Dipartimento di Ingegneria Industriale, Università di Roma Tor Vergata, Roma 00133, Italy.
Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Torino 10129, Italy.
J Theor Biol. 2019 Jul 7;472:46-53. doi: 10.1016/j.jtbi.2019.04.012. Epub 2019 Apr 15.
Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Gröbner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.
在可以通过布尔网络建模的各种现象中，即具有二进制状态变量的离散时间动力系统，基因调控相互作用尤其为人所知。因此，布尔网络的分析至关重要，例如，确定遗传途径和预测突变对细胞功能的影响。最近提出了两种方法（即半张量积和有限域上的 Gröbner 基）来解决确定此类系统的循环和吸引子（以及相应的吸引域）的问题。在这里，通过适当结合来自这两个领域的方法（即线性代数和代数几何），不仅可以确定循环和吸引子，还可以找到布尔网络的闭式解。通过找到将布尔动力学重构为线性形式的浸入和计算后者系统的闭式解，可以实现这一目标。通过完全计算建模人类适应性免疫系统中涉及的白细胞 T 淋巴细胞分化的布尔网络的解，证明了该技术的有效性。
