论文标题
代数实验设计：理论与计算
Algebraic Experimental Design: Theory and Computation
论文作者
论文摘要
在过去的几十年中，代数几何形状为生物实验设计提供了创新的方法，从而解决了理论问题并提高了计算效率。但是，确保独特性和完美的模型恢复仍然是空旷的问题。在这项工作中，我们研究了接线图独特性的问题。我们用作建模框架多项式动力学系统，并利用史坦利 - 赖斯纳理论的简单复合物和无方便的单一理想之间的对应关系，以开发理论并构建一种算法，用于识别输入数据集$ v \ subset \ subset \ subbb f_p f_p f_p f_p^n $，这些算法对应于唯一的无关紧要的无线接线，该算法对应于唯一的接线效果。我们将结果应用于表皮衍生的生长因子受体介导的肿瘤抑制网络，并证明了仔细的实验设计决策如何导致独特的最小接线图识别。理论工作的见解之一是给定$ v \ subset \ mathbb f_p^n $的接线图的独特性与多项式理想$ i（v）\ subset subset \ subset \ subbb f_p f_p f_p f_1，\ ldots，x__n]的独特性之间的连接。我们讨论了现有的结果，并在$ v $中的点上引入了一种新的必要条件，以减少$ i（v）$的唯一性。这些结果还表明，实验输入点相对接近对最小接线图的数量的重要性，然后我们在计算上研究它们。我们发现，有一种具有具体的启发式方法来生成数据，该数据往往会导致最小的接线图。
Over the past several decades, algebraic geometry has provided innovative approaches to biological experimental design that resolved theoretical questions and improved computational efficiency. However, guaranteeing uniqueness and perfect recovery of models are still open problems. In this work we study the problem of uniqueness of wiring diagrams. We use as a modeling framework polynomial dynamical systems and utilize the correspondence between simplicial complexes and square-free monomial ideals from Stanley-Reisner theory to develop theory and construct an algorithm for identifying input data sets $V\subset \mathbb F_p^n$ that are guaranteed to correspond to a unique minimal wiring diagram regardless of the experimental output. We apply the results on a tumor-suppression network mediated by epidermal derived growth factor receptor and demonstrate how careful experimental design decisions can lead to a unique minimal wiring diagram identification. One of the insights of the theoretical work is the connection between the uniqueness of a wiring diagram for a given $V\subset \mathbb F_p^n$ and the uniqueness of the reduced Gröbner basis of the polynomial ideal $I(V)\subset \mathbb F_p[x_1,\ldots, x_n]$. We discuss existing results and introduce a new necessary condition on the points in $V$ for uniqueness of the reduced Gröbner basis of $I(V)$. These results also point to the importance of the relative proximity of the experimental input points on the number of minimal wiring diagrams, which we then study computationally. We find that there is a concrete heuristic way to generate data that tends to result in fewer minimal wiring diagrams.
