A Theoretical Graph Method for Search and Analysis of Critical Phenomena in Biochemical Systems. II. Kinetic Models of Biochemical Oscillators Including Two and Three Substances

Four kinetic models of hypothetical complex reactions containing minimal two-substance or three-substance oscillators were constructed on the basis of the graphical rules suggested in the preceding work. The kinetic models are thought to be a part of one of four general biochemical systems: 1) system of mutual protein phosphorylation/dephosphorylation; 2) autophosphorylation of multisubunit protein; 3) association/dissociation of proteins or protein-containing structures during protein–protein or protein–ligand interaction; and 4) two-substrate enzymatic reaction with substrate inhibition by one substrate. Graphical rules of oscillator association with surrounding medium were considered. The graphical criteria of the oscillation generator elimination and criteria of oscillation damping were obtained. Both damped and undamped oscillations of reaction components were obtained by numerical integration of the mathematical models of these reactions. The areas of changes of model parameters and variables, within which the oscillations exist, were found.     

New Theoretical Schemes of the Simplest Chemical Oscillators

Based on the sufficient conditions for existence of concentration oscillations of the components of bicomponent reaction, all possible chemical reactions with two, three, or four intermediate stages proceeding via the mass-action expression are considered. About forty new schemes of chemical/biochemical oscillators are developed.     

Simplest Kinetic Schemes for Biochemical Oscillators

The topological structure of the simplest critical fragments in biochemical systems has been characterized. The conditions are considered where the critical fragments induce oscillations of the concentrations of the system participants. To illustrate, three biochemical systems (transport of ions through a membrane, protein phosphorylation, and two-substrate reaction) are discussed. The kinetic schemes of these systems contain one of the discovered critical fragments. Relaxation oscillations of the concentrations of the system participants were demonstrated using the numerical integration method.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Robust Concentration and Frequency Control in Oscillatory Homeostats

Homeostatic and adaptive control mechanisms are essential for keeping organisms structurally and functionally stable. Integral feedback is a control theoretic concept which has long been known to keep a controlled variable robustly (i.e. perturbation-independent) at a given set-point by feeding the integrated error back into the process that generates . The classical concept of homeostasis as robust regulation within narrow limits is often considered as unsatisfactory and even incompatible with many biological systems which show sustained oscillations, such as circadian rhythms and oscillatory calcium signaling. Nevertheless, there are many similarities between the biological processes which participate in oscillatory mechanisms and classical homeostatic (non-oscillatory) mechanisms. We have investigated whether biological oscillators can show robust homeostatic and adaptive behaviors, and this paper is an attempt to extend the homeostatic concept to include oscillatory conditions. Based on our previously published kinetic conditions on how to generate biochemical models with robust homeostasis we found two properties, which appear to be of general interest concerning oscillatory and homeostatic controlled biological systems. The first one is the ability of these oscillators (“oscillatory homeostats”) to keep the average level of a controlled variable at a defined set-point by involving compensatory changes in frequency and/or amplitude. The second property is the ability to keep the period/frequency of the oscillator tuned within a certain well-defined range. In this paper we highlight mechanisms that lead to these two properties. The biological applications of these findings are discussed using three examples, the homeostatic aspects during oscillatory calcium and p53 signaling, and the involvement of circadian rhythms in homeostatic regulation.     

Stability analysis of an artificial biomolecular oscillator with non-cooperative regulatory interactions

Oscillators are essential to fuel autonomous behaviours in molecular systems. Artificial oscillators built with programmable biological molecules such as DNA and RNA are generally easy to build and tune, and can serve as timers for biological computation and regulation. We describe a new artificial nucleic acid biochemical reaction network, and we demonstrate its capacity to exhibit oscillatory solutions. This network can be built in vitro using nucleic acids and three bacteriophage enzymes, and has the potential to be implemented in cells. Numerical simulations suggest that oscillations occur in a realistic range of reaction rates and concentrations.     

Positive Feedback Promotes Oscillations in Negative Feedback Loops

A simple three-component negative feedback loop is a recurring motif in biochemical oscillators. This motif oscillates as it has the three necessary ingredients for oscillations: a three-step delay, negative feedback, and nonlinearity in the loop. However, to oscillate, this motif under the common Goodwin formulation requires a high degree of cooperativity (a measure of nonlinearity) in the feedback that is biologically “unlikely.” Moreover, this recurring negative feedback motif is commonly observed augmented by positive feedback interactions. Here we show that these positive feedback interactions promote oscillation at lower degrees of cooperativity, and we can thus unify several common kinetic mechanisms that facilitate oscillations, such as self-activation and Michaelis-Menten degradation. The positive feedback loops are most beneficial when acting on the shortest lived component, where they function by balancing the lifetimes of the different components. The benefits of multiple positive feedback interactions are cumulative for a majority of situations considered, when benefits are measured by the reduction in the cooperativity required to oscillate. These positive feedback motifs also allow oscillations with longer periods than that determined by the lifetimes of the components alone. We can therefore conjecture that these positive feedback loops have evolved to facilitate oscillations at lower, kinetically achievable, degrees of cooperativity. Finally, we discuss the implications of our conclusions on the mammalian molecular clock, a system modeled extensively based on the three-component negative feedback loop.     

Kinetic asymmetry as a key source of functional diversity in biochemical networks

From the analysis of the dynamic properties of various symmetric and asymmetric kinetic schemes, the present report demonstrates that all kinetic schemes, which can be hypothetically divided into two equal halves about an axis of mirror symmetry, are endowed with structural metastability under mass-closed conditions. In mass-closed symmetric schemes, absolute symmetry in reaction conditions in two halves is essential for the occurrence of ordered dynamic behaviour. Even an infinitesimal deviation from the symmetry relations instantaneously drives such systems from limit-cycles to turbulence. Reaction schemes with no axes of symmetry may exhibit a large variety of complex, structurally stable temporal order for wide ranges of values of system parameters and variables. Kinetic asymmetry, therefore, may confer to biochemical networks the functional diversity as well as stability against environmental perturbations.     

Analytical approximations for the amplitude and period of a relaxation oscillator

Background  

Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters.     

Implications of Control Theory for Homeostasis and Phosphorylation of Transport Molecules*

Homeostasis of a cytosolic substrate (e.g. H⁺) can be achieved by transmembrane transport. Control theory implies that the involved activation and deactivation of a transport molecule (as observed in patch clamp experiments) requires input of energy. This energy can be provided by a so-called non-consuming binding site or by other sources and is necessary in order to achieve asymmetric rate-constants required for an efficient integral controller. Another important prediction of control theory is the involvement of two different binding sites for the substrate and its antagonist in order to define a non-zero set-point. The kinetic behavior of a homeostatic feed-back loop is calculated. The system can generate weakly damped oscillations or exponential responses. Numerical calculations using data from the pH-controller of Nitella show that a buffer is necessary in order to explain the observed long periods of 1 hour/cycle. Theoretically, the damping factor can originate from shunting transporters or from special terms of the biochemical reaction schemes. The effect of shunting transport systems is demonstrated in the case of data from the light-effect on H⁺-fluxes in Nitella.     

Bounds for generalized Dreitlein-Smoes models

The analysis of a previous paper obtaining bounds on the total population number of species (chemical or biological) described by the recently proposed Dreitlein-Smoes model of oscillatory kinetic systems, including diffusion, is extended to generalized models of the Dreitlein-Smoes type, describing a system ofS components withS>2. The results for such generalized models are analogous to those of the previous case. It is found that the effects of diffusion serve to restrict the region in the concentration space available to limitcycle type oscillations.     

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.     

Integrating Biosystem Models Using Waveform Relaxation

Modelling in systems biology often involves the integration of component models into larger composite models. How to do this systematically and efficiently is a significant challenge: coupling of components can be unidirectional or bidirectional, and of variable strengths. We adapt the waveform relaxation (WR) method for parallel computation of ODEs as a general methodology for computing systems of linked submodels. Four test cases are presented: (i) a cascade of unidirectionally and bidirectionally coupled harmonic oscillators, (ii) deterministic and stochastic simulations of calcium oscillations, (iii) single cell calcium oscillations showing complex behaviour such as periodic and chaotic bursting, and (iv) a multicellular calcium model for a cell plate of hepatocytes. We conclude that WR provides a flexible means to deal with multitime-scale computation and model heterogeneity. Global solutions over time can be captured independently of the solution techniques for the individual components, which may be distributed in different computing environments.     

Identification of possible two-reactant sources of oscillations in the Calvin photosynthesis cycle and ancillary pathways

A systematic search for possible sources of experimentally observed oscillations in the photosynthetic reaction system has been performed by application of recent theoretical results characterizing the transient-state rate behaviour of metabolic reactions involving two independent concentration variables. All subsystems involving two independent reactants in metabolically fundamental parts of the Calvin cycle and the ancillary pathways of starch and sucrose synthesis have been examined in order to decide on basis of their kinetic and stoichiometric structure whether or not they may trigger oscillations. The results show that no less than 20 possible oscillators can be identified in the examined reaction system, only three of which have been previously considered as potential sources of experimentally observed oscillations. This illustrates the superiority of the method now applied over those previously used to identify possible two-reactant sources of metabolic oscillations and indicates that there should be no difficulty in complex metabolic pathways to point to a multitude of interactions that may trigger an oscillatory rate behaviour of the system.     

The Goodwin Oscillator and its Legacy

In the 1960’s Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin’s work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin’s findings.

     

Modulation analysis of forced non-linear oscillators for biological modelling.

The synchronization of self-sustained oscillations by a forcing oscillation is of interest in a number of biological models. It has been considered for circadian rhythm modelling, heart-rate variability studies and forced breathing experiments. Outside the range of synchronization, conditions of almost-entrainment occur in which changes in amplitude and/or frequency are apparent. It is shown in this paper that such conditions can be analysed as modulation phenomena using the analytical method of harmonic balance. The degree of non-linearity in the self-sustained oscillation affects the nature of modulation, in that increasing distortion gives a trend towards frequency rather than amplitude modulation. The analytical results compare favourably with spectral analysis of simulated oscillators.     

A group-theoretic approach to rings of coupled biological oscillators

In this paper, a general approach for studying rings of coupled biological oscillators is presented. This approach, which is group-theoretic in nature, is based on the finding that symmetric ring networks of coupled non-linear oscillators possess generic patterns of phaselocked oscillations. The associated analysis is independent of the mathematical details of the oscillators' intrinsic dynamics and the nature of the coupling between them. The present approach thus provides a framework for distinguishing universal dynamic behaviour from that which depends upon further structure. In this study, the typical oscillation patterns for the general case of a symmetric ring of n coupled non-linear oscillators and the specific cases of three- and five-membered rings are considered. Transitions between different patterns of activity are modelled as symmetry-breaking bifurcations. The effects of one-way coupling in a ring network and the differences between discrete and continuous systems are discussed. The theoretical predictions for symmetric ring networks are compared with physiological observations and numerical simulations. This comparison is limited to two examples: neuronal networks and mammalian intestinal activity. The implications of the present approach for the development of physiologically meaningful oscillator models are discussed.     

A Structural Classification of Candidate Oscillatory and Multistationary Biochemical Systems

Molecular systems are uncertain: The variability of reaction parameters and the presence of unknown interactions can weaken the predictive capacity of solid mathematical models. However, strong conclusions on the admissible dynamic behaviors of a model can often be achieved without detailed knowledge of its specific parameters. In systems with a sign-definite Jacobian, for instance, cycle-based criteria related to the famous Thomas’ conjectures have been largely used to characterize oscillatory and multistationary dynamic outcomes. We build on the rich literature focused on the identification of potential oscillatory and multistationary behaviors using parameter-free criteria. We propose a classification for sign-definite non-autocatalytic biochemical networks, which summarizes several existing results in the literature. We call weak (strong) candidate oscillators systems which can possibly (exclusively) transition to instability due to the presence of a complex pair of eigenvalues, while we call weak (strong) candidate multistationary systems those which can possibly (exclusively) transition to instability due to the presence of a real eigenvalue. For each category, we provide a characterization based on the exclusive or simultaneous presence of positive and negative cycles in the associated sign graph. Most realistic examples of biochemical networks fall in the gray area of systems in which both positive and negative cycles are present: Therefore, both oscillatory and bistable behaviors are in principle possible. However, many canonical example circuits exhibiting oscillations or bistability fall in the categories of strong candidate oscillators/multistationary systems, in agreement with our results.