On Bifurcations of Equilibria of Intrinsically Curved, Electrically Charged, Rod-like Structures that Model DNA Molecules in Solution |
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Authors: | Yoav Y Biton Bernard D Coleman David Swigon |
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Institution: | (1) Department of Mechanics & Materials Science, Rutgers University, Piscataway, NJ 08854-8058, USA;(2) Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA |
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Abstract: | DNA molecules in the familiar Watson–Crick double helical B form can be treated as though they have rod-like structures obtained
by stacking dominoes one on top of another with each rotated by approximately one-tenth of a full turn with respect to its
immediate predecessor in the stack. These “dominoes” are called base pairs. A recently developed theory of sequence-dependent DNA elasticity (Coleman, Olson, & Swigon, J. Chem. Phys. 118:7127–7140, 2003) takes into account the observation that the step from one base pair to the next can be one of several
distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step. In the present
paper, which is based on that theory, emphasis is placed on the fact that, as each base in a base pair is attached to the
sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson–Crick structure, and
each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair,
which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base
pairs but also to long range electrostatic forces that, because they are only partially screened out by positively charged
counter ions, can render the molecule’s equilibrium configurations sensitive to changes in the concentration c of salt in the medium. When these electrostatic forces are taken into account, the equations of mechanical equilibrium for
a DNA molecule with N + 1 base pairs are a system of μN non-linear equations, where μ, the number of kinematical variables describing the relative displacement and orientation of adjacent pairs is in general
6; it reduces to 3 when base-pair steps are assumed to be inextensible and non-shearable. As a consequence of the long-range
electrostatic interactions of base pairs, the μN × μN Jacobian matrix of the equations of equilibrium is full. An efficient numerically stable computational scheme is here presented
for solving those equations and determining the mechanical stability of the calculated equilibrium configurations. That scheme
is employed to compute and analyze bifurcation diagrams in which c is the bifurcation parameter and to show that, for an intrinsically curved molecule, small changes in c can have a strong effect on stable equilibrium configurations. Cases are presented in which several stable configurations
occur at a single value of c.
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Keywords: | Sequence dependent DNA elasticity Electrically charged rod-like structures Bifurcation theory |
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