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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


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



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



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)


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


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


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


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 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


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


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


Anopheles gam

biae A


Bos taurus B

Canis fam

iliaris BD


Gallus gallus B

Danio rerio B

Macaca m

ulatta A M

us musculus A

Hom

o Sapiens A

Pan troglodytes A

1

2

3


Anopheles gam

biae B


Bos taurus A

Canis fam

iliaris A D


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


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


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


Anopheles gam

biae A


Bos taurus B

Canis fam

iliaris BD


Gallus gallus B

Danio rerio B

Macaca m

ulatta A M

us musculus A

Hom

o Sapiens A

Pan troglodytes A

1

2

3


Anopheles gam

biae B


Bos taurus A

Canis fam

iliaris A D


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


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


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


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

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