Transcript E(r 1 )

Bioinformatics 2 -- lecture 9
Ramachandran angles
Sidechain chi angles
Rotamers
Dead End Elimination Theorem
Backbone angles phi and psi
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In 1968, G.N.Ramachandran built a model like this, ala-ala-ala,
to explore the relationship between interatomic distnces and the
two freely rotatable packbone angles phi and psi. Atom-atom
distances that were too close were not permissible. What angles
were permissible?
Best
Allowed
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This plot is for
all amino acids
except Pro and
Gly.
Use SEQ:Measure-->Ramachandran Plot
to view all residues in your protein plotted phi
versus psi.
Ramachandran Plot
The regions labeled alpha and beta represent valleys of stability, surrounded
by a high energy plateau. Values of phi are limited primarily to the range
between -60 degrees and -150 degrees. For psi, the range is limited to regions
centered about -60 degrees and +120 degrees
Backbone angle statistics
Colors represent the
frequency (in bins of
10°x10°) of phi/psi
angles. E,B and H are
most common. L, l
and x are found most
often in Gly.
Allowed regions are
islands. Are bonds
really "freely
rotatable"?
Sidechain angle space -- rotamers
A random sampling of Phenylalanine sidechains, when
superimposed, fall into three classes: rotamers.
This simplifies the problem of sidechain modeling.
All we have to do is select the right rotamers and we're close to the
right answer.
Sidechain modeling
Given a backbone conformation and the sequence,
can we predict the sidechain conformations?
≠
Energy calculations are sensitive to small changes. So
the wrong sidechain conformation will give the wrong
energy.
Goal of sidechain modeling
Given the sequence and
only the backbone atom
coordinates, accurately
model the positions of the
sidechains.
fine lines = true structure
think lines = sidechain predictions
using the method of Desmet et al.
Desmet et al, Nature v.356, pp339-342 (1992)
Steric interactions determine allowed rotamers
3-bond or 1-4 interactions define the preferred angles, but these may
differ greatly in energy depending on the atom groups involved.
N
N
N
CA
CA
CA
CB
CB
CB
CG
H
O=C
H
"m"
-60° gauche
H
O=C
H
"t"
180° anti/trans
O=C
H
"p"
+60° gauche
Exercise: measure a rotamer
select these atoms
5
4
3
2
1
Create a tripeptide TWV, using Protein Builder
Now, create "meters" for the chi1 and chi2 angles
Dihedral (from right side menu)
Select N-CA-CB-CG (1-2-3-4)
Select CA-CB-CG-CD1 (2-3-4-5)
Trp sidechain is hard to rotate
W sidechain is
shown here lying
over Thr
backbone
Rotamers of W*:
p-90 +60 -90
p90 +60 +90
t-105 180
-105
t90
180
90
m0
-65
5
m95 -65
95
Rendering the molecule as space filling
(Render-->Space filling) allows you to better visualize the
contacts.
Rotamer Libraries
Rotamer libraries have been compiled by clustering the
sidechains of each amino acid over the whole database. Each
cluster is a representative conformation (or rotamer), and is
represented in the library by the best sidechain angles (chi
angles), the "centroid" angles, for that cluster.
Two commonly used rotamer libraries:
*Jane & David Richardson:
http://kinemage.biochem.duke.edu/databases/rotamer.php
Roland Dunbrack: http://dunbrack.fccc.edu/bbdep/index.php
*rotamers of W on the previous page are from the Richardson library.
Exploring Rotamers using MOE
The environment of a buried leucine in 1A07. The interior of a
protein is tightly packed. Bad packing produces voids or collisions.
Exercise: Rotamer explorer
Open 1A07 from the Protein Database
Edit-->Add hydrogens
Compute-->partial charges
Select an amino acid in the interior.
SE: Edit-->Rotamer Explorer (get from MOE)
Select rotamer with the lowest energy. Are the current chi
angles close to the angles of a rotamer? How close? Is it the
lowest energy rotamer?
Select “Mutate”. (The coordinates are permanently changed.)
Exercise: Rotamer explorer
Select an amino acid on the surface.
SE: Edit-->Rotamer Explorer (get from MOE)
Are the current angles close to a rotamer? Is it the lowest
energy rotamer?
What interactions does the best rotamer have?
Mutate.
Then select a nearby sidechain and do the same thing.
How many times would you have to mutate before you could
be sure of having the lowest energy rotamer set?
Dead end elimination theorem
•There is a global minimum energy conformation (GMEC),
where each residue has a unique rotamer.
In other words: GMEC is the set of rotamers that has the
lowest energy.
•Energy is a pairwise thing. Total energy can be broken down
into pairwise interactions. Each atom is either fixed (backbone)
or movable (sidechain).
fixed-fixed
fixed-movable
E is a constant,
=Etemplate
E depends on rotamer,
but independent of
other rotamers
movable-movable
E depends on rotamer,
and depends on
surrounding rotamers
Theoretical complexity of
sidechain modeling
The Global Minimum Energy Configuration (GMEC) is one,
unique set of rotamers.
How many possible sets of rotamers are there?
n1 n2 n3 n4 n5 … nL
where n1 is the number of rotamers for residue 1, and so on.
Estimated complexity for a protein of 100 residue, with an
average of 5 rotamers per position: 5100 = 8*1069
DEE reduces the complexity of the problem from 5L to
approximately (5L)2
Dead end elimination theorem
•Each residue is numbered (i or j) and each residue has a set of
rotamers (r, s or t). So, the notation ir means "choose rotamer r
for position i".
•The total energy is the sum of the three components:
fixed-movable
fixed-fixed
movable-movable
Eglobal = Etemplate + iE(ir) + ijE(ir,js)
where r and s are any choice of rotamers.
NOTE:
Eglobal ≥ EGMEC
for any choice of rotamers.
Dead end elimination theorem
•If ig is in the GMEC and it is not, then we can separate the
terms that contain ig or it and re-write the inequality.
EGMEC = Etemplate + E(ig) + jE(ig,jg) + jE(jg) + jkE(jg,kg)
...is less than...
EnotGMEC = Etemplate + E(it) + jE(it,jg) + jE(jg) + jkE(jg,kg)
Canceling all terms in black, we get:
E(ir) + j E(irjs) > E(ig) + j E(ig,js)
So, if we find two rotamers ir and it, and:
E(ir) + j mins E(irjs) > E(it) + j maxs E(it,js)
Then ir cannot possibly be in the GMEC.
Dead end elimination theorem
E(ir) + j mins E(irjs) > E(it) + j maxs E(it,js)
This can be translated into plain English as follows:
If the "worst case scenario" for rotamer t is better than
the "best case scenario" for rotamer r, then you can
eliminate r.
Exercise: Dead End Elimination
Using the DEE worksheet:
(1) Find a rotamer that satisfies the DEE theorem.
(2) Eliminate it.
(3) Repeat until each residue has only one rotamer.
What is the final GMEC energy?
DEE exercise
Three sidechains. Each with three rotamers. Therefore, there are
3x3x3=27 ways to arrange the sidechains. • Each rotamer has an
energy E(r), which is the non-bonded energy between sidechain and
template. • Each pair of rotamers has an interaction energy E(r1,r2),
which is the non-bonded energy between sidechains.
3
a
b
c
1
2
DEE exercise
a
1
r2
2
3
1
b
r1
c a
a
b
c
a -1 3
b 1 5
c 1 1
2
b
c
a
b
c
-1 1
1
-2 2
5
0
3
5
1
0
5
-1
0
5
5
-1
0
0
0
5
0
0
1
0
12 5
0
0
4
0
0
5
5
3
E(r1,r2)
-1
3
a -2 0
b 2 5
c 5 -1
0
0
12
4
0
0
0
5
3
0
0
1
0
0
10
0 0
E(r1)
5
0
0
0
0
0 10
E(r2)
DEE exercise: instructions
If the “best case scenario” for r1 is worse than the “worst case
scenario” for r2 you can eliminate r1.
(1) The best (worst) energies are found using the worksheet:
Add E(r1) to the sum of the lowest (highest) E(r1,r2) that have
not been previously eliminated.
(2) There are 9 possible DEE comparisons to make: 1a versus
1b, 1a versus 1c, 1b versus 1c, 2a versus 2b, etc. etc. For each
comparison, find the minimum and maximum energy
choices of the other rotamers. If the maximum energy of r1
is less than the minimum energy of r2, eliminate r2.
(3) Scratch out the eliminated rotamer and repeat until one
rotamer per position remains.
Sequence design using DEE
•Did you notice that Rotamer Explorer in MOE allows
you to choose a different sidechain?
Choosing an amino acid for each position, based on
the backbone structure and the energy function, is
called Protein Design.