Population balance models of autonomous microbial oscillations

From single cells to microbial population dynamics: modelling in biotechnology based on measurements of individual cells

The development of dynamic modelling of microbial populations in bioprocesses is reviewed. In the 1960s Arnold Fredrickson established the theoretical basis of such models, and other researchers have subsequently advanced them. This review explores the relationships that describe cell proliferation and evaluates the importance of the application of flow cytometry to the fundamental parameterisation of the models for their use in bioprocess engineering. The section "Individual-Based Modelling" discusses recent theoretical developments. Delay-differential equations are demonstrated to describe accurately temporal variation of the cell proliferation cycle and specialised approaches and related iconography are applied to stochastic and deterministic modelling of stages of cellular development. Synchronised cultures of the bacterium Cupriavidus necator were prepared and monitored using a flow cytometer. The data obtained demonstrate that cell proliferation could be simulated quantitatively using the developed model.

ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast

Biologists have long observed periodic-like oxygen consumption oscillations in yeast populations under certain conditions, and several unsatisfactory explanations for this phenomenon have been proposed. These ‘autonomous oscillations’ have often appeared with periods that are nearly integer divisors of the calculated doubling time of the culture. We hypothesize that these oscillations could be caused by a form of cell cycle synchronization that we call clustering. We develop some novel ordinary differential equation models of the cell cycle. For these models, and for random and stochastic perturbations, we give both rigorous proofs and simulations showing that both positive and negative growth rate feedback within the cell cycle are possible agents that can cause clustering of populations within the cell cycle. It occurs for a variety of models and for a broad selection of parameter values. These results suggest that the clustering phenomenon is robust and is likely to be observed in nature. Since there are necessarily an integer number of clusters, clustering would lead to periodic-like behaviour with periods that are nearly integer divisors of the period of the cell cycle. Related experiments have shown conclusively that cell cycle clustering occurs in some oscillating yeast cultures.

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.

Modeling the synchronization of yeast respiratory oscillations

The budding yeast Saccharomyces cerevisiae exhibits autonomous oscillations when grown aerobically in continuous culture with ethanol as the primary carbon source. A single cell model that includes the sulfate assimilation and ethanol degradation pathways recently has been developed to study these respiratory oscillations. We utilize an extended version of this single cell model to construct large cell ensembles for investigation of a proposed synchronization mechanism involving hydrogen sulfide. Ensembles with as many as 10,000 cells are used to simulate population synchronization and to compute transient number distributions from asynchronous initial cell states. Random perturbations in intracellular kinetic parameters are introduced to study the synchronization of single cells with small variations in their unsynchronized oscillation periods. The cell population model is shown to be consistent with available experimental data and to provide insights into the regulatory mechanisms responsible for the synchronization of yeast metabolic oscillations.

Dynamic modeling of microbial cell populations

Microbial cultures are comprised of heterogeneous cells that differ according to their size and intracellular concentrations of DNA, proteins and other constituents. Recent advances have been made in cell population modeling, which allow the effects of cell heterogeneity on culture dynamics and metabolite production to be predicted. If the intracellular state can be captured with a few variables, the population balance equation framework is a viable modeling approach. The cell ensemble modeling technique is better suited for the development of population models that include more detailed descriptions of cellular metabolism and/or cell-cycle progression.

Free intracellular amino acid pools during autonomous oscillations in Saccharomyces cerevisiae

In the present work dynamic changes of free intracellular amino acid pools during autonomous oscillations of Saccharomyces cerevisiae were quantified in glucose-limited continuous cultivations. At a dilution rate of D = 0.22 h(-1) cyclic changes with a period of 120 min were found for many variables such as carbon dioxide production rate, dissolved oxygen, pH, biomass content, and various metabolite concentrations. On the basis of the observed dynamic patterns, free intracellular amino acids were classified to show oscillatory, stationary, or chaotic behavior. Amino acid pools such as serine, alanine, valine, leucine, or lysine were subjected to clear oscillations with a frequency of 120 min, identical to that of other described cultivation variables, indicating that there is a direct correlation between the periodic changes of amino acid concentrations and the metabolic oscillations on the cellular level. The oscillations of these amino acids were unequally phase-delayed and had different amplitudes of oscillation. Accordingly, they exhibited different patterns in phase plane plots vs. intracellular trehalose. Despite the complex and marked metabolic changes during oscillation, selected intracellular amino acids such as histidine, threonine, isoleucine, or arginine remained about constant. Concentrations of glutamate and glutamine showed a chaotic behavior. However, the ratio of glutamate to glutamine concentration was found to be oscillatory, with a period of 60 min and a corresponding figure eight-shaped pattern in a plot vs. trehalose concentration. Considering the described diversity, it can be concluded that the observed periodic changes are neither just the consequence of low or high rates of protein biosynthesis/degradation nor correlated to changing cell volumes during oscillation. The ratio between doubling time (189 min) and period of oscillation of intracellular amino acids (120 min) was 1:6. The fact that there is a close relationship between doubling time and period of oscillation underlines that the described autonomous oscillations are cell-cycle-associated.

Cyclic AMP mediates the cell cycle dynamics of energy metabolism in Saccharomyces cerevisiae

We have investigated the role of 3',5'-cyclic-adenosine-monophosphate (cAMP) in mediating the coupling between energy metabolism and cell cycle progression in both synchronous cultures and oscillating continuous cultures of Saccharomyces cerevisiae. For the first time, a peak in intracellular cAMP was shown to precede the observed breakdown of trehalose and glycogen during cell cycle-related oscillations. Measurements in synchronous cultures demonstrated that this peak can be associated with the cell cycle dynamics of cAMP under conditions of glucose-limited growth, which was found to differ significantly from that observed in synchronous glucose-repressed cultures. Our results support the notion that cAMP plays a major role in mediating the integration of energy metabolism and cell cycle progression, both in the single cell and during cell cycle-related oscillations in continuous culture, respectively. Evidence is presented that the dynamic behaviour of intracellular cAMP during the cell cycle is modulated depending on nutrient supply. The implications of these findings regarding the role of cAMP in regulating cell cycle progression and energy metabolism are discussed.

Cell population modelling of yeast glycolytic oscillations

We investigated a cell-population modelling technique in which the population is constructed from an ensemble of individual cell models. The average value or the number distribution of any intracellular property captured by the individual cell model can be calculated by simulation of a sufficient number of individual cells. The proposed method is applied to a simple model of yeast glycolytic oscillations where synchronization of the cell population is mediated by the action of an excreted metabolite. We show that smooth one-dimensional distributions can be obtained with ensembles comprising 1000 individual cells. Random variations in the state and/or structure of individual cells are shown to produce complex dynamic behaviours which cannot be adequately captured by small ensembles.

Multiple stable states and hysteresis in continuous, oscillating cultures of budding yeast

The conditions that precede the onset of autonomous oscillations in continuous yeast cultures were studied in three different types of experiments. It was found that the final state of the culture depended on the protocol used to start up the reactor. Batch cultures, switched to continuous operation at different stages of the batch growth curve, all exhibited similar dynamics-ethanol depletion followed by autonomous oscillations. Small perturbations of the distribution of states in the reactor, achieved by addition of externally grown cells, were able to quench the oscillatory dynamics. Reaching the desired operating point by slow dilution rate changes gave rise to different final states, two oscillatory states and one steady state, depending on the rate of change in dilution rate. The multiplicity of stable states at a single operating point is not explained by any current distributed model and points toward a segregated mechanism of these oscillations.

Effect of the dilution rate on the mode of oscillation in continuous cultures of Saccharomyces cerevisiae

The growth properties of the asymmetric budding yeast Saccharomyces cerevisiae were analysed during spontaneous oscillations in continuous cultures at varying dilution rates D. The length of the oscillation period changed between 1.4 and 14 h in response to the decrease of dilution rate from 0.15 to 0.05 h-1. The distribution of parent and daughter cells in the population was determined microscopically after staining the bud scars and DNA. Most of the data obtained fits a theoretical population balance model assuming two-classes of subpopulations and integer ratios between the generation times of both classes. Some data has to be described by an extended population model assuming there is one parent and two daughter cell classes. How changes of dilution rate may cause an accidental switch of the mode of oscillation is demonstrated. Glucose consumption and metabolite production were measured off-line by enzymatic methods and gas exchange was monitored on-line. All these data of one period point to internal and external signals responsible for the synchronisation of the cell cycle.

Solution and properties of age population balance models which assume discrete division ages

A solution procedure for models of the transient age distribution which employ the assumption of discrete, constant division ages is developed. The solutions for the final state of the age distribution, the solutions obtained in the limit as time goes to infinity, are found to have properties which do not make biological sense. In particular, the solutions will only approach the steady-state solution for very special initial conditions. For most initial conditions, the solution for the final state will instead exhibit an oscillatory behavior. In addition, the oscillatory solutions are unstable with respect to changes in values of the model parameters and solutions with very different periods of oscillation are found arbitrarily close to each other in the parameter space. Models which assume discrete division ages must therefore be used with caution and may be unsuitable as models of autonomous microbial oscillations.