Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960's as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers.
An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper we consider cooperative systems, i.e. systems with positive feedback interconnections among all variables, which in a continuous setting guarantees a very stable dynamics. In Part I of this paper we presented a construction that shows that for an arbitrary constant 0<c<2 and sufficiently large n there exist n-dimensional Boolean cooperative networks in which both the indegree and outdegree of each for each variable is bounded by two (bi-quadratic networks) and which nevertheless contain periodic orbits of length at least c^n.
In this part, we prove an inverse result showing that for sufficiently large n and for 0<c<2 sufficiently close to 2, any n-dimensional cooperative, bi-quadratic Boolean network with a cycle of length at least c^n must have a large proportion of variables with indegree 1. Such systems therefore share a structural similarity to the systems constructed in Part I.
Boolean networks are a popular modeling framework in computational biology to capture the dynamics of molecular networks, such as gene regulatory networks. It has been observed that many published models of such networks are defined by regulatory rules driving the dynamics that have certain so-called canalizing properties. In this paper, we investigate the dynamics of a random Boolean network with such properties using analytical methods and simulations. From our simulations, we observe that Boolean networks with higher canalizing depth have generally fewer attractors, the attractors are small…
Effective control of biological systems can often be achieved through the control of a surprisingly small number of distinct variables. We bring clarity to such results using the formalism of Boolean dynamical networks, analyzing the effectiveness of external control in selecting a desired final state when that state is among the original attractors of the dynamics. Analyzing 49 existing biological network models, we find strong numerical evidence that the average number of nodes that must be forced scales logarithmically with the number of original attractors. This suggests that biological ne…
Permutation theory governs long-term dynamics of critical Boolean networks
University of California, Los Angeles（加州大学洛杉矶分校）；University of Florida（佛罗里达大学）；Watchung Hills Regional High School（瓦丘山地区高中）；Ponte Vedra High School（蓬特韦德拉高中）；Iowa State University（爱荷华州立大学）
We study the target control problem of asynchronous Boolean networks, to identify a set of nodes, the perturbation of which can drive the dynamics of the network from any initial state to the desired steady state (or attractor). We are particularly interested in temporary perturbations, which are applied for sufficient time and then released to retrieve the original dynamics. Temporary perturbations have the apparent advantage of averting unforeseen consequences, which might be induced by permanent perturbations. Despite the infamous state-space explosion problem, in this work, we develop an e…
Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions between Boolean components. They reproduce, in different degrees, the behaviours emerging in more quantitative models. In particular, regulatory conflicts can prevent the standard asynchronous dynamics from reproducing some trajectories that might be expected upon inspection of more detailed models. We introduce and study the class of networks with linear cuts, wh…
The regulatory network of E. coli metabolism as a boolean dynamical system exhibits both homeostasis and flexibility of response
阐明细胞的大规模遗传调控网络的结构和动力学是系统生物学的重要目标。我们研究了大肠杆菌的遗传网络，调节其代谢的系统级动力学特性，并展示了它的设计如何导致生物有用的细胞特性。我们的研究使用数据库（Covert et al.，Nature 2004），包含583个基因和96个外部代谢物，它不仅描述了网络连接，还描述了每个基因节点处的布尔规则，该规则控制基因的开关作为其输入的函数。我们已经研究了如何从这个数据库构建的布尔动力系统的吸引子依赖于基因的初始条件和各种环境条件对应的缓冲最小媒体。我们发现，该系统表现出稳态，它的吸引子，原来是固定点或低周期周期，是高度不敏感的初始条件或扰动的基因配置为任何给定的固定环境。同时，当外部介质变化时，吸引子显示出很大的变化，这意味着系统对变化的环境条件具有高度灵活的响应。调节动力学作用于在改变的培养基下提高细胞生长速率。我们的研究表明，在{ \it E. coli}是分层的、模块化的，并且在很大程度上是非循环的，环境变量控制层次结构的根。这种结构使细胞对基因构型的扰动具有高度鲁棒性，并对环境变化具有高度响应性。该动力系统虽然没有接近混沌的边缘，但仍然具有稳态和响应灵活性的双重特性。
We present BoolSi, an open-source cross-platform command line tool for distributed simulations of deterministic Boolean networks with synchronous update. It uses MPI standard to support execution on computational clusters, as well as parallel processing on a single computer. BoolSi can be used to model the behavior of complex dynamic networks, such as gene regulatory networks. In particular, it allows for identification and statistical analysis of network attractors. We perform a case study of the activity of a cambium cell to demonstrate the capabilities of the tool.
Understanding control mechanisms in biological systems plays a crucial role in important applications, for instance in cell reprogramming. Boolean modeling allows the identification of possible efficient strategies, helping to reduce the usually high and time-consuming experimental efforts. Available approaches to control strategy identification usually focus either on attractor or phenotype control, and are unable to deal with more complex control problems, for instance phenotype avoidance. They also fail to capture, in many situations, all possible minimal strategies, finding instead only su…
The control of biological systems presents interesting applications such as cell reprogramming or drug target identification. A common type of control strategy consists in a set of interventions that, by fixing the values of some variables, force the system to evolve to a desired state. This work presents a new approach for finding control strategies in biological systems modeled by Boolean networks. In this context, we explore the properties of trap spaces, subspaces of the state space which the dynamics cannot leave. Trap spaces for biological networks can often be efficiently computed, and …
Genetic regulatory networks control ontogeny. For fifty years Boolean networks have served as models of such systems, ranging from ensembles of random Boolean networks as models for generic properties of gene regulation to working dynamical models of a growing number of sub-networks of real cells. At the same time, their statistical mechanics has been thoroughly studied. Here we recapitulate their original motivation in the context of current theoretical and empirical research. We discuss ensembles of random Boolean networks whose dynamical attractors model cell types. A sub-ensemble is the cr…
LRI, CNRS, U. Paris-Sud U. Paris-Saclay（巴黎-萨克雷大学南巴黎校区法国国家科学研究中心计算机研究所）；LaBRI, CNRS, U. Bordeaux Bordeaux INP（波尔多大学波尔多国立理工学院法国国家科学研究中心计算机与自动化研究所）；Institut Curie, INSERM U. PSL, Mines ParisTech（巴黎文理研究大学法国国家健康与医学研究院巴黎高等矿业学院居里研究所）
Boolean networks model finite discrete dynamical systems with complex behaviours. The state of each component is determined by a Boolean function of the state of (a subset of) the components of the network. This paper addresses the synthesis of these Boolean functions from constraints on their domain and emerging dynamical properties of the resulting network. The dynamical properties relate to the existence and absence of trajectories between partially observed configurations, and to the stable behaviours (fixpoints and cyclic attractors). The synthesis is expressed as a Boolean satisfiability…
