Структурная и Вычислительная Биология
Соотношение между Структурой и Динамикой Белков
Ярослав Рябов
Protein Domain Dynamics
Predictions of Protein Diffusion Tensor
Assembling Protein Structures
Statistics of genomes
Caenorhabditis elegans A
Anopheles gam
biae A
Tetraodon nigroviridis A
Bos taurus B
Canis fam
iliaris BD
rosophila Melanogaster A
Gallus gallus B
Danio rerio B
Macaca m
ulatta A M
us musculus A
Hom
o Sapiens A
Pan troglodytes A
1
2
3
Caenorhabditis elegans B
Anopheles gam
biae B
Tetraodon nigroviridis B
Bos taurus A
Canis fam
iliaris A D
rosophila Melanogaster B
Gallus gallus A
Danio rerio A
Macaca m
ulatta B M
us musculus B
Hom
o Sapiens B
Pan troglodytes B
Headlines
•Protein Domain Dynamics a model with applications to NMR
•Diffusion Properties of Proteins from ellipsoid model
•Assembling Structures of Multi-Domain Proteins and Protein Complexes guided by protein diffusion tensor
•Genome Evolution from statistics of proteins’ properties to Exon size distribution
Experimental data
NMR relaxation R1, R2, NOE,RDC Spin Labeling dataetc.
Theoretical model C(t)
Also a function of model parameters like
Protein structure, Protein diffusion tensor Parameters of internal motions etc.
a very General Concept
Fitting routine
Protein Domain Dynamics a model with applications to NMR
Ryabov & Fushman, Proteins (2006), MRC (2006), JACS (2007)
Time Correlation Function Wigner rotation matrixes
)}(),(),({)( tttt
Euler Angles
20 0
20
Instantaneous Residue orientation (I)Laboratory frame (L)
z
x
yz
x
y
IL
tILqILq DDtC
)()()( )2(0 ,
0*)2(0 , )exp()()exp()( )(
,)(, indimD l
nml
nm
Abragam, 1961 Principles of Nuclear Magnetism
LI
Laboratory frame (L)
Instantaneous Residue orientation (I)
Protein orientation (P)
Domain orientation (D)
Averaged Residue orientation (R)
LP
PD
DR
RI
))(())0((
))(())0((
))(())0((
))(())0(()(
2,''
*2
2''
*2
2''
*2
2
2',
2
2',
2
2',
2
2'.
2'
*2
tDD
tDD
tDD
tDDtC
RIknRInk
DRnmDRmn
PDmlPDlm
ll mm nn kkLPqlLPql
P0B
L
D RI
Assumption
All dynamic modes are statistically independent from each other
))(())0((
))(())0((
))(())0((
))(())0(()(
2,''
*2
2''
*2
2''
*2
2
2',
2
2',
2
2',
2
2'.
2'
*2
tDD
tDD
tDD
tDDtC
RIknRInk
DRnmDRmn
PDmlPDlm
ll mm nn kkLPqlLPql
))(())0((
))(())0((
))(())0((
))(())0(()(
2,''
*2
2''
*2
2''
*2
2
2',
2
2',
2
2',
2
2'.
2'
*2
tDD
tDD
tDD
tDDtC
RIknRInk
DRnmDRmn
PDmlPDlm
ll mm nn kkLPqlLPql
Protein orientation (P)
Laboratory frame (L)
LPL P
Dz
Dy
Dx
Rotation diffusion of an ellipsoid
Favro, Phys. Rev. 1960Woessner, J. Chem. Phys. 1962
2
2,
*,
)2(,
0*)2(,, 5
1)()()(
znzmz
tE
PL
tPLnqPLmq
LPqnqm aaeDDtC z
mza , Decomposition coefficients andEigenvaluesareFunctions of Diffusion tensor Principal values only
zE
zyx DDD , ,
P coincides with Diffusion tensor principal vectors
Huntress Jr, Adv. Magn. Reson. 1970
Domain orientation (D)
Protein orientation (P)
PD P
state “–” state “+”
)()( ), ()(
)()()(
)2(,
0*)2(,
, ,
00
)2(,
0*)2(,,
0
tDPlnDPkm
tDPDPDPeq
DP
tDPlnDPkm
PDnlmk
DDtpp
DDtC
DPDPt
DPDP
}/exp{), (
}/exp{1), (
}/exp{1), (
}/exp{), (
jDPDP
jDPDP
jDPDP
jDPDP
tpptp
tptp
tptp
tpptp
pp
pp
DPeq
DPeq
)(
)(
)/( j
Domain mobility correlation time
Model of Interconversion between Two States
Assumption
Averaged Residue orientation (R)
Domain orientation (D)
DR
Averaged orientation of a residue within a domain is constant and is given in PDB conventionDR
)()())(())0(()( )2(,
*)2(,
)2(,
*)2(,, DRhlDRskDRDRhlDRsk
DRlhks DDtDDtC
PDB 1D3Z: Ubiquitin
Instantaneous Residue orientation (I)
Averaged Residue orientation (R)
RI
Lipari & Szabo – “Model Free” J. Am. Chem. Soc. 1982
Wobbling in a Cone model
}/exp{)1()( 220,0,0,0 lhs
RIhs tSStC
)1)(cos3(2
1)( 20)*2(
0,0 IRIRDS
)cos(12
)cos( S
Order Parameter
R
I
NMR 15N Relaxation and NOE
for Ubiquitin dimer
17 fitting parameters including
And all 12 Euler angles for both Domain orientations in two conformations
zyx DDD , ,
pp ,
+ +
- -
K 48
H 68
L 8
V 70I 44
Derived Structures and Dynamics Parameters
Crystal Ub2
PDB 1AAR
Docked:Ub2 + UBA
pH 6.8 pH 4.5
+ +
- -
K 48
H 68
L 8
V 70I 44
Derived Structures and Dynamics Parameters
Crystal Ub2
PDB 1AAR
Docked:Ub2 + UBA
pH 6.8 pH 4.5
Both protonated 0.002 0.826
One protonated 0.091 0.165
Non protonated 0.907 0.009
His 68 pKa=5.5Fujiwara et al, J. Bio. Chem. 2004
pH 6.8 pH 4.5
Conclusions
• Suggested model captures essential features of domain dynamics in Ubiquitin Dimer and provides information about domain mobility and orientations in this protein that cannot be derived from other approaches
• The results derived from this model agree with independent experimental data such as crystal and docked structures, chemical shift perturbation, and spin labeling data
• However, it should be noted that this approach can report only about mutual domain orientations - not positioning - in multidomain proteins
Connections with the next Project
• Is it possible to use derived information about protein dynamics for further characterization of protein structure?
State of the art: Bead algorithms
Modeling Diffusion Properties of Proteins
J. Garcia de la Torre et al., J Mag Res 147, 138–146 (2000)
Extrapolation for Bead size zero
10 000 beads
Why an ellipsoid model ?
Diffusion Properties of Proteins from ellipsoid model
Diffusion Tensor Ellipsoid Shell
z
y
x
D
D
D
00
00
00
3 Euler angles forDiffusion Tensor PAF
Az
Ay
Ax
3 Euler angles forEllipsoid orientation
One-to-One mapping
Main problem:
How to build the ellipsoid shell for a protein structure ?
Inertia is irrelevant for protein diffusion
Diffusion Properties of Proteins from ellipsoid model
State of the art: Inertia-equivalent ellipsoids
01.0ForcesFriction
Forces Inertia
LV
Re
srelaxationinertial13
10 1.0 relaxationinertiald Å
Diffusion process related to friction
The friction occurs at protein surface
Diffusion Properties of Proteins from ellipsoid model
Ryabov & Fushman, JACS (2006)
Proposal
Let us use topology of protein surface to derive Equivalent ellipsoid for protein
Mapping protein surfaces
Mapping protein surfacesSURF program
Build equivalent ellipsoidPrincipal Component Analysis (PCA)
N
j
nnj
mmjnm XXXX
NCov
1, ))((
1
N
j
mj
m XN
X1
1
nnn E SSCov
nn Ea 3
Hydration shell
Hydration shell
Hydration shellEquivalent ellipsoid is approximately twice bigger
Build equivalent ellipsoid with PCA
N
j
nnj
mmjnm XXXX
NCov
1, ))((
1
N
j
mj
m XN
X1
1
nnn E SSCov nn Ea 3
Evaluate diffusion tensor components with Perrin’s Equations
02232 ))(()( sasasa
dsP
zyx
02232 ))(()( sasasa
dsQ
xzy
02232 ))(()( sasasa
dsR
yxz
)(3
)(1622
22
RaQa
aaC
zy
zyx
)(3
)(1622
22
PaRa
aaC
xz
zxy
)(3
)(1622
22
QaPa
aaC
yx
yxz
ll C
kTD
Perrin J. Phys. Radium (1934, 1936)
ELM Algorithm
Complexity of the algorithms
ELM Nat
HYDRONMR Nat
2
ELM : HYDRONMR
1 : 500
Comparison with experimental data proteins from 2.9 to 82 kDa
Ryabov & Fushman, JACS (2006)
Comparison with the experimental data
Correlation times for 841 protein structuresHydration shell effect
Power law qHPCA M
Fractal surface dimension
q ~ 0.923
qd f /2 ~2.2 2.3
10 20 30 40 50 60 70
10
20
30
40
50
60
70
2 PC
A(0
.0)
, [n
s]
H (3.2) , [ns]
Corr [ 2 PCA(0.0), H (3.2) ] = 0.994
Correlations for diffusion tensor components
Correlations of diffusion tensors orientations
• ELM provides at least the same level or accuracy and precision in description of experimental data as the other approaches, which represent protein surface with large number of friction elements (HYDRONMR). This results 500 times speed up in calculation time
• Hydration shell makes apparent diffusion correlation time of real proteins approximately twice longer
• Most of the diffusion tensors of real protein structures analyzed in our work are axially symmetric. In general rhombicities of diffusion tensors for real proteins are small and below the precisions of existing calculation methods predicting protein diffusion properties
Summary
a very General Concept
Assembling Structures of Multi-Domain Proteins and Protein Complexes guided by protein diffusion tensor
Ryabov & Fushman, JACS (2006)
Positioning Procedure
Matches components of diffusion tensor
Minimizing
3,3,1
2
,,2
iji
expji
calcji DD
Use to be impossible with bead and shell algorithms like HYDRONMR
NOW possible with ELM
Tests for Proteins with Known Structureand Predicted Diffusion Tensor
b) 1A22a) 1BRS c) 1LP1
Fit with components of Diffusion tensor Predicted with ELM
= 7.510 -8
RMSD=0.0034 [A]
= 4.410 -9
RMSD=0.0085 [A]
= 6.110 -9
RMSD=0.0008 [A]
Tests for Proteins with Known Structureand Predicted Diffusion Tensor : Mapping 2 space
a) 1BRS
b) 1A22
c) 1LP1
Tests for Proteins with Known Structureand Experimental Diffusion Tensor : HIV-1 protease
X[-1.8;1] Y[-2;1] Z[-1;0.3]
RMSD=0.36 [A]
Tests for Proteins with Known Structureand Experimental Diffusion Tensor : MBP
RMSD=1.34 [A]
X[-3.1;1.3] Y[-1.7;1.7] Z[-1;1.1]
Protein with UnKnown Structure
Ub2
X[-2.5;4]Y[-3.9; 3.7]Z[-2;1.9]
• Diffusion tensor can provide significant portion of new information encoded in principal values of its components. This information can be successfully used for assembling of multi domain protein structures
• Tested for a number of examples the suggested procedure of multi-domain proteins assembling showed very high performance and accuracy
• The method provides the unique possibility to assemble dynamic protein structures when traditional X-ray, NMR NOE based, or docking methods are inappropriate
• Method shall be further developed to take explicitly into account rotational degrees of freedom, potentials of intermolecular interaction and improved method for evaluation the hydration shell effect
Conclusions
Genome Evolution from statistics of proteins’ properties to Exon size distribution
obeys Lognormal Distribution
Why ?
Hypothesis
~ Mass ~Length ~ Gene
Exons
Introns
1 2 3 4 5 6 7 8 9 10
5
10
15
20
25
30
Ln(Exon Size)
Homo Sapiens
Num
ber
of C
ount
s (t
hous
ands
)
Kolmogoroff process (1941)
-28 -24 -20 -16 -12 -8 -4 0
200
400
600
800
Num
ber
of C
ount
s (t
hous
ands
)
Ln(Exon Size)
tNME ~ln~ln
tN ~ln~
Real Genomes: Two peaks Pattern
2 4 6 8 100
5
10
15
20
25
30
35
peak A M = 4.81 ± 0.01 = 0.82 ± 0.02
peak B M = 5.08 ± 0.07 = 3.13 ± 0.18
Nu
mbe
r o
f C
oun
ts (
thou
san
ds)
Ln(Exon Size)
Pan troglodytes (chimpanzee)
Two peaks Pattern from Soft Bifurcation
-35 -30 -25 -20 -15 -10 -5 0
100
200
300
400
500
600peak A M = -20.69 ± 0.02 = 6.83 ± 0.03
peak B M = -12.60 ± 0.02 = 6.61 ± 0.04
Num
ber
of C
ount
s (t
hous
ands
)
Ln(Exon Size)
-35 -30 -25 -20 -15 -10 -5 0
100
200
300
400
500
600peak A M = -20.29 ± 0.01 = 6.82 ± 0.01
peak B M = -13.61 ± 0.02 = 6.70 ± 0.04
Num
ber
of C
ount
s (t
hous
ands
)
Ln(Exon Size)
2 4 6 8 100
1
2
3
4
5
6
7
peak A M = 5.07 ± 0.01 = 0.55 ± 0.03
peak B M = 5.53 ± 0.01 = 1.89 ± 0.02
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Anopheles gambiae (mosquito)
2 4 6 8 100
5
10
15
20
25
30
35
peak A M = 4.59 ± 0.05 = 2.06 ± 0.12
peak B M = 4.83 ± 0.01 = 0.77 ± 0.03
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Bos taurus (cow)
2 4 6 8 100
2
4
6
8
10
12
14
16
peak A M = 4.75 ± 0.02 = 0.71 ± 0.05
peak B M = 5.22 ± 0.03 = 1.46 ± 0.03
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Caenorhabditis elegans (worm)
2 4 6 8 100
5
10
15
20
25
30
35
peak A M = 4.62 ± 0.05 = 2.10 ± 0.12
peak B M = 4.84 ± 0.01 = 0.77 ± 0.02
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Canis familiaris (dog)
2 4 6 8 100
5
10
15
20
25
30
35
40
peak A M = 4.50 ± 0.05 = 2.49 ± 0.11
peak B M = 4.83 ± 0.01 = 0.77 ± 0.02
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Danio rerio (zebrafish)
2 4 6 8 100
5
10
15
20
25
30
peak A M = 4.56 ± 0.06 = 2.21 ± 0.13
peak B M = 4.83 ± 0.01 = 0.76 ± 0.03
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Gallus gallus (chiken)
2 4 6 8 100
5
10
15
20
25
30
35
peak A M = 4.81 ± 0.01 = 0.82 ± 0.02
peak B M = 4.86 ± 0.05 = 2.73 ± 0.16
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Macaca mulatta (rhesus macaque)
2 4 6 8 100
5
10
15
20
25
30
35
peak A M = 4.82 ± 0.01 = 0.84 ± 0.02
peak B M = 5.38 ± 0.10 = 3.09 ± 0.20
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Mus musculus (house mouse)
2 4 6 8 100
5
10
15
20
25
30
35
peak A M = 4.85 ± 0.01 = 0.76 ± 0.02
peak B M = 5.00 ± 0.03 = 1.95 ± 0.10
Nu
mb
er
of
Co
un
ts (
tho
usa
nd
s)
Ln(Exon Size)
Tetraodon nigroviridis (spotted green pufferfish)
Evolutionary Ladder
Caenorhabditis elegans A
Anopheles gam
biae A
Tetraodon nigroviridis A
Bos taurus B
Canis fam
iliaris BD
rosophila Melanogaster A
Gallus gallus B
Danio rerio B
Macaca m
ulatta A M
us musculus A
Hom
o Sapiens A
Pan troglodytes A
1
2
3
Caenorhabditis elegans B
Anopheles gam
biae B
Tetraodon nigroviridis B
Bos taurus A
Canis fam
iliaris A D
rosophila Melanogaster B
Gallus gallus A
Danio rerio A
Macaca m
ulatta B M
us musculus B
Hom
o Sapiens B
Pan troglodytes B
The role of alternative splicing
• Statistical distribution of exon sizes in general obey the log-normal law that can be originated from random Kolmogoroff fractioning process
• Concept of Kolmogoroff process suggests that the process of intron gaining is independent of exon size but, hypothetically, related to a mechanism dependent of intron-exon boundaries
• The concept of random exon breaking supports the hypothesis that at the initial stages of evolution simpler organisms had lower fraction of introns – Introns Late Hypothesis
• Two distinctive classes of exons presented in exon length statistics are still not fully understood. However, it is clear that one of these classes contain exons that are in general conserved during evolution while another class undergoes intensive diversification
• Distribution of exons between these classes correlates with phenomenon of alternative splicing which is, presumably, responsible for the existing variety of living species.
Conclusions
Directions for Future
Researches Protein Domain Dynamics
• Theoretical Models for the case of explicit coupling between overall tumbling and domain reorientations
• Search for the other possible applications like for the case of Insulin
• Molecular Dynamics investigations of domain mobility. The first target probe correlations between charged state of His 68 in UB2 and domain dynamics for MD trajectories
• Elaboration of data treatment software for other than NMR experimental techniques
Directions for Future
Researches
Assembling Protein Structures
• Improvements of the method to include exhaustive search for all degrees of freedom including rotation angles.
• Adding parameters of intermolecular interactions with implementations in docking algorithms
• Investigations of Hydration Layer effect
• Molecular Dynamics investigations of coupling between overall tumbling and large scale mobility in proteins
Predictions of Protein Diffusion Tensor
Directions for Future
Researches
• Statistics of other parts of genomes: Introns is the first target
• Investigations of correlations between observed two exon peaks and other known classes of genes: ortologous and paralogous gene families, major and minor spliceosomal pathways are the first targets
• Plants and other classes of organisms
• Alternative splicing from the point of view of statistical distributions
• Correlations with real evolutionary mechanisms involving exon-intron boundaries
Statistics of genomes
Caenorhabditis elegans A
Anopheles gam
biae A
Tetraodon nigroviridis A
Bos taurus B
Canis fam
iliaris BD
rosophila Melanogaster A
Gallus gallus B
Danio rerio B
Macaca m
ulatta A M
us musculus A
Hom
o Sapiens A
Pan troglodytes A
1
2
3
Caenorhabditis elegans B
Anopheles gam
biae B
Tetraodon nigroviridis B
Bos taurus A
Canis fam
iliaris A D
rosophila Melanogaster B
Gallus gallus A
Danio rerio A
Macaca m
ulatta B M
us musculus B
Hom
o Sapiens B
Pan troglodytes B
Acknowledgements
Yuri Feldman
Alexei Sokolov
Michel Gribskov
Stephen Mount
Natalia Grishina
David Fushman
Nikolai Skrynnikov
Amitabh Varshney
Ranjanee Varadan
Jenifer Hall
Laboratory frame (L) Instantaneous Residue orientation (I)
z
x
yz
x
y
Protein orientation (P)
z
y
xLP
PI
))(())0((
))(())0(()(
20'
*20
2
2',
2'
*2
tDD
tDDtC
PIlPIl
llLPqlLPql
Wigner, 1959 Group theory and its application to the quantum mechanics of atomic spectra
Appendix
ns 9.2dpH 6.8 pH 4.5 ns 9.1d
ns 9.25j ns 31.95j
+ +
- -
K 48
H 68
L 8
V 70I 44
Derived Structures and Dynamics Parameters
Crystal Ub2
PDB 1AAR
Docked:Ub2 + UBA
pH 6.8 pH 4.5
Both protonated 0.002 0.826
One protonated 0.091 0.165
Non protonated 0.907 0.009
His 68 pKa=5.5Fujiwara et al, J. Bio. Chem. 2004
- 730
+ 730
- 790
+ 790
B
D
6.8 pH 4.5 pH
- 650
+ 650
- 870
+ 870
A
C
E F
AppendixStructures from relaxation fit
Ub2 conformations. For all presented dimers proximal domains are on the right. Orientations of diffusion tensor principle axes are shown on the left. On the left panel of the figure are conformations obtained for 6.8 pH data; on the right panel are conformations for 4.5 pH data. A) and B) each show two distinct conformations of a dimer obtained by jumping domain model: red axis are rotation axis for every domain. C) and D) show conformations obtained for anisotropic diffusion model applied separately to each domain. E) and F) represent conformations obtained by jumping domain model with artificially repressed dynamics.
Appendix NMR 15N Relaxation and NOE for Ubiquitin dimer
Local fast motions subtracted by relaxation rates ratio approach
)0(4
)(3
''2
'
12
1
J
J
RR
R N
Experimental data from: Varadan et al, JMB 2002
Fushman et al, Prog NMR 2004
Appendix NMR 15N Relaxation, NOE and RDC
Experimental data from: Varadan et al, JMB 2002
5.5
6.0
6.5
7.0
7.5
8.0
8.5
r = 0.95
r = 0.96
calc
Fitting without RDCProximal Distal
-15
-10
-5
0
5
10
15Fitting without RDCProximal Distal
r = 0.62Q = 0.58
5.5 6.0 6.5 7.0 7.5 8.0 8.5
5.5
6.0
6.5
7.0
7.5
8.0
8.5
Fitting with RDCProximal Distal
x102
x102
x102
calc
exp
-15 -10 -5 0 5 10 15
-15
-10
-5
0
5
10
15Fitting with RDCProximal Distal
r = 0.94Q = 0.23
[Hz]
[Hz]
RD
C ca
lc
RDC exp[Hz]
RD
C ca
lc
AssumptionAlignment processoccurs on the time scale that is much longerthan the time scales of
all considered dynamic modes
22 fitting parameters including
12 Euler angels for both Domain orientations in two conformationsAnd5 components of alignment tensor
zyx DDD , ,
pp ,
Only at pH 6.8
zyxjijijiij
NH ppSdd,,,
max coscoscoscos
p+=0.9 p-=0.1
p+=0.8 p-=0.2
Appendix Derived Structures and Dynamic Parameters
Relaxation VS Relaxation + RDC
Relaxation data only
Relaxation + RDC data
ns 9.25j
ns 0.63j
Appendix
Validation with Spin Labeling data
“+”
“-”
Blue ball for “+” conformation only
Red ball for “+” and “-” together
Fitted position of Spin Label
AppendixSpectral density for the model of Interconversion between Two States
)()(
)()()()(
)()()()()()1(
)1(
)()()()(
)()( )()(
5
2)(
)2(0,
)*2(0
)2(,
)*2(,
)2(,
)*2(,
)2(,
)*2(,
)2(,
)*2(,22
)2(,
)*2(,
)2(,
)*2(,
)2(,
)*2(,
2)2(,
)*2(,
222
2
2
2
2
2
2
2
2
2
2,
*,
RDlRDk,
DPlnDPkmDPlnDPkm
DPlnDPkmDPlnDPkmjjz
jzj
DPlnDPkmDPlnDPkm
DPlnDPkmDPlnDPkmz
z
m n k l znzmz
DD
DDDD
DDDDE
ppE
DDppDDpp
DDpDDpE
E
aaJ
z
2 1 0 -1 -2
g
2 0 0
1 0 0 0
0 0 0 0
-1 0 0 0
-2 0 02N
w
2N
u
2
1
2
1
2
1
N
u
N
w
2
1
2
1
2N
w
2N
u
2
1
az,g
AppendixDefinitions of diffusion related parameters
2
2
2
2,
0*,
0 )()(),(z g
tEtgzgz
t zetP
yx DDu 3
wN 2
yxzyxz
yxz
DDDDDD
DDD
w
case, oblate for the 22
22
z
2
-2
1
-1
0
zE )(, gz
26 sD
)()(
2)(
2
5
2
1 *)2(2,
*)2(2,
*)2(0,,2 gggg DD
wuD
N
)(3 sz DD )()(4
5 *)2(2,
*)2(2,,2 ggg DD
)(3 sx DD )()(4
5 *)2(1,
*)2(1,,1 ggg DD
)(3 sy DD )()(4
5 *)2(1,
*)2(1,,1 ggg DD
26 sD
)()(
2)(
2
5
2
1 *)2(2,
*)2(2,
*)2(0,,0 gggg DD
wuD
N
AppendixDefinitions of diffusion related parameters
case, oblate for the ))(()(
))(()(
2
2
yxzyzxzxx
yzxzxx
DDDDDDDDD
DDDDDD zyxs DDDD 3
1
pH Dx Dy Dz j p+ Domain + + + - - -
6.81.53
(0.32)1.73
(0.06)2.20
(0.08)9.3
(4.8)0.90
(0.06)
Proximal218 (35)
109 (11)
140 (4)
203 (38)
110 (9)
72 (8)
Distal91 (28)
58 (7)
321 (19)
156
(33)
96 (38)
356
(33)
4.51.61
(0.13)1.71
(0.06)2.20
(0.06)31.9(9.8)
0.82(0.06)
Proximal147 (30)
112 (16)
322 (8)
191 (25)
122 (14)
45 (13)
Distal213 (24)
80 (8)
350 (12)
151
(29)
51 (20)
328
(23)
AppendixFitted parameters for the model of Interconversion between Two States
Relaxation data only
D in 107 s-1
in [ns]
Dx Dy Dz j p+ Domain
1.57(0.12)
1.75(0.06)
2.39 (0.08)
36.0(9.8)
0.76(0.07)
Proximal287 (22)
131 (8)
153 (7)
258 (28)
115 (10)
88 (12)
Distal72
(15)52 (8)
327 (8)
147 (21)
87 (21)
356 (16)
PS PS PS
7.3 (1.3) 18.5 (3.2) -52 (23) 95 (8) 11 (6)
AppendixAnalysis of Relaxation data together with RDC
77.12 relax DP
DP
DP
DP
DP DP
Parameters of the alignment tensor
xxNH Sd max yy
NH Sd max
49.02 RDCParameters of the ITS model
D in 107 s-1
in [ns]
AppendixModel free ideas in application to domain mobility
)()()1()()()( )2(,0
)*2(,0
/22)2(0,
)*2(0,, DMFlDMFk
tMFPnMFPm
PDnlmk DDeSSDDtC
Correlation function for domain mobility in MF form
)()()()(
)1()()(5
1)(
)2(0,
*)2(0,
)2(,0
*)2(,0
2
2
2
2
2
2
2
2
2
2
/22)2(0,
*)2(0,,
*,
RDlRDkDMFlDMFk
m n k l z
tMFPnMFPmnzmz
tE
DDDD
eSSDDaaetC z
Total correlation function and spectral density
)()()()(
)()1(
)1)(1()()(
5
2)(
)2(0,
*)2(0,
)2(,0
*)2(,0
2
2
2
2
2
2
2
2
2
222
2
22
2)2(0,
*)2(0,,
*,
RDlRDkDMFlDMFk
m n k l z z
z
z
zMFPnMFPmnzmz
DDDD
E
ES
E
ESDDaaJ
)()()()( )2(0,
)*2(0,
)2(,0
)*2(,0 RDlRDkDMFlDMFk DDDD
)2/( '1
'2
'1 RRR NOTE that for this spectral density relaxation rates ratio,
becomes independent from the term, which means absolute independence of the residue orientation. In other words this representation leads to
quasi-isotropic form of
AppendixUsing Extended Model Free for domain motions
Clore et al, JACS 1990Chang and Tjandra, JACS 2001
)/exp()1(
)/exp()1()()()(
222
20,0,
)2(0,
0*)2(0,0,0
sssf
ffhsIR
tIRhIRs
RIhs
tSSS
tSDDtC
)()()( )2(,
*)2(,, DPlnDPkm
PDnlmk DDtC
)/exp()1(
)()()()(5
1
)()()(
22
)2(0,
*)2(0,
2
2
2
2
2
2
2
2
2
2
)2(,
*)2(,,
*,
)2(0,
0*)2(0,
sss
RDlRDkm n k l z
DPlnDPkmnzmztE
IL
tILqILq
tSS
DDDDaae
DDtC
z
)()1(
)1)(1(
)()()()(5
2)(
22
2
22
2
)2(0,
*)2(0,
2
2
2
2
2
2
2
2
2
2
)2(,
*)2(,,
*,
ssz
szs
z
zs
RDlRDkm n k l z
DPlnDPkmnzmz
E
ES
E
ES
DDDDaaJ
Apparently NO domain motions
Overall fitting parameters pH 6.8
0.70 0.88 1.42
Domain fitting parameters (all angles are in degrees)
Proximal Distal
2.0800 11.2334
0.3197 0.0164
266.5950 25.6547
42.6660 54.4322
327.6032 348.3614
Overall fitting parameters pH 4.5
1.36 1.50 1.83
Domain fitting parameters (all angles are in degrees)
Proximal Distal
2.5784 8.2130
0.5556 0.1970
111.2829 166.3341
72.0742 111.9480
151.4625 176.9133
xD /srad 10 27yD /srad 10 27
zD/srad 10 27
][nss2sS
DP
DP
DP
AppendixUsing Extended Model Free fitting Ubiquitin dimer data
xD /srad 10 27yD /srad 10 27
zD/srad 10 27
][nss2sS
DP
DP
DP
AppendixBootstrap method for estimation of Confidence Intervals
Press et al, Numerical recipes in C, 1992
1/e ~ 37% of original data substituted by duplicates of the rest data.
Then
Data fitted in the same way as the original data
Procedure repeated 200 times
Confidence intervals estimated from resulted distributions of fitted parameters
Dx 107 [rad/s] Dy 107 [rad/s] Dz 107 [rad/s] Tau [ns]
Relaxation fit @ pH 6.8
Fitted values 1.53 1.73 2.20 9.1575
Closed Structure used in the paper
HydroNMR 1.211 1.222 2.107 11.01
Surf-PCA a) 1.2763 1.2782 2.2900 10.32
Surf-PCA b) 1.5034 1.510 2.3432 9.3342
Open Structure used in the paper
HydroNMR 1.312 1.326 2.213 10.31
Surf-PCA a) 1.2817 1.2876 2.2390 10.399
Surf-PCA b) 1.5088 1.5192 2.288 9.4057
Closed Structure fitted by surf-PCA routine
HydroNMR 1.415 1.426 2.266 9.796
Surf-PCA a) 1.3769 1.3877 2.2587 9.9536
Surf-PCA b) 1.6011 1.6123 2.331 9.0147
Open Structure fitted by surf-PCA routine
HydroNMR 1.414 1.438 2.280 9.742
Surf-PCA a) 1.3647 1.3747 2.2353 10.051
Surf-PCA b) 1.628 1.6372 2.3379 8.9236
AppendixHydrodynamics Calculations HYDRONMR and PCA
De La Torre et al, J Mag. Res, 2000
Settings for HYDRONMR
Temperature 297 K,
eta=0.0091 [sp], AER=3.2 Å
Settings for Surf-PCA based algorithm
Temperature 297 K,
a) was evaluated for dry protein and then the eigenvalues of diffusion tensor were divided by factor of 2.
b) Sell parameter was 2.8 Å.
No any additional factors
9.83142.30571.39341.3867Surf-PCA b)
10.952.1721.2071.189HydroNMR
Open Structure used in the paper
9.65362.30871.43871.4320Surf-PCA b)
11.002.1551.2011.189HydroNMR
Closed Structure used in the paper
9.05802.201.711.61Fitted values
Relaxation fit @ pH 4.5
Tau [ns]Dz 107 [rad/s]Dy 107 [rad/s]Dx 107 [rad/s]
AppendixHydrodynamics Calculations HYDRONMR and PCA
De La Torre et al, J Mag. Res, 2000
Settings for HYDRONMR
Temperature 297 K,
eta=0.0091 [sp], AER=3.2 Å
Settings for Surf-PCA based algorithm
Temperature 297 K,
a) was evaluated for dry protein and then the eigenvalues of diffusion tensor were divided by factor of 2.
b) Sell parameter was 2.8 Å.
No any additional factors
Complexity of the algorithms
ELM
HYDRONMR
FastHYDRONMR
Nat
Nat
Nat
2
4/3
ELM : FastHYDRONMR : HYDRONMR
1 : 2 : 500
Comparison with the experimental data
Comparison with experimental data proteins from 2.9 to 82 kDa
Ryabov & Fushman, JACS (2006)
0
10
20
30
40
50
Nu
mb
er
of c
ou
nts
0.1 1 10 100
angle between ZH and Z
PCA
0.1 1 10 100
angle between XH and X
PCA
0.1 1 10 100
angle between YH and Y
PCA
Appendix
0.00 0.04 0.08 0.120
20
40
60
80
Nu
mb
er
of C
ou
nts
RmH
0.00 0.04 0.08 0.12 0.16
RmPCA
Rhombicity statisticsNormal distribution
Appendix
Appendix
1 2 3 40
1000
2000
3000
4000
5000
6000Data: Data2_BModel: gauss_twoEquation: y=A1*exp(-0.5*((x-xc1)/w1)^2)+A2*exp(-0.5*((x-xc2)/w2)^2)+y0Weighting: y No weighting Chi^2/DoF = 5483.30527R^2 = 0.99814 A1 2131.98794 ±71.33435xc1 2.20154 ±0.00386w1 0.11216 ±0.0047A2 3890.72527 ±43.64846xc2 2.50936 ±0.00582w2 0.45371 ±0.00417y0 0 ±0
Num
ber
of C
oun
ts
Log10
(Exon Bp)
Drosophyla Melanogaster from EnsEMBLFlyExon statistics (removed duplicates)A
1=12% A
2=88%
1 2 3 40
5000
10000
15000
20000
25000
30000Data: CStatDat_BModel: gauss_twoEquation: y=A1*exp(-0.5*((x-xc1)/w1)^2)+A2*exp(-0.5*((x-xc2)/w2)^2)+y0Weighting: y No weighting Chi^2/DoF = 327707.35374R^2 = 0.99504 A1 26122.36006 ±524.86156xc1 2.08596 ±0.00331w1 0.18617 ±0.00436A2 3823.28544 ±447.08133xc2 2.33151 ±0.0516w2 0.64822 ±0.04831y0 0 ±0
Nu
mb
er
of C
ou
nts
Log10
(Exon Bp)
Homo Sapience from EnsEMBLMan (remooved duplicated data)A
1=66% A
2=34%
Major and Minor Spliseosomal Pathway ???
Exons
Introns
Protein Rotation Diffusion Tensor
z
y
x
D
D
D
00
00
00
3 Euler angles forDiffusion Tensor PAF
DxDy
Dz