The effect of negative feedback loops on the dynamics of boolean networks

Minimal Pinning Control for Oscillatority of Boolean Networks

In this article, minimal pinning control for oscillatority (i.e., instability) of Boolean networks (BNs) under algebraic state space representations method is studied. First, two criteria for oscillatority of BNs are obtained from the aspects of state transition matrix (STM) and network structure (NS) of BNs, respectively. A distributed pinning control (DPC) from these two aspects is proposed: one is called STM-based DPC and the other one is called NS-based DPC, both of which are only dependent on local in-neighbors. As for STM-based DPC, one arbitrary node can be chosen to be controlled, based on certain solvability of several equations, meanwhile a hybrid pinning control (HPC) combining DPC and conventional pinning control (CPC) is also proposed. In addition, as for NS-based DPC, pinning control nodes (PCNs) can be found using the information of NS, which efficiently reduces the high computational complexity. The proposed STM-based DPC and NS-based DPC in this article are shown to be simple and concise, which provide a new direction to dramatically reduce control costs and computational complexity. Finally, gene networks are simulated to discuss the effectiveness of theoretical results.

Characterizing attitudinal network graphs through frustration cloud

Attitudinal network graphs are signed graphs where edges capture an expressed opinion; two vertices connected by an edge can be agreeable (positive) or antagonistic (negative). A signed graph is called balanced if each of its cycles includes an even number of negative edges. Balance is often characterized by the frustration index or by finding a single convergent balanced state of network consensus. In this paper, we propose to expand the measures of consensus from a single balanced state associated with the frustration index to the set of nearest balanced states. We introduce the frustration cloud as a set of all nearest balanced states and use a graph-balancing algorithm to find all nearest balanced states in a deterministic way. Computational concerns are addressed by measuring consensus probabilistically, and we introduce new vertex and edge metrics to quantify status, agreement, and influence. We also introduce a new global measure of controversy for a given signed graph and show that vertex status is a zero-sum game in the signed network. We propose an efficient scalable algorithm for calculating frustration cloud-based measures in social network and survey data of up to 80,000 vertices and half-a-million edges. We also demonstrate the power of the proposed approach to provide discriminant features for community discovery when compared to spectral clustering and to automatically identify dominant vertices and anomalous decisions in the network.

Synthesis of the Dynamical Properties of Feedback Loops in Bio-Pathways

Feedback loops regulate various biological functions such as oscillations, bistability, and robustness. They play a significant role in developmental signalling and failure of feedback can lead to disease. Systematic analysis of feedback loops could be useful in understanding their properties and biological effects. We propose here a method to automatically analyze feedback loops in bio-pathways and synthesize temporal logic properties which describe their dynamics. Starting with an ordinary differential equations (ODEs) based model of a bio-pathway, for a chosen feedback loop present in the pathway, we use a convolutional neural network to classify the behaviours of the key components of the feedback according to templates specified in bounded linear temporal logic (BLTL). Once a template has been identified, we instantiate the symbolic variables appearing in the template and synthesize properties using a parameter estimation procedure based on sequential hypothesis testing. We have applied this framework to a number of bio-pathway models and validated that the synthesized properties faithfully describe the behaviours of the feedback loops.

Evolution of Cellular Differentiation: From Hypotheses to Models

Cellular differentiation is one of the hallmarks of complex multicellularity, allowing individual organisms to capitalize on among-cell functional diversity. The evolution of multicellularity is a major evolutionary transition that allowed for the increase of organismal complexity in multiple lineages, a process that relies on the functional integration of cell-types within an individual. Multiple hypotheses have been proposed to explain the origins of cellular differentiation, but we lack a general understanding of what makes one cell-type distinct from others, and how such differentiation arises. Here, we describe how the use of Boolean networks (BNs) can aid in placing empirical findings into a coherent conceptual framework, and we emphasize some of the standing problems when interpreting data and model behaviors.

Concepts in Boolean network modeling: What do they all mean?

Boolean network models are one of the simplest models to study complex dynamic behavior in biological systems. They can be applied to unravel the mechanisms regulating the properties of the system or to identify promising intervention targets. Since its introduction by Stuart Kauffman in 1969 for describing gene regulatory networks, various biologically based networks and tools for their analysis were developed. Here, we summarize and explain the concepts for Boolean network modeling. We also present application examples and guidelines to work with and analyze Boolean network models.

Attractor Stability in Finite Asynchronous Biological System Models

We present mathematical techniques for exhaustive studies of long-term dynamics of asynchronous biological system models. Specifically, we extend the notion of [Formula: see text]-equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order (Macauley and Mortveit in Nonlinearity 22(2):421, 2009). We extend earlier work by Veliz-Cuba and Stigler (J Comput Biol 18(6):783-794, 2011), Goles et al. (Bull Math Biol 75(6):939-966, 2013), and others by comparing long-term dynamics up to topological conjugation: rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems from Macauley and Mortveit (Proc Am Math Soc 136(12):4157-4165, 2008. https://doi.org/10.1090/S0002-9939-09-09884-0 ; 2009; Electron J Comb 18:197, 2011a; Discret Contin Dyn Syst 4(6):1533-1541, 2011b. https://doi.org/10.3934/dcdss.2011.4.1533 ; Theor Comput Sci 504:26-37, 2013. https://doi.org/10.1016/j.tcs.2012.09.015 ; in: Isokawa T, Imai K, Matsui N, Peper F, Umeo H (eds) Cellular automata and discrete complex systems, 2014. https://doi.org/10.1007/978-3-319-18812-6_6 ) to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to (i) the lac operon model for Escherichia coli proposed by Veliz-Cuba and Stigler (2011), and (ii) the regulatory network involved in the control of the cell cycle and cell differentiation in the Caenorhabditis elegans vulva precursor cells proposed by Weinstein et al. (BMC Bioinform 16(1):1, 2015). In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all [Formula: see text] sequential update orders, we demonstrate that it is sufficient to consider 344 representative update orders, and, more notably, that these 344 representatives give rise to 4 distinct attractor structures. A similar analysis performed for the C. elegans model demonstrates that it has precisely 125 distinct attractor structures. We conclude with observations on the variety and distribution of the models' attractor structures and use the results to discuss their robustness.

Using Sub-Network Combinations to Scale Up an Enumeration Method for Determining the Network Structures of Biological Functions

Deduction of biological regulatory networks from their functions is one of the focus areas of systems biology. Among the different techniques used in this reverse-engineering task, one powerful method is to enumerate all candidate network structures to find suitable ones. However, this method is severely limited by calculation capability: due to the brute-force approach, it is infeasible for networks with large number of nodes to be studied using traditional enumeration method because of the combinatorial explosion. In this study, we propose a new reverse-engineering technique based on the enumerating method: sub-network combinations. First, a complex biological function is divided into several sub-functions. Next, the three-node-network enumerating method is applied to search for sub-networks that are able to realize each of the sub-functions. Finally, complex whole networks are constructed by enumerating all possible combinations of sub-networks. The optimal ones are selected and analyzed. To demonstrate the effectiveness of this new method, we used it to deduct the network structures of a Pavlovian-like function. The whole Pavlovian-like network was successfully constructed by combining robust sub-networks, and the results were analyzed. With sub-network combination, the complexity has been largely reduced. Our method also provides a functional modular view of biological systems.

A powerful weighted statistic for detecting group differences of directed biological networks

Complex disease is largely determined by a number of biomolecules interwoven into networks, rather than a single biomolecule. Different physiological conditions such as cases and controls may manifest as different networks. Statistical comparison between biological networks can provide not only new insight into the disease mechanism but statistical guidance for drug development. However, the methods developed in previous studies are inadequate to capture the changes in both the nodes and edges, and often ignore the network structure. In this study, we present a powerful weighted statistical test for group differences of directed biological networks, which is independent of the network attributes and can capture the changes in both the nodes and edges, as well as simultaneously accounting for the network structure through putting more weights on the difference of nodes locating on relatively more important position. Simulation studies illustrate that this method had better performance than previous ones under various sample sizes and network structures. One application to GWAS of leprosy successfully identifies the specific gene interaction network contributing to leprosy. Another real data analysis significantly identifies a new biological network, which is related to acute myeloid leukemia. One potential network responsible for lung cancer has also been significantly detected. The source R code is available on our website.

Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.

Optimization-based inference for temporally evolving networks with applications in biology

The problem of identifying dynamics of biological networks is of critical importance in order to understand biological systems. In this article, we propose a data-driven inference scheme to identify temporally evolving network representations of genetic networks. In the formulation of the optimization problem, we use an adjacency map as a priori information and define a cost function that both drives the connectivity of the graph to match biological data as well as generates a sparse and robust network at corresponding time intervals. Through simulation studies of simple examples, it is shown that this optimization scheme can help capture the topological change of a biological signaling pathway, and furthermore, might help to understand the structure and dynamics of biological genetic networks.

Boolean modeling of biological regulatory networks: a methodology tutorial

Given the complexity and interactive nature of biological systems, constructing informative and coherent network models of these systems and subsequently developing efficient approaches to analyze the assembled networks is of immense importance. The integration of network analysis and dynamic modeling enables one to investigate the behavior of the underlying system as a whole and to make experimentally testable predictions about less-understood aspects of the processes involved. In this paper, we present a tutorial on the fundamental steps of Boolean modeling of biological regulatory networks. We demonstrate how to infer a Boolean network model from the available experimental data, analyze the network using graph-theoretical measures, and convert it into a predictive dynamic model. For each step, the pitfalls one may encounter and possible ways to circumvent them are also discussed. We illustrate these steps on a toy network as well as in the context of the Drosophila melanogaster segment polarity gene network.

Boolean models can explain bistability in the lac operon

The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. Our model includes the two main glucose control mechanisms of catabolite repression and inducer exclusion. We show that this Boolean model is capable of predicting the ON and OFF steady states and bistability. Further, we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model exhibits the same dynamics. This work suggests that the key to model qualitative dynamics of gene systems is the topology of the network and Boolean models are well suited for this purpose.

Determining the distance to monotonicity of a biological network: a graph-theoretical approach

The authors use ideas from graph theory in order to determine how distant is a given biological network from being monotone. On the signed graph representing the system, the minimal number of sign inconsistencies (i.e. the distance to monotonicity) is shown to be equal to the minimal number of fundamental cycles having a negative sign. Suitable operations aiming at computing such a number are also proposed and shown to outperform all algorithms that are so far existing for this task. [Includes supplementary material].

Attractor analysis of asynchronous Boolean models of signal transduction networks

Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes' states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.

Monotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks

BACKGROUND

For large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations.

RESULTS

In this paper we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). A possible interpretation of this result is in terms of energetic cost of an interaction: an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that unlike for transcriptional networks, in the signaling/metabolic networks the probability of finding the system in its least frustrated configuration tends to be high also in correspondence of a moderate energetic regime, meaning that, in spite of the higher frustration, these networks can achieve a globally ordered response to perturbations even for moderate values of the strength of the interactions. Furthermore, an analysis of the energy landscape shows that signaling and metabolic networks lack energetic barriers around their global optima, a property also favouring global order.

CONCLUSION

In conclusion, transcriptional and signaling/metabolic networks appear to have systematic differences in both the index of frustration and the transition to global order. These differences are interpretable in terms of the different functions of the various classes of networks.

Inference of gene regulatory networks using boolean-network inference methods

The modeling of genetic networks especially from microarray and related data has become an important aspect of the biosciences. This review takes a fresh look at a specific family of models used for constructing genetic networks, the so-called Boolean networks. The review outlines the various different types of Boolean network developed to date, from the original Random Boolean Network to the current Probabilistic Boolean Network. In addition, some of the different inference methods available to infer these genetic networks are also examined. Where possible, particular attention is paid to input requirements as well as the efficiency, advantages and drawbacks of each method. Though the Boolean network model is one of many models available for network inference today, it is well established and remains a topic of considerable interest in the field of genetic network inference. Hybrids of Boolean networks with other approaches may well be the way forward in inferring the most informative networks.

Distributions for negative-feedback-regulated stochastic gene expression: dimension reduction and numerical solution of the chemical master equation

In this work we introduce a novel approach to study biochemical noise. It comprises a simplification of the master equation of complex reaction schemes (via an adiabatic approximation) and the numerical solution of the reduced master equation. The accuracy of this procedure is tested by comparing its results with analytic solutions (when available) and with Gillespie stochastic simulations. We further employ our approach to study the stochastic expression of a simple gene network, which is subject to negative feedback regulation at the transcriptional level. Special attention is paid to the influence of negative feedback on the amplitude of intrinsic noise, as well as on the relaxation rate of the system probability distribution function to the steady solution. Our results suggest the existence of an optimal feedback strength that maximizes this relaxation rate.

Identification of a topological characteristic responsible for the biological robustness of regulatory networks

Attribution of biological robustness to the specific structural properties of a regulatory network is an important yet unsolved problem in systems biology. It is widely believed that the topological characteristics of a biological control network largely determine its dynamic behavior, yet the actual mechanism is still poorly understood. Here, we define a novel structural feature of biological networks, termed ‘regulation entropy’, to quantitatively assess the influence of network topology on the robustness of the systems. Using the cell-cycle control networks of the budding yeast (Saccharomyces cerevisiae) and the fission yeast (Schizosaccharomyces pombe) as examples, we first demonstrate the correlation of this quantity with the dynamic stability of biological control networks, and then we establish a significant association between this quantity and the structural stability of the networks. And we further substantiate the generality of this approach with a broad spectrum of biological and random networks. We conclude that the regulation entropy is an effective order parameter in evaluating the robustness of biological control networks. Our work suggests a novel connection between the topological feature and the dynamic property of biological regulatory networks.

Living organisms exert very complicated control on the functionality of their components. Such control systems can often operate in a surprisingly robust manner, in spite of constant perturbations from fluctuating internal conditions and a volatile external environment. What feature makes such control mechanisms robust? Is there a general way to achieve robustness? Here, we address these questions by investigating the wiring of interaction networks, which contains the most condensed information about the control mechanisms of biological systems. We suggest that one of the most important factors in the realization of biological robustness rests in the global coherency of the control strategy, i.e., the consistency of commands flowing through different routes in the network to the same destination. To implement this idea, we propose an order parameter termed ‘regulation entropy’ to quantitatively describe this control consistency of networks. We find that this order parameter correlates with the resistance of the system to external perturbations as well as internal fluctuations. Our results suggest that the self-consistency of the control strategy is important for the vitality and robustness of living organisms.