Robustness elasticity in complex networks

Network robustness refers to a network's resilience to stress or damage. Given that most networks are inherently dynamic, with changing topology, loads, and operational states, their robustness is also likely subject to change. However, in most analyses of network structure, it is assumed that interaction among nodes has no effect on robustness. To investigate the hypothesis that network robustness is not sensitive or elastic to the level of interaction (or flow) among network nodes, this paper explores the impacts of network disruption, namely arc deletion, over a temporal sequence of observed nodal interactions for a large Internet backbone system. In particular, a mathematical programming approach is used to identify exact bounds on robustness to arc deletion for each epoch of nodal interaction. Elasticity of the identified bounds relative to the magnitude of arc deletion is assessed. Results indicate that system robustness can be highly elastic to spatial and temporal variations in nodal interactions within complex systems. Further, the presence of this elasticity provides evidence that a failure to account for nodal interaction can confound characterizations of complex networked systems.     

Robustness and evolvability in genetic regulatory networks

Living organisms are robust to a great variety of genetic changes. Gene regulation networks and metabolic pathways self-organize and reaccommodate to make the organism perform with stability and reliability under many point mutations, gene duplications and gene deletions. At the same time, living organisms are evolvable, which means that these kind of genetic perturbations can eventually make the organism acquire new functions and adapt to new environments. It is still an open problem to determine how robustness and evolvability blend together at the genetic level to produce stable organisms that yet can change and evolve. Here we address this problem by studying the robustness and evolvability of the attractor landscape of genetic regulatory network models under the process of gene duplication followed by divergence. We show that an intrinsic property of this kind of networks is that, after the divergence of the parent and duplicate genes, with a high probability the previous phenotypes, encoded in the attractor landscape of the network, are preserved and new ones might appear. The above is true in a variety of network topologies and even for the case of extreme divergence in which the duplicate gene bears almost no relation with its parent. Our results indicate that networks operating close to the so-called "critical regime" exhibit the maximum robustness and evolvability simultaneously.     

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.     

Robustness of a gene regulatory circuit.

Complex interacting systems exhibit system behavior that is often not predictable from the properties of the component parts. We have tested a particular system property, that of robustness. The behavior of a system is termed robust if that behavior is qualitatively normal in the face of substantial changes to the system components. Here we test whether the behavior of the phage lambda gene regulatory circuitry is robust. This circuitry can exist in two alternative patterns of gene expression, and can switch from one regulatory state to the other. These states are stabilized by the action at the O(R) region of two regulatory proteins, CI and Cro, which bind with differential affinities to the O(R)1 and O(R)3 sites, such that each represses the synthesis of the other one. In this work, this pattern of binding was altered by making three mutant phages in which O(R)1 and O(R)3 were identical. These variants had the same qualitative in vivo patterns of gene expression as wild type. We conclude that the behavior of the lambda circuitry is highly robust. Based on these and other results, we propose a two-step pathway, in which robustness plays a key role, for evolution of complex regulatory circuitry.     

Chaotic motifs in gene regulatory networks

Chaos should occur often in gene regulatory networks (GRNs) which have been widely described by nonlinear coupled ordinary differential equations, if their dimensions are no less than 3. It is therefore puzzling that chaos has never been reported in GRNs in nature and is also extremely rare in models of GRNs. On the other hand, the topic of motifs has attracted great attention in studying biological networks, and network motifs are suggested to be elementary building blocks that carry out some key functions in the network. In this paper, chaotic motifs (subnetworks with chaos) in GRNs are systematically investigated. The conclusion is that: (i) chaos can only appear through competitions between different oscillatory modes with rivaling intensities. Conditions required for chaotic GRNs are found to be very strict, which make chaotic GRNs extremely rare. (ii) Chaotic motifs are explored as the simplest few-node structures capable of producing chaos, and serve as the intrinsic source of chaos of random few-node GRNs. Several optimal motifs causing chaos with atypically high probability are figured out. (iii) Moreover, we discovered that a number of special oscillators can never produce chaos. These structures bring some advantages on rhythmic functions and may help us understand the robustness of diverse biological rhythms. (iv) The methods of dominant phase-advanced driving (DPAD) and DPAD time fraction are proposed to quantitatively identify chaotic motifs and to explain the origin of chaotic behaviors in GRNs.     

Cardiac gene regulatory networks in Drosophila

The Drosophila system has proven a powerful tool to help unlock the regulatory processes that occur during specification and differentiation of the embryonic heart. In this review, we focus upon a temporal analysis of the molecular events that result in heart formation in Drosophila, with a particular emphasis upon how genomic and other cutting-edge approaches are being brought to bear upon the subject. We anticipate that systems-level approaches will contribute greatly to our comprehension of heart development and disease in the animal kingdom.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Moonlighting kinases with guanylate cyclase activity can tune regulatory signal networks

Guanylate cyclase (GC) catalyzes the formation of cGMP and it is only recently that such enzymes have been characterized in plants. One family of plant GCs contains the GC catalytic center encapsulated within the intracellular kinase domain of leucine rich repeat receptor like kinases such as the phytosulfokine and brassinosteroid receptors. In vitro studies show that both the kinase and GC domain have catalytic activity indicating that these kinase-GCs are examples of moonlighting proteins with dual catalytic function. The natural ligands for both receptors increase intracellular cGMP levels in isolated mesophyll protoplast assays suggesting that the GC activity is functionally relevant. cGMP production may have an autoregulatory role on receptor kinase activity and/or contribute to downstream cell expansion responses. We postulate that the receptors are members of a novel class of receptor kinases that contain functional moonlighting GC domains essential for complex signaling roles.