1
1
Biological Overview:
Sequence-Structure Asymmetry
1DWR 2JHO
Sequence Identity ~ 85%
Horse Sperm Whale
2
Biological Overview:
Sequence-Structure Asymmetry
1LH1 2JHO
Sequence Identity ~ 20%
Sperm WhaleLupinus luteus
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Sequence-Structure Asymmetry: Understanding Protein Evolution and Capacity, Papers of Electrical and Electronics Engineering
The concept of sequence-structure asymmetry in biology, focusing on the relationship between sequence identity and structural compatibility. Topics include sequence capacity, flow between structures, and measuring protein fitness. The document also discusses methods for estimating sequence capacity, such as telescoping ratios and umbrella sampling, and the use of energy functions to measure protein fitness.
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1
Biological Overview:
Sequence-Structure Asymmetry
1DWR 2JHO
Sequence Identity ~ 85%
Horse Sperm Whale
2
Biological Overview:
Sequence-Structure Asymmetry
1LH1 2JHO
Sequence Identity ~ 20%
Lupinus luteus Sperm Whale
3
Structures are better conserved
than sequences during evolution
-> Homology modeling of structures
-> Protein design and evolution
1LH1:_ 2/3 ALTESQAALVKSSWEEFNANIPKHTHRFFILVLEIAPAAKDLFSFLKGTSE--VPQNNPE
1MBC:_ 1/2 VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASED
1LH1:_ 60/61 LQAHAGKVFKLVYEAAIQLEVTGVVV--TDATLKNLGSVHVSKGVADAHFPVVKEAILKT
1MBC:_ 61/62 LKKHGVTVLTALGAILKKK---GHHEAELKPLAQSHATKHK---IPIKYLEFISEAIIHV
1LH1:_ 118/119 IKEVVGAKWSEELNSAWTIAYDELAIVIKKEMDDAA
1MBC:_ 115/116 LHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELG
4
Biological Overview:
Sequence-Structure Asymmetry
7
Physical (stability based) network
model for sequence capacity of
structures & structural flips
Detailed (whole PDB), efficiently computable and
Detailed (whole PDB), efficiently computable and
experimentally testable model (the set of PDB
experimentally testable model (the set of PDB
structures was argued to be complete)
structures was argued to be complete)
Design of protein structures and protein switches
Design of protein structures and protein switches
Zero order model of the evolution of protein
Zero order model of the evolution of protein
sequences & structures (no selection due to
sequences & structures (no selection due to
function). function).
8
Related work on capacity of
specific protein models
ShakhnovichShakhnovich
Dill
Dill
WolynesWolynes
Thirumalai
Thirumalai
LevittLevitt
…
So far no global view of capacity (and
So far no global view of capacity (and
thermodynamics) of the PDB, no flow.
thermodynamics) of the PDB, no flow.
9
Is capacity relevant to biology?
Capacity shows weak correlation with the
Capacity shows weak correlation with the
number of sequences that
number of sequences that are found for a
are found for a
particular fold in the NR database (correlationparticular fold in the NR database (correlation
coefficient 0.2)coefficient 0.2)
Capacity correlates with mutation ratesCapacity correlates with mutation rates
measured experimentallymeasured experimentally ((J. D. Bloom, D. A.J. D. Bloom, D. A.
Drummond, F. H. Arnold, and C. O.Drummond, F. H. Arnold, and C. O. WilkeWilke. Structural. Structural
determinants of the rate of protein evolution in yeast. Mol.determinants of the rate of protein evolution in yeast. Mol. BiolBiol..
EvolEvol. 23:1751-1761, 2006. And C.. 23:1751-1761, 2006. And C. WilkeWilke, private, private
communication).communication).
10
Experimental tests
Collaboration with ThomasCollaboration with Thomas MaglieryMagliery onon
Lambda repressor - 160K mutants
Lambda repressor - 160K mutants
Protein flipsProtein flips - Bryan lab (flips are well- Bryan lab (flips are well
known for RNA)
known for RNA)
13
Approximate Energy Functions
TE13 (Toby, Elber)
X
X
N
E D
X
Q
( )
1
n
X
i i
i
E S X c!
=
THOM2 (Meller, Elber)
( )
X
i j ij
i j
E S X c!! r
<
A
R
N
E D
C
Q
Ergodic/well mixed model
NP complete
14
Specifying Sequence-Structure
Fitness Criteria
H
I
L
M K
F
P
A
R
N
E D
C
Q
Original (Native)
Sequence:
S
nat
: HILKMFP
Candidate Sequence:
S : ARNEDCQ
We say S is fit for X if ( ) ( )
nat
E S! X " E S! X
15
Estimating the Sequence
Capacity of a Fold
In general, we would like to compute the function N(E) :
Specifically, we want to compute N ( E
nat
).
The number of all possible sequences is 20The number of all possible sequences is 20
nn
, for proteins, for proteins
of length of length nn. For small proteins,. For small proteins, nn ≈≈ 50.50.
Random sampling of sequence space does not work:
Random sampling of sequence space does not work:
since since NN (( EE
natnat
) / 20) / 20
nn
can be exponentially small.can be exponentially small.
Need a more sophisticated counting method.Need a more sophisticated counting method.
N ( E ) = S : E ( S! X )" E
16
Estimating N(E)
Express
Express
N
N
E
E
S
S
E
E
S
S → X ) < E
k
as
as
telescoping ratios/ umbrella sampling:
telescoping ratios/ umbrella sampling:
1 2
1
( ) ( ) ( )
( ) ( ) ...
( ) ( ) ( )
ref
ref m
N E N E N E
N E N E
N E N E N E
=!!!!
EE
refref
:: Pre-selected reference energyPre-selected reference energy
N
N (
( E
E
refref
) : Number of sequences below
) : Number of sequences below E
E
refref
EE
11
…… EE
mm
: Values above ratios are: Values above ratios are itermediatesitermediates..
19
Choosing Intermediate E
k
Values
E mean
E
min
E nat
20
Algorithm Summary: N ( E )
Given a structureGiven a structure XX , compute, compute EE
meanmean
andand NN (( EE
meanmean
Pick
Pick E
E
11
E
E
mm
s.t.
s.t. E
E
kk
E
E
k+k+ 11
and (
and ( E
E
kk
E
E
kk +1+
) is
) is
decreasing withdecreasing with k.k.
For
For k
k = 1
m
m , run the Markov chain for
, run the Markov chain for t
t steps.
steps.
Compute
Compute l
l
tt
k
k +1) /
l
l
tt
k
k )
N
N
E
E
kk +1+
N
N
E
E
kk
1 2
1
mean
mean m
N E N E N E
N E N E
N E N E N E
21
Counting With THOM2:
Markov Chain Convergence
State space is connected: all states communicate
State space is connected: all states communicate
via the minimum-energy state
via the minimum-energy state
Mixing time (the Markov chain is
Mixing time (the Markov chain is ergodic
ergodic ) is
) is
polynomial in sequence length.
polynomial in sequence length.
Generalizes Morris-Sinclair algorithm (1999) for
Generalizes Morris-Sinclair algorithm (1999) for
counting knapsack solutions to arbitrary alphabets counting knapsack solutions to arbitrary alphabets
22
Sequence capacity without
competition
Remain at a particular fold
Remain at a particular fold
X
X
and perform
and perform
counting for this single structure
counting for this single structure
Compute
Compute
N(E),
N(E),
Ω
Ω
(E)
(E)
=dN/dE
=dN/dE
, S(E)=log(
, S(E)=log(
Ω
Ω
(E))
(E))
The temperature of sequence selection is defined
The temperature of sequence selection is defined
as: as: T=(T=(dS/dEdS/dE))
-1-
Compute for a representative set of PDB
structures (~3000 folds)
25
Counting without competition:
Different folds, same length
26
Sequence Capacity and Flow
27
Coarse description of fold
connectivity
Different temperatures forDifferent temperatures for alternate foldsalternate folds
suggests lack of connectivity. suggests lack of connectivity.
T=(
T=(
dS/dE
dS/dE
-1-
28
Temperature distribution for the
potential TE-
31
Differentiating Between
Competing Folds
For a foldFor a fold XX , how do we count only sequences that, how do we count only sequences that
both are both compatible with both are both compatible with XX and preferand prefer XX to allto all
other folds?
other folds?
Given a structureGiven a structure XX and a set ofand a set of competingcompeting
structures structures YY = {= { YY
1
1
,, …… YY
K
K
},}, we wish to estimatewe wish to estimate
the function
the function C
C
E
E
) which gives the size of the set
) which gives the size of the set
: & , 1
j
S! " E S # X < E E S # X < E S # Y j = … K
32
Differentiating Between
Competing Folds
( X )
min
E
( X )
E
( X )
nat
E
( Y )
nat
E
( Y )
min
E
( Y )
E
E
(
X
)
=
E
(
Y
)
X Y
33
Differentiating Between
Competing Folds
( X )
min
E
( X )
E
( X )
nat
E
( Y )
nat
E
( Y )
min
E
( Y )
E
E
(
X
)
=
E
(
Y
)
X Y
34
Counting with Competition
E
min
E
k
E
k+ 1
k k ret k
C E = N E! f E
37
Maximal Retention Energy E
E
E
**
is the first-encountered energy where
is the first-encountered energy where
f
f
retret
is
is
maximum (mostly 1). In general,
maximum (mostly 1). In general,
E
E
*
*
E
E
min
min
Between
Between
E
E
*
*
and
and
E
E
min
min
the protein evolves in
the protein evolves in
structure and sequence spaces. structure and sequence spaces.
Below
Below
E
E
*
*
only the sequence evolves.
only the sequence evolves.
Native
Native
proteins are always found above proteins are always found above EE
**
38
E
and Behavior of f
ret
( E )
E
mean
E
nat
E
*
f
ret
39
E
nat
-E
and contact density
40
E
and f
ret