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Research Papers

# A Model for Highly Strained DNA Compressed Inside a Protein Cavity

[+] Author and Article Information
Andrew D. Hirsh

Research Assistant
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109

Todd D. Lillian

Assistant Professor
Department of Mechanical Engineering,
Texas Tech University,
Lubbock, TX 79409
e-mail: todd.lillian@ttu.edu

Troy A. Lionberger

Postdoctoral Research Associate
Howard Hughes Medical Institute,
Department of Physics,
University of California,
Berkeley, Berkeley, CA 94720
e-mail: talionberger@gmail.com

Maryna Taranova

Research Assistant
e-mail: taranova.maryna@uci.edu

Ioan Andricioaei

Associate Professor
e-mail: andricio@uci.edu
Department of Chemistry,
University of California,
Irvine, Irvine, CA 92697

N. C. Perkins

Professor
Fellow ASME
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ncp@umich.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 31, 2011; final manuscript received March 9, 2012; published online October 30, 2012. Assoc. Editor: Aki Mikkola.

J. Comput. Nonlinear Dynam 8(3), 031001 (Oct 30, 2012) (8 pages) Paper No: CND-11-1121; doi: 10.1115/1.4007535 History: Received July 31, 2011; Revised March 09, 2012

## Abstract

Deoxyribonucleic acid (DNA) is a long and flexible biopolymer that contains genetic information. Building upon the discovery of the iconic double helix over 50 years ago, subsequent studies have emphasized how its biological function is related to the mechanical properties of the molecule. A remarkable system which highlights the role of DNA bending and twisting is the packing and ejection of DNA into and from viral capsids. A recent 3D reconstruction of bacteriophage $φ29$ reveals a novel toroidal structure of highly bent/twisted DNA contained in a small cavity below the viral capsid. Here, we extend an elastic rod model for DNA to enable simulation of the toroid as it is compacted and subsequently ejected from a small volume. We compute biologically-relevant forces required to form the toroid and predict ejection times of several nanoseconds.

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Topics: Cavities , DNA , Proteins

## Figures

Fig. 1

A schematic of mature bacteriophage φ29 based on the cryo-electron micoscopy reconstruction [6,28]. The toroidal DNA is contained within a cavity just below the capsid and suspected to contain 30–40 bp of DNA. The cavity is formed from the void between the connector and the lower collar. Shown inside the capsid is a cross sectional view of concentric hoops of DNA which ultimately fill the entire capsid.

Fig. 2

The atomic structure of DNA superimposed with a an elastic rod with equivalent elastic properties. R→(s,t) tracks the position of the helical axis as a function of contour length s and time t with respect to the inertial frame e. We also define a body-fixed reference frame a(s,t) which is also a function of s and t. Figure adaped from [29].

Fig. 3

Side view of the 3-D cavity structure estimated from the connector and lower collar geometry in the cryo-EM images [6] with relevant dimensions labeled. We assume the cavity is symmetric about the vertical axis. Cavity grid points are spaced in stacked rings of points separated by dc above and below one another and dc along the circumference of each ring (see red arrows). Here, we have set dc to 4 Å to reduce the number of grid points by an order of magnitude (over 1 Å separation) and thereby gain computational efficiency. The shaded blue box indicates the approximate area A=dc2 surrounding each grid point.

Fig. 4

The interaction forces are dependent on all pairwise vectors between rod grid points p and points q representing the cavity surface which are fixed in space

Fig. 5

(a) Computational snapshots (aligned to the dots on the plot directly below) from simulating 90 bp of DNA compressed within the cavity. (b) Internal (compressive) force (pN) and elastic energy (kT) following Eq. (6) as functions of the shortening of the rod δ = L – d where L is the length of DNA (306 Å) and d is the distance between the upper and lower rod boundaries. The green box illustrates where we report internal force along the contour length. (c) Top and side view of the final toroidal structure.

Fig. 6

Equilibrium conformation from Fig. 5(c) upon altering the cavity grid spacing parameter dc to (a) 8 Å, (b) 4 Å, (c) 2 Å, (d) 1 Å

Fig. 7

(a) Dynamic ejection and toroid collapse. Snapshots illustrate DNA conformation at each of the 5 ns increments denoted in the figure below. (b) DNA reaction force (solid green line and circles) and torque (solid red line and circles) on the remaining capsid/DNA (upper end) as a function of time. The solid blue line represents the component of the viscous drag force (integrated over the length of the rod) acting along the z-axis (vertical axis).

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