Intermediate presentation at the group seminar July 18th 2012 Maximilian Thaller 1 Contents Molecular Motors Connection between rates and cumulants Simulation Discussion of a special case ID: 934993
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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
Slide2Contents
Molecular MotorsConnection between rates and cumulants
Simulation
Discussion of a special case
Further discussion and outlook
2
Slide3Molecular 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)
Slide4Molecular Motors
4
The Motor-Position
is
a Stochastic Process
Slide5Molecular
Motors
5
Distribution
of
the
Motors Position
at
Different Times
Slide6Usually
,
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.
Slide7Molecular 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
Slide8Define 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.
Slide9On 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
Slide10Consider 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
!
Slide11Connection 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)
Slide12Connection between Rates and
Cumulants
One
obtains
12
Effective
Rates
All
cumulants
of
an
arbitrary
molecular
motor can be calculated without knowing
a single eigenvalue!
Slide13Simulation
Chosen rates:
13
The
Two
State Motor
2
1
1
1
2
2
1
Slide14Simulation
14
Example
–
Three
simulated
motors
Slide15Simulation
15
Measuring
the
Cumulants
t
‘
One
molecular
motor
Many
molecular
motors
Probability distribution at time t‘
→
Yields the
cumulants
Slide16Simulation
16
Convergence
of the Cumulants
Only
a
few
series
are
necessary
to obtain relable
values for κ1.
Slide17Simulation
17
Convergence
of the Cumulants
Reliable values for
κ
2
can be achieved with a small number of series.
Slide18Simulation
18
Convergence
of the Cumulants
A large
number
of
series
is
necessary
to obtain
relable values for κ3.
Slide19The
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!
Slide20Simulation
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)
Slide21Simulation
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.
Slide22Special 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
Slide232
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!
Slide24Special 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)
Slide25For
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
Slide26Discussion
& 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)
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
Slide28Discussion & 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