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.     

Dynamical roles of biological regulatory circuits

Regulatory circuits are found at the basis of all non-trivial dynamical properties of biological networks. More specifically, positive circuits are involved in the generation of multiple differentiated states, whereas negative circuits can generate cyclic or homeostatic behaviours. These notions are briefly reviewed, from initial biological formulations to mathematical formalisations, further encompassing their application to the design of synthetic regulatory systems. Finally, current challenges for the analysis of increasingly complex regulatory networks are indicated, as well as prospects for our understanding of development and evolution.     

Coupled feedback loops form dynamic motifs of cellular networks

Cellular networks are composed of complicated interconnections among components, and some subnetworks of particular functioning are often identified as network motifs. Among such network motifs, feedback loops are thought to play important dynamical roles. Intriguingly, such feedback loops are very often found as a coupled structure in cellular circuits. Therefore, we integrated all the scattered information regarding the coupled feedbacks in various cellular circuits and investigated the dynamical role of each coupled structure. Finally, we discovered that coupled positive feedbacks enhance signal amplification and bistable characteristics; coupled negative feedbacks realize enhanced homeostasis; coupled positive and negative feedbacks enable reliable decision-making by properly modulating signal responses and effectively dealing with noise.     

A model of sequential branching in hierarchical cell fate determination

Multipotent stem or progenitor cells undergo a sequential series of binary fate decisions, which ultimately generate the diversity of differentiated cells. Efforts to understand cell fate control have focused on simple gene regulatory circuits that predict the presence of multiple stable states, bifurcations and switch-like transitions. However, existing gene network models do not explain more complex properties of cell fate dynamics such as the hierarchical branching of developmental paths. Here, we construct a generic minimal model of the genetic regulatory network controlling cell fate determination, which exhibits five elementary characteristics of cell differentiation: stability, directionality, branching, exclusivity, and promiscuous expression. We argue that a modular architecture comprising repeated network elements reproduces these features of differentiation by sequentially repressing selected modules and hence restricting the dynamics to lower dimensional subspaces of the high-dimensional state space. We implement our model both with ordinary differential equations (ODEs), to explore the role of bifurcations in producing the one-way character of differentiation, and with stochastic differential equations (SDEs), to demonstrate the effect of noise on the system. We further argue that binary cell fate decisions are prevalent in cell differentiation due to general features of the underlying dynamical system. This minimal model makes testable predictions about the structural basis for directional, discrete and diversifying cell phenotype development and thus can guide the evaluation of real gene regulatory networks that govern differentiation.     

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.     

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.     

Regulatory dynamics of synthetic gene networks with positive feedback

Biological processes are governed by complex networks ranging from gene regulation to signal transduction. Positive feedback is a key element in such networks. The regulation enables cells to adopt multiple internal expression states in response to a single external input signal. However, past works lacked a dynamical aspect of this system. To address the dynamical property of the positive feedback system, we employ synthetic gene circuits in Escherichia coli to measure the rise-time of both the no-feedback system and the positive feedback system. We show that the kinetics of gene expression is slowed down if the gene regulatory system includes positive feedback. We also report that the transition of gene switching behaviors from the hysteretic one to the graded one occurs. A mathematical model based on the chemical reactions shows that the response delay is an inherited property of the positive feedback system. Furthermore, with the aid of the phase diagram, we demonstrate the decline of the feedback activation causes the transition of switching behaviors. Our findings provide a further understanding of a positive feedback system in a living cell from a dynamical point of view.     

Genetic regulation networks: circuits,regulons and attractors

We deal in this paper with the concept of genetic regulation network. The genes expression observed through the bio-array imaging allows the geneticist to obtain the intergenic interaction matrix W of the network. The interaction graph G associated to W presents in general interesting features like connected components, gardens of Eden, positive and negative circuits (or loops), and minimal components having 1 positive and 1 negative loop called regulons. Depending on parameters values like the connectivity coefficient K(W) and the mean inhibition weight I(W), the genetic regulation network can present several dynamical behaviours (fixed configuration, limit cycle of configurations) called attractors, when the observation time increases. We give some examples of such genetic regulation networks and analyse their dynamical properties and their biological consequences.     

The modularity of biological regulatory networks

A useful approach to complex regulatory networks consists of modeling their elements and interactions by Boolean equations. In this context, feedback circuits (i.e. circular sequences of interactions) have been shown to play key dynamical roles: whereas positive circuits are able to generate multistationarity, negative circuits may generate oscillatory behavior. In this paper, we principally focus on the case of gene networks. These are represented by fully connected Boolean networks where each element interacts with all elements including itself. Flexibility in network design is introduced by the use of Boolean parameters, one associated with each interaction or group of interactions affecting a given element. Within this formalism, a feedback circuit will generate its typical dynamical behavior (i.e. multistationarity or oscillations) only for appropriate values of some of the logical parameters. Whenever it does, we say that the circuit is 'functional'. More interestingly, this formalism allows the computation of the constraints on the logical parameters to have any feedback circuit functional in a network. Using this methodology, we found that the fraction of the total number of consistent combinations of parameter values that make a circuit functional decreases geometrically with the circuit length. From a biological point of view, this suggests that regulatory networks could be decomposed into small and relatively independent feedback circuits or 'regulatory modules'.     

Positive and negative feedback: striking a balance between necessary antagonists

Most biological regulation systems comprise feedback circuits as crucial components. Negative feedback circuits have been well understood for a very long time; indeed, their understanding has been the basis for the engineering of cybernetic machines exhibiting stable behaviour. The importance of positive feedback circuits, considered as "vicious circles", has however been underestimated. In this article, we give a demonstration based on degree theory for vector fields of the conjecture, made by René Thomas, that the presence of positive feedback circuits is a necessary condition for autonomous differential systems, covering a wide class of biologically relevant systems, to possess multiple steady states. We also show ways to derive constraints on the weights of positive and negative feedback circuits. These qualitative and quantitative results provide, respectively, structural constraints (i.e. related to the interaction graph) and numerical constraints (i.e. related to the magnitudes of the interactions) on systems exhibiting complex behaviours, and should make it easier to reverse-engineer the interaction networks animating those systems on the basis of partial, sometimes unreliable, experimental data. We illustrate these concepts on a model multistable switch, in the context of cellular differentiation, showing a requirement for sufficient cooperativity. Further developments are expected in the discovery and modelling of regulatory networks in general, and in the interpretation of bio-array hybridization and proteomics experiments in particular.     

Cooperative differentiation through clustering in multicellular populations

The coordinated development of multicellular organisms is driven by intercellular communication. Differentiation into diverse cell types is usually associated with the existence of distinct attractors of gene regulatory networks, but how these attractors emerge from cell-cell coupling is still an open question. In order to understand and characterize the mechanisms through which coexisting attractors arise in multicellular systems, here we systematically investigate the dynamical behavior of a population of synthetic genetic oscillators coupled by chemical means. Using bifurcation analysis and numerical simulations, we identify various attractors and attempt to deduce from these findings a way to predict the organized collective behavior of growing populations. Our results show that dynamical clustering is a generic property of multicellular systems. We argue that such clustering might provide a basis for functional differentiation and variability in biological systems.     

"Immunetworks", intersecting circuits and dynamics

This paper proposes a study of biological regulation networks based on a multi-level strategy. Given a network, the first structural level of this strategy consists in analysing the architecture of the network interactions in order to describe it. The second dynamical level consists in relating the patterns found in the architecture to the possible dynamical behaviours of the network. It is known that circuits are the patterns that play the most important part in the dynamics of a network in the sense that they are responsible for the diversity of its asymptotic behaviours. Here, we pursue further this idea and argue that beyond the influence of underlying circuits, intersections of circuits also impact significantly on the dynamics of a network and thus need to be payed special attention to. For some genetic regulation networks involved in the control of the immune system (“immunetworks”), we show that the small number of attractors can be explained by the presence, in the underlying structures of these networks, of intersecting circuits that “inter-lock”.     

Dose-response aligned circuits in signaling systems

Cells use biological signal transduction pathways to respond to environmental stimuli and the behavior of many cell types depends on precise sensing and transmission of external information. A notable property of signal transduction that was characterized in the Saccharomyces cerevisiae yeast cell and many mammalian cells is the alignment of dose-response curves. It was found that the dose response of the receptor matches closely the dose responses of the downstream. This dose-response alignment (DoRA) renders equal sensitivities and concordant responses in different parts of signaling system and guarantees a faithful information transmission. The experimental observations raise interesting questions about the nature of the information transmission through DoRA signaling networks and design principles of signaling systems with this function. Here, we performed an exhaustive computational analysis on network architectures that underlie the DoRA function in simple regulatory networks composed of two and three enzymes. The minimal circuits capable of DoRA were examined with Michaelis-Menten kinetics. Several motifs that are essential for the dynamical function of DoRA were identified. Systematic analysis of the topology space of robust DoRA circuits revealed that, rather than fine-tuning the network's parameters, the function is primarily realized by enzymatic regulations on the controlled node that are constrained in limiting regions of saturation or linearity.     

Analysis of Discrete Bioregulatory Networks Using Symbolic Steady States

A discrete model of a biological regulatory network can be represented by a discrete function that contains all available information on interactions between network components and the rules governing the evolution of the network in a finite state space. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem. In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in some region of state space. We state rules to explicitly construct attractors of the system from subnetwork attractors. Using the results, we formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the graph representing the structure of the system. In addition, we discuss approaches to finding symbolic steady states. We focus both on dynamics derived via synchronous as well as asynchronous update rules. Lastly, we illustrate the results by analyzing a model of T helper cell differentiation.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

The regulatory network that controls the differentiation of T lymphocytes

There is a vast amount of molecular information regarding the differentiation of T lymphocytes, in particular regarding in vitro experimental treatments that modify their differentiation process. This publicly available information was used to infer the regulatory network that controls the differentiation of T lymphocytes into CD4⁺ and CD8⁺ cells. Hereby we present a network that consists of 50 nodes and 97 regulatory interactions, representing the main signaling circuits established among molecules and molecular complexes regulating the differentiation of T cells. The network was converted into a continuous dynamical system in the form of a set of coupled ordinary differential equations, and its dynamical behavior was studied. With the aid of numerical methods, nine fixed point attractors were found for the dynamical system. These attractors correspond to the activation patterns observed experimentally for the following cell types: CD4⁻CD8⁻, CD4⁺CD8⁺, CD4⁺ naive, Th1, Th2, Th17, Treg, CD8⁺ naive, and CTL. Furthermore, the model is able to describe the differentiation process from the precursor CD4⁻CD8⁻ to any of the effector types due to a specific series of extracellular signals.     

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.