Metabolic modeling of Saccharomyces cerevisiae using the optimal control of homeostasis: a cybernetic model definition

A model is presented to describe the observed behavior of microorganisms that aim at metabolic homeostasis while growing and adapting to their environment in an optimal way. The cellular metabolism is seen as a network with a multiple controller system with both feedback and feedforward control, i.e., a model based on a dynamic optimal metabolic control. The dynamic network consists of aggregated pathways, each having a control setpoint for the metabolic states at a given growth rate. This set of strategies of the cell forms a true cybernetic model with a minimal number of assumptions. The cellular strategies and constraints were derived from metabolic flux analysis using an identified, biochemically relevant, stoichiometry matrix derived from experimental data on the cellular composition of continuous cultures of Saccharomyces cerevisiae. Based on these data a cybernetic model was developed to study its dynamic behavior. The growth rate of the cell is determined by the structural compounds and fluxes of compounds related to central metabolism. In contrast to many other cybernetic models, the minimal model does not consist of any assumed internal kinetic parameters or interactions. This necessitates the use of a stepwise integration with an optimization of the fluxes at every time interval. Some examples of the behavior of this model are given with respect to steady states and pulse responses. This model is very suitable for describing semiquantitatively dynamics of global cellular metabolism and may form a useful framework for including structured and more detailed kinetic models.     

A simultaneous saccharification and fermentation model for dynamic growth environments

Many mathematical models by researchers have been formulated for Saccharomyces cerevisiae which is the common yeast strain used in modern distilleries. A cybernetic model that can account for varying concentrations of glucose, ethanol and organic acids on yeast cell growth dynamics does not exist. A cybernetic model, consisting of 4 reactions and 11 metabolites simulating yeast metabolism, was developed. The effects of variables such as temperature, pH, organic acids, initial inoculum levels and initial glucose concentration were incorporated into the model. Further, substrate and product inhibitions were included. The model simulations over a range of variables agreed with hypothesized trends and to observations from other researchers. Simulations converged to expected results and exhibited continuity in predictions for all ranges of variables simulated. The cybernetic model did not exhibit instability under any conditions simulated.     

A simple structured model describing the growth of Streptomyces lividans

The growth of Streptomyces lividans in defined media was modeled using a simple structured growth model. Conventional unstructured models like Monod kinetics, substrate inhibition kinetics, and the logistic equation were also used in an attempt to fit the data, but the results were all unsatisfactory. The main reason for failure in applying simple unstructured models is that they cannot describe the long lag phases sometimes observed during growth of S. lividans. The simple structured growth model was derived along similar principles to cybernetic growth models. This model quite accurately describes the growth of S. lividans. It assumes that the rate of assimilation of a substrate depends on the concentration of a specific key enzyme. This key enzyme is only produced in the presence of the substrate, and it is broken down at a steady rate. An enzyme synthesis allocation variable, w, similar to the cybernetic variable, u, described in cybernetic growth models, is proposed to control enzyme synthesis. Until the key enzyme concentration approaches its maximum level, very little substrate is consumed. And consequently, the lag phase is sustained.     

Unveiling steady-state multiplicity in hybridoma cultures: the cybernetic approach

Mammalian cells grown in suspension produce waste metabolites such as lactate, alanine, and ammonia, which reduce the yield of cell mass and the desired product on the nutrients supplied. Previous studies (Cruz et al., 1999; Europa et al., 2000; Follstad et al., 1999) have shown that the cells can be made to alter their metabolism by starving them on their nutrients in continuous cultures at low dilution rates or starting the culture as a fed-batch. This leads to multiple steady states in continuous reactors, with some states being more favorable than others. Mathematical models that take into account the metabolic regulation that leads to these multiple steady states are invaluable tools for bioreactor control. In this article we present a cybernetic modeling strategy in which Metabolic Flux Analysis (MFA) is used to guide the cybernetic formulation. The hybridoma model presented as a result of this strategy considers the partially substitutable, partially complementary nature of glucose and glutamine. The choice of competitions within the network is guided by MFA and the model is successful in explaining the three multiple steady states observed. The cybernetic model though identified for the hybridoma experiments of Hu and others (Europa et al., 2000) seem generally applicable to mammalian systems as it captures the pathways that are common to mammalian cells grown in suspension. The model presented here could be used for start-up strategies for continuous reactors and model-based feedback control for maintaining high productivity of the reactor.     

Integrating cybernetic modeling with pathway analysis provides a dynamic, systems-level description of metabolic control

Cybernetic modeling strives to uncover the inbuilt regulatory programs of biological systems and leverage them toward computational prediction of metabolic dynamics. Because of its focus on incorporating the global aims of metabolism, cybernetic modeling provides a systems-oriented approach for describing regulatory inputs and inferring the impact of regulation within biochemical networks. Combining cybernetic control laws with concepts from metabolic pathway analysis has culminated in a systematic strategy for constructing cybernetic models, which was previously lacking. The newly devised framework relies upon the simultaneous application of local controls that maximize the net flux through each elementary flux mode and global controls that modulate the activities of these modes to optimize the overall nutritional state of the cell. The modeling concepts are illustrated using a simple linear pathway and a larger network representing anaerobic E. coli central metabolism. The E. coli model successfully describes the metabolic shift that occurs upon deleting the pta-ackA operon that is responsible for fermentative acetate production. The model also furnishes predictions that are consistent with experimental results obtained from additional knockout strains as well as strains expressing heterologous genes. Because of the stabilizing influence of the included control variables, the resulting cybernetic models are more robust and reliable than their predecessors in simulating the network response to imposed genetic and environmental perturbations.     

Revised enzyme synthesis rate expression in cybernetic models of bacterial growth

A revised enzyme synthesis rate expression for cybernetic models of bacterial growth is presented. The rate expression, which is comprised of inducible and constitutive contributions, provides for a basal enzyme level that is necessary to predict certain types of commonly observed continuous culture transients. The response of a continuous culture to a step change in feed stream composition is simulated using both the old and new formulations, and the ramifications for the "matching-law" formulation are discussed.     

Cybernetic modelling of growth and ethanol production in a recombinant Saccharomyces cerevisiae strain secreting a bifunctional fusion protein

Recombinant Saccharomyces cerevisiae YPB-G strain secreting a fusion protein displaying both BsAAase/GAase activities was grown in 1.5 l YPS media containing single (starch) and mixed carbon sources (glucose+starch) using a 2.5 l New Brunswick BiofloIII fermenter. Ethanol and biomass formation, starch utilisation, secretion of the amylolytic enzymes (-amylase and glucoamylase), accumulation of reducing sugars and glucose were followed during the fermentation of YPB-G under different conditions. Moreover, a model has been developed for the growth of recombinant yeast on substitutable substrates using cybernetic framework principles and incorporating product formation. In the present work, both the biphasic and the diauxic growth patterns observed experimentally in batch culture of recombinant yeast cells were simulated successfully by modifying the cybernetic framework to include ethanol formation and the degradation kinetics of starch which is not directly utilised by yeast. The model can further be expanded to fed-batch systems.     

Simulation of cell behavior: Normal and abnormal growth

A mathematical model simulating a cell growing in a culture medium is obtained. Using this model, various behavioral patterns of the cell are obtained under different types of disturbances, in particular when (i) a Mg2+ deficiency experiment and, (ii) a split-dose ionizing radiation experiment are carried out, (iii) when disturbances on the rate constants of the biochemical reactions taking place in the nucleus of the cell are applied, and (iv) when the cell's interior components are perturbed. The cell model results obtained agree well with experimental results for the Mg2+ and split dose experiments, and explain the mechanism of the split dose radiation experiment without the need to introduce additional axioms (e.g. healing processes) into the dynamics of the cell. Conditions are obtained which cause the cell to behave in a rapidly growing 'tumor-like' mode; it is shown that once the cell moves into this 'tumor-like; mode, its behavior is irreversible, i.e. if a disturbance of opposite type is then applied to the 'tumor cell, the cell will not revert back to its original normal behavior.     

On enhancing productivity of bioethanol with multiple species

The present work is initiated to investigate whether a defined culture comprising a mixture of three yeast species, Kluyveromyces marxianus, Saccharomyces cerevisiae, and Pichia stipitis can ferment a mixture of sugars to produce bioethanol at rates higher than those achieved by pure cultures of the same. For this purpose, we develop models of single species based on the hybrid cybernetic model framework, and simulate fermentations in the mixed culture by combining individual models. An underlying assumption is that the behavior of each species is determined only by the common environment independently of the presence and metabolism of other species. Model performance is thoroughly assessed using the experimental data available in the literature. The dynamic behavior of mixed cultures in mixed culture experiments are accurately predicted by the model reflecting faithfully the simultaneous/sequential uptake patterns of mixed substrates. This model is then used to investigate performance of various possible reactor configurations. With the foregoing species of organisms, mixed culture itself does not lead to a significant increase of bioethanol productivity. Rather, the model shows that substantial improvement is acquired by sequential use of different, properly chosen organisms during fermentation. Thus, the successive use of K. marxianus and P. stipitis is shown to increase bioethanol productivity up to about 58% in comparison to fermentation by single species alone.     

The re-incarnation, re-interpretation and re-demise of the transition probability model.

There are two classes of models for the cell cycle that have both a deterministic and a stochastic part; they are the transition probability (TP) models and sloppy size control (SSC) models. The hallmark of the basic TP model are two graphs: the alpha and beta plots. The former is the semi-logarithmic plot of the percentage of cell divisions yet to occur, this results in a horizontal line segment at 100% corresponding to the deterministic phase and a straight line sloping tail corresponding to the stochastic part. The beta plot concerns the differences of the age-at-division of sisters (the beta curve) and gives a straight line parallel to the tail of the alpha curve. For the SC models the deterministic part is the time needed for the cell to accumulate a critical amount of some substance(s). The variable part differs in the various variants of the general model, but they do not give alpha and beta curves with linear tails as postulated by the TP model. This paper argues against TP and for an elaboration of SSC type of model. The main argument against TP is that it assumes that the probability of the transition from the stochastic phase is time invariant even though it is certain that the cells are growing and metabolizing throughout the cell cycle; a fact that should make the transition probability be variable. The SSC models presume that cell division is triggered by the cell's success in growing and not simply the result of elapsed time. The extended model proposed here to accommodate the predictions of the SSC to the straight tailed parts of the alpha and beta plots depends on the existence of a few percent of the cell in a growing culture that are not growing normally, these are growing much slower or are temporarily quiescent. The bulk of the cells, however, grow nearly exponentially. Evidence for a slow growing component comes from experimental analyses of population size distributions for a variety of cell types by the Collins-Richmond technique. These subpopulations existence is consistent with the new concept that there are a large class of rapidly reversible mutations occurring in many organisms and at many loci serving a large range of purposes to enable the cell to survive environmental challenges. These mutations yield special subpopulations of cells within a population. The reversible mutational changes, relevant to the elaboration of SSC models, produce slow-growing cells that are either very large or very small in size; these later revert to normal growth and division. The subpopulations, however, distort the population distribution in such a way as to fit better the exponential tails of the alpha and beta curves of the TP model.     

Metabolic engineering from a cybernetic perspective: aspartate family of amino acids

Using the modular cybernetic framework developed by Varner and Ramkrishna (Varner and Ramkrishna; 1998a, b) a cybernetic model is formulated that describes the time evolution of the aspartate family of amino acids in Corynebacterium lactofermentum ATCC 21799. The network model formulation is employed in the role of a diagnostic tool for the overproduction of threonine. More precisely, having determined a parameter set that describes the time evolution of a base strain (lysine producer), the model predicted response to genetic perturbations, designed to enhance the level of threonine, are simulated using an appropriately modified cybernetic model and compared with the experimental results of Stephanopoulos and Sinskey (Colón et al., 1995a, Appl. Environ. Microbiol. 61, 74-78) for identical genetic perturbations. It is found that the model predicted response to enzymatic over-expression in the aspartate pathway agrees, for the most part, with experimental observations within the experimental error bounds. This result lends credence to the hypothesis that cybernetic models can be employed to predict the local response of a metabolic network to genetic perturbation, thereby, affording cognizance of the potential pitfalls of a particular genetic alteration strategy a priori.     

Metabolic regulation in bacterial continuous cultures: II

The transient behavior of a continuous culture of Klebsiella pneumoniae with mixed feed of glucose and xylose arising from step-up and step-down in dilution rates and from a feed-switching experiment is presented. he organism gradually switches from simultaneous utilization of the substrates at low growth rates to preferred utilization of the faster substrate (i.e, supporting a higher growth rate) at high dilution rates. The metabolic lags following a step increase in dilution rate and a significant accumulation of the slower substrate during the transient period result from the effects of metabolic regulation. The cybernetic modeling approach that successfully described the foregoing situations with single-substrate feeds is employed to describe mixed substrate behavior. The parameters in the mixed-substrate (glucose and xylose) model are the same as those in the single-substrate models with the singular exception of the rate constant for the xylose growth enzyme synthesis. The reason for this discrepancy is discussed in detail. It appears that the constitutive rate of enzyme synthesis for growth on a given substrate may be related to the past history of the organism in regard to whether or not the organism has been exposed to the particular substrate. Thus, the results further demonstrate the ability of the framework to effectively describe metabolic regulation in batch, fedbatch, and continuous microbial cultures.     

Cybernetic modeling of growth in mixed, substitutable substrate environments: Preferential and simultaneous utilization

Growth of microorganisms on substitutable substrate mixtures display diverse growth dynamics characterized by simultaneous or preferential uptake of carbon sources. This article shows that cybernetic modeling concepts which were successful in predicting diauxic growth patterns can be extended to describe simultaneous consumption of substrates. Thus the growth of Escherichia coli on mixtures of glucose and organic acids such as pyruvate, fumarate, and succinate has been described successfully by the cybernetic model presented here showing both diauxic and simultaneous uptake when observed. The model also describes the changes in utilization patterns that occur under changing dilution rates, substrate concentrations, and models of preculturing. The model recognizes the importance of the synthesis of biosynthetic precursors in cell growth through a kinetic structure that is quite general for any mixture of carbon-energy sources. (c) 1996 John Wiley & Sons, Inc.     

Effect of root tip amputation on spiral oscillations of the growing hypocotyl with radicle of the pea (Pisum sativum L.)

In order to interpret the mechanism of elongation growth of the hypocotyl with radicle in the first stages of germination of the pea seed (Pisum sativum L.) a cybernetic model utilizing feed-back as a mechanism of correcting the wrong direction of growth was proposed (Spurný 1966, 1967). In the present study, the effect of amputation of the root tip as the control centre on the trajectory of the growing radicle was investigated. The results have shown that the hypocotyl grows, elongates—the rate of growth being slightly lower than that of the standard, but that no spiral oscillations at all are executed by the organ after amputation of the root tip. This finding appears to confirm the applicability of the proposed cybernetic model, for amputation of the root tip means that not only the control block is eliminated, but also that the channel of feed-back impulses to zone of elongation is interrupted.     

Perspectives in the modeling and optimization of PHB production by pure and mixed cultures

Poly(beta-hydroxybutyrate) or PHB is an important member of the family of polyhydroxyalkanoates with properties that make it potentially competitive with synthetic polymers. In addition, PHB is biodegradable. While the biochemistry of PHB synthesis by microorganisms is well known, improvement of large-scale productivity requires good fermentation modeling and optimization. The latter aspect is reviewed here. Current models are of two types: (i) mechanistic and (ii) cybernetic. The models may be unstructured or structured, and they have been applied to single cultures and co-cultures. However, neither class of models expresses adequately all the important features of large-scale non-ideal fermentations. Model-independent neural networks provide faithful representations of observations, but they can be difficult to design. So hybrid models, combining mechanistic, cybernetic and neural models, offer a useful compromise. All three kinds of basic models are discussed with applications and directions toward hybrid model development.     

Mitotic autoregulation, growth control and neoplasia

A model is proposed for the mitotic autoregulation of the haemopoietic stem cell population. Conditions are derived for the model population to exhibit spontaneous self-limitation; these consist of the satisfaction of algebraic inequality relationships between mathematical parameters referring to the global porperties of the model cell population. Violation of these conditions precludes the possibility of spontaneous self-limitation of the population, which therefore grows indefinitely, but is not synonymous with abolition of all responsiveness to homeostatic regulation. Non-limiting growth modes exist for which “cybernetic” responsiveness to perturbations occur.It is suggested that certain features of the model accord well with some observed characteristics of neoplasia, such as the non-localized “field phenomena” frequently seen in tumour growth and the continued responsiveness of some tumours to hormones and other physiological signals.     

Growth of microbial populations in nonminimal media: some considerations for modeling

Ramkrishna and his co-workers have developed so-called cybernetic models which purport to describe, among other things, how microorganisms make choices when presented with two or more functionally equivalent, or substitutable, nutrients that are sources of carbon and available energy. In general, however, organisms are presented with choices not just between nutrients that are substitutable for one another, but also between sets of nutrients some of which are by no means substitutable for one another. It is postulated herein that the main ideas of cybernetic modeling apply to these more general choices as they seem to apply to the choices considered by Ramkrishna and his co-workers. Some consequences of the postulate are worked out for steady-state growth situations where two, or in one case three, nutrients limit or potentially limit growth rate. If predicted phenomena are observed experimentally so as to verify the postulate, a significantly improved basis for understanding growth of microorganisms in practical fermentation media as well as in natural situations will be provided by this application of cybernetic modeling ideas.     

On the matching and proportional laws of cybernetic models

The Matching and Proportional Laws are heuristic control policies that have found widespread use in cybernetic models of biological systems. Within this context, the laws serve as optimization surrogates for predicting the response of metabolic control circuits that modulate enzyme levels and activities. The key result of the current contribution is to demonstrate clearly the optimality properties of these laws and the assumptions that underlie their development. In doing so, we arrive at generalized versions of the Matching and Proportional Laws that are shown to collapse to the forms originally derived by Kompala et al. (Biotechnol. Bioeng. 1986, 28, 1044-1055) when certain simplifications are applied. As a further line of investigation, we show how Kompala et al.'s cybernetic laws compare with alternative control policies in their ability to describe diauxic growth behavior of microbial cultures. We find that Kompala et al.'s model describes the experimental observations more accurately than other limiting-case models that are either too aggressive or too passive in capturing the mixed-substrate growth rates and intermediate lag periods. Monte Carlo analysis of computational growth experiments in which strains obeying different regulatory policies directly compete for available nutrients reveals that the Matching and Proportional Law policy does not maximize the average growth rate of the culture. However, it allocates metabolic resources more frugally than other policies that outperform it and may be more realistic in reflecting the cell's true fitness-to-cost tradeoff as judged by its agreement with experimental growth data.     

Perspectives in the Modeling and Optimization of PHB Production by Pure and Mixed Cultures

ABSTRACT

Poly(β-hydroxybutyrate) or PHB is an important member of the family of polyhydroxyalkanoates with properties that make it potentially competitive with synthetic polymers. In addition, PHB is biodegradable. While the biochemistry of PHB synthesis by microorganisms is well known, improvement of large-scale productivity requires good fermentation modeling and optimization. The latter aspect is reviewed here.

Current models are of two types: (i) mechanistic and (ii) cybernetic. The models may be unstructured or structured, and they have been applied to single cultures and co-cultures. However, neither class of models expresses adequately all the important features of large-scale non-ideal fermentations. Model-independent neural networks provide faithful representations of observations, but they can be difficult to design. So hybrid models, combining mechanistic, cybernetic and neural models, offer a useful compromise. All three kinds of basic models are discussed with applications and directions toward hybrid model development.