Rachdi Mustapha, Waku Jules, Hazgui Hana, Demongeot Jacques
Team AGIM (Autonomy, Gerontechnology, Imaging, Modelling & Tools for e-Gnosis Medical), Laboratory AGEIS, EA 7407, University Grenoble Alpes, Faculty of Medicine, 38700 La Tronche, France.
LIRIMA-UMMISCO, Université de Yaoundé, Faculté des Sciences, BP 812 Yaoundé, Cameroun.
Entropy (Basel). 2020 Feb 25;22(3):260. doi: 10.3390/e22030260.
Genetic regulatory networks have evolved by complexifying their control systems with numerous effectors (inhibitors and activators). That is, for example, the case for the double inhibition by microRNAs and circular RNAs, which introduce a ubiquitous double brake control reducing in general the number of attractors of the complex genetic networks (e.g., by destroying positive regulation circuits), in which complexity indices are the number of nodes, their connectivity, the number of strong connected components and the size of their interaction graph. The stability and robustness of the networks correspond to their ability to respectively recover from dynamical and structural disturbances the same asymptotic trajectories, and hence the same number and nature of their attractors. The complexity of the dynamics is quantified here using the notion of attractor entropy: it describes the way the invariant measure of the dynamics is spread over the state space. The stability (robustness) is characterized by the rate at which the system returns to its equilibrium trajectories (invariant measure) after a dynamical (structural) perturbation. The mathematical relationships between the indices of complexity, stability and robustness are presented in case of Markov chains related to threshold Boolean random regulatory networks updated with a Hopfield-like rule. The entropy of the invariant measure of a network as well as the Kolmogorov-Sinaï entropy of the Markov transition matrix ruling its random dynamics can be considered complexity, stability and robustness indices; and it is possible to exploit the links between these notions to characterize the resilience of a biological system with respect to endogenous or exogenous perturbations. The example of the genetic network controlling the kinin-kallikrein system involved in a pathology called angioedema shows the practical interest of the present approach of the complexity and robustness in two cases, its physiological normal and pathological, abnormal, dynamical behaviors.
遗传调控网络通过用众多效应器（抑制剂和激活剂）使控制系统复杂化而得以进化。也就是说，例如，微小RNA和环状RNA的双重抑制就是这种情况，它们引入了一种普遍存在的双重制动控制，通常会减少复杂遗传网络的吸引子数量（例如，通过破坏正调控回路），其中复杂性指标包括节点数量、它们的连通性、强连通分量的数量以及它们相互作用图的大小。网络的稳定性和鲁棒性分别对应于它们从动态和结构干扰中恢复相同渐近轨迹的能力，从而恢复相同数量和性质的吸引子的能力。这里使用吸引子熵的概念来量化动力学的复杂性：它描述了动力学的不变测度在状态空间中的分布方式。稳定性（鲁棒性）的特征在于系统在动态（结构）扰动后恢复到其平衡轨迹（不变测度）的速率。在与用类似霍普菲尔德规则更新的阈值布尔随机调控网络相关的马尔可夫链的情况下，给出了复杂性、稳定性和鲁棒性指标之间的数学关系。网络不变测度的熵以及支配其随机动力学的马尔可夫转移矩阵的柯尔莫哥洛夫 - 西奈熵可以被视为复杂性、稳定性和鲁棒性指标；并且可以利用这些概念之间的联系来表征生物系统对内源性或外源性扰动的恢复力。控制参与一种称为血管性水肿的病理学的激肽 - 激肽释放酶系统的遗传网络的例子，在其生理正常和病理、异常的动态行为这两种情况下，展示了当前复杂性和鲁棒性方法的实际意义。
