We introduce an analytical method to generate the pathway of a closed protein-bound DNA minicircle. We develop an analytical equation to connect two open curves smoothly and use the derived expressions to join the ends of two helical pathways and form models of nucleosome-decorated DNA minicircles. We find that the simplest smooth connector which satisfies the boundary conditions at the end points and the length requirement for such connections to be a quartic function on the xy-plane and linear along the z-direction. This is a general method which can be used to connect any two open curves with well defined mathematical definitions as well as pairs of discrete systems found experimentally. We used this method to describe the configurations of torsionally relaxed, 360-base pair DNA rings with two evenly-spaced, ideal nucleosomes. We considered superhelical nucleosomal pathways with different levels of DNA wrapping and allowed for different inter-nucleosome orientations. We completed the DNA circles with the smooth connectors and studied the associated bending and electrostatic energies for different configurations in the absence and presence of salt. The predicted stable states bear close resemblance to reconstituted minicircles observed under low and high salt conditions.

我们引入了一种分析方法来生成闭合蛋白质结合 DNA 小环的途径。我们开发了一个解析方程来平滑连接两条开放曲线，并使用导出的表达式连接两条螺旋路径的末端并形成核小体修饰的 DNA 微环模型。我们发现满足端点边界条件和这种连接的长度要求的最简单的平滑连接器是xy平面上的四次函数，并且沿z方向是线性的。这是一种通用方法，可用于连接任意两条具有明确数学定义的开曲线以及通过实验发现的离散系统对。我们使用这种方法来描述具有两个均匀间隔的理想核小体的扭转松弛的 360 碱基对 DNA 环的构型。我们考虑了具有不同水平 DNA 包裹的超螺旋核小体途径，并允许不同的核小体间方向。我们用光滑的连接器完成了 DNA 环，并研究了在不存在和存在盐的情况下不同配置的相关弯曲能和静电能。预测的稳定状态与在低盐和高盐条件下观察到的重构小圆非常相似。