On the robustness of update schedules in Boolean networks

Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems.     

Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.     

Random simulation and confiners: Their application to neural networks

Random simulation of complex dynamical systems is generally used in order to obtain information about their asymptotic behaviour (i.e., when time or size of the system tends towards infinity). A fortunate and welcome circumstance in most of the systems studied by physicists, biologists, and economists is the existence of an invariant measure in the state space allowing determination of the frequency with which observation of asymptotic states is possible. Regions found between contour lines of the surface density of this invariant measure are called confiners. An example of such confiners is given for a formal neural network capable of learning. Finally, an application of this methodology is proposed in studying dependency of the network's invariant measure with regard to: 1) the mode of neurone updating (parallel or sequential), and 2) boundary conditions of the network (searching for phase transitions).     

Robustness analysis and behavior discrimination in enzymatic reaction networks

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics.     

Dynamical analysis of a generic Boolean model for the control of the mammalian cell cycle

MOTIVATION: To understand the behaviour of complex biological regulatory networks, a proper integration of molecular data into a full-fledge formal dynamical model is ultimately required. As most available data on regulatory interactions are qualitative, logical modelling offers an interesting framework to delineate the main dynamical properties of the underlying networks. RESULTS: Transposing a generic model of the core network controlling the mammalian cell cycle into the logical framework, we compare different strategies to explore its dynamical properties. In particular, we assess the respective advantages and limits of synchronous versus asynchronous updating assumptions to delineate the asymptotical behaviour of regulatory networks. Furthermore, we propose several intermediate strategies to optimize the computation of asymptotical properties depending on available knowledge. AVAILABILITY: The mammalian cell cycle model is available in a dedicated XML format (GINML) on our website, along with our logical simulation software GINsim (http://gin.univ-mrs.fr/GINsim). Higher resolution state transitions graphs are also found on this web site (Model Repository page).     

Evolving learning rules and emergence of cooperation in spatial prisoner's dilemma

In the evolutionary Prisoner's dilemma (PD) game, agents play with each other and update their strategies in every generation according to some microscopic dynamical rule. In its spatial version, agents do not play with every other but, instead, interact only with their neighbours, thus mimicking the existing of a social or contact network that defines who interacts with whom. In this work, we explore evolutionary, spatial PD systems consisting of two types of agents, each with a certain update (reproduction, learning) rule. We investigate two different scenarios: in the first case, update rules remain fixed for the entire evolution of the system; in the second case, agents update both strategy and update rule in every generation. We show that in a well-mixed population the evolutionary outcome is always full defection. We subsequently focus on two-strategy competition with nearest-neighbour interactions on the contact network and synchronised update of strategies. Our results show that, for an important range of the parameters of the game, the final state of the system is largely different from that arising from the usual setup of a single, fixed dynamical rule. Furthermore, the results are also very different if update rules are fixed or evolve with the strategies. In these respect, we have studied representative update rules, finding that some of them may become extinct while others prevail. We describe the new and rich variety of final outcomes that arise from this co-evolutionary dynamics. We include examples of other neighbourhoods and asynchronous updating that confirm the robustness of our conclusions. Our results pave the way to an evolutionary rationale for modelling social interactions through game theory with a preferred set of update rules.     

The Arabidopsis thaliana flower organ specification gene regulatory network determines a robust differentiation process

The Arabidopsis thaliana flower organ specification gene regulatory network (FOS-GRN) has been modeled previously as a discrete dynamical system, recovering as steady states configurations that match the genetic profiles described in primordial cells of inflorescence, sepals, petals, stamens and carpels during early flower development. In this study, we first update the FOS-GRN by adding interactions and modifying some rules according to new experimental data. A discrete model of this updated version of the network has a dynamical behavior identical to previous versions, under both wild type and mutant conditions, thus confirming its robustness. Then, we develop a continuous version of the FOS-GRN using a new methodology that builds upon previous proposals. The fixed point attractors of the discrete system are all observed in the continuous model, but the latter also contains new steady states that might correspond to genetic activation states present briefly during the early phases of flower development. We show that both the discrete and the continuous models recover the observed stable gene configurations observed in the inflorescence meristem, as well as the primordial cells of sepals, petals, stamens and carpels. Additionally, both models are subjected to perturbations in order to establish the nature of additional signals that may suffice to determine the experimentally observed order of appearance of floral organs. Our results thus describe a possible mechanism by which the network canalizes molecular signals and/or noise, thus conferring robustness to the differentiation process.     

<Emphasis FontCategory="SansSerif">Investigation on changes of modularity and robustness by edge-removal mutations in signaling networks</Emphasis>

Background

Biological networks consisting of molecular components and interactions are represented by a graph model. There have been some studies based on that model to analyze a relationship between structural characteristics and dynamical behaviors in signaling network. However, little attention has been paid to changes of modularity and robustness in mutant networks.

Results

In this paper, we investigated the changes of modularity and robustness by edge-removal mutations in three signaling networks. We first observed that both the modularity and robustness increased on average in the mutant network by the edge-removal mutations. However, the modularity change was negatively correlated with the robustness change. This implies that it is unlikely that both the modularity and the robustness values simultaneously increase by the edge-removal mutations. Another interesting finding is that the modularity change was positively correlated with the degree, the number of feedback loops, and the edge betweenness of the removed edges whereas the robustness change was negatively correlated with them. We note that these results were consistently observed in randomly structure networks. Additionally, we identified two groups of genes which are incident to the highly-modularity-increasing and the highly-robustness-decreasing edges with respect to the edge-removal mutations, respectively, and observed that they are likely to be central by forming a connected component of a considerably large size. The gene-ontology enrichment of each of these gene groups was significantly different from the rest of genes. Finally, we showed that the highly-robustness-decreasing edges can be promising edgetic drug-targets, which validates the usefulness of our analysis.

Conclusions

Taken together, the analysis of changes of robustness and modularity against edge-removal mutations can be useful to unravel novel dynamical characteristics underlying in signaling networks.

     

Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

Dynamical properties of min-max networks

In this paper we study the dynamical behavior of a class of neural networks where the local transition rules are max or min functions. We prove that sequential updates define dynamics which reach the equilibrium in O(n2) steps, where n is the size of the network. For synchronous updates the equilibrium is reached in O(n) steps. It is shown that the number of fixed points of the sequential update is at most n. Moreover, given a set of p < or = n vectors, we show how to build a network of size n such that all these vectors are fixed points.     

From network-to-antibody robustness in a bio-inspired immune system

Behavioural robustness at antibody and immune network level is discussed. The robustness of the immune response that drives an autonomous mobile robot is examined with two computational experiments in the autonomous mobile robots trajectory generation context in unknown environments. The immune response is met based on the immune network metaphor for different low-level behaviours coordination. These behaviours are activated when a robot sense the appropriate conditions in the environment in relation to the network current state. Results are obtained over a case study in computer simulation as well as in laboratory experiments with a Khepera II microrobot. In this work, we develop a set of tests where such an immune response is externally perturbed at network or low-level behavioural modules to analyse the robust capacity of the system to unexpected perturbations. Emergence of robust behaviour and high-level immune response relates to the coupling between behavioural modules that are selectively engaged with the environment based on immune response. Experimental evidence leads discussions on a dynamical systems perspective of behavioural robustness in artificial immune systems that goes beyond the isolated immune network response.     

A robust model to describe the differentiation of T-helper cells

There is a wealth of information regarding the differentiation of T-helper cells. Nevertheless, there is no general agreement on the topology and dynamical properties of the molecular network controlling the differentiation of these cells. This paper presents a continuous dynamical system to model the signaling network that controls the differentiation process of T-helper cells. The model is able to represent the differentiation from the precursor Th0 cell to any of the four effectors types (Th1, Th2, Th17, and Treg), as well as the phenotype of single null mutants. We present the first sensitivity analysis of the equations defining the Th model, showing that the qualitative dynamical behavior of the model is very robust against changes in three out of four tested parameters. The robustness of the model is in agreement with our claim that the qualitative behavior of the system is to a large extent independent of the methodological framework used for modeling.     

Robustness: confronting lessons from physics and biology

The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, self‐organisation) developed in physics are relevant to biology, or whether specific extensions and novel frameworks are required to account for the robustness properties of living systems. To clarify this issue, the different meanings covered by this unique term are discussed; it is argued that they crucially depend on the kind of perturbations that a robust system should by definition withstand. Possible mechanisms underlying robust behaviours are examined, either encountered in all natural systems (symmetries, conservation laws, dynamic stability) or specific to biological systems (feedbacks and regulatory networks). Special attention is devoted to the (sometimes counterintuitive) interrelations between robustness and noise. A distinction between dynamic selection and natural selection in the establishment of a robust behaviour is underlined. It is finally argued that nested notions of robustness, relevant to different time scales and different levels of organisation, allow one to reconcile the seemingly contradictory requirements for robustness and adaptability in living systems.     

Analysis of the Robustness of Network-Based Disease-Gene Prioritization Methods Reveals Redundancy in the Human Interactome and Functional Diversity of Disease-Genes

Complex biological systems usually pose a trade-off between robustness and fragility where a small number of perturbations can substantially disrupt the system. Although biological systems are robust against changes in many external and internal conditions, even a single mutation can perturb the system substantially, giving rise to a pathophenotype. Recent advances in identifying and analyzing the sequential variations beneath human disorders help to comprehend a systemic view of the mechanisms underlying various disease phenotypes. Network-based disease-gene prioritization methods rank the relevance of genes in a disease under the hypothesis that genes whose proteins interact with each other tend to exhibit similar phenotypes. In this study, we have tested the robustness of several network-based disease-gene prioritization methods with respect to the perturbations of the system using various disease phenotypes from the Online Mendelian Inheritance in Man database. These perturbations have been introduced either in the protein-protein interaction network or in the set of known disease-gene associations. As the network-based disease-gene prioritization methods are based on the connectivity between known disease-gene associations, we have further used these methods to categorize the pathophenotypes with respect to the recoverability of hidden disease-genes. Our results have suggested that, in general, disease-genes are connected through multiple paths in the human interactome. Moreover, even when these paths are disturbed, network-based prioritization can reveal hidden disease-gene associations in some pathophenotypes such as breast cancer, cardiomyopathy, diabetes, leukemia, parkinson disease and obesity to a greater extend compared to the rest of the pathophenotypes tested in this study. Gene Ontology (GO) analysis highlighted the role of functional diversity for such diseases.     

Dynamic properties of network motifs contribute to biological network organization

Biological networks, such as those describing gene regulation, signal transduction, and neural synapses, are representations of large-scale dynamic systems. Discovery of organizing principles of biological networks can be enhanced by embracing the notion that there is a deep interplay between network structure and system dynamics. Recently, many structural characteristics of these non-random networks have been identified, but dynamical implications of the features have not been explored comprehensively. We demonstrate by exhaustive computational analysis that a dynamical property—stability or robustness to small perturbations—is highly correlated with the relative abundance of small subnetworks (network motifs) in several previously determined biological networks. We propose that robust dynamical stability is an influential property that can determine the non-random structure of biological networks.     

Node vulnerability under finite perturbations in complex networks

A measure to quantify vulnerability under perturbations (attacks, failures, large fluctuations) in ensembles (networks) of coupled dynamical systems is proposed. Rather than addressing the issue of how the network properties change upon removal of elements of the graph (the strategy followed by most of the existing methods for studying the vulnerability of a network based on its topology), here a dynamical definition of vulnerability is introduced, referring to the robustness of a collective dynamical state to perturbing events occurring over a fixed topology. In particular, we study how the collective (synchronized) dynamics of a network of chaotic units is disrupted under the action of a finite size perturbation on one of its nodes. Illustrative examples are provided for three systems of identical chaotic oscillators coupled according to three distinct well-known network topologies. A quantitative comparison between the obtained vulnerability rankings and the classical connectivity/centrality rankings is made that yields conclusive results. Possible applications of the proposed strategy and conclusions are also discussed.     

Representational switching by dynamical reorganization of attractor structure in a network model of the prefrontal cortex

The prefrontal cortex (PFC) plays a crucial role in flexible cognitive behavior by representing task relevant information with its working memory. The working memory with sustained neural activity is described as a neural dynamical system composed of multiple attractors, each attractor of which corresponds to an active state of a cell assembly, representing a fragment of information. Recent studies have revealed that the PFC not only represents multiple sets of information but also switches multiple representations and transforms a set of information to another set depending on a given task context. This representational switching between different sets of information is possibly generated endogenously by flexible network dynamics but details of underlying mechanisms are unclear. Here we propose a dynamically reorganizable attractor network model based on certain internal changes in synaptic connectivity, or short-term plasticity. We construct a network model based on a spiking neuron model with dynamical synapses, which can qualitatively reproduce experimentally demonstrated representational switching in the PFC when a monkey was performing a goal-oriented action-planning task. The model holds multiple sets of information that are required for action planning before and after representational switching by reconfiguration of functional cell assemblies. Furthermore, we analyzed population dynamics of this model with a mean field model and show that the changes in cell assemblies' configuration correspond to those in attractor structure that can be viewed as a bifurcation process of the dynamical system. This dynamical reorganization of a neural network could be a key to uncovering the mechanism of flexible information processing in the PFC.     

Analysis of Homeostatic Mechanisms in Biochemical Networks

Cell metabolism is an extremely complicated dynamical system that maintains important cellular functions despite large changes in inputs. This “homeostasis” does not mean that the dynamical system is rigid and fixed. Typically, large changes in external variables cause large changes in some internal variables so that, through various regulatory mechanisms, certain other internal variables (concentrations or velocities) remain approximately constant over a finite range of inputs. Outside that range, the mechanisms cease to function and concentrations change rapidly with changes in inputs. In this paper we analyze four different common biochemical homeostatic mechanisms: feedforward excitation, feedback inhibition, kinetic homeostasis, and parallel inhibition. We show that all four mechanisms can occur in a single biological network, using folate and methionine metabolism as an example. Golubitsky and Stewart have proposed a method to find homeostatic nodes in networks. We show that their method works for two of these mechanisms but not the other two. We discuss the many interesting mathematical and biological questions that emerge from this analysis, and we explain why understanding homeostatic control is crucial for precision medicine.