Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Biophysics with Single
DNA Molecules
Jens-Christian Meiners
University of Michigan
Why Biophysics with
DNA-Molecules?
W e use biomolecules and molecular biology
techniques to solve problems in physics:
- DNA is an ideal model system for
polymer physics
W e use physics to shed light on problems in biology:
- DNA is the most important biopolymer
- Its physical properties and statistical mechanics
are important for many biological functions (e.g.
replication or gene expression)
Dr. Jens-Christian Meiners, ITP & University of Michigan
1
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Why Single Molecules?
‘Traditional’ experiments probe an equilibrium distribution of
an ensemble of molecules:
Single-molecule experiments allow the preparation and
observation of individual non-equilibrium states:
Our primary technique: Optical Tweezers
How Do Optical Tweezers Work?
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
How Optical Tweezers Work
• A gradient force pulls
a particle into the
focus of the laser
beam
• Typical spring
constant: 20 pN/µm
• Typical time constant:
0.3 ms
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Tying a Knot into a
Single DNA Molecule
Force Measurement by Video Analysis
∆x = ∆F / ktrap
Force-extension curve of a single
λ-phage DNA molecule
Limitations:
- Speed
- Sensitivity
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Optical Tweezer Calibration
Histogram of bead position in optical trap
Number
100
 − 1 k x2 
trap 

P( x ) ∝ exp 2

 k BT 


10
1
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Spring constant
ktrap = 7.37 pN/µm
Position (micrometers)
What is the Force Sensitivity?
Limited by Brownian motion of bead in trap; time constant τ =
Equipartition theorem : < x 2 >=
ζ
k
k BT
; f = − kx ⇒ < f 2 >= kk B T
k
Averaging force for time δt , the error is
2k B Tζ
< f2>
=
.
δ t 2τ
δt
fN .
For a 1 µm bead, sensitivity is 12
In 1 kHz bandwidth, error is ~0.4pN Hz
rms.
Molecular motors exert ~2 pN in 1-10 msec.
DNA force fluctuations are ~10 fN in 1 msec
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Measuring femtoNewton Forces
Brownian motion of beads in trap dominates DNA force fluctuations,
but… we can measure cross correlation function of beads!
V(x)
x1
x2
<x1(t)x2(0)> … bead motion cancels. Only the correlated motion due to
DNA fluctuations remains.
We have measured ~6 femtoNewton forces with millisecond time
resolution.
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Hydrodynamic Coupling Between
Two Brownian Particles
The Brownian motion of the two trapped spheres is
coupled through hydrodynamic interaction.
This is a toy model for the motion of peptide fragments on
a large protein complex.
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Hydrodynamic Interaction
between Two Trapped Spheres
100000
80000
60000
4000
40000
2000
20000
0
0
-20000
-2000
-40000
-4000
-60000
-6000
auto correlation (fN2)
cross correlation (fN2)
6000
-80000
-100000
0
1
2
3
4
time (ms)
Phys. Rev. Lett. 82, 2211 (1999).
The Quantitative Picture
Time scale from the Brownian motion of the beads
in their traps:
τ=
ζ bead
≈ 0.5 ms
ktrap
Using Smoluchowski’s equation and the Oseen
tensor to describe the hydrodynamic interaction,
we find:
(
)
auto-correlation
(
)
cross-correlation
1 k BT −t (1+δ ) /τ
+ e −t (1−δ ) / τ
e
2 k
1 k BT − t (1+δ ) / τ
− e −t (1−δ ) /τ
x1 (0) x2 (t ) =
e
2 k
x1 (0) x1 (t ) =
with δ=3r/E
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
The Physical Picture
Isn't this time-delayed anti-correlation counterintuitive?
I.e. if one of the beads moves to the left, the other one follows
later to the right, instead of just being dragged along?
Look at the system as a superpositon of two types of fluctuations:
correlated
anti-correlated
fast
slow
The correlated fluctuations are fast, because one bead drags the
other one along in its wake. The anti-correlated fluctuations are slow,
because the fluid between the spheres has to be displaced. At equal
amplitudes, the slower fluctuations dominate the correlation function.
Thermal Fluctuations of an
Extended DNA Molecule
We can measure:
- The relaxation time τDNA
- The amplitude /
spring constant kDNA
We expect the fluctuations to
decay exponentially, perhaps
showing some non-linear
behavior:
- Friction Coefficients
- Anisotropy
ν
f1 (0) f 2 (t ) = k DNA k BT e − ( t /τ DNA )
Dr. Jens-Christian Meiners, ITP & University of Michigan
- Nonlinear Effects (υ ≠ 1)
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
2
25000
20000
15000
10000
5000
0
-5000
cross correlation (fN )
2
cross correlation (fN )
Relaxation of a Stretched DNA molecule
longitudinal
92% extension
3000
1000
0
0 1 2 3 4 5 6 7 8 9 10
time (ms)
2
cross correlation (fN )
2
cross correlation (fN )
0 1 2 3 4 5 6 7 8 9 10
time (ms)
1500
transverse
92% extension
1000
500
0
0
10
20 30 40
time (ms)
longitudinal
82% extension
2000
1500
transverse
88% extension
1000
500
0
50
0
10
20 30 40
time (ms)
50
Phys. Rev. Lett. 84, 5014 (2000)
The relaxation times are highly anisotropic:
25
spring constant (N/m)
relaxation time (ms)
6e-6
20
transverse
15
10
5
longitudinal
0
0.70
5e-6
4e-6
3e-6
2e-6
longitudinal
1e-6
transverse
0
0.75
0.80
0.85
0.90
0.95
0.70
0.75
0.80
0.85
0.90
0.95
nonlinear exponent
2.0
1.5
The cross correlation functions
are compressed exponentials
at high extension.
1.0
0.5
0.0
0.70
0.75
0.80
0.85
0.90
0.95
relative extension
Dr. Jens-Christian Meiners, ITP & University of Michigan
10
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
The Model
From cross-correlation: kc, τc
(mostly DNA)
From auto-correlation: ka, τa
(mostly beads)
Correct for mixing:
ζ bead
τ a (k a + k c )
k (1 − τ a / τ c ) − kc (1 + τ a / τ c )
=τc ⋅ a
(k a + kc )(1 − τ a / τ c )
k DNA = kc
τ DNA
ζ DNA =
τ DNA k DNA
π2
The Theory
Extension introduces anisotropy:
(e.g. in a stretched violin string – or a DNA molecule)
In the spring constant:
Longitudinal:
kl =
dF
dx
Transverse:
kt = F ( E ) / E
E
In the friction coefficient:
ζl
<
ζt
In the limit of a rigid rod:
Longitudinal:
ζl =
2πη s L
ln( L / d )
4πη L
Transverse: ζ t = ln( L /s d )
Dr. Jens-Christian Meiners, ITP & University of Michigan
11
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Using the Marko-Siggia approximation for the wormlike
chain, we can predict spring constants and relaxation time
for the extended DNA molecule:
−3
kl =
τl =
k BT  E 
1 −  + 1
2b  L 
kt =
2η s Lb
1
⋅
π k BT ln( L / d ) 1  E  − 3
1 −  + 1
2 L
τt =
−2
k BT  1  E 
1 E
1−  − + 


b 4 L
4 L 
4η s Lb
E
π k BT ln( L / d ) 1  E  − 2 1 E
1 −  − +
4 L
4 L
The relaxation times are highly anisotropic:
25
spring constant (N/m)
relaxation time (ms)
6e-6
20
transverse
15
10
5
longitudinal
0
0.70
5e-6
4e-6
3e-6
2e-6
longitudinal
1e-6
transverse
0
0.75
0.80
0.85
0.90
0.95
0.70
0.75
0.80
0.85
0.90
0.95
nonlinear exponent
2.0
1.5
The cross correlation functions
are compressed exponentials
at high extension.
1.0
0.5
0.0
0.70
0.75
0.80
0.85
0.90
0.95
relative extension
Dr. Jens-Christian Meiners, ITP & University of Michigan
12
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Friction Coefficient of the DNA Molecule
friction coefficient (Ns/m)
transverse
rigid rod
2.0e-8
transverse
data
1.5e-8
1.0e-8
longitudinal
rigid rod
longitudinal
data
5.0e-9
0.0
0.70
0.75
0.80
0.85
0.90
0.95
relative extension
What have we learned?
We understand thermal fluctuations in extended flexible
polymers.
• The amplitude is determined by the spring constant and
highly anisotropic. The wormlike-chain model, modified for
the violin-string geometry gives an accurate quantitative
description.
• The relaxation time decreases with increasing extension
and its behavior is dominated by the spring constant, not the
friction coefficient.
• The friction coefficient is consistent with a rigid-rod model,
hydrodynamic screening through adjacent segments is
insignificant.
• We observe evidence for unusual nonlinear effects at high
extensions.
Dr. Jens-Christian Meiners, ITP & University of Michigan
13
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Biological Relevance?
Thermal fluctuations bring distant sites on a
DNA molecule temporarily in close contact.
Important for
• Regulation of Gene Expression
• Control of Replication
• Site-specific Recombination
Transcriptional Control
through DNA Looping
(Lac Z)
(Lac I)
In the presence of sugar, binding
of the regulatory protein is
inhibited, the gene is tuned on
Dr. Jens-Christian Meiners, ITP & University of Michigan
In the absence of sugar, the
regulatory protein bends the
DNA into a loop, the gene is
tuned off
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Transcriptional Control
through Mechanical Forces?
->Tension affects
transcription!
50
10 kb
3 kb
1 kb
40
2
600
500
400
30
300
20
1
200
10
0
0
0
50
100
150
100
Loop Formation Rate (s-1)
->Tension affects
loop formation rates
3
Loop Formation Rate (s-1)
Tension affects the
thermal motion of
the DNA molecule
0
200
Force (fN)
Forces as low as 50 fN may be sufficient to prevent looping
RNAP can exert forces of up to 20 pN! -> Molecular Lever?
Transcription under Tension
Can we turn transcription on and off by
pulling on the DNA substrate?
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Microfluidics for
Single-Molecule Experiments
Soft lithography in a silicone
elastomer allows the fabrication
of flow channels and valves
open valve
closed valve
Low-Force and Constant-Force
Measurements with Line
Tweezers
Triangular optical potentials
allow for measurements in
a constant-force mode:
The force acting on the
micro spheres is independent
of their separation.
Dr. Jens-Christian Meiners, ITP & University of Michigan
16
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
High-speed Measurements with
Line Tweezers
DNA Condensation through
Multivalent Cations
The speed of the collapse can
tell us something about its
dynamics (one nucleation site
vs. several sites that merge)
Dr. Jens-Christian Meiners, ITP & University of Michigan
17
Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
The Role of other Mechanical
Constraints - Supercoiling
Three-dimensional diffusion vs. one-dimensional diffusion
Is a one-dimensional search more efficient than a threedimensional one?
Not necessarily, if you take the increased hydrodynamic
drag into account
Active Structural Biology with
Optical Tweezers?
Can we study the effect of tension in the substrate DNA
on the conformation of the repressor protein with optical
tweezers and FRET?
Dr. Jens-Christian Meiners, ITP & University of Michigan
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Dynamics of Single DNA Molecules Dynamics of Single DNA Molecules (ITP Complex Fluids Program
2/12/02)
Conclusion
Single-molecule experiments are a powerful
new tool to shed light on old problems in
physics and biology
Acknowledgments
• Dhruv Acharya
• Raj Nambiar
• Ben Liesfeld
• Liz McDowell
• Kajal Parikh
• Steven Quake (Caltech)
• Joel Stavans (Weizmann)
• Funding from NSF,
Alfred. P Sloan Foundation,
Research Corporation
Dr. Jens-Christian Meiners, ITP & University of Michigan
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