6.581J / 20.482J
Foundations of Algorithms and Computational Techniques in Systems Biology
Professor Bruce Tidor
Professor Jacob K. White
9 February 2006
Thursday
MIT 6.581/20.482J
FOUNDATIONS OF ALGORITHMS AND COMPUTATIONAL
TECHNIQUES IN SYSTEMS BIOLOGY
Spring 2006
LECTURE 2 : MODELS OF PROTEINS
?
selection of
problems & phrasings
selection of methods, algorithms, &
techniques
BIOLOGY
COMPUTATION
Fundamental role of models:
Understanding
Prediction
Design
side chain
DNA
(genome)
i–1
backbone
- mRNA
i
-
Protein
i+1
amino acid residue
H
Ri – 1
Ca
N
C
H
O
H
O
N
C
H
Ca
Ca
Ri
Ri + 1
H
N
C
H
O
The “Ri” groups are chosen from the common 20 amino acid side chains - chemical
diversity
H
(1) size:
small - large
N
RGly : –H
RTrp
(2)
polarity:
hydrophobic
-
polar
-
charged
-
RArg :
O
-
RLeu :
(3)
uniformity of character
(4)
local backbone flexibility
Gly
(flexible)
RAsn :
-
C
NH2
Pro
(rigidity)
+
N
H
NH2
NH2
6.581J / 20.482J
Foundations of Algorithms and Computational Techniques in Systems Biology
Professor Bruce Tidor
Professor Jacob K. White
Coordinate systems:
1) Absolute Cartesian Coordinates
 x1 
 x 
st
 2  cartesian coordinates of 1 atom
 x3 


x4 

r
X =  x5 


 x6 
 M 


 x3 N −1  Nth atom
x 
 3N 
2) Relative Coordinates – Internal
Think of the molecules as graphs where
- atoms are vertices
bond lengths & bond angles – rigid
- bonds are edges
torsions – soft
χ21
χ11
H
+
1
1
1
H3N φ ψ C ω
χ22
generally 180º
sometimes 0º O
–
Desire : Mapping
O
r 3N
r 3N
X →EX
57 cartesian
degrees of freedom
–
10 torsional
degrees of freedom
H
2
N φ
Cα
19 atoms
COO
Cα ψ 2
χ12
H
χ32
O
( )
“energy”
scalar value
⇒ Bias toward mechanistic basis for model
nuclear & electrons
Chemistry − Physics (Quantum Mechanics)
∂Ψ
h2 2
Schrödinger Equation: ih
=−
∇ Ψ + V ( x )Ψ (x, t ) ≡ Hˆ Ψ ( x, t )
2m
∂t
Linus Pauling
Observations:
• bond lengths, angles – fixed
• torsions – “soft” & sinusoidal
• atoms appear to have a fixed spherical size & approach to contact neighbors
• complementary electrostatics
2
6.581J / 20.482J
Foundations of Algorithms and Computational Techniques in Systems Biology
Professor Bruce Tidor
Professor Jacob K. White
9 February 2006
Thursday
MIT 6.581/20.482J
FOUNDATIONS OF ALGORITHMS AND COMPUTATIONAL
TECHNIQUES IN SYSTEMS BIOLOGY
Spring 2006
δ+
δ–
δ+
δ–
O
H
N
hydrogen bond
O
+
–
H3N
O
salt bridge
Molecular Mechanics Potential:
r
E X3 N = U COVALENT + U NON-COVALENT
( )
bonded
U COVALENT =
through space
∑
1
2
i : bonds
+
kb ,i (bi − bo,i ) +
2
∑
1
2
i : angles
k [1 + cos(niφi − δ i )]
∑
2
∑
1
2 Φ ,i
i : impropers
k
(Φ
− Φ o,i )
2
i
1
2 φ ,i
i : torsions
U NON-COVALENT =
Cij 
qq
+ ∑ i j
6 
rij 
i >j
i > j ε rij
123
144244
3
 Bij
∑  r
12
ij
−
Electrostatics
→Coulombic
van der Waals
→Lennard-Jones
UvdW
kθ ,i (θ i − θ o,i ) +
electron
cloud
repulsion
induced
dipole –dipole
interaction
rij
3