Oscillatory behavior of Saccharomyces cerevisiae in continuous culture: I. Effects of pH and nitrogen levels

The appearance of sustained oscillations in bioreactor variables (biomass and nutrient concentrations) in continuous cultures of Saccharomyces cerevisiae indicates the complex nature of microbial systems, the inadequacy of current growth kinetic models, and the difficulties which may arise in bioprocess control and optimization. In this study we investigate continuous bioreactor behavior over a range of operating conditions (dilution rate, feed glucose concentration, feed ammonium concentration, dissolved oxygen, and pH) to determine the process requirements which lead to oscillatory behavior. We present new results which indicate that high feed ammonium concentrations may eliminate oscillations and that under oscillatory conditions ammonium levels are generally low and oscillatory as well. The effects of pH are complex and oscillations were only observed at pH values 5.5 and 6.5; no oscillations were observed at a pH of 4.5. Under our nominal operating conditions (feed glucose concentration 10 g/L, dilution rate 0.145 h(-1), feed ammonium concentration 0.0303M, dissolved oxygen level 50%, pH 5.5, and T = 30 degrees C) we found two possible final bioreactor states depending on the transient used to reach the nominal operating conditions. One of the states was oscillatory and characteristic of oxidative metabolism and the other was nonoscillatory and fermentative.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.     

RhoA Regulates Calcium-Independent Periodic Contractions of the Cell Cortex

When microtubules are depolymerized in spreading cells, they experience morphological oscillations characterized by a period of about a minute, indicating that normal interactions between the microfilament and microtubule systems have been significantly altered. This experimental system provides a test bed for the development of both fine- and coarse-grained models of complex motile processes, but such models need to be adequately informed by experiment. Using criteria based on Fourier transform analysis, we detect spontaneous oscillations in spreading cells. However, their amplitude and tendency to operate at a single frequency are greatly enhanced by microtubule depolymerization. Knockdown of RhoA and addition of various inhibitors of the downstream effector of RhoA, Rho kinase, block oscillatory behavior. Inhibiting calcium fluxes from endoplasmic reticulum stores and from the extracellular medium does not significantly affect the ability of cells to oscillate, indicating that calcium plays a subordinate regulatory role compared to Rho. We characterized the dynamic structure of the oscillating cell by light, fluorescence, and electron microscopy, showing how oscillating cells are dynamically polarized in terms of their overall morphology, f-actin and phosphorylated myosin light chain distribution, and nuclear position and shape. Not only will these studies guide future experiments, they will also provide a framework for the development of refined mathematical models of the oscillatory process.     

Cyclic Negative Feedback Systems: What is the Chance of Oscillation?

Many biological oscillators have a cyclic structure consisting of negative feedback loops. In this paper, we analyze the impact that the addition of a positive or a negative self-feedback loop has on the oscillatory behavior of the three negative feedback oscillators proposed by Tsai et al. (Science 231:126–129, 2008) where, in contrast with numerous oscillator models, the interactions between elements occur via the modulation of the degradation rates. Through analytical and computational studies we show that an additional self-feedback affects the oscillatory behavior. In the high-cooperativity limit, i.e., for large Hill coefficients, we derive exact analytical conditions for oscillations and show that the relative location between the dissociation constants of the Hill functions and the ratio of kinetic parameters determines the possibility of oscillatory activities. We compute analytically the probability of oscillations for the three models and show that the smallest domain of periodic behavior is obtained for the negative-plus-negative feedback system whereas the additional positive self-feedback loop does not modify significantly the chance to oscillate. We numerically investigate to what extent the properties obtained in the sharp situation applied in the smooth case. Results suggest that a switch-like coupling behavior, a time-scale separation, and a repressilator-type architecture with an even number of elements facilitate the emergence of sustained oscillations in biological systems. An additional positive self-feedback loop produces robustness and adaptability whereas an additional negative self-feedback loop reduces the chance to oscillate.     

Immune network behavior—II. From oscillations to chaos and stationary states

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.     

Influence of parameter values on the oscillation sensitivities of two p53–Mdm2 models

Biomolecular networks that present oscillatory behavior are ubiquitous in nature. While some design principles for robust oscillations have been identified, it is not well understood how these oscillations are affected when the kinetic parameters are constantly changing or are not precisely known, as often occurs in cellular environments. Many models of diverse complexity level, for systems such as circadian rhythms, cell cycle or the p53 network, have been proposed. Here we assess the influence of hundreds of different parameter sets on the sensitivities of two configurations of a well-known oscillatory system, the p53 core network. We show that, for both models and all parameter sets, the parameter related to the p53 positive feedback, i.e. self-promotion, is the only one that presents sizeable sensitivities on extrema, periods and delay. Moreover, varying the parameter set values to change the dynamical characteristics of the response is more restricted in the simple model, whereas the complex model shows greater tunability. These results highlight the importance of the presence of specific network patterns, in addition to the role of parameter values, when we want to characterize oscillatory biochemical systems.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9173-y) contains supplementary material, which is available to authorized users.     

Quantitative assessment of regulation in metabolic systems

We show how metabolic regulation as commonly understood in biochemistry can be described in terms of metabolic control analysis. The steady-state values of the variables of metabolic systems (fluxes and concentrations) are determined by a set of parameters. Some of these parameters are concentrations that are set by the environment of the system; they can act as external regulators by communicating changes in the environment to the metabolic system. How effectively a system is regulated depends both on the degree to which the activity of the regulatory enzyme with which a regulator interacts directly can be altered by the regulator (its regulability) and on the ability of the regulatory enzyme to transmit the changes to the rest of the system (its regulatory capacity). The regulatory response of a system also depends on its internal organisation around key variable metabolites that act as internal regulators. The regulatory performance of the system can be judged in terms of how sensitivity the fluxes respond to the external stimulus and to what degree homeostasis in the concentrations of the internal regulators is maintained. We show how, on the level of both external and internal regulation, regulability can be quantified in terms of an elasticity coefficient and regulatory capacity in terms of a control coefficient. Metabolic regulation can therefore be described in terms of metabolic control analysis. The combined response relationship of control analysis relates regulability and regulatory capacity and allows quantification of the regulatory importance of the various interactions of regulators with enzymes in the system. On this basis we propose a quantitative terminology and analysis of metabolic regulation that shows what we should measure experimentally and how we should interpret the results. Analysis and numerical simulation of a simple model system serves to demonstrate our treatment.     

Improved Estimation of Human Lipoprotein Kinetics with Mixed Effects Models

Context

Mathematical models may help the analysis of biological systems by providing estimates of otherwise un-measurable quantities such as concentrations and fluxes. The variability in such systems makes it difficult to translate individual characteristics to group behavior. Mixed effects models offer a tool to simultaneously assess individual and population behavior from experimental data. Lipoproteins and plasma lipids are key mediators for cardiovascular disease in metabolic disorders such as diabetes mellitus type 2. By the use of mathematical models and tracer experiments fluxes and production rates of lipoproteins may be estimated.

Results

We developed a mixed effects model to study lipoprotein kinetics in a data set of 15 healthy individuals and 15 patients with type 2 diabetes. We compare the traditional and the mixed effects approach in terms of group estimates at various sample and data set sizes.

Conclusion

We conclude that the mixed effects approach provided better estimates using the full data set as well as with both sparse and truncated data sets. Sample size estimates showed that to compare lipoprotein secretion the mixed effects approach needed almost half the sample size as the traditional method.     

In vitro DNA-based predator-prey system with oscillatory kinetics

A coupled system of two isothermal in vitro DNA/RNA amplification reactions using different primers is modeled kinetically with realistic rate parameters and shown to exhibit oscillatory behavior in a flow reactor. One of the two isothermal amplification reactions acts as a predator of the other, the prey. The mechanism of the oscillatory behavior is analyzed in terms of a hierarchy of kinetic models. The work provides an insight into the choice of parameters for experiments. The latter are important in providing detailed insight into the complex processes of ecological interactions and their evolution.     

Modeling of signaling networks

Biochemical networks, including those containing signaling pathways, display a wide range of regulatory properties. These include the ability to propagate information across different time scales and to function as switches and oscillators. The mechanisms underlying these complex behaviors involve many interacting components and cannot be understood by experiments alone. The development of computational models and the integration of these models with experiments provide valuable insight into these complex systems-level behaviors. Here we review current approaches to the development of computational models of biochemical networks and describe the insights gained from models that integrate experimental data, using three examples that deal with ultrasensitivity, flexible bistability and oscillatory behavior. These types of complex behavior from relatively simple networks highlight the necessity of using theoretical approaches in understanding higher order biological functions.     

Conservation of glycolytic oscillations in Saccharomyces cerevisiae and human pancreatic beta-cells: a study of metabolic robustness

The present study compares two computer models of the first part of glucose catabolism in different organisms in search of evolutionarily conserved characteristics of the glycolysis cycle and proposes the main parameters that define the stable steady-state or oscillatory behavior of the glycolytic system. It is suggested that in both human pancreatic beta-cells and Saccharomyces cerevisiae there are oscillations that, despite differences in wave form and period of oscillation, share the same robustness strategy: the oscillation is not controlled by only one but by at least two parameters that will have more or less control over the pathway flux depending on the initial state of the system as well as on extra-cellular conditions. This observation leads to two important interpretations: the first is that in both S. cerevisiae and human beta-cells, despite differences in enzyme kinetics and mechanism of feedback control, evolution seems to have kept an oscillatory behavior coupled to the glucose concentration outside the cytoplasm, and the second is that the development of drugs to regulate metabolic dysfunctions in more complex systems may require further study, not only determining which enzyme is controlling the flux of the system but also under which conditions and how its control is maintained by the enzyme or transferred to other enzymes in the pathway as the drug starts acting.     

The independence principle. A reconsideration

The electrodiffusion model presented in the previous paper, which specifically excludes ion-ion interactions, is analyzed for the ratio of one-way fluxes (flux ratio) as a function of the ionic driving force across the membrane. Significant deviations from the behavior expected on the basis of the Ussing relation are found. These are sufficient to explain the “nonindependent” ion movement noted in some biological flux ratio data. One-way fluxes are dependent on the ionic concentration on both sides of the membrane. The coupling of these fluxes to ionic concentrations comes from the dependence of ionic mobility and the diffusion coefficient on the equilibrium potential. It is concluded that nonindependent behavior in experimental data is not sufficient to implicate ion-ion interaction as the source of the discrepancy.     

Static and dynamic behavior of a class of unstructured models of continuous bioreactors with growth associated product

The static and dynamic behavior of a class of unstructured models of continuous bioprocesses, for which the product is growth associated, are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which both the rates of utilization of limiting substrate and product formation are linearly proportional to the specific cell growth rate. The kinetic expressions are allowed to assume general forms of substrate and nonbiomass product. The steady-state analysis allows the derivation of analytical results and the construction of a useful picture in the models' parameter space delineating the different static behavior these models can predict, including unique steady states and bistability. The analysis of the dynamic behavior allows the derivation of general analytical conditions for the occurrence of periodic behavior in the models. It is also shown that the subclass of these models for which the specific cell growth rate expression is monotonic with respect to the nonbiomass product is unable to predict a stable oscillatory behavior regardless of the expression of the growth rate. These results illustrate the fundamental weakness of this class of unstructured models in predicting transient behavior in continuous cultures. The effect of kinetic and operating parameters on the stability characteristics of these models is also investigated.     

Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics

This paper examines the validity of the linlog approach, which was recently developed in our laboratory, by comparison of two different kinetic models for the metabolic network of Escherichia coli. The first model is a complete mechanistic model; the second is an approximative model in which linlog kinetics are applied. The parameters of the linlog model (elasticities) are derived from the mechanistic model. Three different optimization cases are examined. In all cases, the objective is to calculate the enzyme levels that maximize a certain flux while keeping the total amount of enzyme constant and preventing large changes of metabolite concentrations. For an average variation of metabolite levels of 10% and individual changes of a factor 2, the predicted enzyme levels, metabolite concentrations and fluxes of both models are highly similar. This similarity holds for changes in enzyme level of a factor 4-6 and for changes in fluxes up to a factor 6. In all three cases, the predicted optimal enzyme levels could neither have been found by intuition-based approaches, nor on basis of flux control coefficients. This demonstrates that kinetic models are essential tools in Metabolic Engineering. In this respect, the linlog approach is a valuable extension of MCA, since it allows construction of kinetic models, based on MCA parameters, that can be used for constrained optimization problems and are valid for large changes of metabolite and enzyme levels.     

Cytosolic potassium homeostasis revisited: <Superscript>42</Superscript>K-tracer analysis in<Emphasis Type="Italic"> Hordeum vulgare</Emphasis> L. reveals set-point variations in [K<Superscript>+</Superscript>]

Current models of potassium acquisition and cytochemical processes in plants assume that potassium concentrations in the cytosol ([K+]cyt) are maintained homeostatically at approximately 100 mM. Here, we use 42K radiotracer data in the model plant species Hordeum vulgare L. (barley) to show that this assumption is incorrect. Our study reveals that [K+]cyt in root cells of intact barley seedlings is held at a minimum of two physiological set points, coinciding with two fundamentally distinct modes of K+ transport, each of which is characterized by a unique network of fluxes to and from the cytosol, and reflects variations in mechanisms and energetics of K+ transport, cytosolic K+ turnover, flux partitioning, and sensitivity to NH4+. Increased external potassium or ammonium concentrations caused a substantial drop in [K+]cyt, as well as a switch from a transport mode dominated by high-affinity, energy-dependent, influx to a mode dominated by channel-mediated fluxes in both directions across the plasma membrane. Our study provides the first subcellular demonstration of the flexibility, rather than strict homeostasis, of cellular K+ maintenance, and of the dynamic interaction between plant membrane fluxes of the two major nutrient cations K+ and NH4+.     

Mechanobiology in cortical waves and oscillations

Cortical actin waves have emerged as a widely prevalent phenomena and brought pattern formation to many fields of cell biology. Cortical excitabilities, reminiscent of the electric excitability in neurons, are likely fundamental property of the cell cortex. Although they have been mostly considered to be biochemical in nature, accumulating evidence support the role of mechanics in the pattern formation process. Both pattern formation and mechanobiology approach biological phenomena at the collective level, either by looking at the mesoscale dynamical behavior of molecular networks or by using collective physical properties to characterize biological systems. As such they are very different from the traditional reductionist, bottom-up view of biology, which brings new challenges and potential opportunities. In this essay, we aim to provide our perspectives on what the proposed mechanochemical feedbacks are and open questions regarding their role in cortical excitable and oscillatory dynamics.     

Classical versus stochastic kinetics modeling of biochemical reaction systems

We study fundamental relationships between classical and stochastic chemical kinetics for general biochemical systems with elementary reactions. Analytical and numerical investigations show that intrinsic fluctuations may qualitatively and quantitatively affect both transient and stationary system behavior. Thus, we provide a theoretical understanding of the role that intrinsic fluctuations may play in inducing biochemical function. The mean concentration dynamics are governed by differential equations that are similar to the ones of classical chemical kinetics, expressed in terms of the stoichiometry matrix and time-dependent fluxes. However, each flux is decomposed into a macroscopic term, which accounts for the effect of mean reactant concentrations on the rate of product synthesis, and a mesoscopic term, which accounts for the effect of statistical correlations among interacting reactions. We demonstrate that the ability of a model to account for phenomena induced by intrinsic fluctuations may be seriously compromised if we do not include the mesoscopic fluxes. Unfortunately, computation of fluxes and mean concentration dynamics requires intensive Monte Carlo simulation. To circumvent the computational expense, we employ a moment closure scheme, which leads to differential equations that can be solved by standard numerical techniques to obtain more accurate approximations of fluxes and mean concentration dynamics than the ones obtained with the classical approach.     

Oscillations in plant membrane transport: model predictions, experimental validation, and physiological implications

Although oscillations in membrane-transport activity are ubiquitous in plants, the ionic mechanisms of ultradian oscillations in plant cells remain largely unknown, despite much phenomenological data. The physiological role of such oscillations is also the subject of much speculation. Over the last decade, much experimental evidence showing oscillations in net ion fluxes across the plasma membrane of plant cells has been accumulated using the non-invasive MIFE technique. In this study, a recently proposed feedback-controlled oscillatory model was used. The model adequately describes the observed ion flux oscillations within the minute range of periods and predicts: (i) strong dependence of the period of oscillations on the rate constants for the H+ pump; (ii) a substantial phase shift between oscillations in net H+ and K+ fluxes; (iii) cessation of oscillations when H+ pump activity is suppressed; (iv) the existence of some 'window' of external temperatures and ionic concentrations, where non-damped oscillations are observed: outside this range, even small changes in external parameters lead to progressive damping and aperiodic behaviour; (v) frequency encoding of environmental information by oscillatory patterns; and (vi) strong dependence of oscillatory characteristics on cell size. All these predictions were successfully confirmed by direct experimental observations, when net ion fluxes were measured from root and leaf tissues of various plant species, or from single cells. Because oscillatory behaviour is inherent in feedback control systems having phase shifts, it is argued from this model that suitable conditions will allow oscillations in any cell or tissue. The possible physiological role of such oscillations is discussed in the context of plant adaptive responses to salinity, temperature, osmotic, hypoxia, and pH stresses.