biology Iowa
State University Ames, Iowa
2
Folding an Open Chain
Graph Embedding
Polynomial Time Solvable
3
Folding a Closed Chain
Graph Embedding
NP-complete (Saxe 1979) (More and Wu 1997)
4
Biological Building Blocks DNA, RNA, Protein
DNA
GAA GTT GAA AAT CAG GCG AAC CCA CGA CTG
RNA
GAA GUU GAA AAU CAG GCG AAC CCA CGA CUG
Protein
GLU GAL GLU ASN GLN ALA ASN PRO ARG LEU
5
Protein Folding
LEU
ARG
ASN
PRO
ALA
ASN
GLN
GLU
GLU
VAL
GLU
GLU
ASN
VAL
LEU
ARG
PRO
ASN
ALA
GLN
. . .
6
Examples
Myoglobin, John Kendrew, 1962, Nobel Prize in
Chemistry
Examples
Prion, Stanley B. Prusiner, 1997, Nobel Prize in
Physiology and Medicine
7
Folding Mechanism
Bond Constraints Bond Angle Constraints
Dihedral Angle Constraints Non-Bond Constraints
xi
aijk
xj
xl
ajkl
aijkl
aklm
xm
xk
ajklm
8
Folding Mechanism
xi
aijk
xj
xl
ajkl
aijkl
aklm
xm
xk
ajklm
9
Folding Mechanism
xi
aijk
xj
xl
ajkl
aijkl
aklm
xm
xk
ajklm
10
Mathematical Model
Initial-Value Problem
11
Numerical Solutions
x
x(t)
xk1
xk
t
tk
tk1
Verlet 1967
12
Time Scales for Protein Motion
Bond vibration
Isomeris- ation
Water dynamics
Helix forms
Fastest folders
Typical folders
Slow folders
10-15 femto
10-12 pico
10-9 nano
10-6 micro
10-3 milli
100 seconds
13
Folding of Villin Headpiece Subdomain (HP-36)
Duan and Kollman 1998
14
Alternative Approaches
Boundary-Value Formulation
Ron Elber 1996
15
Single Shooting
x1
x
x1 ?(v0) f(v0) ?(v0)-x1 f(v0) 0
x1
v0
x0
v0
t0
t1
t
Newtons Method
16
Multiple Shooting
x
fj(xj-1, vj-1, xj) ?j(xj-1, vj-1) - xj fj(
xj-1, vj-1, xj) 0 j 1, , m
?j
xm
(xj-1,vj-1)
x0
t0
tm
t
(Vedell and Wu 2005)
Newtons Method
17
Alternative Approaches
Energy Minimization
min E (x1, x2, , xn)
Scheraga, et al.
18
Energy Landscape
Peter Wolynes, et al.
19
Energy Transformation
Scheraga et al. 1989, Shalloway 1992, Straub
1996, Wu 1996, More Wu 1997
20
Having puzzled the scientists for decades, the
protein folding problem remains a grand challenge
of modern science. The protein folding problem
may be studied through MD simulation under
certain boundary conditions. An efficient
optimization algorithm may be developed to obtain
a fast fold by exploiting the special structure
of protein energy landscape. The successful
simulation of protein folding requires correct
physics, efficient and accurate algorithms, and
sufficient computing power.