Friedrich Srienc

Numerical solution of a mass structured cell population balance model in an environment of changing substrate concentration

Cell population balance models are deterministic formulations which describe the dynamics of cell growth and take into account the biological fact that cell properties are distributed among the cells of a population, due to the operation of the cell cycle. Such models, typically consist of a partial integro-differential equation, describing cell growth, and an ordinary integro-differential equation, accounting for substrate consumption. A numerical solution of the mass structured cell population balance in an environment of changing substrate concentration has been developed. The presented method is general. It can be applied for any type of single-cell growth rate expression, equal or unequal cell partitioning at cell division, and constant or changing substrate concentration. It consists of a time-explicit, one-step, finite difference scheme which is characterized by limited requirements in memory and computational time. Simulations were made and conclusions were drawn by applying this numerical method to several different single-cell growth rate expressions. A periodic behavior was observed in the case of linear growth rate, equal partitioning and constant substrate concentration. The periodicity was equal to the average doubling time of the population. In all other cases examined, a state of balanced growth was reached. Unequal partitioning resulted in broader balanced growth distributions which are reached faster. For the specific types of growth rate dependence on the substrate concentration considered, the changing substrate concentration did not affect the balanced growth-normalized distributions, except for the case of linear growth rate and equal partitioning, where the depletion of the substrate destroyed the periodic behavior observed for constant substrate concentration, and forced the system to reach a steady state.

Saccharomyces cerevisiae expressing bacterial polyhydroxybutyrate synthase produces poly-3-hydroxybutyrate

The polyhydroxybutyrate (PHB) synthase gene of the bacterium Alcaligenes eutrophus was used to construct a yeast plasmid which enabled expression of the functional synthase enzyme in Saccharomyces cerevisiae. Cells transformed with the synthase plasmid accumulated up to 0.5% of cell dry weight as PHB, with accumulation occurring in the stationary phase of batch growth. The identity of PHB in recombinant yeast cells was confirmed with 1H-NMR spectra of chloroform-extracted cell material. In addition, freeze-fracture electron microscopy revealed cytoplasmic granules exhibiting plastic deformations characteristic for PHB. GC results indicated a low background level of PHB in the wild-type strain, but intact polymer could not be detected by 1H-NMR. Formation of PHB in the recombinant strain implies the participation of native yeast enzymes in the synthesis of D-3-hydroxybutyryl-CoA (3-HB-CoA). Inhibition studies with cerulenin indicated that the fatty acid synthesis pathway is not involved in PHB precursor formation. Wild-type cell-free extracts showed D-3-HB-CoA dehydrogenase activity [150-200 nmol min-1 (mg protein)-1] and acetoacetyl-CoA thiolase activity [10-20 nmol min-1 (mg protein)-1], which together could synthesize monomer from acetyl-CoA. PHB accumulation was simultaneous with ethanol production, suggesting that PHB can act as an alternate electron sink in fermentative metabolism. We propose that PHB synthesis in recombinant yeast is catalysed by native cytoplasmic acetoacetyl-CoA thiolase, a native beta-oxidation protein possessing D-3-HB-CoA dehydrogenase activity and heterologous PHB synthase.

Quantitative analysis of transient gene expression in mammalian cells using the green fluorescent protein

The green fluorescent protein (Gfp) has been used as a reporter, along with flow cytometric analysis, to follow the dynamics of gene expression in transiently transfected mammalian cells. Gene transfer conditions for lipofection were optimized. The highest fraction of transfectants were obtained when lipid-DNA complexes were formed with 6 microliters lipid and 1 microgram DNA for chinese hamster ovary (CHO) cells and with 9 microliters lipid and 2 micrograms DNA for NIH/3T3 cells. Chinese hamster ovary cells were monitored for Gfp expression and growth for 6 days following transfection. An initial decrease in viability for 36 h was observed after which cell growth followed exponential kinetics with increasing viability. Intracellular accumulation of recombinant protein peaked at 24 h post-transfection and then decreased with first order kinetics at a rate comparable to the specific growth rate. It appears that dilution by growth accounts for the decrease of Gfp in the biomass. Immunofluorescent staining of Gfp and subsequent flow cytometric analysis of transfected cells revealed a linear correlation between the green fluorescence and immunofluorescence. This indicates that green fluorescence is a quantitative measure of intracellular Gfp in single cells in spite of the dynamics of post-translational modifications involved in the conversion of expressed protein into its fluorescent form. A structured model has been formulated to describe the observed kinetics of gene expression and fluorophore formation. The model accurately predicts experimental trends and suggests that the fraction of non-fluorescent Gfp is significant only during the initial period of gene expression.

A flow injection flow cytometry system for on-line monitoring of bioreactors.

For direct and on-line study of the physiological states of cell cultures, a robust flow injection system has been designed and interfaced with flow cytometry (FI-FCM). The core of the flow injection system includes a microchamber designed for sample processing. The design of this microchamber allows not only an accurate on-line dilution but also on-line cell fixation, staining, and washing. The flow injection part of the system was tested by monitoring the optical density of a growing E.coli culture on-line using a spectrophotometer. The entire growth curve, from lag phase to stationary phase, was obtained with frequent sampling. The performance of the entire FI-FCM system is demonstrated in three applications. The first is the monitoring of green fluorescent protein fluorophore formation kinetics in E.coli by visualizing the fluorescence evolution after protein synthesis is inhibited. The data revealed a subpopulation of cells that do not become fluorescent. In addition, the data show that single-cell fluorescence is distributed over a wide range and that the fluorescent population contains cells that are capable of reaching significantly higher expression levels than that indicated by the population average. The second application is the detailed flow cytometric evaluation of the batch growth dynamics of E.coli expressing Gfp. The collected single-cell data visualize the batch growth phases and it is shown that a state of balanced growth is never reached by the culture. The third application is the determination of distribution of DNA content of a S. cerevisiae population by automatically staining cells using a DNA-specific stain. Reproducibility of the on-line staining reaction shows that the system is not restricted to measuring the native properties of cells; rather, a wider range of cellular components could be monitored after appropriate sample processing. The system is thus particularly useful because it operates automatically without direct operator supervision for extended time periods.

Numerical solution of multi-variable cell population balance models. II. Spectral methods

Abstract Several Galerkin, Tau and Collocation (pseudospectral) approximations have been developed for the solution of the multi-variable cell population balance model in its most general formulation, i.e. for any set of single-cell physiological state functions. Time-explicit methods were found to be more efficient than time-implicit methods for the time integration of the system of ordinary differential equations that results after the spectral approximation in space. The Legendre and Tchebysheff polynomials that were used in Tau algorithms were shown to have significantly worse convergence and stability properties than the Galerkin and collocation algorithms that were applied with sinusoidal trial functions. The collocation method that was implemented with discrete fast Fourier transforms was found to be the most efficient from all the Galerkin and Tau algorithms that were developed. However, the method was inferior to the best finite difference algorithm that was presented in our earlier work.

Solutions of population balance models based on a successive generations approach

Abstract Microbial and cell cultures are composed of discrete organisms, each of which goes through a cell cycle that terminates in production of new cells. The internal state of an individual cell changes as the cell progresses through the cell cycle, and randomness in various features of the cell cycle always produces a distribution of cell states in the culture. Rigorous models of this situation lead to the so-called population balance equations, which are integropartial differential equations. These equations are notoriously difficult to solve, and the difficulties increase as the number of parameters needed to describe cell state increases. The cells in a culture are of different generations, and cells of the (k + 1)th generation originate only from divisions of cells of the kth generation. A population balance equation written for the (k + 1)th generation is therefore not an integral equation, although it contains a source term which is an integral over the distribution of states of the kth generation. If competition of coexisting generations for environmental resources does not affect growth and reproduction rates, the population balance equations for the various generations in a culture do not have to be solved simultaneously but rather can be solved successively, and thus, some of the major difficulties of population balance equations written for entire populations are circumvented. In this paper, the successive generations approach to modeling is illustrated by its application to two problems where cell state is described by a single parameter, either cell age or cell mass. It is then applied to a problem where two parameters, namely cell age and cell mass, are used to describe cell state at the same time. Analytical solutions of the population balance equations for the successive generations are found for the cases discussed, and the solutions are used to calculate the evolutions of the distributions of cell states with time for the single parameter cases.

Metabolic modeling of polyhydroxybutyrate biosynthesis

A mathematical model describing intracellular polyhydroxybutyrate (PHB) synthesis in Alcaligenes eutrophus has been constructed. The model allows investigation of issues such as the existence of rate-limiting enzymatic steps, possible regulatory mechanisms in PHB synthesis, and the effects different types of rate expressions have on model behavior. Simulations with the model indicate that activities of all PHB pathway enzymes influence overall PHB flux and that no single enzymatic step can easily be identified as rate limiting. Simulations also support regulatory roles for both thiolase and reductase, mediated through AcCoA/CoASH and NADPH/NADP+ ratios, respectively. To make the model more realistic, complex rate expressions for enzyme-catalyzed reactions were used which reflect both the reversibility of the reactions and the reaction mechanisms. Use of the complex kinetic expressions dramatically changed the behavior of the system compared to a simple model containing only Michaelis-Menten kinetic expressions; the more complicated model displayed different responses to changes in enzyme activities as well as inhibition of flux by the reaction products CoASH and NADP+. These effects can be attributed to reversible rate expressions, which allow prediction of reaction rates under conditions both near and far from equilibrium.

Peroxisomes as Sites for Synthesis of Polyhydroxyalkanoates in Transgenic Plants

Bacterial genes responsible for poly(3‐hydroxybutyrate) (PHB) biosynthesis were targeted to plant peroxisomes by adding a carboxy‐terminal targeting sequence. The enzymes evidently were transported into peroxisomes, retained their catalytic activity, and reacted with peroxisomally available precursors because PHB synthesis in transgenic plant cells was localized to peroxisomes. Up to 2 mg/g fresh weight PHB was produced in suspension cultures of Black Mexican Sweet maize cells after biolistic transformation with three peroxisomally targeted bacterial genes. An equilibrium effect is proposed to explain the unexpected existence of (R)‐3‐hydroxybutyryl‐CoA in plant peroxisomes.

Multistaged corpuscular models of microbial growth: Monte Carlo simulations

A new framework is developed by extending the existing population balance framework for modeling the growth of microbial populations. The new class of multistaged corpuscular models allows further structuring of the microbial life cycle into separate phases or stages and thus facilitates the incorporation of cell cycle phenomena to population models. These multistaged models consist of systems of population balance equations coupled by appropriate boundary conditions. The specific form of the equations depend on the assumed forms for the transition rate functions, the growth rate functions, and the partitioning function, which determines how the biological material is distributed at division. A growth model for ciliated protozoa is formulated to demonstrate the proposed framework. To obtain a solution to the system of the partial integro differential equations that results from such formulation, we adopted a Monte Carlo simulation technique which is very stable, versatile, and insensitive to the complexity of the model. The theory and implementation of the Monte Carlo simulation algorithm is analyzed and results from the simulation of the ciliate growth model are presented. The proposed approach seems to be promising for integrating single-cell mechanisms into population models.

Effect of lactic acid on the kinetics of growth and antibody production in a murine hybridoma : secretion patterns during the cell cycle

The effects of elevated lactic acid concentration on the cell cycle kinetics of hybridoma cell growth and antibody production in batch culture were studied using conventional methods based on population-average data analysis and using flow cytometry based on single-cell data analysis. When 33 mM lactic acid was initially present, the true specific growth rate was reduced by 37% and the cell specific antibody production rate increased by a factor of 2.6 relative to a control culture with no additional lactic acid. DNA content distribution measured during balanced exponential growth were not affected by lactic acid concentration indicating lactic acid has a uniform effect on cell growth throughout the cell cycle. There was little or no effect on single-cell distributions of intracellular antibody content measured for the total population and for each cell cycle phase. The net rate of total antibody synthesis was found to be independent of specific growth rate. This implies that the balance of the total amount of antibody synthesized is shifted from cellular accumulation towards secretion when specific growth rate decreases. Our data predict that a maximum specific secretion rate of 2.7 pg per cell per h could be achieved if the specific growth rate was reduced to zero. The rates of secretion in the G1 and S phases increased with decreasing specific growth rate, while the rate of secretion in the G2+M phase remained relatively constant. Under the assumptions that (a) at the fastest growth rate, secretion in the G1 phase is negligible and (b) the rate of synthesis increases exponentially as cells proceed from the S phase to the G2+M phase, our data predict that for the slowest growth rate, the rate of secretion in G2+M is approx. 3-times that in the G1 phase and 5-times that in the S phase.