The Dependence of Polymer Dynamics on Its Length
Undergraduate Project Summary
Shai Kinast
Supervisor: Dr. Oleg Krichevsky
Abstract
Previous measurements showed that the dynamics of dsDNA polymers follows closely the prediction of the Rouse model, which is the first model of polymer dynamics. According to this model,
the mean square displacement (MSD) of the polymer is proportional to the square root of the time.
Moreover, the results
obtained on DNA of different lengths exhibit a certain kind of correlation:
√
the prefactor in t dependence grows with increasing DNA length. The correlation is not perfect
and it is not clear at this point whether the changes in the prefactor are indeed due to the changes
in DNA length, or e.g. are related to differences in DNA sequences in theses experiments.
The aim of the project was to design and prepare DNA polymers of different lengths which have
identical genetic sequences, and to measure the internal dynamics of these samples in order to see
whether there is any effect of the polymer length on its kinetics.
Brief Introduction to Polymer Dynamics
The complex dynamical properties of polymers arise from the frustrated dynamics of their monomers
as their diffusion is subject to the connectivity constraints within the polymer chain. The first theoretical description of random motions of monomers within an isolated polymer coil was presented
by Rouse as the classical beads-springs model. Rouse model accounted for the entropic elasticity
of the polymer segments and the viscous friction of the solvent. Furthermore, it was recognized
rather early that the hydrodynamic interactions between the monomers play a major role in the
polymer dynamics. These effects were incorporated into the Rouse model by Zimm. Zimm theory
is generally believed to hold for all flexible polymers in the dilute regime, while Rouse theory fits
the behavior of semi-flexible polymers.
When investigating the time-dependence of the polymer MSD, the relaxation time of the slowest
mode (τ ) is of high importance. For times that are smaller
√ than the relaxation time of the slowest
mode (t τ ), Rouse model suggests that < r2 (t) ∝ t, while in Zimm model the relation is
< r2 (t) ∝ t2/3 . In both models, however, for t τ the MSD is directly proportional to the time,
as in a ’simple’ diffusion equation.
Regarding the influence of polymer length on its dynamics, in both models the length has no
influence on the dynamics in the short time regime (t τ ), while in the long time regime (t τ )
the length of the polymer determines the prefactor of t.
The polymer we used for this experiments was dsDNA, a polymer considered as a semi-flexible
polymer: the length characterizing its rigidity with respect to thermal fluctuations, or Kuhn length,
b ∼ 100nm (∼ 340bp), is much larger than dsDNA width d ≈ 2nm.
Fluorescence Correlation Spectroscopy Technique
In our experiments the sensitivity to the monomer motions is obtained through the specific fluorescent labeling of a single base on the long DNA molecules. The motion of the labeled monomers is
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then monitored by Fluorescence Correlation spectroscopy (FCS). FCS measures the diffusion kinetics of fluorescent species by monitoring the fluctuations δI(t) in emission intensity I(t) in an inhomogeneous illumination beam (typically a confocal volume). As fluorescent molecules move in and
out of the sampling volume, their fluorescence excitation and thus emission fluctuate: therefore the
shape of the autocorrelation function of the fluorescence fluctuations G(t) = hδI(0)δI(t)i / hI(t)i2
reflects the kinetics of diffusion. For independent point
sources
of fluorescence randomly moving in
a Gaussian beam, G(t) is directly related to the MSD r2 (t) of the fluorophore:
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G(t) =
N
!−1
2 r2 (t)
1+
2
3 wxy
!−1/2
2 r2 (t)
1+
3 wz2
(1)
where wxy and wz define the dimensions of the sampling volume in lateral and transversal dimensions respectively, and N is the average number of independently
moving fluorophores in the
sampling volume. Through Eq.1, the mean-square displacement r2 (t) of a labeled monomer (in
this case the end monomer) can be extracted from the measured G(t).
Target of Experiment
We are interested to measure the influence of polymer length on its internal dynamics. We will
use dsDNA of different lengths as model of semi-flexible polymers. DNA fragments will be labeled
specifically at a single position by fluorescent carboxyrhodamine6G and will be measured in the
confocal FCS setup. Fluctuations in detected
intensity will reveal the time dependence of the DNA
monomer’s mean-square displacement r2 (t) .
Experimental Setup
The exciting radiation provided by a laser beam is directed into a microscope objective via a
dichroic mirror and focused on the sample (see figure 1). The fluorescence light from the sample is
collected by the water immersion objective and passed through the dichroic mirror and the emission
filter. The pinhole in the image plane (field aperture) blocks any fluorescence light not originating
from the focal region, thus providing axial resolution. Afterwards, the light is focused onto the
detector, preferably an avalanche photodiode or a photomultiplier with single photon sensitivity.
Figure 1: Scheme of a confocal FCS setup. Green arrow is direction of excited laser beam. Red
arrow is direction of emitted laser beam. DM - Dicroic Mirror; OB high resolution objective; S sample, sampling volume 0.5 fl; NF notch filter; L lens (40mm focus length); PH - pinhole; APD
avalanche photodiode; CORR correlator.
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Sample Preparation
pUC18 DNA (∼ 3000bp, dsDNA only) was linearized with HindIII, and then ligated (using T4
DNA Ligase) while oligonucleotides of various (low) concentrations were introduced. In this way we
made DNA polymers of different lengths (3000bp, 6000bp, 9000bp etc). The polymers were labeled
by ligating prelabeled oligonucleotides to their end. Using gel separation the samples were separated
by length, and then put in the experimental buffer (10mM Na2HPO4-NaH2PO4, 1mM EDTA, 0.1M
NaCl, pH 7.6). The dynamics of the samples was measured within the FCS setup.
Experimental Procedure
The dynamics of the samples was measured within the FCS setup described above. The measurements were conducted at 20o C. The excitation power of the laser beam was chosen to be 12µW weak enough to prevent photobleach of the labeled ploymers, yet strong enough to get an appropriate results. The concentration of the samples was within the dilute regime and corresponded to 0.1
to 1 molecules in the confocal volume (0.5fl) on average (from 0.3 to 9mg/L, depending on DNA
length). The correlation functions were collected by short runs of 20-30s over 40 times and then
averaged.
The mean-square displacement r2 (t) of the labeled
monomers was obtained from the correlation
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curves G(t) by solving Eq.1 with respect to r (t) for every time lag t. The confocal volume
dimensions wxy and wz were obtained by measuring the diffusion of free Rh6G fluorophores and
fitting the resulting FCS correlation functions with:
G(t) =
1
1
q
N 1+ t 1+
τ
(2)
t
τ ω2
2 /(4D) is the characteristic time of diffusion of Rh6G molecule across the sampling
where τ = wxy
volume and ω = wz /wxy . Typically, τ = 44±µs and ω = 5±1. We used the value of Rh6G diffusion
constant D = 2.810−6 cm2 s−1 at
20o C. Eq. 2 is a standard formula for the diffusing objects, and it
can be obtained from Eq. 1 by r2 (t) = 6Dt substitution.
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Results and Discussion
We present in Fig. 2 typical measured correlation function for 3000bp, 6000bp, 9000bp and 12000bp
(for 1, 2, 3 and 4 ligated segments of pUC18, respectively). The main decay of the measured G(t)
in the millisecond
scale characterizes the kinetics of monomer motion. Then through Eq. (1), the
MSD ( r2 (t) ) of a labeled monomer (i.e. end monomer) can be extracted from the measured G(t).
Figure 2: FCS correlation functions for the end monomers of 3000bp, 6000bp, 9000bp and 12000bp
fragments.
In Fig. 3 we present the monomer’s MSD for the above mentioned samples. The measurements
span over 5 orders of magnitude in time, from ∼ 5µs to ∼ 0.5s. Two different regimes
can
2of motion
be clearly distinguished
for each of the samples. At long time lags (t >∼ 10ms), r (t) is almost
linear with time ( r2 (t) ∼ t0.95±0.03 for 3000bp). This regime corresponds to the monomer
motions
above the polymer coil size and reflects the trivial diffusion of the coil as a whole ( r2 (t) ∼ t).
The other regime occupies shorter time scales and corresponds to the monomer motions
inside
the polymer coil. In this regime, the time dependence of the MSD significantly different ( r2 (t) ∼
t0.46±0.02
√ 3000bp). This behavior is consistent with the prediction of the original Rouse model
for
( r2 (t) ∼ t).
As we see in Fig. 3, the results are consistent with previous measurements, as it can be seen
clearly that increasing of the length of the polymer results in increasing of a prefactor in the short
time scale regime (an exception is the 12000bp fragments). Having the same genetic sequence in all
samples, this experiment cancels the possibility of different genetic sequence as a reason for the yet
unexplained prefactor.
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Figure 3: The mean square displacement as a function of time for the end monomers of
3000bp, 6000bp, 9000bp and 12000bp fragments, as extracted from measured FCS correlation functions.
Finally, we conclude that the results do not agree with Rouse model. This model suggests that
length-dependent prefactor should be added to the long time scale, while in the other regime no such
pre-factor is in place. The results, however, are just the opposite: the length-dependent prefactor
appears clearly in the short time scale, and it reduces significantly in the second regime.
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