1. Background
It is well known that DNA is a double helix structure composed of four base
pairs, adenine, guanine, cytosine, and thymine which are linked to ribose
phosphate [1]. DNA is an immensely long molecule with tens of millions of base
pairs, it is required to compact itself to a micron-sized scale in order to wrap
around histones, creating chromatin, and fit within a nucleus [1]. These
structures typically contain around three meters of DNA [2]. Most of this
compaction is achieved through superlooping, a loop created by excessive
torsion either mechanically or chemically applied as a means of relieving strain
energy.
Due to its applications in targeted drug therapies, gene expression is currently
a highly researched topic. It has been determined that gene expression can, in
some cases, be mediated by superlooping in specific regions of DNA.
Superlooping can cause these regions to destabilize locally, thus allowing
specific promoters to become active, resulting in varied protein synthesis [3-6].
Currently, research is being conducted to explore torsional properties of DNA
and subsequent superlooping using micromanipulation techniques such as
optical tweezing. In these experiments, DNA is constrained at one end, and
torsion with a specific force is applied to the other to induce superlooping [1, 6].
The major consideration with this method of experimentation lies in that it
considers the entire length of the DNA strand being tested, while actual DNA is
superlooped only in specific regions [5]. Because of restriction, there is still a
lack of understanding of how twisting directly affects gene expression.
In order to more fully understand the biological purpose of superlooping, it has
been experimentally examined from a mechanical standpoint; the theoretical
understandings are also mechanical expressions. Currently, several
mathematical models exist to explain the relationship between superloops and
the energy they relieve; however, there is limited experimental data to verify
them. Many of these models propose a correlation between torsion applied to
the DNA and the physical dimensions of the resultant superloop, but fail to
back up their propositions with hard data. As such, the primary goal of our
experimentation is to investigate the mechanics of superlooping using an
isotropic polymer line as a simulation for DNA. Specifically, we will be focusing
on the model1 of T. Strick et al. which proposes a relationship between the
applied force, along with the physical properties of the polymer line, and the
radius of the first superloop (induced by reaching critical torque) [1]. At this
point, the energy, ԑ, due to bending and work against the force, F, is equal to
πB/R + 2πRF where B is the bending modulus, a function of elastic modulus
and the polymer line dimensions [1]. According to the model, this “critical”
radius, Rc,b, can be derived from the energy equation to yield should Rc,b=
(B/2F)1/2 [1], where B=(EIFA)1/2 [1]. Theoretical values of Rc,b (calculated from
the equation) will be calculated and compared to the experimental results.
Defining the relationship between the radius of the first superloop and the twist
and applied force will help increase the understanding of the mechanism of
superlooping in terms of energy release and provide a more theoretical
explanation of its role in DNA mechanics.
Experimental Methods
Project Description
References
Marlen Mahendraratnam, Allison Stiller, and Stephen J. Burns
Dept. of Biomedical Engineering, Dept. of Mechanical Engineering Materials Science, University of Rochester
Experimental Analysis of Superlooping in Twisted Polymer Line and its Applications to Mathematical Models of DNA
50 lb polypropylene fishing line was purchased from Danielson company and
used in the experiments in order to simulate DNA as an homogeneous polymer
line. Using a screw driven Instron test frame with cross-head rate controls with a
load cell, a tensile test was done to determine the elastic modulus of the polymer
line.
Four sets of twisting-looping data were taken using masses of 0.06lb, 0.16lb,
0.26lb, and 0.36lb hanging from the double braided line. The line was looped
through the masses so that the fishing line would overlap itself when torsion was
applied, mocking the double helix structure of DNA. The exact same procedure
was followed with all four weights and all measurements were taken using a
caliper. First, the original length (L0) of the fishing line from the hook to the weight
was measured. The weights were then turned 360 degrees an n number of times
until the first loop of buckling occurred. The new length (L1) from the hook to the
weight and the diameter of the first loop were recorded. Lastly, the weight was
released, allowing the torsional energy to decrease and the turns to unwind. The
length (L2) from the hook to the unwounded weight was recorded. Using Excel
and MATLAB, graphs were created to show the relationships between L0 and
twist, radii, L1, and L2.
Experimental Data
Superlooping in DNA is a mechanism thought to relieve torsional strain energy; however, it also serves a
biological purpose as it allows for compaction of DNA which consequently affects gene expression. Unfortunately,
despite nano-torsional testing, the mechanics of superlooping are still relatively unknown. Several mathematical
models exist to provide an explanation, but nearly all of them lack experimental evidence to verify their claims.
Our goal is to verify the equations provided by T. Strick et. al. via torsional tests on an isotropic 50 lb fishing line
to mimic the mechanism of superlooping in DNA. Primarily, we are investigating the relationship between applied
twist (turns per unit length), applied force, radius of the first resultant superloop, and length of the polymer line
used. We hope that by understanding the mechanics behind superlooping, researchers will be able to better
direct future mechanical experimentation of DNA.
Equations Analyzed
Equation 1: Energy in the polymer line due to bending and work against the force, F.
Equation 2: By taking the derivative of Equation 1 with respect to R, the radius of critical buckling
(dԑ = 0) is found. We can see that this radius is a function of B, bending modulus, and applied force.
Equation 3: It was determined that B was a function of modulus, E, moment of inertia, I, applied
force, F, and cross-sectional area, A based on unit analysis of the energy function.
[1] Strick, T., Allemand, J., Croquette, V., Bensimon, D. 2000. “Twisting and stretching single DNA molecules.” Progress in Biophysics and Molecular Biology. 74: 115-140.
[2] Strick, T.R., Allemand, J.F., Bensimon, D., Croquette, V., 1998. “Behavior of Supercoiled DNA.” Biophysical Journal. 74: 2016-2028.
[3]Hatfield, G.W., Benham, C.J., 2002. “DNA Topology-Mediated Control of Global Gene Expression in Escherichia Coli.” Annual Review of Genetics. 36: 175-203.
[4] Gilbert, N., Allan, J., 2014. “Supercoiling in DNA and chromatin.” Current Opinion in Genetics and Development. 25:15-21.
[5] van Loenhout, M. T. J., de Grunt, M.V., Dekker, C., 2012. “Dynamics of DNA Supercoils.” Science. 338:94-97.
[6] Mosconi., F., Allemand, J. F., Bensimon, D., Croquette, V., 2009. “Measurement of the Torque on a Single Stretched and Twisted DNA Using Magnetic Tweezers.” Physical
Review Letters.
Comparison of theoretical and experimental radii values for the four applied masses.
References Conclusion
• According to our experimental data, there is a high correlation between the twist at which the first
superloop occurs and the initial length of the polymer line. Using curve fitting techniques, it was
determined that this relationship follows the expression: y=Axb.
• There was no correlation found between the radius of the first superloop and the initial length of the
polymer line.
• There is a clear correlation between the original length, L0, and the final length, L2 which appeared to
follow a linear trend: y=mx+b.
• Experimental results of the radius of critical buckling were compared with the results obtained using
Equation 2. It was found that the experimental results differ from the theoretical results by a factor of
1.304.
Graph of length after reaching first superloop (L1) vs initial
length of polymer line (L0).
Graph of length after unwinding polymer line (L2) vs initial
length of polymer line (L0).
Graph of radius of first superloop vs initial length of polymer line
(L0).
Graph of radius of twist (turns/ length) at first superloop vs
initial length of polymer line (L0).
Comparisons of each test between different applied weights.
Fester, T. “Beyond the Double Helix”. http://www.scivit.de/blog/wp-content/uploads/2010/07/test7b.jpg