Fractional Gaussian Noise, Fractional Gaussian Noise, Subdiffusion - - PowerPoint PPT Presentation

SLIDE 1

Fractional Gaussian Noise, Fractional Gaussian Noise, Subdiffusion Subdiffusion and Stochastic and Stochastic Networks in Biophysics Networks in Biophysics

Samuel Kou Department of Statistics Harvard University

SLIDE 2

Single-Molecule Experiments

Statistics & probability experienced fundamental

change in the past 20 years

Biophysics & chemistry also witnessed dramatic

progress: single-molecule experiments

Using nanotechnology, scientists can study

biological processes on a single-molecule basis (eg. enzymatic kinetics, protein/DNA dynamics)

“Seeing images of single atoms is a religious experience”

• -- Richard Feynman

SLIDE 3

New Aspects for Scientific Discovery

Can measure molecular properties

individually, instead of inferring from population statistics

If the reaction/kinetic time is slow, ensemble

experiments become almost impossible due to the difficulty of synchronization

Single-molecule trajectory provides detailed

dynamic information

Understanding the dynamics of individual is

essential to unlock their biofunctions

SLIDE 4

Statistical & Probabilistic Challenges Statistical & Probabilistic Challenges

Require new stochastic modeling

Subdiffusion Enzymatic reaction

Data are noisier, and efficient inference

methodology is needed

SLIDE 5

Brownian diffusion

Since Einstein's 1905 paper, the theory of Brownian

diffusion has revolutionized not only natural sciences but also social sciences.

Brownian motion described by Langevin equation

where F(t) is white noise process satisfying by fluctuation-dissipation theorem.

Solution: Ornstein-Uhlenbeck process v(t) Gaussian Displacement (location)

• -- Corner stone for statistical mechanics

) (t F v dt dv m

t t

+ − = ζ

) ( )} ( ) ( { t t T k t F t F E

B

′ − ⋅ = ′ δ ζ

ds s v t x

t

) ( ) ( ∫ =

t t T k t x E

B

large for , 2 } ) ( {

2

ζ ∼

SLIDE 6

Subdiffusion

Brownian diffusion, however, cannot explain so

called subdiffusion:

Distance fluctuation within a single protein

molecule (Yang et al. 2003, Science; Min et al. 2005, Physical

Review Letters).

Need tools beyond Langevin equation and BM.

1 , } ) ( {

2

< < ∝ α

α

t t x E

SLIDE 7

Single-molecule fluorescence experiment on protein complex

Yang et al. (2003, Science) studied a protein-enzyme

complex Fre, catalyzes the reduction of flavin

Fre contains two substructures:

a flavin adenine dinucleotide (FAD) and a tyrosine (Tyr).

Fluorescence lifetime of FAD varies

due to distance fluctuation

Relationship between fluorescence

lifetime and distance

FAD Tyr

1 ) ( 1

] [ ) (

− + − −

=

t eq

X X

e k t

β

γ

SLIDE 8

Autocorrelation function

)} ( ) ( {

1 1

t E

− −

∆ ∆ γ γ

)} ( { ) ( ) (

1 1 1

t E t t

− − −

− = ∆ γ γ γ

0.001 0.01 0.1 1 10 100 1000

time (second)

0.00 0.02 0.04 0.06

Cγ (t)

• Need tools beyond Langevin

equation and BM.

• The model: Generalized

Langevin equation with fractional Gaussian noise (GLE with fGn).

SLIDE 9

Generalized Langevin Equation with fGn

Langevin equation Generalized Langevin equation Fluctuation-dissipation theorem links memory

kernel K(t) with fluctuating force

Key question: How to introduce the noise structure? Understand the white noise: White noise is the

derivative of the Wiener process

B(t) is the unique process: (i) Gaussian, (ii)

independent increment, (iii) stationary increment, (iv) self-similar

) (t F v dt dv m

t t

+ − = ζ , ) (

t u t t

G du u t K v dt dv m + − − =

∫ ∞

−

ζ

) ( } { s t K T k G G E

B s t

− ⋅ = ζ

t dt d t

B F =

SLIDE 10

} {

) (

=

H t

B E

. , for 2 / ) ( } {

2 2 2 ) ( ) (

≥ − − + = s t s t s t B B E

H H H H s H t

dt dB H

H t

t F

) (

) (

) (

=

for , | | ) 1 2 ( )} ( ) ( { ) (

2 2 ) ( ) (

≠ − = =

−

t t H H t F F E t K

H H H H

H H it H

H H dt t K e K

2 1

| | ) sin( ) 1 2 ( ) ( ) ( ~

− ∞ ∞ −

+ Γ = = ∫ ϖ π ϖ

ϖ

Natural generalization: (i) Gaussian (ii) stationary

increment (iii) self-similar.

The ONLY candidate fBM B(H)(t) , 0 < H < 1

, and covariance function when H = 1/2, reduces to B(t).

Fractional Gaussian noise :

Gaussian & stationary.

Memory kernel Spectral density

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Toward subdiffusion

Applying Fourier transform on

v(t) Gaussian E{v(t)} = 0,

For displacement H > 1/2

leads to subdiffusion!

, ) (

) (H t H u t t

F du u t K v dt dv m + − − =

∫ ∞

−

ζ

2

) ( ~ ) ( ~ ) ( ~ ) ( ~ 2 1 )} ( ) ( { ) ( m i K K T k C d C e t v v E t C

H H B it

ϖ ϖ ζ ϖ ζ ϖ ϖ ϖ π

ϖ

− = = =

+ ∞ ∞ −

∫

( )

ds s v t X

t

) ( ∫ =

ds du u v s v E t X E

t t

)} ( ) ( { } ) ( {

2

∫ ∫

=

H H B

t t H H H H T k t X E

2 2 2 2 2

) 2 2 )( 1 2 ( ) 2 sin( 2 ~ } ) ( {

− −

∝ − − π π ζ

SLIDE 12

Harmonic potential

GLE: For external field U(x), changes to

where , . For harmonic potential

Fourier method gives E{X(t)X(s)}, E{X(t)v(s)}

Overdamped condition

Acceleration negligible, GLE reads

, ) (

) (H t H u t t

F du u t K v dt dv m + − − =

∫ ∞

−

ζ

) ( ) ( ) ( ) ( ) (

) (

t F X U du u t K u X t X m

H t H t

+ ′ − − − =

∫ ∞

−

& & & ζ

ds s v t X

t

) ( ) ( ∫ =

dt t dv

t X

) (

) ( = & &

2 2 2 1

) ( x m x U ω =

) ( ) ( ) ( ) ( ) (

) ( 2

t F t X m du u t K u X t X m

H H t

+ − − − =

∫ ∞

−

ω ζ & & & ) ( ) ( ) ( ) (

) ( 2

t F du u t K u X t X m

H H t

+ − − =

∫ ∞

−

& ζ ω

SLIDE 13

Solution: X(t) stationary Gaussian E{X(t)}=0

where

For all H,

the thermal equilibrium value.

For H = 1/2, recovers the Brownian diffusion result

) ( ) ( ) ( ) (

) ( 2

t F du u t K u X t X m

H H t

+ − − =

∫ ∞

−

& ζ ω

) ) 1 2 ( ( )} ( ) ( { ) (

2 2 2 2 2 2 H H B xx

t H m E m T k t X X E t C

− −

+ Γ − = = ζ ω ω

) 1 ( / ) ( + Γ = ∑

∞ =

k z z E

k k

α

α

2

)} ( ) ( { ) ( ω m T k X X E C

B xx

= =

t B xx

m

e m T k t X X E t C

ζ ω

ω

2

2 2

)} ( ) ( { ) (

−

= =

SLIDE 14

The Hamiltonian for GLE with fGn

GLE with fGn can be derived from the interaction

between a particle and a harmonic oscillator heat bath.

Start from system Hamiltonian

and the heat bath Hamiltonian where mb the media molecule mass, ωj individual frequency, and γj coupling strength of jth oscillator.

Equation of motion for Hs + HB

mv p x m m p x m mv H s = + = + = , 2 1 2 2 1 2 1

2 2 2 2 2 2

ω ω

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = ∑

2 2 2 2

) ( 2 1 2 x q m m p H

j j j j b b j j B

ω γ ω ) ( ), (

B s j j B s j j

H H q dt dp H H p dt dq + ∂ ∂ − = + ∂ ∂ = ) ( ), (

B s B s

H H x dt dp H H p dt dx + ∂ ∂ − = + ∂ ∂ =

SLIDE 15

Solve it. Leads to (i) GLE

where and (ii) the fluctuation-dissipation theorem

Furthermore, if we take

) ( ) ( ) ( ) ( ) (

2

t F t X m du u t K u X t X m

t

+ − − − =

∫ ∞

−

ω ζ & & &

) sin( ) ( ) cos( )) ( ) ( ( ) ( ) ( ) cos( ) ( cos ) (

2 2 2 2 2

t p t x q m t F d g t m t m t K

j j j j j j j j j j b j b j j j j b

ω ω γ ω ω γ γ ϖ ϖ ϖ ϖ ϖ γ ω ω γ

∑ ∑ ∫ ∑

+ − = = =

) ( } { s t K T k F F E

B s t

− ⋅ = ζ

2g  1  Γ2H  1sinH||3−2H Kt  H2H − 1t2H−2

fGn memory kernel

SLIDE 16

Back to experiment

FAD Tyr

1 ) ( 1

] [ ) (

− + − −

=

t eq

X X

e k t

β

γ

Model X(t) by GLE with fGn under harmonic potential easy calculation of lifetime autocorrelation Fluorescence lifetime of FAD depends on the distance between FAD and Tyr

) 1 ( )} ( ), ( {

) ( ) ( 2 2 1 1

2 2

− =

+ − − t C C X

xx xx eq

e e k t Cov

β β β

γ γ

SLIDE 17

Fitting experimental autocorrelation

H 0.74 0.40 0.81

2

ω ζ m

2

2 β

ω m T kB

0.001 0.01 0.1 1 10 100 1000

time (second)

0.00 0.02 0.04 0.06

Cγ (t) Kou and Xie, Phys. Rev. Lett., 93, 180603 (2004).

SLIDE 18

Higher order autocorrelation functions

0.001 0.01 0.1 1 10

t (sec)

0.0 0.2 0.4 0.6 0.8 1.0

〈δγ−1(0) δγ−1(t) δγ−1(2t)〉 〈γ−1〉3

0.001 0.01 0.1 1 10

t (sec)

1 2 3 4 5

〈δγ−1(0) δγ−1(t) δγ−1(2t) δγ−1(3t)〉 〈γ−1〉3

Three-point Four-point

〉 〈 − =

− − −

) ( ) ( : ) (

1 1 1

t t t γ γ δγ

H 0.74 0.40 0.81

2

ω ζ m

2

2 β

ω m T kB

Same parameters:

SLIDE 19

A prediction for three-point autocorrelation

GLE with fGn predicts time symmetry In particular Check with experiments:

} ) ( ) ( ) ( { } ) ( ) ( ) ( {

1 2 1 1 2 1 1 2 1 1 1 1 − − − − − −

+ = + t t t E t t t E δγ δγ δγ δγ δγ δγ

t t t E t t E all for )} 3 ( ) 2 ( ) ( { )} 3 ( ) ( ) ( {

1 1 1 1 1 1 − − − − − −

= δγ δγ δγ δγ δγ δγ

SLIDE 20

0.00 0.01 0.02 0.03

〈δγ−1(0)δγ−1(2t)δγ−1(3t)〉

0.00 0.01 0.02 0.03

〈δγ−1(0)δγ−1(t)δγ−1(3t)〉

)} 3 ( ) ( ) ( {

1 1 1

t t E

− − −

δγ δγ δγ )} 3 ( ) 2 ( ) ( {

1 1 1

t t E

− − −

δγ δγ δγ

versus

0.001 0.01 0.1 1 10 time 0.00 0.01 0.02 0.03 correlation

〈δγ−1(0)δγ−1(2t)δγ−1(3t)〉

0.001 0.01 0.1 1 10 time 0.00 0.01 0.02 0.03 correlation

〈δγ−1(0)δγ−1(t)δγ−1(3t)〉

SLIDE 21

Another system

Tyr37 Fluorescein 3.6Å Tyr37 Fluorescein 3.6Å

A protein complex formed between fluorescein (FL) and monoclonal anti-fluorescein (anti-FL)

Min, et al. Phys. Rev. Lett. 94, 198302 (2005).

SLIDE 22

Obtain distance fluctuation from Since

Potential function

0 100 150 200

• 1

1 0.00 0.05 0.10 P(x) x(t) (Å) t (sec) 0 100 150 200

• 1

1 0.00 0.05 0.10 0 100 150 200

• 1

1 0.00 0.05 0.10 P(x) x(t) (Å) t (sec)

1 ) ( 1

] [ ) (

− + − −

=

t eq

X X

e k t

β

γ

) / ) ( exp( ) ( T k x U x P

B

− ∝

) ( ˆ ln ) ( ˆ x P T k x U

B

− =

• 1

1 1 2 3 U(x) (kBT) x (Å)

• 1

1 1 2 3 U(x) (kBT) x (Å)

• 1

1 1 2 3 U(x) (kBT) x (Å)

SLIDE 23

Autocorrelation of distance fluctuation

10

• 3

10

• 2

10

• 1

10 10

1

10

2

10

3

0.01 0.1 1 Cx(t) (Å

2)

t (sec)

Data Mittag-Leffler fitting Error Bounds

10

• 3

10

• 2

10

• 1

10 10

1

10

2

10

3

0.01 0.1 1 10

• 3

10

• 2

10

• 1

10 10

1

10

2

10

3

0.01 0.1 1 Cx(t) (Å

2)

t (sec)

Data Mittag-Leffler fitting Error Bounds

• ne-to-one correspondence

) ( ~ ) ( ) ( ~ ) ( ~

2

s C s C s C m s K

x x x

− = ζ ω

{

) ( ) ( ) ( ) (

2

t F du u t K u x t x m

t

+ − − =

∫ ∞

− &

ζ ω

SLIDE 24

Experimental Memory kernel

SLIDE 25

Brief Recap

Propose Generalized Langevin Equation with

fGn to explain subdiffusion

Explains the observed conformational dynamics One set of parameters fits all Key model assumptions verified from

experiments

SLIDE 26

Michaelis-Menten Mechnism

,

2 1 1

P E ES S E

k k k

+ ⎯→ ⎯ ⎯⎯ ← ⎯→ ⎯ +

−

E E

k

⎯→ ⎯ 3

] [ ] ][ [ ] [

1 1

ES k S E k dt E d

−

+ − = ] )[ ( ] ][ [ ] [

2 1 1

ES k k S E k dt ES d + − =

−

] [ ] [ ] [

2

ES k dt P d dt E d = =

Lineweaver-Burke plot [E]T / v (ms) 1/[S](mM-1)

Classical Michaelis-Menten equation

, ] [ ] [

max M

K S S v v + =

1 2 1 2 max

/ ) ( ]) [ ] ([ k k k K ES E k v

M

+ = + =

−

SLIDE 27

Single-molecule case

P E ES S E

k k k

+ ⎯→ ⎯ ⎯⎯ ← ⎯→ ⎯ +

− 2 1 1

) ( ) ( ] [ ) (

1 1

t P k t P S k dt t dP

ES E E −

+ − = ) ( ) ( ) ( ] [ ) (

2 1 1

t P k k t P S k dt t dP

ES E ES

+ − =

−

) ( ) (

2

t P k dt t dP

ES E

=

Turnover time distribution

0.00 0.02 0.04 0.06 0.08 0.10 10

• 2

10

• 1

10 10

1

10

2

f (t) t (s)

[S]=0.020 mM [S]=0.050 mM [S]=0.100 mM

M

K S S k T E v + = = ] [ ] [ ) ( 1

2

: Still obey the hyperbolic form

[ ]

t A B t B A A S k k t f ) exp( ) exp( 2 ] [ ) (

2 1

− − + = 2 / ) ] [ ( , ] [

2 1 1 2 1 2

k k S k B S k k B A + + − = − =

−

Reaction rate:

SLIDE 28

Single Molecule Ensemble

[E]T / v (ms)

<t> (ms)

1/[S](mM-1) 1/[S](mM-1)

English et al., Nature Chem. Biol., 2, 87 (2006)

• E. coli ß-gal catalyzes Hydrolysis of Lactose

β-galactosidase

The experiment uses photogenic substrate

P E ES S E

k k k

+ ⎯→ ⎯ ⎯⎯ ← ⎯→ ⎯ +

− 2 1 1

SLIDE 29

Single Molecule Turnover Experiment of Single Molecule Turnover Experiment of ß ß-

• galactosidase

galactosidase

Each enzymatic turnover creates a fluorescent burst

SLIDE 30

A

Low Substrate Concentration 20µM

Adding inhibitor Off the enzyme

High Substrate Concentration 100µM

SLIDE 31

Multi-exponential Distributions of Turnover Times

[S]=10 µM [S]=20 µM [S]=50 µM [S]=100 µM

• Skewed decay at high substrate concentration
• Single exponential decay at low substrate concentration

SLIDE 32

Memory between successive turnover times

P E ES S E

k k k

+ ⎯→ ⎯ ⎯⎯ ← ⎯→ ⎯ +

− 2 1 1

Under Michaelis-Menten Mechnism three-state continuous-time Markov chain

10

• 2

10

• 1

10 10

1

0.05 0.1 0.15 0.2 0.25 0.3

t (s) C(t)

Successive turnover times should have NO correlation Experimental data

SLIDE 33

Single Molecule

<t> (ms)

10

• 2

10

• 1

10 10

1

0.05 0.1 0.15 0.2 0.25 0.3

t (s) C(t)

?

SLIDE 34

Dynamic disorder – the fluctuation of enzyme

An enzyme is a dynamic entity with constant

spontaneous conformation fluctuation Conformation fluctuation occurs on a board range of time scales Different conformation could have different enzymatic reaction rate constants.

SLIDE 35

Dynamic disorder – the fluctuation of enzyme

All the E1, E2, … are experimentally indistinguishable

Kou et al., J. Phys. Chem. B, 109, 19068 (2005)

SLIDE 36

Dynamic disorder – the fluctuation of enzyme

Rugged Energy Landscape

SLIDE 37

Explain the memory

If parallel transition rates are small Transitions will stay in one channel for a quite while before going to the next Naturally give rise to the strong correlation

SLIDE 38

Turnover time: first passage time Can be solved via Laplace transform and matrix analysis Under various conditions

M

C S S T E v + = = ] [ ] [ ) ( 1

2

γ

1 2 2 2 2

) (

− ∞

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∫ dk k k p γ

Weighted harmonic mean

Single Molecule Michaelis-Menten Equation

1 1 2

k k CM

−

+ = γ

SLIDE 39

M

C S S T E v + = = ] [ ] [ ) ( 1

2

γ

Reaction rate If one of the following conditions holds: (a) slow interconversion between Ei (b) slow interconversion between ESi (c) fast interconversion between Ei (d) fast interconversion between ESi (e) k-1i >> k2i (f) k2i/k-1i = const

Quasi equilibrium Conformational equilibrium

} Quasi static

SLIDE 40

Model fitting

[S]=10 µM [S]=20 µM [S]=50 µM [S]=100µM

• Skewed decay at high substrate concentration
• Single exponential decay at low concentration

k11 = k12 = … = k1n = k1 k-11 = k-12 = … = k-1n = k-1

SLIDE 41

Turnover Time distribution for fluctuating enzymes

Quasi Static Limit:

[ ]

t A B t B A A S k k k w dk t f ) exp( ) exp( 2 ] [ ) ( ) (

2 1 2 2

− − + = ∫

∞

2 / ) ] [ ( , ] [

2 1 1 2 1 2

k k S k B S k k B A + + − = − =

−

SLIDE 42

Multi-exponential Distributions of Turnover Times

[S]=10 µM [S]=20 µM [S]=50 µM [S]=100 µM

) / exp( ) ( 1 ) (

2 1 2 2

b k k a b k w

a a

− Γ =

−

k1= 5 ×107 M-1s-1 a = 4.2, b = 224 s-1 k-1= 1.83 ×104 s-1

Gamma Distribution

SLIDE 43

Single Molecule Ensemble

[E]T / v (ms)

<t> (ms)

1/[S](mM-1) 1/[S](mM-1)

Consequences: (a) Single Molecule reconciled with ensemble study (b) Dynamic disorder might be masked in ensemble studies! (c) The apparent k2 and KM of the Michaelis-Menten equation are complex functions of the k2 and KM of a large distribution of conformers, different from their conventional interpretations.

SLIDE 44

Summary

Michaelis-Menten with dynamic disorder

Introduce a stochastic network model Explains experimental puzzle Derive single-molecule Michaelis-Menten equation Dynamic disorder might be masked in ensemble

studies!

Generalized Langevin Equation with fGn

Explains the observed conformational dynamics One set of parameters fits all Prediction confirmed by experimental data Each model assumption verified from experiments

SLIDE 45

Summary (Continued)

The connection

Simple underlying picture behind both An enzyme is a dynamic entity with conformational

fluctuation on a board range of time scales.

The interconverting conformations have different

enzymatic reaction rate constants.

SLIDE 46

Acknowledgement

Xie group at Harvard Chemistry & Chemical

Biology

Binny Cherayil Attila Szabo Hong Qian NSF grant DMS-02-04674 NSF Career Award