BACKGROUND This paper addresses the problem of finding attractors in biological regulatory networks. We focus here on non-deterministic synchronous and asynchronous multi-valued networks, modeled using automata networks (AN). AN is a general and well-suited formalism to study complex interactions between different components (genes, proteins,...). An attractor is a minimal trap domain, that is, a part of the state-transition graph that cannot be escaped. Such structures are terminal components of the dynamics and take the form of steady states (singleton) or complex compositions of cycles (non-singleton). Studying the effect of a disease or a mutation on an organism requires finding the attractors in the model to understand the long-term behaviors. RESULTS We present a computational logical method based on answer set programming (ASP) to identify all attractors. Performed without any network reduction, the method can be applied on any dynamical semantics. In this paper, we present the two most widespread non-deterministic semantics: the asynchronous and the synchronous updating modes. The logical approach goes through a complete enumeration of the states of the network in order to find the attractors without the necessity to construct the whole state-transition graph. We realize extensive computational experiments which show good performance and fit the expected theoretical results in the literature. CONCLUSION The originality of our approach lies on the exhaustive enumeration of all possible (sets of) states verifying the properties of an attractor thanks to the use of ASP. Our method is applied to non-deterministic semantics in two different schemes (asynchronous and synchronous). The merits of our methods are illustrated by applying them to biological examples of various sizes and comparing the results with some existing approaches. It turns out that our approach succeeds to exhaustively enumerate on a desktop computer, in a large model (100 components), all existing attractors up to a given size (20 states). This size is only limited by memory and computation time.

中文翻译：

---

基于 ASP 的非确定性同步和异步多值网络中吸引子枚举方法。


背景本文解决了在生物调节网络中寻找吸引子的问题。我们在这里重点关注使用自动机网络（AN）建模的非确定性同步和异步多值网络。 AN 是一种通用且非常适合研究不同成分（基因、蛋白质等）之间复杂相互作用的形式。吸引子是一个最小陷阱域，即状态转移图中无法逃脱的部分。这种结构是动力学的终端组成部分，并采取稳态（单例）或复杂的循环组合（非单例）的形式。研究疾病或突变对生物体的影响需要找到模型中的吸引子以了解长期行为。结果我们提出了一种基于答案集编程（ASP）的计算逻辑方法来识别所有吸引子。该方法无需任何网络缩减即可执行，可以应用于任何动态语义。在本文中，我们提出了两种最广泛的非确定性语义：异步和同步更新模式。逻辑方法通过对网络状态的完整枚举来找到吸引子，而无需构建整个状态转换图。我们实现了广泛的计算实验，显示出良好的性能并符合文献中预期的理论结果。结论 我们方法的独创性在于通过 ASP 的使用，对所有可能的（组）状态进行了详尽的枚举，从而验证了吸引子的属性。我们的方法应用于两种不同方案（异步和同步）中的非确定性语义。 我们的方法的优点通过将其应用于各种大小的生物示例并将结果与​​一些现有方法进行比较来说明。事实证明，我们的方法成功地在台式计算机上的大型模型（100 个组件）中详尽枚举了给定大小（20 个状态）的所有现有吸引子。该大小仅受内存和计算时间的限制。