Research Papers

Electrostatics and Self-Contact in an Elastic Rod Approximation for DNA

[+] Author and Article Information
Todd D. Lillian1

Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409todd.lillian@ttu.edu

N. C. Perkins

Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109ncp@umich.edu


Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011008 (Oct 04, 2010) (6 pages) doi:10.1115/1.4002267 History: Received June 09, 2009; Revised June 21, 2010; Published October 04, 2010; Online October 04, 2010

Deoxyribonucleic acid (DNA) is an essential molecule that enables the storage and retrieval of genetic information. In its role during cellular processes, this long flexible molecule is significantly bent and twisted. Previously, we developed an elastodynamic rod approximation to study DNA deformed into a loop by a gene regulatory protein (lac repressor) and predicted the energetics and topology of the loops. Although adequate for DNA looping, our model neglected electrostatic interactions, which are essential when considering processes that result in highly supercoiled DNA including plectonemes. Herein, we extend the rod approximation to account for electrostatic interactions and present strategies that improve computational efficiency. Our calculations for the stability for a circularly bent rod and for an initially straight rod compare favorably to existing equilibrium models. With this new capability, we are now well-positioned to study the dynamics of transcription and other dynamic processes that result in DNA supercoiling.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

At a short length scale (inset), DNA is composed of two nucleotide chains that bond together (basepairing) and twist around each other to form the familiar double helix. During transcription, an enzyme (RNA polymerase) induces DNA supercoiling which may result in long length scale bending and twisting of the DNA helical axis (supercoiling).

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Figure 2

The all-atom structure of DNA as represented by a flexible rod with equivalent averaged elastic properties. The position vector R(s,t) locates the helical axis of DNA as a function of the contour length coordinate s and time t with respect to the inertial frame e. a(s,t) represents a body-fixed frame of the rod as a function of s and t.

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Figure 3

The electrostatic force is dependent on all pairwise vectors rp/q (the position (Rp) of an electric charge at gridpoint p on the DNA axis relative to another at q). Also depicted are the fixed inertial reference frame e and a body-fixed frame aq for gridpoint q.

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Figure 4

Electrostatic self contact in circular twisted rod. (a) The time evolution of Tw and Lk. Tw is plotted using blue and green to distinguish times when Lk is decreasing and increasing respectively. (b) Tw versus Lk. The critical linking numbers for transitions between the circular and figure-8, as reported in Ref. 31, are shown with dashed vertical lines: blue for decreasing Lk and green for increasing Lk. The solid blue and green lines represent Tw as a function of Lk for decreasing and increasing Lk, respectively. The additional dotted blue line is attained by lowering the rate of end rotation to 0.2% of its original value in order to show convergence to the critical value of Lk. (c) Two deformed states calculated with the slower loading rate and corresponding to the points 1 and 2 in (b).

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Figure 5

Plectoneme formation in a twisted, but otherwise clamped, elastic rod. (a) A series of snapshots of the rod configuration with increasing end rotation. (b) A plot of the nondimensional torque at one end of the rod as a function of the net end rotation. The solid black curve represents the calculated equilibrium curve of Fig. 10(a) in Ref. 32. The other two dashed curves are calculated from our dynamic model and present results for increasing (blue) and decreasing (red) end rotation. The numbered points correspond to the numbered series of snapshots in (a).




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