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 and alternative stable states in inbreeding, cooperation, sex roles and other evolutionary processes

A large proportion of studies in systems science focus on processes involving a mixture of positive and negative feedbacks, which are also common themes in evolutionary ecology. Examples of negative feedback are density dependence (population regulation) and frequency-dependent selection (polymorphisms). Positive feedback, in turn, plays a role in Fisherian 'runaway' sexual selection, the evolution of cooperation, selfing and inbreeding tolerance under purging of deleterious alleles, and the evolution of sex differences in parental care. All these examples feature self-reinforcing processes where the increase in the value of a trait selects for further increases, sometimes via a coevolutionary feedback loop with another trait. Positive feedback often leads to alternative stable states (evolutionary endpoints), making the interpretation of evolutionary predictions challenging. Here, we discuss conceptual issues such as the relationship between self-reinforcing selection and disruptive selection. We also present an extension of a previous model on parental care, focusing on the relationship between the operational sex ratio and sexual selection, and the influence of this relationship on the evolution of biparental or uniparental care.     

Hardware (DNA) circuits

A scheme is presented whereby a new genetic control circuit can be introduced into an organism, permitting the experimenter to turn the expression of a given gene (or set of genes) on or off at will. The proposed scheme involves a positive feedback loop--here, a positive regulator, the CII protein of phage lambda, with its structural gene engineered so as to require CII for its expression. This feedback loop creates the possibility of two stable steady states, with gene cII ON or OFF. Genes added downstream of cII and lacking a promoter will follow the same expression as cII. Two additional circuits allow the experimenter to switch at will between the ON and OFF states.     

Quantifying the relative contributions of divisive and subtractive feedback to rhythm generation

Biological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes: divisive (decreases the gain of the positive feedback loop) and subtractive (increases the input threshold) that both contribute to slowly move the system between the high- and low-activity states. Can we determine the relative contribution of each type of negative feedback process to the rhythmic activity? Does one dominate? Do they control the active and silent phase equally? To answer these questions we use a neural network model with excitatory coupling, regulated by synaptic depression (divisive) and cellular adaptation (subtractive feedback). We first attempt to apply standard experimental methodologies: either passive observation to correlate the variations of a variable of interest to system behavior, or deletion of a component to establish whether a component is critical for the system. We find that these two strategies can lead to contradictory conclusions, and at best their interpretive power is limited. We instead develop a computational measure of the contribution of a process, by evaluating the sensitivity of the active (high activity) and silent (low activity) phase durations to the time constant of the process. The measure shows that both processes control the active phase, in proportion to their speed and relative weight. However, only the subtractive process plays a major role in setting the duration of the silent phase. This computational method can be used to analyze the role of negative feedback processes in a wide range of biological rhythms.     

Does mutual interference always stabilize predator–prey dynamics? A comparison of models

Based on a qualitative analysis of ODE systems, the dynamic properties of alternative predator-prey models with predator-dependent functional response have been compared in order to study the role that predator interference plays in the stabilisation of trophic systems. The models considered for interference have different mathematical expressions and different conceptual foundations. Despite these differences, they give essentially the same qualitative results: when interference is low, increasing it has a positive effect on asymptotic stability and thus on the resilience of the biological system. When it is high, it is the contrary (with logistic prey growth, increasing the interference parameter ensures stability but leads to very small predator densities). Possible consequences on the evolution of the interference level in real ecosystems are discussed.     

Cell cycle dynamics in a response/signalling feedback system with a gap

We consider a dynamical model of cell cycles of n cells in a culture in which cells in one specific phase (S for signalling) of the cell cycle produce chemical agents that influence the growth/cell cycle progression of cells in another phase (R for responsive). In the case that the feedback is negative, it is known that subpopulations of cells tend to become clustered in the cell cycle; while for a positive feedback, all the cells tend to become synchronized. In this paper, we suppose that there is a gap between the two phases. The gap can be thought of as modelling the physical reality of a time delay in the production and action of the signalling agents. We completely analyse the dynamics of this system when the cells are arranged into two cell cycle clusters. We also consider the stability of certain important periodic solutions in which clusters of cells have a cyclic arrangement and there are just enough clusters to allow interactions between them. We find that the inclusion of a small gap does not greatly alter the global dynamics of the system; there are still large open sets of parameters for which clustered solutions are stable. Thus, we add to the evidence that clustering can be a robust phenomenon in biological systems. However, the gap does effect the system by enhancing the stability of the stable clustered solutions. We explain this phenomenon in terms of contraction rates (Floquet exponents) in various invariant subspaces of the system. We conclude that in systems for which these models are reasonable, a delay in signalling is advantageous to the emergence of clustering.     

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.     

A minimal model of light-induced circadian rhythms of nitrate reductase activity in leaves of barley

Observed circadian rhythms of nitrate reductase (NR) (EC 1.6.6.1) activity in leaves of barley ( Hordeum vulgare L. cv. Herta) under continuous light conditions are described by a simple kinetic model. The oscillatory mechanism has been decomposed into the negative and positive feedback loops which are necessary according to present theories of chemical oscillating systems. Our results indicate that the decrease of NR activity in darkness can be considered as a reversible unimolecular conversion of the active form of NR into an inactive form, forming a negative stabilizing feedback loop. The light-induced increase of NR activity is related to a positive destabilizing feedback loop. In our treatment this process is represented as an autocatalytical reaction.     

The principle of homeostasis in the hypothalamus-pituitary-adrenal system: new insight from positive feedback

Feedback control, both negative and positive, is a fundamental feature of biological systems. Some of these systems strive to achieve a state of equilibrium or "homeostasis". The major endocrine systems are regulated by negative feedback, a process believed to maintain hormonal levels within a relatively narrow range. Positive feedback is often thought to have a destabilizing effect. Here, we present a "principle of homeostasis," which makes use of both positive and negative feedback loops. To test the hypothesis that this homeostatic concept is valid for the regulation of cortisol, we assessed experimental data in humans with different conditions (gender, obesity, endocrine disorders, medication) and analyzed these data by a novel computational approach. We showed that all obtained data sets were in agreement with the presented concept of homeostasis in the hypothalamus-pituitary-adrenal axis. According to this concept, a homeostatic system can stabilize itself with the help of a positive feedback loop. The brain mineralocorticoid and glucocorticoid receptors-with their known characteristics-fulfill the key functions in the homeostatic concept: binding cortisol with high and low affinities, acting in opposing manners, and mediating feedback effects on cortisol. This study supports the interaction between positive and negative feedback loops in the hypothalamus-pituitary-adrenal system and in this way sheds new light on the function of dual receptor regulation. Current knowledge suggests that this principle of homeostasis could also apply to other biological systems.     

Delay model of circadian gene expression with two negative feedback loops

In this theoretical paper we propose a quantitative minimal model for circadian gene expression based on two negative feedback loops. We perform numerical simulations to analyse its dynamics and parameter sensitivities in free-running conditions, and verify the entrainability by a single periodic driver. We furthermore apply two simultaneously acting external drivers, leading to aperiodic oscillations in the case of a single-loop system. These can be turned into regular periodic oscillations by introduction of a second loop. Our studies confirm the increasing evidence that multiple feedback loops increase the robustness of regulatory systems, and stress the particular situation of systems that are close to transition from free-running oscillation to steady-state behaviour. We discuss possible molecular realisations of the featured feedback loops and suggest the application of complex patterns of external stimulation as a generally useful approach to assess the functionality of models of circadian systems.     

Noise Propagation in Gene Regulation Networks Involving Interlinked Positive and Negative Feedback Loops

It is well known that noise is inevitable in gene regulatory networks due to the low-copy numbers of molecules and local environmental fluctuations. The prediction of noise effects is a key issue in ensuring reliable transmission of information. Interlinked positive and negative feedback loops are essential signal transduction motifs in biological networks. Positive feedback loops are generally believed to induce a switch-like behavior, whereas negative feedback loops are thought to suppress noise effects. Here, by using the signal sensitivity (susceptibility) and noise amplification to quantify noise propagation, we analyze an abstract model of the Myc/E2F/MiR-17-92 network that is composed of a coupling between the E2F/Myc positive feedback loop and the E2F/Myc/miR-17-92 negative feedback loop. The role of the feedback loop on noise effects is found to depend on the dynamic properties of the system. When the system is in monostability or bistability with high protein concentrations, noise is consistently suppressed. However, the negative feedback loop reduces this suppression ability (or improves the noise propagation) and enhances signal sensitivity. In the case of excitability, bistability, or monostability, noise is enhanced at low protein concentrations. The negative feedback loop reduces this noise enhancement as well as the signal sensitivity. In all cases, the positive feedback loop acts contrary to the negative feedback loop. We also found that increasing the time scale of the protein module or decreasing the noise autocorrelation time can enhance noise suppression; however, the systems sensitivity remains unchanged. Taken together, our results suggest that the negative/positive feedback mechanisms in coupled feedback loop dynamically buffer noise effects rather than only suppressing or amplifying the noise.     

The dynamics of production and destruction: Analytic insight into complex behavior

This paper analytically explores the properties of simple differential-difference equations that represent dynamic processes with feedback dependent on prior states of the system. Systems with pure negative and positive feedback are examined, as well as those with mixed (positive/negative) feedback characteristics. Very complex time dependent behaviors may arise from these processes. Indeed, the same mechanism may, depending on system parameters and initial conditions, produce simple, regular, repetitive patterns and completely irregular random-like fluctuations.For the differential-delay equations considered here we prove the existence of: (i) stable and unstable limit cycles, where the stable cycles may have an arbitrary number of extrema per period; and (ii) chaos, meaning the presence of infinitely many periodic solutions of different period and of infinitely many irregular and mixing solutions.     

A dichotomy for a class of cyclic delay systems

Two complementary analyses of a cyclic negative feedback system with delay are considered in this paper. The first analysis applies the work by Sontag, Angeli, Enciso and others regarding monotone control systems under negative feedback, and it implies the global attractiveness towards an equilibrium for arbitrary delays. The second one concerns the existence of a Hopf bifurcation with respect to the delay parameter, and it implies the existence of nonconstant periodic solutions for special delay values. A key idea is the use of the Schwarzian derivative, and its application for the study of Hill function nonlinearities. The positive feedback case is also addressed.     

A Critical Quantity for Noise Attenuation in Feedback Systems

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as “signed activation time”. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.