Bifurcations of Equilibria in DNA Elasticity

对于DNA弹性理论中的许多问题，DNA分子可以被看作是一种棒状结构，它是通过将多米诺骨牌一个接一个地堆叠起来而获得的，每个多米诺骨牌相对于堆栈中的前一个骨牌旋转大约十分之一圈。在分子生物学中，这些“多米诺骨牌”被称为碱基对，因为每个碱基对都是由两个几乎平面互补的核苷酸碱基用氢键连接在一起形成的。固有几何（例如，无应力状态下的曲率）和弹性特性（例如，控制弯曲、扭曲、剪切和这些变形模式之间的耦合的模量）对DNA分子中的核苷酸序列都很敏感。每对碱基对都以共价的方式连接在两条DNA链中的一条上，这两条DNA链形成了沃森-克里克结构。由于主链上的每个磷酸基团都带一个电子电荷，因此每对碱基对都带两个电子电荷。施加在每个碱基对上的电作用力取决于介质中盐的浓度和碱基对在同一DNA分子中的位置，即使碱基对的位置很遥远。在先前的NSF资助下，基于该模型进行的计算表明，溶液中固有弯曲DNA分子的平衡构型对介质中盐的浓度c非常敏感。以c为参数的分岔图可能非常复杂，并且在适当的情况下，包含在单个c值下出现几个局部稳定平衡构型（每个构型都赋予分子不同的形状）的区域。本项目的目标是发展模型的数学理论，使其结论易于进行实验测试(例如，通过预测对于特定的DNA序列，在实验中可检测到的结构变化应该发生在c值接近计算的临界值时)。关于远程静电力以及盐浓度变化对内在弯曲（通常是非均匀的）DNA分子结构的影响的完善理论的实现，是生物物理学中普遍感兴趣的问题，对生物工程具有影响。该项目的研究有望应用于成像和分选基因组长度DNA分子的微设备。在一个这样的装置中，DNA被限制在一个宽度为0.1微米的通道中被拉长，待开发的计算方法将有助于理解被限制在这样一个通道中的DNA所经历的延伸量与通道直径、盐浓度和DNA的固有曲率有关的方式。该理论的另一个应用是研究由几百个碱基对的适当序列形成的环状DNA分子的可能性，这种分子可以作为中尺度的机械化学开关，在盐浓度的微小变化中发生大的结构变化。

For many problems in the theory of DNA elasticity, a DNA molecule can be treated as though it is a rod-like structure 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. In molecular biology these "dominoes" are called base pairs, because each is formed by joining together with hydrogen bonds two nearly planar complementary nucleotide bases. Both the intrinsic geometry (e.g., curvature in the stress-free state) and the elastic properties (e.g., moduli governing bending, twisting, shearing, and coupling between such modes of deformation) are sensitive to the nucleotide sequence in the DNA molecule. Each base pair is covalently 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 as each phosphate group in the backbone chain bears one electronic charge, two such charges are associated with each base pair. The electrical force exerted on each base pair depends on the concentration of salt in the medium and the position in space of even remotely placed base pairs in the same DNA molecule. Calculations based on this model performed under a previous NSF grant indicate that the equilibrium configurations of an intrinsically curved DNA molecule in solution are very sensitive to the concentration c of salt in the medium. Bifurcation diagrams with c as the parameter can have great complexity and, under appropriate circumstances, contain regions in which several locally stable equilibrium configurations (each giving the molecule a distinct shape) occur at a single value of c. The goal of this project is to develop the mathematical theory of the model to the point where its conclusions are easily capable of experimental testing (e.g., by predicting that for particular DNA sequences experimentally detectable changes in configuration should occur at values of c near to calculated critical values).The attainment of a well developed theory of the influence of long-range electrostatic forces, and hence of changes in salt concentration, on the configurations of intrinsically curved (and in general non-homogeneous) DNA molecules is a matter of general interest in biophysics that has implications in bioengineering. The research in this project is expected to have applications to microdevices for imaging and sorting genomic-length DNA molecules. In one such device the DNA is elongated by confinement to a channel with a width of 0.1 microns, and the computational methods to be developed will aid in the attainment of an understanding of the way the amount of extension experienced by DNA confined in such a channel is related to the channel diameter, the concentration of salt, and the intrinsic curvature of the DNA. Another application of the theory is the investigation of the possibility that circularized molecules of DNA formed from appropriate sequences of several hundred base-pairs can serve as mesoscale mechano-chemical switches that undergo large changes in configuration upon small changes in salt concentration.