A description of dynamical graphs associated to elementary regulatory circuits

The biological and dynamical importance of feedback circuits in regulatory graphs has often been emphasized. The work presented here aims at completely describing the dynamics of isolated elementary regulatory circuits. Our analytical approach is based on a discrete formal framework, built upon the logical approach of R. Thomas. Given a regulatory circuit, we show that the structure of synchronous and asynchronous dynamical graphs depends only on the length of the circuit (number of genes) and on its sign (which depends on the parity of the number of negative interactions). This work constitutes a first step towards the analytical characterisation of discrete dynamical graphs for more complex regulatory networks in terms of contributions corresponding to their embedded elementary circuits.     

A logical model provides insights into T cell receptor signaling

Cellular decisions are determined by complex molecular interaction networks. Large-scale signaling networks are currently being reconstructed, but the kinetic parameters and quantitative data that would allow for dynamic modeling are still scarce. Therefore, computational studies based upon the structure of these networks are of great interest. Here, a methodology relying on a logical formalism is applied to the functional analysis of the complex signaling network governing the activation of T cells via the T cell receptor, the CD4/CD8 co-receptors, and the accessory signaling receptor CD28. Our large-scale Boolean model, which comprises 94 nodes and 123 interactions and is based upon well-established qualitative knowledge from primary T cells, reveals important structural features (e.g., feedback loops and network-wide dependencies) and recapitulates the global behavior of this network for an array of published data on T cell activation in wild-type and knock-out conditions. More importantly, the model predicted unexpected signaling events after antibody-mediated perturbation of CD28 and after genetic knockout of the kinase Fyn that were subsequently experimentally validated. Finally, we show that the logical model reveals key elements and potential failure modes in network functioning and provides candidates for missing links. In summary, our large-scale logical model for T cell activation proved to be a promising in silico tool, and it inspires immunologists to ask new questions. We think that it holds valuable potential in foreseeing the effects of drugs and network modifications.     

Boolean dynamics of biological networks with multiple coupled feedback loops

Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixed-point attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limit-cycle attractors as they have more coupled negative feedback loops.     

Mapping multivalued onto Boolean dynamics

This paper deals with the generalized logical framework defined by René Thomas in the 70's to qualitatively represent the dynamics of regulatory networks. In this formalism, a regulatory network is represented as a graph, where nodes denote regulatory components (basically genes) and edges denote regulations between these components. Discrete variables are associated to regulatory components accounting for their levels of expression. In most cases, Boolean variables are enough, but some situations may require further values. Despite this fact, the majority of tools dedicated to the analysis of logical models are restricted to the Boolean case. A formal Boolean mapping of multivalued logical models is a natural way of extending the applicability of these tools.Three decades ago, a multivalued to Boolean variable mapping was proposed by P. Van Ham. Since then, all works related to multivalued logical models and using a Boolean representation rely on this particular mapping. We formally show in this paper that this mapping is actually the sole, up to cosmetic changes, that could preserve the regulatory structures of the underlying graphs as well as their dynamical behaviours.     

Emergence of complex behaviour from simple circuit structures

The set of (feedback) circuits of a complex system is the machinery that allows the system to be aware of the levels of its crucial constituents. Circuits can be identified without ambiguity from the elements of the Jacobian matrix of the system. There are two types of circuits: positive if they comprise an even number of negative interactions, negative if this number is odd. The two types of circuits play deeply different roles: negative circuits are required for homeostasis, with or without oscillations, positive circuits are required for multistationarity, and hence, in biology, for differentiation and memory. In non-linear systems, a circuit can positive or negative (an 'ambiguous circuit', depending on the location in phase space. Full circuits are those circuits (or unions of disjoint circuits) that imply all the variables of the system. There is a tight relation between circuits and steady states. Each full circuit, if isolated, generates steady state(s) whose nature (eigenvalues) is determined by the structure of the circuit. Multistationarity requires the presence of at least two full circuits of opposite Eisenfeld signs, or else, an ambiguous circuit. We show how a significant part of the dynamical behaviour of a system can be predicted by a mere examination of its Jacobian matrix. We also show how extremely complex dynamics can be generated by such simple logical structures as a single (full and ambiguous) circuit.     

Positive feedback circuits and memory

The concept of regulatory feedback circuit refers to oriented cyclic interactions between elements of a system. There are two classes of circuits, positive and negative, whose properties are in striking contrast. Positive circuits are a prerequisite for the occurrence of multiple steady states (multistationarity), and hence, they are involved in all processes showing hysteresis or memory. Endogenous or exogenous perturbations can lead the system to exhibit or to evoke one particular stable regime. The role of positive circuits in cell differentiation and in immunology is well documented. Negative circuits are involved in homeostatic regulation, with or without oscillations. The aim of this paper is to show: a) that positive circuits account for many features of memory stricto sensu (i.e., neural memory and mnesic evocation) as well as largo sensu (e.g. differentiation or immunological memory); and b) that simple combinations of positive and negative circuits provide powerful regulatory modules, which can also be associated in batteries. These entities have vast dynamical possibilities in the field of neurobiology, as well as in the fields of differentiation and immunology. Here we consider a universal minimal regulatory module, for which we suggest to adopt the term 'logical regulon', which can be considered as an atom of Jacob's integron. It comprises a positive and a negative circuit in its interaction matrix, and we recall the main results related to the simultaneous presence of these circuits. Finally, we give three applications of this type of interaction matrix. The first two deal with the coexistence of multiple stable steady states and periodicity in differentiation and in an immunological system showing hysteretic properties. The third deals with the dual problems of synchronization and desynchronization of a neural model for hippocampus memory evocation processes.     

Maximum Number of Fixed Points in Regulatory Boolean Networks

Boolean networks (BNs) have been extensively used as mathematical models of genetic regulatory networks. The number of fixed points of a BN is a key feature of its dynamical behavior. Here, we study the maximum number of fixed points in a particular class of BNs called regulatory Boolean networks, where each interaction between the elements of the network is either an activation or an inhibition. We find relationships between the positive and negative cycles of the interaction graph and the number of fixed points of the network. As our main result, we exhibit an upper bound for the number of fixed points in terms of minimum cardinality of a set of vertices meeting all positive cycles of the network, which can be applied in the design of genetic regulatory networks.     

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.     

Identification of all steady states in large networks by logical analysis

The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.     

Formal Methods for Hopfield-Like Networks

Building a meaningful model of biological regulatory network is usually done by specifying the components (e.g. the genes) and their interactions, by guessing the values of parameters, by comparing the predicted behaviors to the observed ones, and by modifying in a trial-error process both architecture and parameters in order to reach an optimal fitness. We propose here a different approach to construct and analyze biological models avoiding the trial-error part, where structure and dynamics are represented as formal constraints. We apply the method to Hopfield-like networks, a formalism often used in both neural and regulatory networks modeling. The aim is to characterize automatically the set of all models consistent with all the available knowledge (about structure and behavior). The available knowledge is formalized into formal constraints. The latter are compiled into Boolean formula in conjunctive normal form and then submitted to a Boolean satisfiability solver. This approach allows to formulate a wide range of queries, expressed in a high level language, and possibly integrating formalized intuitions. In order to explore its potential, we use it to find cycles for 3-nodes networks and to determine the flower morphogenesis regulatory network of Arabidopsis thaliana. Applications of this technique are numerous and concern the building of models from data as well as the design of biological networks possessing specified behaviors.     

Criticality Is an Emergent Property of Genetic Networks that Exhibit Evolvability

Accumulating experimental evidence suggests that the gene regulatory networks of living organisms operate in the critical phase, namely, at the transition between ordered and chaotic dynamics. Such critical dynamics of the network permits the coexistence of robustness and flexibility which are necessary to ensure homeostatic stability (of a given phenotype) while allowing for switching between multiple phenotypes (network states) as occurs in development and in response to environmental change. However, the mechanisms through which genetic networks evolve such critical behavior have remained elusive. Here we present an evolutionary model in which criticality naturally emerges from the need to balance between the two essential components of evolvability: phenotype conservation and phenotype innovation under mutations. We simulated the Darwinian evolution of random Boolean networks that mutate gene regulatory interactions and grow by gene duplication. The mutating networks were subjected to selection for networks that both (i) preserve all the already acquired phenotypes (dynamical attractor states) and (ii) generate new ones. Our results show that this interplay between extending the phenotypic landscape (innovation) while conserving the existing phenotypes (conservation) suffices to cause the evolution of all the networks in a population towards criticality. Furthermore, the networks produced by this evolutionary process exhibit structures with hubs (global regulators) similar to the observed topology of real gene regulatory networks. Thus, dynamical criticality and certain elementary topological properties of gene regulatory networks can emerge as a byproduct of the evolvability of the phenotypic landscape.     

Computational design of synthetic regulatory networks from a genetic library to characterize the designability of dynamical behaviors

The engineering of synthetic gene networks has mostly relied on the assembly of few characterized regulatory elements using rational design principles. It is of outmost importance to analyze the scalability and limits of such a design workflow. To analyze the design capabilities of libraries of regulatory elements, we have developed the first automated design approach that combines such elements to search the genotype space associated to a given phenotypic behavior. Herein, we calculated the designability of dynamical functions obtained from circuits assembled with a given genetic library. By designing circuits working as amplitude filters, pulse counters and oscillators, we could infer new mechanisms for such behaviors. We also highlighted the hierarchical design and the optimization of the interface between devices. We dissected the functional diversity of a constrained library and we found that even such libraries can provide a rich variety of behaviors. We also found that intrinsic noise slightly reduces the designability of digital circuits, but it increases the designability of oscillators. Finally, we analyzed the robust design as a strategy to counteract the evolvability and noise in gene expression of the engineered circuits within a cellular background, obtaining mechanisms for robustness through non-linear negative feedback loops.     

Positive circuits and d-dimensional spatial differentiation: application to the formation of sense organs in Drosophila

We discuss a rule proposed by the biologist Thomas according to which the possibility for a genetic network (represented by a signed directed graph called a regulatory graph) to have several stable states implies the existence of a positive circuit. This result is already known for different models, differential or discrete formalism, but always with a network of genes contained in a single cell. Thus, we can ask about the validity of this rule for a system containing several cells and with intercellular genetic interactions. In this paper, we consider the genetic interactions between several cells located on a d-dimensional lattice, i.e., each point of lattice represents a cell to which we associate the expression level of n genes contained in this cell. With this configuration, we show that the existence of a positive circuit is a necessary condition for a specific form of multistationarity, which naturally corresponds to spatial differentiation. We then illustrate this theorem through the example of the formation of sense organs in Drosophila.     

Multisite phosphorylation and network dynamics of cyclin-dependent kinase signaling in the eukaryotic cell cycle

Multisite phosphorylation of regulatory proteins has been proposed to underlie ultrasensitive responses required to generate nontrivial dynamics in complex biological signaling networks. We used a random search strategy to analyze the role of multisite phosphorylation of key proteins regulating cyclin-dependent kinase (CDK) activity in a model of the eukaryotic cell cycle. We show that multisite phosphorylation of either CDK, CDC25, wee1, or CDK-activating kinase is sufficient to generate dynamical behaviors including bistability and limit cycles. Moreover, combining multiple feedback loops based on multisite phosphorylation do not destabilize the cell cycle network by inducing complex behavior, but rather increase the overall robustness of the network. In this model we find that bistability is the major dynamical behavior of the CDK signaling network, and that negative feedback converts bistability into limit cycle behavior. We also compare the dynamical behavior of several simplified models of CDK regulation to the fully detailed model. In summary, our findings suggest that multisite phosphorylation of proteins is a critical biological mechanism in generating the essential dynamics and ensuring robust behavior of the cell cycle.     

On the long-run sensitivity of probabilistic Boolean networks

Boolean networks and, more generally, probabilistic Boolean networks, as one class of gene regulatory networks, model biological processes with the network dynamics determined by the logic-rule regulatory functions in conjunction with probabilistic parameters involved in network transitions. While there has been significant research on applying different control policies to alter network dynamics as future gene therapeutic intervention, we have seen less work on understanding the sensitivity of network dynamics with respect to perturbations to networks, including regulatory rules and the involved parameters, which is particularly critical for the design of intervention strategies. This paper studies this less investigated issue of network sensitivity in the long run. As the underlying model of probabilistic Boolean networks is a finite Markov chain, we define the network sensitivity based on the steady-state distributions of probabilistic Boolean networks and call it long-run sensitivity. The steady-state distribution reflects the long-run behavior of the network and it can give insight into the dynamics or momentum existing in a system. The change of steady-state distribution caused by possible perturbations is the key measure for intervention. This newly defined long-run sensitivity can provide insight on both network inference and intervention. We show the results for probabilistic Boolean networks generated from random Boolean networks and the results from two real biological networks illustrate preliminary applications of sensitivity in intervention for practical problems.     

From minimal signed circuits to the dynamics of Boolean regulatory networks

It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (involved in homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a nontrivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally minimal. We then apply these results to the vertebrate immune system, and show that the two minimal functional positive circuits of the model indeed behave as modules which combine to explain the presence of the three stable states corresponding to the Th0, Th1 and Th2 cells. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.