Modeling fluctuations in the force-extension single-molecule experiments

Alexander Vologodskii

New York University

Diagram of the force-extension experiment

F

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80- Experimental data of Smith et al.

chainWorm-like

Extension, <x>/L

The force-extension dependence for DNA is well studied

The force has entropic nature

From: J. W. Shaevitz1, E. A. Abbondanzieri, R. Landick & S. M. Block. Backtracking by single RNA polymerase molecules observed at near-base-pair resolution. Nature, 426, 684-687.

The entropic force from extended DNA molecule is used in many single-molecule experiments

DNA model for Brownian dynamics simulations

The intersegment interaction is specified by the Debye-Hückel potential

Segments are stretchable

Virtual beads of a certain diameter placed at chain vertices specify hydrodynamic interaction with solution and between the beads

Discrete wormlike chain with some modifications:

Dynamics of the chain is described by the Langevin equations

midvi

dt ij

j v j Fi ij

j f j

ij

where

is a configuration-dependent friction tensor

is a force acting on beadFi i

represents the randomly fluctuating force resulting from the thermal motion of the surrounding fluid

ijj f j

is the mass of bead imi

How accurate is Brownian dynamics simulation of DNA properties?

Comparison of measured and simulated diffusion coefficients of knots along stretched DNA moleculeshows that simulation is quite accurate

Tying knots by optical tweezers

Experimental measurement of knot diffusion

X. R. Bao, H.J. Lee and S.R. Quake, Phys. Rev. Lett., 91, 265506 (2003)

Brownian dynamics simulation of knot diffusion

Typical simulated conformations of knotted model chains

Comparison of the measured and computed diffusion coefficients of knots

Knot type

Computed diffusion

coefficient, m2/s

Measured diffusion

coefficient, m2/s

8.6 ± 1

12.5 ± 0.5

6.0 ± 1

7.9 ± 0.3

2.5 ± 0.3

4.8 ± 0.2

Simulated values of the force fluctuate strongly

Time, ns

0.0 0.2 0.4 0.6 0.8 1.0

For

ce,

Fb/

kT

-10

-5

0

5

10

Time, ns

0.0 0.2 0.4 0.6 0.8 1.0

x/L

0.79

0.80

0.81

0.82

0.83

0.84

Time, ns

0 20 40 60 80 100

For

ce,

pn

-20

-10

0

10

The force fluctuations do not depend on its average value

Each point is the averaging over 1 ns

The force averaging does not occur over 0.1 s

Each point is the averaging over 100 ns

Time, s

0 2 4 6 8 10

For

ce,

pn

-4

-2

0

2

4dt = 400 psdt = 4 ps

Time, ms

0.0 0.2 0.4 0.6 0.8 1.0

For

ce,

pn

-1.0

-0.5

0.0dt = 400 ps

The force averaging does not occur over 10 s

Time, ms

0 20 40 60 80 100

For

ce,

Fb/

kT

-0.7

-0.6

-0.5

-0.4

-0.3

A good averaging of the force is achieved by averaging over 1 ms

Fluctuations of the force do not depend on DNA length

DNA length, bp

1000 10000

x, p

n0

20

40

60

80

f = 0.5 pn

f = 2.5 pn

f = 8.3 pn

DNA length, bp

1000 10000

f, pn

0

1

2

3

4

5

f = 0.1 pn

f = 0.5 pn

f = 2.5 pnf = 8.3 pn

Presence of a protein-induced bend decreases DNA extension

Can the extension measurement be used to determine the bend angle?

Simulated values of the extension reduction resultingfrom DNA bending by angle

Time

Ext

ensi

on

No bound protein No bound proteinOne protein is bound

Large fluctuations of the extension and a finite time of the protein-bound state create a problem

DNA length, bp

0 1000 2000 3000

, n

m

0

50

100

150F = 0.1 pn

F = 1 pn

The variations of the extensions are large

Extension of a single DNA molecule by force

ForceForce

These are actual proportions for 1500 bp DNA

time, ms

0 20 40 60 80

Ext

ensi

on,

nm

0

100

200

300

400L = 500 nmF = 0.1 pn

No beadBead radius 500 nm

Fluctuations of DNA extension averaged over 0.4 ms

Fluctuations of DNA extension averaged over 40 ms

Time, ms

200 400 600 800

Ext

ensi

on, n

m

200

400L = 500 nmF = 0.1 pn

No beadBead radius is 500 nm

Interval of averaging, ms

0.01 0.1 1 10 100 1000

Ext

ensi

on v

aria

tion,

nm

10

100

L = 500 nmF = 0.1 pn

With bead R = 500 nmNo bead

What averaging interval do we need?

The work was supported by NIH

Displacement of unknotted part of the model chain eliminates the chain length restriction