Resultanta and loop closure
Resultanta and loop closure
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DOI：
10.1002/qua.20751
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发表时间：
2006-01-01
影响因子：
2.2
通讯作者：
Dill, KA
中科院分区：
文献类型：
--
作者：
Coutsias, EA;Seok, C;Dill, KA
三肽环闭合问题是用角{Ti}(I)(3)(=1)表示的，它描述了每个肽单元围绕连接Cα原子的虚轴的取向。在两个这样的单元的交界处施加这样的约束，即键Cα-N和Cα-C之间的键角被固定在某个规定值0，得到了一个由u(I)中的三个二元多项式组成的系统：=每个变量的2次tan tau(I)/2。利用结式，即由两个(或多个)多项式的系数组成的矩阵的行列式，分析了系统公共解的存在性，其消失是多项式有公共根的充要条件。比较了两个结式：经典的西尔维斯特结式和狄克逊结式。结果表明，当其中两个变量被消去后，得到一个16次多项式。对于它的每个实根，系统都有一个对应的公零点。每一个这样的零都对应着链的一致构象。西尔维斯特方法可以在24×24矩阵的特征值中找到这些零点。对于Dixon方法，在去掉外部因素后，得到了大小为16×16的最优特征值问题。最后，给出了对更一般的三轴闭合问题的简单推广，并给出了在任意链上实现该方法的算法。(C)2005年威利期刊公司。
The problem of tripeptide loop closure is formulated in terms of the angles {Ti}(i)(3)(=1), describing the orientation of each peptide unit about the virtual axis joining the C alpha atoms. Imposing the constraint that at the junction of two such units the bond angle between the bonds C alpha-N and C alpha-C is fixed at some prescribed value 0 results in a system of three bivariate polynomials in u(i) : = tan tau(i)/2 of degree 2 in each variable. The system is analyzed for the existence of common solutions by making use of resultants, determinants of matrices composed of the coefficients of two (or more) polynomials, whose vanishing is a necessary and sufficient condition for the polynomials to have a common root. Two resultants are compared: the classical Sylvester resultant and the Dixon resultant. It is shown that when two of the variables are eliminated in favor of the third, a polynomial of degree 16 results. To each one of its real roots, there is a corresponding common zero of the system. To each such zero there corresponds a consistent conformation of the chain. The Sylvester method can find these zeros among the eigenvalues of a 24 X 24 matrix. For the Dixon approach, after removing extraneous factors, an optimally sized eigenvalue problem of size 16 X 16 results. Finally, the easy extension to the more general problem of triaxial loop closure is presented and an algorithm for implementing the method on arbitrary chains is given. (c) 2005 Wiley Periodicals, Inc.