I. Dynamics of living systems
• Understanding the dynamics at the molecular level.
• Understanding the dynamics at the cellular level
• Filling the gap between these two levels
Dynamics  Function
Life’s complexity pyramid
Oltvai & Barabasi, Science 2002, 298, 763-764
“The complexity pyramid might not be specific only to cells”
Increasing specificity/chemistry)
Dominance of molecular machinery
Different levels of structural organization:
microtubules
Challenge: to understand the long-time
dynamics of large systems
Model: Coarse-grained
Method: Analysis of principal modes of motion
(Frame transformation: Cartesian  collective coordinates)
What is the optimal (realistic, but computationally
efficient) model for a given scale (length and time) of
representation?
Which level of details is needed for representing
global (collective) motions?
How much specificity we need for modeling large
scale systems and/or motions?
What should be the minimal ingredients of a
simplified (reductionist) model?
Protein dynamics
C
(a)
Passage over one or more energy barriers
Transitions between infinitely many
conformations
Fluctuations near the folded
state
Local conformational changes
Fluctuations near a global minimum
(b)
Energy
Folding/unfolding dynamics
U
N'
N
Conformational space coordinate
Can we predict fluctuations dynamics
from native state topology only?
Gaussian Network Model
FOR MORE INFO…
Bahar, I., Atilgan, A.R., & Erman, B. (1997) Folding & Des. 2, 173.
Flory, P.J. (1976) Proc. Roy. Soc. London A. 351, 351.
Detailed specific potentials
Approximate uniform potential
“A single parameter potential is sufficient to reproduce
the slow dynamics in good detail”
Rouse chain
Connectivity matrix
1
-1
=
-1
2
-1
-1
2
-1
..
...
-1
Vtot = (g/2) [ (DR12)2 + (DR23)2 + ........ (DRN-1,N)2 ]
= (g/2) [ (DR1 - DR2)2 + (DR2 - DR3)2 + ........
2
-1
-1
1
Kirchhoff matrix of contacts
=
1 if rik < rcut
ik=
0 if rik > rcut
ii = - Sk ik
Vtot = (g/2) DRT  DR
Comparison with X-ray Temperature Factors
100
(b) 1omf
theory
75
Debye-Waller factors:
Bk = 8 2 <DRk  DRk> /3
experiments
50
25
3
0
0
50
100
150
200
250
300
80
(a) 2ccya
60
40
20
0
0
20
40
60
80
100
FOR MORE INFO...
Bahar, I., Atilgan, A.R., & Erman, B. (1997) Folding & Design 2, 173-181
120
350
Comparison with H/D Exchange – NMR data
DSi = k ln W(DRi) = - g (DRi)2/ (2T [-1]ii)
20
BPTI
15
Free energy change/RT
10
5
0


0
10


20
30
40

50
60
SNase
12
8
4
0
  
0
20

40

60
 
80
100

120
140
residue
Bahar, I., Wallquist, A., Covell, D.G., and Jernigan, R.L. (1998) Biochemistry 37,
1067.
Covariance matrix
(directly found from MD or MC trajectories)
<DR1 . DR1>
<DR1 . DR2>
<DR2. DR1>
<DR2 . DR2>
<DR1 . DR3>
C=
<DRN . DRN>
DRi = instantaneous fluctuation in the position vector Ri of atom i= Ri - <Ri>
<DR1 . DR1> = ms fluctuation of site 1 averaged over all snapshots.
Eigenvalue decomposition of C
C = U L U-1
U is the matrix of eigenvectors, L is the diagonal matrix of
eigenvalues. The ith column (eigenvector) of U is given by a linear
combination of Cartesian coordinates and represents the axis of the
ith collective coordinate (principal axis) in the conformational
space.
The ith eigenvalue represents the mean-square fluctuation along the ith
principal axis. The motion along the ith principal axis is the ith mode.
Decomposition into normal modes
• Slowest (global) modes  function
• Fastest (local) modes  stability
FOR MORE INFO...
Bahar, I., Atilgan, AR, Demirel MC, Erman B. (1998) Physical Review Lett. 80, 2733.
Demirel MC, Atilgan AR, Jernigan RL, Erman B. & Bahar, I. (1999) Protein Science 7, 2522.
Compare experimental B-factors with theoretical B-factors
theoretical B-factor
experimental B-factor
50
40
30
20
10
0
0
50
100
150
residue number
http://www.ccbb.pitt.edu/CCBBResearchDynHemRel.htm
200
250
300
Comparison of the slowest modes of T and R2
 chain
 chain
T
92-100
R2
35-40
b2
84-94
2
145-146
37-44
1
132-141
0
50
100
150
residue number
200
250
300
T  R transitions in Hb
Experimental T
Experimental R2
Reference...
Xu & Bahar, submitted.
Computed (R2)
Si = 3/2 <cos2Di> - 1/2
Order parameter
Order parameters for CO-bound and unliganded Hb
CO-Hb ()
deoxygenated ( )
50
Order parameter
0
residue index
100
150
-1 deoxygenated Hb
-1 CO-Hb
For details on theory see...
0
Haliloglu & Bahar, Proteins 1999, 37, 654-667
30
60
90
residue index
120
150
Fluctuations of the nevirapine-bound (A) and unliganded (B) forms of RT
(I)
(A)
(II)
• Fluctuating conformations of the
nevirapine-bound (A) and unliganded
(B) forms of RT. The p66 subdomains
are colored cyan (fingers), yellow
(palm), red (thumb), green
(connection) and pink (RNase H).
(I)
(B)
(II)
• See the difference in the mechanism
of global fluctuations for the liganded
and unliganded RTs. This difference is
significant given that the sizes or
distributions of fluctuations are
unaffected by ligand binding
http://www.ccbb.pitt.edu/CCBBResearchDomMot.htm
Two hinge bending sites
• (A) Two hinge-bending centers on RT
forming minima in Figure 1: (I) near the
NNRTI binding site, involving residues 107110 (cyan), 161-165 (green), 180-188 (red)
and 219-231 (blue), and (II) near the p66
connection and RNase H interface,
comprising residues 363-366 (cyan), 394-408
(green), 410-423 (loop, magenta), 424-429
(interdomain linker, red), and 504-512
(yellow).
•(B) A closer view of region II, showing
explicitly the side chains near the hinge site.
Close tertiary contacts are indicated by the
yellow dots.
Topology-based models
•
•
•
•
•
•
•
•
Near-native fluctuations
•
Ben-Avraham (1993)
Tirion (1996)
Bahar et al. (1997)
Hinsen (1998)
Sanejouand, Tama (2000)
Wriggers, Brooks (2001)
Ma (2002)
•
•
•
•
•
•
•
•
Folding/unfolding processes
(folding  loss of configurational entropy)
(springs acting on effective
centroids, usually C atoms)
Micheletti et al, PRL (1999)
Cecconi et al. Proteins (2001)
Go & Scheraga Macromolecules (1976)
Galzitskaya & Finkelstein, PNAS (1999)
Munoz et al. PNAS (1999)
Alm & Baker, PNAS (1999)
Klimov & Thirumalai, PNAS (2000)
Clementi et al (Onuchic), JMB (2000)
“Native topology determines forceinduced unfolding pathways”
Protein folding kinetics examined by a Go-like model
Koga, N. & Takada, S. J Mol. Biol. 2001, 313, 171-180
Topological and Energetic Factors: What determines the
transition state ensemble, and folding intermediates?
Simulations with Go-like potential
Applied to CI2, SH3 (2-state folders) and barnase, RNase H and CheY (have intermediates)
“ Topology plays a central role in determining folding mechanisms”
Can we use such simplified approaches for estimating amyloidogenic intermediates?
Clementi, C. Nyemeyer, H. & Onuchic, J. N. J Mol. Biol 2000, 298, 937.
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Lecture 12 - Computational & Systems Biology