A Theoretical Graph Method for Search and Analysis of Critical Phenomena in Biochemical Systems. I. Graphical Rules for Detecting Oscillators

Sufficient conditions for existence of concentration oscillations of components of complex reactions are considered on the basis of graph theory. Graphical rules were developed for detecting oscillators in kinetic schemes of complex biochemical systems. The main types of topology and principles of construction of kinetic schemes of oscillators containing two, three, and four substances are considered. The resulting oscillators might be a part of the biochemical systems of the n-th order. Under certain conditions they were found to be capable of generating oscillations of the system components.     

Rhythms of slow-wave sleep and wakefulness in fluctuations of the potential of the oxidative-reductive status of the cerebral cortex

A method of recording slow ROSP's changes of the brain in chronic animals was used. It has been shown that wakefulness was accompanied by quasisinusoidal oscillations of ROSP (periods--several seconds). During slow sleep, oscillations of ROSP became more complicated and their periods were longer. It is suggested that transitions from sleep to wakefulness and vice versa are connected with removal of maximums of oxidative metabolism tension between biochemical systems characterized by different rhythms of self-regulation. Rhythmic oscillations of ROSP reveal the possibility of separate functional system of the ROSP of the brain cortex to synchronize their oscillations in the brain tissue (biochemical synergism).     

On the dynamics of controlled metabolic network and cellular behavior

The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negetive and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.     

Nonlinear PI controllers for continuous bioreactors using population balance models

Continuous bioreactors are critical unit operations in many biological systems, but the unique modeling is very complicated due to the underlying biochemical reactions and the distributed properties of cell population. The scope of this paper considers a popular modeling method for microbial cell cultures by population balance equation models, and the control objective aims to attenuate undesired oscillations appeared in the nonlinear distributed parameter system. In view of pursuing the popular/practical control configuration and the lack of on-line sensors, an approximate technique by exploiting the “pseudo-steady-state” approach constructs a simple nonlinear control model. Through an off-line estimation mechanism for the system having self-oscillating behavior, two kinds of nonlinear PI configurations are developed. Closed-loop simulation results have confirmed that the regulatory and tracking performances of the control system proposed are good.     

Simple critical schemes in non-autocatalytic systems of biochemical reactions

Kinetic behavior of various reaction systems consisting of three compounds was analyzed in order to find and study the origin of multiple-steady-state and auto-oscillatory behavior in biochemical systems. More than 3000 such reactions were considered. Using the bigraph method, six simple critical reaction schemes were found. It is shown that the presence of these fragments in the composition of more complex biochemical systems may be the origin of the multiple steady-states and autooscillations.     

Unraveling the mechanisms of the vertebrate circadian clock: zebrafish may light the way

Most organisms display oscillations of approximately 24 hours in their physiology. In higher organisms, these circadian oscillations in biochemical and physiological processes ultimately control complex behavioral rhythms that allow an organism to thrive in its natural habitat. Daily and seasonal light cycles are mainly responsible for keeping the circadian system properly aligned with the environment. The molecular mechanisms responsible for the control of the circadian clock have been explored in a number of systems. Interestingly, the circadian oscillations that are responsive to environmental stimuli are present very early during development. This review focuses on the advantages of using the zebrafish to study the development of the vertebrate circadian system and light-dependent signaling to the clock.     

Oscillations in peroxidase-catalyzed reactions and their potential function in vivo

The peroxidase-oxidase reaction has become a model system for the study of oscillations and complex dynamics in biochemical systems. In the present paper we give an overview of previous experimental and theoretical studies of the peroxidase-oxidase reaction. Recent in vitro experiments have raised the question whether the reaction also exhibits oscillations and complex dynamics in vivo. To investigate this possibility further we have undertaken new experimental studies of the reaction, using horseradish extracts and phenols which are widely distributed in plants. The results are discussed in light of the occurrence and a possible functional role of oscillations and complex dynamics of the peroxidase-oxidase reaction in vivo.     

Roles of protein ubiquitination and degradation kinetics in biological oscillations

Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.     

Simple kinetic models explaining critical phenomena in enzymatic reactions with isomerization of the enzyme and substrate

Kinetic models for enzyme reactions are considered which take into account enzyme and substrate isomerization. Application of graph-theoretic methods allows to reveal fragments in schemes which may induce multiple stead-states or concentrational selfoscillations. The role of substrate isomers in the inhibition of enzyme isomers to produce critical phenomena is considered. The boundaries of parameter domains for critical phenomena are estimated. It is shown that the controlled change in concentrations of substrate and enzyme isomers may be important in regulation of enzyme systems, if different enzyme isomers are inhibited mainly by different substrate isomers. The models are used for interpretation of possible critical phenomena in the open reaction catalyzed by lactate dehydrogenase. It is shown that lactate dehydrogenase may act as a trigger in carbohydrate metabolism by changing "critically" its activity in relation to changes in pH and pyruvate fluxes. Slow enzyme inhibition by enolpyruvate is suggested as a possible reason for glycolytic oscillations.     

钙离子振荡对肝糖磷酸化酶激活的随机效应研究

在复杂生化系统的研究过程中,仿真与建模变得越来越重要.对于参与分子数量比较大的生化系统,通常可以采用常微分方程来解决这一问题.对于分子数量比较小的系统,离散粒子基础上的随机模拟方法更精确.然而目前还没有明确的理论方法来确定,对于实际问题用哪种方法能得到更合理的结果.因此需要在一个普遍研究的体系中,通过Ca~(2+)振荡传导信号来研究从随机行为到确定行为的过渡过程.本文以肝细胞中Ca~(2+)振荡对肝糖磷酸化酶激活随机效应为例,讨论了利用随机微分方程来解决分子数量比较小的生化系统的仿真与建模问题,利用细胞内Ca~(2+)有关的Li-Rinzel随机模型,研究了在磷酸化酶降解肝糖的磷酸化-去磷酸化作用循环过程中,三磷酸肌醇受体通道(IP_3R)释放Ca~(2+)的调控作用.结果表明,肝糖磷酸化酶的激活率随受体通道IP_3R的总数增大而减弱,而且三磷酸肌醇浓度比较小时出现相干共振.     

Hormonal regulation of 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase: kinetic models

A graphical method to reveal the so-called 'critical fragments' in schemes of biochemical systems is considered. These fragments produce multiple steady states or self-oscillations in systems. As an example, the bifunctional enzyme, 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase, regulated by glucagon through enzyme phosphorylation, is discussed. It is shown that this enzyme may act as a metabolic switching mechanism in discontinuous or oscillatory regimes, depending on the specific structure of its kinetic scheme. The boundaries of concentrational and parameter domains for these critical phenomena are also predicted.     

State selection in coupled identical biochemical systems with coexisting stable states

This work examines state selection for coupled biochemical systems with coexisting stable states. For biochemically identical biochemical systems, different coupled systems are examined for the coexistence of (a) one steady state and one oscillatory state or (b) two oscillatory states. For case (a), it is revealed that state selection is always governed by two key factors: the values of kinetic parameters and the coupling strength. When the coupling strength is small, the coupled systems remain in the basin of attraction of their original states. When it is sufficiently large, all coupled systems are always entrained, independently of their original states. Furthermore, for the entrainment, which of the two coexisting states is selected depends sensitively on the activity of recycling enzyme (one of kinetic parameters). It is shown that this is because changing the activity of recycling enzyme alters the size of basin of attraction of each state. When both systems in the same oscillatory state are coupled, an additional factor, namely phase shift between two oscillations, may also affect state selection, and coupling may cause the systems to select either the original oscillatory state or the coexisting steady state. In addition to the features of case (a), case (b) also supports quasiperiodic oscillations and synchronisation of two periodic oscillations. Implications of the results for understanding state selection during the evolution of coupled biochemical systems with coexisting stable states are discussed.     

Is a constant low-entropy process at the root of glycolytic oscillations?

We measured temporal oscillations in thermodynamic variables such as temperature, heat flux, and cellular volume in suspensions of non-dividing yeast cells which exhibit temporal glycolytic oscillations. Oscillations in these variables have the same frequency as oscillations in the activity of intracellular metabolites, suggesting strong coupling between them. These results can be interpreted in light of a recently proposed theoretical formalism in which isentropic thermodynamic systems can display coupled oscillations in all extensive and intensive variables, reminiscent of adiabatic waves. This interpretation suggests that oscillations may be a consequence of the requirement of living cells for a constant low-entropy state while simultaneously performing biochemical transformations, i.e., remaining metabolically active. This hypothesis, which is in line with the view of the cellular interior as a highly structured and near equilibrium system where energy inputs can be low and sustain regular oscillatory regimes, calls into question the notion that metabolic processes are essentially dissipative.     

Mechanogenetic Coupling of Hydra Symmetry Breaking and Driven Turing Instability Model

The freshwater polyp Hydra can regenerate from tissue fragments or random cell aggregates. We show that the axis-defining step (“symmetry breaking”) of regeneration requires mechanical inflation-collapse oscillations of the initial cell ball. We present experimental evidence that axis definition is retarded if these oscillations are slowed down mechanically. When biochemical signaling related to axis formation is perturbed, the oscillation phase is extended and axis formation is retarded as well. We suggest that mechanical oscillations play a triggering role in axis definition. We extend earlier reaction-diffusion models for Hydra regrowth by coupling morphogen transport to mechanical stress caused by the oscillations. The modified reaction-diffusion model reproduces well two important experimental observations: 1), the existence of an optimum size for regeneration, and 2), the dependence of the symmetry breaking time on the properties of the mechanical oscillations.     

Hysteresis,oscillations, and pattern formation in realistic immobilized enzyme systems

Summary Hysteresis, oscillations, and pattern formation in realistic biochemical systems governed by P.D.E.s are considered from both numerical and mathematical points of view. Analysis of multiple steady states in the case of hysteresis, and bifurcation theory in the cases of oscillations and pattern formation, account for the observed numerical results. The possibility to realize these systems experimentally is their main interest, thus bringing further arguments in favor of theories explaining basic biological phenomena by diffusion and reaction.     

Allosteric regulation, cooperativity, and biochemical oscillations

Allosteric regulation is associated with a number of periodic phenomena in biochemical systems. The cooperative nature of such regulatory interactions provides a source of nonlinearity that favors oscillatory behavior. We assess the role of cooperativity in the onset of biochemical oscillations by analyzing two specific examples. First, we consider a model for a product-activated allosteric enzyme which has previously been proposed to account for glycolytic oscillations. While enzyme cooperativity plays an important role in the occurrence of oscillations, we show that these may nevertheless occur in the absence of cooperativity when the reaction product is removed in a Michaelian rather than linear manner. The second model considered was recently proposed to account for signal-induced oscillations of intracellular calcium. This phenomenon originates from a nonlinear process of calcium-induced calcium release. Here also, the cooperative nature of that positive feedback favors the occurrence of oscillations but is not absolutely required for periodic behavior. Besides underlining the importance of cooperativity, the results highlight the role of diffuse nonlinearities distributed over several steps within a regulated system: even in the absence of cooperativity, such mild nonlinearities (e.g., of the Michaelian type) may combine to raise the overall degree of nonlinearity up to the level required for oscillations.     

A class of parametrically excited calcium oscillation detectors.

Intracellular Ca2+ oscillations are often a response to external signals such as hormones. Changes in the external signal can alter the frequency, amplitude, or form of the oscillations suggesting that information is encoded in the pattern of Ca2+ oscillations. How might a cell decode this signal? We show that an excitable system whose kinetic parameters are modulated by the Ca2+ concentration can function as a Ca2+ oscillation detector. Such systems have the following properties: (1) They are more sensitive to an oscillatory than to a steady Ca2+ signal. (2) Their response is largely independent of the signal amplitude. (3) They can extract information from a noisy signal. (4) Unlike other frequency sensitive detectors, they have a flat frequency response. These properties make a Ca(2+)-sensitive excitable system nearly ideal for detecting and decoding Ca2+ oscillations. We suggest that Ca2+ oscillations, in concert with these detectors, can act as cellular timekeepers to coordinate related biochemical reactions and enhance their overall efficiency.     

Communication and control in reproduction: the ubiquity of periodic phenomena

Two statements will be presented and defended in this paper. First, it is claimed that oscillations are common in all biological systems. From the subcellular to the organismic levels of organization, oscillations are the normal operational mode for many biochemical and physiological control networks. Second, it will be demonstrated that periodic control offers several functional advantages over steady-state control.     

Transition from stochastic to deterministic behavior in calcium oscillations

Simulation and modeling is becoming more and more important when studying complex biochemical systems. Most often, ordinary differential equations are employed for this purpose. However, these are only applicable when the numbers of participating molecules in the biochemical systems are large enough to be treated as concentrations. For smaller systems, stochastic simulations on discrete particle basis are more accurate. Unfortunately, there are no general rules for determining which method should be employed for exactly which problem to get the most realistic result. Therefore, we study the transition from stochastic to deterministic behavior in a widely studied system, namely the signal transduction via calcium, especially calcium oscillations. We observe that the transition occurs within a range of particle numbers, which roughly corresponds to the number of receptors and channels in the cell, and depends heavily on the attractive properties of the phase space of the respective systems dynamics. We conclude that the attractive properties of a system, expressed, e.g., by the divergence of the system, are a good measure for determining which simulation algorithm is appropriate in terms of speed and realism.     

Oscillatory behavior of a simple kinetic model for proteolysis during cell invasion.

Extracellular proteolysis during cell invasion is thought to be tightly organized, both temporally and spatially. This work presents a simple kinetic model that describes the interactions between extracellular matrix (ECM) proteins, proteinases, proteolytic fragments, and integrins. Nonmonotonous behavior arises from enzyme de novo synthesis consecutive to integrin binding to fragments or entire proteins. The model has been simulated using realistic values for kinetic constants and protein concentrations, with fibronectin as the ECM protein. The simulations show damped oscillations of integrin-complex concentrations, indicating alternation of maximal adhesion periods with maximal mobility periods. Comparisons with experimental data from the literature confirm the similarity between this system behavior and cell invasion. The influences on the system of cryptic functions of ECM proteins, proteinase inhibitors, and soluble antiadhesive peptides were examined. The first critical parameter for oscillation is the discrepancy between integrin affinity for intact ECM proteins and the respective proteolytic fragments, thus emphasizing the importance of cryptic functions of ECM proteins in cell invasion. Another critical parameter is the ratio between proteinase and the initial ECM protein concentration. These results suggest new insights into the organization of the ECM degradation during cell invasion.