Viscoelastic Properties of Entangled DNA Solutions: Dependence on Molecular Length
and Concentration
1Physics
Abstract
We use macroscopic rheology to investigate
the viscoelastic properties of solutions of
monodisperse linear DNA, as a function of DNA
length and concentration. We span from the
unentangled to the entangled regime by using
DNA lengths that vary from 11 to 115
kilobasepairs (3.7 to 39 μm) and solution
concentrations that range between 0.5 and 4.0
mg/ml. We investigate the effects of oscillatory
frequency on the linear elastic (G’) and viscous
(G”) moduli, with frequency values of 0.01 - 30
Hz. In addition, the dependence of viscosity on
strain rate is studied with strain rates ranging
from 0.01 to 100 Hz. Importantly, these studies
are the first to examine the molecular length
dependence of linear viscoelastic properties for
concentrated DNA solutions. Results are
compared to theoretical predictions based on the
Rouse model and Reptation model for
unentangled and entangled polymer solutions,
respectively.
Introduction
Entangled polymer physics is governed by the
Reptation model which represents entangled
polymer solutions as tube networks confining
individual polymers to 1D motion along the length of
the tube. The polymer solution concentration at
which this tube model is expected to be observed,
which is also dependent on polymer length, is known
as the entanglement concentration, corresponding to
when the polymers are all expected to have one
entanglement over their entire length.
Fig. 1. A representation of an
entangled polymer mesh in
which a single polymer of
interest (red) is depicted as
being confined to a tube
according to the Reptation
model1.
It is the viscoelastic characteristics of nearentangled and entangled DNA samples, as a
function of both polymer length and concentration,
that we probed to better understand the nature of
these systems on a macroscopic level. Viscoelastic
characteristics are characterized by the linear elastic
modulus (G’) and the viscous (G”) modulus. They
are so named because G’ is the in-phase response
of the system while G” is the out-of-phase response
of the system corresponding to resistance to and
relaxation in response to an applied stress.
Methods
Sample Preparation
DNA samples were prepared according to a
large-scale DNA extraction procedure2. The
extraction procedure involves growing E. Coli cells
containing the appropriate DNA plasmids, lysing
the cells to release the DNA, and purifying the DNA
by getting rid of other unwanted cellular contents.
Patrick
1
Smith ;
Veslin
2
Dobrev ;
Jeff
3
Urbach ;
2Chemistry
Rae
1
Anderson
Department, University of San Diego, San Diego, CA;
Department, Georgetown University, Washington
3
D.C.; Physics Department, Georgetown University, Washington D.C.
Sample Preparation Continued
For the samples we prepared, there was an additional step
of converting the DNA from a circular to a linear form.
Samples contained DNA of 11 kbp (3.7 µm) or 45 kbp (15
µm) and required BamHI and ApaI enzymes for
linearization, respectively.
Rheological Measurements
Georgetown University
Results
4A, 11 kbp DNA
4B, 45 kbp DNA
ω𝑐 ~ 0.22 rad/s
τ𝐷 ~ 29 s
Instrumentation
Rheological data in this project was taken with an Anton
Paar MCR 301 rheometer using a 25mm cone-plate setup.
The top surface, known as the tool, is lowered until the
sample completely fills the gap between the tool and the
baseplate. Then, the rheometer rotates the tool while
measuring various viscoelastic properties. This setup is
what enables easy variation of strain amplitude and
frequency of oscillation. The geometry of the tool-sample
setup is such that strain is the same everywhere in the
sample in order to simplify the system. Data was collected
using Anton Paar’s RheoPlus software.
Fig. 2. A diagram of a cone-plate
geometry of a rheometer3. Of import
is that the bottom surface is flat
while the top surface slopes
downwards towards the center to
maintain even strain throughout the
sample.
Experimental Procedure
For each sample, we first performed a strain sweep
experiment from 0.1-100% strain to determine the strain
magnitude of the linear regime to ensure subsequent
experiments fell within this regime. We generally found
10% strain to be an appropriate strain amplitude.
Fig. 3. A strain sweep of
45 kbp DNA 45 kbp DNA at 1.5 mg/ml
concentration. As we
hoped to study the linear
regime, we found a strain
amplitude where G’ and
G” had slopes of ~ 0 and
the strain amplitude was
large enough to avoid
experimental shortfalls of
the instrumentation.
ω𝑐 ~ 0.31 rad/s
τ𝐷 ~ 20 s
Fig. 4(A,B). Frequency sweep experiments. Scale bars have been included to compare to theory.
The crossover frequency, ω𝐶 , is the frequency at which G’ is finally equal to G”. It is an indicator of
when entanglements are taking effect as before this time, the period of oscillation is greater than the
time it takes for the polymer to reptate out of its confinement tube, τ𝐷 , so the sample behaves mostly
like a Newtonian fluid. At this point, however, the period of oscillation is now equal to the time it takes
a polymer to leave its tube so it is liable to get caught on neighboring polymers as they are tugged.
Below the crossover frequency, ω𝐶 , the theoretical scaling laws are G’ ~ ω2 and G” ~ ω.4 Above ω𝐶 ,
we expect to see G’ approach a slope of 0 at a region called the plateau modulus, 𝐺0𝑁 . At this point
we would also expect to see G” ~ ω−1/4 . Scaling bars have included for all four mentioned predicted
scaling laws in order to compare our experimental results to theoretical predictions. For the 11 kbp
DNA we find that the disengagement time and the plateau modulus scale with concentration, c, as τ𝐷
~ 𝑐0.5 and 𝐺0𝑁 ~ 𝑐1.9 . For the 45 kbp DNA we find that 𝐺𝑁0 ~ 𝑐1.8 , very similar to our findings for the 11
kbp DNA. The scaling of τ𝐷 agrees with results from a previous study on λ DNA5. These scalings for
the plateau modulus, however, slightly vary from the scaling of 𝐺𝑁0 ~ 𝑐 2.3 predicted by the Reptation
model.
to minimize sample alteration during long data collection
periods (6+ hours), a solvent trap was used.
Conclusions
To our knowledge, this is the first time that
macrorheology is being performed on monodisperse
DNA of different lengths. We find that our results
support the hypothesis that DNA behaves like an
entangled flexible polymer at the concentrations and
lengthscales we study. We hope to continue this
research by taking data at different DNA
concentrations as well as with DNA of greater length
to better probe the effects of polymer length and
concentration on the viscoelastic properties of the
solution. Also, we hope to further explore the
transition from the terminal regime to the entangled
regime to better characterize this transition.
Furthermore, we hope to take videos of individual
polymers in these solutions as we apply stress to the
system using confocal microscopy coupled to a
rheometer.
References
Fig. 3A
1. http://www.nobelprize.org/nobel_prizes/physics/laureates/1991/illpres/polymers.html
2. Laib, S., Robertson, R.M. & Smith, D.E. Preparation and characterization of a set of
linear DNA molecules for polymer physics and rheology studies. Macromolecules 39,
4115-4119 (2006).
Next we performed a frequency sweep from 0.01-100 Hz
(0.0628-618 rad/s) to determine G’(ω) and G”(ω) followed
by a shear rate sweep from 0.01-100 Hz and finishing
again with a frequency sweep over the same frequency
range. While the strain and frequency sweep experiments
are oscillatory measurements, shear rate sweeps are nonoscillatory measurements. Three trials were conducted for
each experiment for each sample. Between all trials and
all experiments there was a five minute delay to ensure
the system had enough time to relax back to equilibrium,
which we will show in the results section. Finally, in order
Fig. 6. Measured shear stress for both 11 kbp and 45 kbp
DNA. These results are in agreement with Reptation
theory augmented with constraint release mechanisms
which predict a positive slope at low shear rate followed
by a plateau which is characteristic of entangled
systems5.
3. Arevalo, Richard Carl. "Shedding Light on the Nonlinear Stiffening Effect of Sheared
Type-I Collagen Networks." (2013).
4. Chapman, Cole D., et al. "Onset of Non-Continuum Effects in Microrheology of Entangled
Polymer Solutions." Macromolecules (2014).
5A, 11 kbp DNA
5B, 45 kbp DNA
5. Teixeira, Rodrigo E., et al. "The individualistic dynamics of entangled DNA in
solution." Macromolecules 40.7 (2007): 2461-2476.
Acknowledgments
Fig. 5(A,B). Complex viscosity at 0.1 strain for 11 kbp (3.7 μm, Fig. 5A) and 45 kbp (15 μm, Fig. 5B)
DNA. In the terminal regime where there is minimal entanglement, a scaling law of η∗ ~ ω−1/2 is
predicted corresponding to a semi-dilute solution while in the entangled regime we expect a scaling
law of η∗ ~ ω−1 indicating a fully entangled solution. As seen, these systems display entanglement
dominated behavior over the majority of the frequency range.
We thank L. Fred and K. Gerhart for assistance with
sample preparation, P. Kumar for assistance with
rheology instrumentation, and C. Chapman for
assistance with procedures and troubleshooting. We
also thank the National Science Foundation for
support through grant number REU DMR-1004268.
Poster produced by Faculty & Curriculum Support (FACS), Georgetown University School of Medicine