Bifurcations of Equilibria in DNA Elasticity

对于DNA弹性理论中的许多问题，DNA分子可以被视为一种棒状结构，它是通过将多米诺骨牌堆叠在另一个多米诺骨牌上，每个多米诺骨牌相对于堆叠中的前一个多米诺骨牌旋转大约十分之一圈而获得的。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接近平面的互补核苷酸碱基通过氢键连接在一起形成的。DNA分子的固有几何结构(如无应力状态下的曲率)和弹性性质(如控制弯曲、扭转、剪切和这些变形模式之间的耦合的模数)都对DNA分子中的核苷酸序列敏感。每个碱基对共价连接到两条DNA链中的一条的糖-磷酸主链上，这两条DNA链结合在一起形成了Watson-Crick结构，由于主链上的每个磷酸基团都带有一个电子，所以每个碱基对都有两个这样的电荷。施加在每个碱基对上的电动力取决于介质中盐的浓度和同一DNA分子中甚至遥远放置的碱基对在空间中的位置。基于该模型的计算表明，溶液中固有弯曲的DNA分子的平衡构型对介质中盐的浓度c非常敏感。以c为参数的分叉图可能具有极大的复杂性，并且在适当的情况下，包含其中在单个c值处出现几个局部稳定的平衡构型(每个给予分子不同形状)的区域。该项目的目标是将该模型的数学理论发展到其结论能够容易地进行实验测试的程度(例如，通过预测对于特定DNA序列，在接近计算的临界值的c值处应发生实验可检测的构型变化)。获得关于长程静电力的影响的完善的理论，从而实现盐浓度的变化，关于本质上弯曲的(通常是非均一的)DNA分子的构型是生物物理学中的一个普遍感兴趣的问题，这在生物工程中具有重要意义。该项目的研究有望应用于对基因组长度的DNA分子进行成像和分类的微型设备。在一种这样的装置中，通过将DNA限制在宽度为0.1微米的通道中来延长DNA，并且将要开发的计算方法将有助于理解限制在这种通道中的DNA所经历的延伸量与通道直径、盐浓度和DNA的本征曲率有关的方式。该理论的另一个应用是研究由数百个碱基对的适当序列形成的环状DNA分子可以作为中尺度机械力化学开关的可能性，这些开关在盐浓度发生微小变化时经历大的构型变化。

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.