Full counting statistics of Markov chains applied to the kinetics of molecular motors - PowerPoint Presentation

Slide1

Full counting statistics of Markov chains applied to the kinetics of molecular motors

Intermediate presentation at the group seminarJuly, 18th 2012 | Maximilian Thaller

1

Slide2

Contents

Molecular MotorsConnection between rates and cumulants

Simulation

Discussion of a special case

Further discussion and outlook

2

Slide3

Molecular Motors

3

Introduction

Enzymes that move along cytoskeletal tracks (microtubules)

Purpose: Transport of molecules and organelles in a cell

A kinesin molecule

Motor domain (heads)

Tail domain (cargo gets connected here)

Kinesin in action

cargo

kinesin

microtubule

Anatoly B.

Kolomeisky

et al.,

Annu

.

Rev

. Phys. Chem. 58:675-95 (2007)

Slide4

Molecular Motors

4

The Motor-Position

is

a Stochastic Process

Slide5

Molecular

Motors

5

Distribution

of

the

Motors Position

at

Different Times

Slide6

Usually

,

people

only

look

at

κ1

(velocity) and κ

2 (variance).But: Concidering higher cumulants yields

much more information

about the enzyme‘s internal

processes!

The cumulant generating function S(a) is defined by

The j-th cumulant

κj is then given by

Molecular Motors

6

Cumulants & Cumulant Generating Function

Cumulants

describe the shape of a

probability

distribution.

Slide7

Molecular Motors

7

Notations

2

3

4

N

…

1

1

2

3

4

Chain of N internal states

Transitions between states with constant rates

q

ij

At the transition the motor moves one step forwards

At the transition the motor moves one step backwards

Rate Equations

N

Slide8

Define the vector

Perform a z-transform on the rate equations:

Molecular Motors

8

Master Equation

Obtain the master equation

with the counting matrix

M

. This matrix contains all the transition rates.

Slide9

On the one hand side, one gets (solution of the master equation):

on the other hand side, it is by definition:

Thus, in the long time limit one obtains

Molecular Motors

9

Getting the

Cumulant

Generating Function

Slide10

Consider a molecular motor with

N

internal states. In general, there are 2

N

different transition rates.

The characteristic polynomial

χ

M

(x(a))

of the counting matrix

M

has the form

Connection

between

Rates

and

Cumulants

We

have

seen:

10Effective Rates

The eigenvalue

λ

max

is fully determined by the coefficients ci

. The 2N rates only appear in N+1 combinations. → All systems with the same ci show the same behaviour

even if they have different rates!

→ The rates can’t be uniquely calculated from the

cumulants

!

Slide11

Connection between Rates and Cumulants

We want

to

express the κi in terms

of

effective

rates

ci.Remember

:Thus, we consider χM

(λ(a)) = 0:

11

The Kumulants in Terms of the Effective Rates

Zbigniew

Koza

, J. Phys. A: Math. Gen. 32 7637-7651 (1999)

Slide12

Connection between Rates and

Cumulants

One

obtains

12

Effective

Rates

All

cumulants

of

an

arbitrary

molecular

motor can be calculated without knowing

a single eigenvalue!

Slide13

Simulation

Chosen rates:

13

The

Two

State Motor

2

1

1

1

2

2

1

Slide14

Simulation

14

Example

–

Three

simulated

motors

Slide15

Simulation

15

Measuring

the

Cumulants

t

‘

One

molecular

motor

Many

molecular

motors

Probability distribution at time t‘

→

Yields the

cumulants

Slide16

Simulation

16

Convergence

of the Cumulants

Only

a

few

series

are

necessary

to obtain relable

values for κ1.

Slide17

Simulation

17

Convergence

of the Cumulants

Reliable values for

κ

2

can be achieved with a small number of series.

Slide18

Simulation

18

Convergence

of the Cumulants

A large

number

of

series

is

necessary

to obtain

relable values for κ3.

Slide19

The

measurement

result

ki of

the

cumulant

κi

is a random variable.

k1 for instance, is given by

n

= number of motor

obersvationsXi = end-position

of the motor at

the i-th observation

Simulation

For high cumulant

measurements lots of data

is necessary. But how many

motor observations to be

accurate?19

The Variance of the

Cumulants

The

random variable ki has an

expectation value and a variance!

Slide20

Simulation

20

The

Variance

of

the

Cumulants

It

can

be

calculated

how many measurements one

has to do in order to

get reliable values for the

cumulants.

Aproximatively, one

obtains:

The variances are inversely proportional

to the number of performed

measurements. Higher cumulants have greater variances.

Maurice G. Kendall, The Advanced

Theory of Statitics,

Charles Griffin & Company, (1945)

Slide21

Simulation

21

„

Measured

“ and Originally Chosen Effective Rates

Measured

cumulants

Calculated

cumulants

k

1 = 114,695

σk1 = 0,037

κ1 = 114,286

σκ1 = 0,039k2

= 153,515σk2 = 0,633

κ2 = 152,77

σ

κ2 = 0,684k

3 = 41,316σ

k3 = 15,024κ3 = 39,46

σκ

3 = 14,652

Measured effective

ratesOriginally

chosen effective rates

c0+ = 0,206σc0+ = 0,006

c

0

+

= 0,2

c

0

-

=

0,041

σ

c0-

= 0,003

c

0

-

= 0,04

c

1

=1,44

σ

c1

= 0,079

c

1

= 1,4

The effective rates can be calculated from the

‟

measured“

cumulants

very precisely.

Slide22

Special Case

22

Description

of

the Case

2

3

4

N

…

1

1

2

3

4

All

rates

in

the

same

direction

are

equal

.

We

thus

get

the

counting

matrix

N

Slide23

2

3

4

N

…

1

1

Special Case

23

Similartiy

Transformation

Step

of

length

x

2

3

4

N

…

1

1

Step

of

length

x/N

Counting

matrix

:

Counting

matrix

:

similar

The

statistics

of

both

systems

is

the

same!

Slide24

Special Case

24

Excursus

:

Tight

binding

model

of

a ring

…

1…

…

9

8

7

6

3

2

1

4

Magnetic

flux

Φ

An

ecelctron

is

hopping

between

the

sites

1, …, N,

gaining

a

phase

due

to

the

magnetic

flux

electron

N

5

Hamiltonian

in

the

tight

binding

approximation

:

t

=

transition

energy

ϕ

=

phase

the

electron

gains

in

one

cycle

The

eigenvalues

of

this

hamiltonian

can

be

found

using

transfer

matrices

!

J. Heinrichs, phys.

Rev

. 63, 165108 (2001)

Slide25

For

the

cumulant generating function,

one

obtains

The

j

-th

cumulant of a system with N internal states

thus is given

byThe

cumulants are decreasing with

j doing a zigzag curve

Special Case

25

Solution

Slide26

Discussion

& Outlook

26

Internal States

and

Rates

of

Real

Molecular

Motors

2

3

4

5

2

3

1

1

Step

of

length

x

Step

of

length

x

Real systems are more complicated than the models

disucssed

above:

Transitions are possible not only between

neighbouring

states.

There is branching in the cycle of internal states.

But still all

cumulants

can be found!

5

4

1

Bason

E.

Clancy

et al.,

Nat

Struct

Mol

Biol.

; 18(9

):

1020-7 (2011)

Slide27

Discussion

& Outlook

27

Internal States

and

Rates

of

Real

Molecular

Motors

1

5

4

1

2

3

1

Bason

E.

Clancy

et al.,

Nat

Struct

Mol

Biol.

; 18(9

):

1020-7 (2011)

Step

backwards

Step

forwards

Slide28

Discussion & Outlook

Comparison with real measured data (cooperation with biologists)Consideration of changing rates (e.g. ATP dependency)Branched chains of internal states

Description of detaching from and reattaching to the track.

28

Outlook