An age-structured population balance model for microbial dynamics

An age-structured population balance model for microbial dynamics

M.V.E.Duarte; J.L.Medeiros; O.Q.F.Araújo; M.A.Z.Coelho

Escola de Química, Universidade Federal do Rio de Janeiro, Fax: [phone], C.P. 68542, CEP 21949-900, Rio de Janeiro, RJ, Brazil

^{Address to correspondence}

ABSTRACT

This work presents an age-structured population balance model (ASPBM) for a bioprocess in a continuous stirred-tank fermentor. It relates the macroscopic properties and dynamic behavior of biomass to the operational parameters and microscopic properties of cells. Population dynamics is governed by two time- and age-dependent density functions for living and dead cells, accounting for the influence of substrate and dissolved oxygen concentrations on cell division, aging and death processes. The ASPBM described biomass and substrate oscillations in aerobic continuous cultures as experimentally observed. It is noteworthy that a small data set consisting of nonsegregated measurements was sufficient to adjust a complex segregated mathematical model.

Keywords: Population balance, population dynamics, yeast, autonomous oscillations, segregated modeling.

INTRODUCTION

Traditional models of microbial industrial bioprocesses describe the biophase as an aggregated lump of identical biological beings, and thus do not account for the effects of heterogeneity of the microbial population. This lump-based approach is mostly due to the serious difficulties inherent in quantification of variables which are microscopic (corpuscular properties that should be quantified on a single-cell scale, such as size, levels of intracellular constituents, etc.) and the straightforward methods available for measuring macroscopic variables (properties of the whole population, such as the dry weight of cells).

Although the growth of the bioprocess industry during the past century was based on process design methodologies using lump-based models, some experimental features and behaviors of well-known processes cannot be predicted by these models. Indeed, it is unusual to find process models that incorporate biochemical and/or genetic insights, even though it is known that cell-cycle operation does have an influence on process performance. Thus, despite the usefulness of traditional models (in particular, Monod-like models), it is of great academic and industrial interest to develop bioprocess models that (i) describe and predict performance under a broad range of operational conditions, (ii) predict dynamic features of populations, which are already known, but not predicted by traditional models, and (iii) rely on data from simple and inexpensive experiments.

Population balance models (PBM's) can describe a broad range of dynamic behaviors and are suited for processes undertaken in groups of corpuscular entities that have individual properties which depend on some variable property (amongst all individuals). Also referred to as statistical models, they use statistical functions to compute rates of "birth" and "death" of these corpuscles, calculating the density of corpuscles per unit volume. Formulation of these microbial models generally includes a partial integro-differential equation and an ordinary integro-differential equation, accounting for density of cells and limiting substrate concentration, respectively (Villadsen and Nielsen, 1992). In addition to time, these equations are functions of another variable. Regarding microbial growth, this other variable is either "mass" (any property or constituent to which a conservation law is applicable – volume, protein content, etc.) or "age" of the cells. Therefore, these PBM's are said to be mass- or age-structured.

This work presents an age-structured population balance model (ASPBM) that computes densities of living and dead cells as a function of substrate and dissolved oxygen concentrations. Model parameters were estimated from common, nonsegregated data, in contrast with other PBM's, which need segregated data for parameter regression. Finally, the model is able to reproduce the periodic behavior of continuous cultures of the yeast Saccharomyces cerevisiae.

THE MODEL

Consider a population of cells inside a continuous stirred-tank fermentor (CSTF – see Fig.1). These cells have features (size, intracellular compositions, etc.) and behaviors (probabilities of division and death) that are age-dependent. By "age" the real time passed since the "birth" of the cell is generally meant, whilst by "genealogical age" the number of cell cycles undergone by the cell is meant (i.e., the number of times one given cell has divided, generating a newborn cell). In the model presented herein, "age" is a continuous variable associated with (discrete) genealogical age.

Here x is the age-associated variable (dimensionless), t is time, N₀ is the total number of living cells when t = 0, M₀ is the single-cell standard mass, g_L is the dimensionless density of living cells, g_D is the dimensionless density of dead cells, is the dimensionless density of living cells at the inlet, is the dimensionless density of dead cells at the inlet, S is the substrate concentration, Sⁱⁿ is the substrate concentration at the inlet, DO is the dissolved oxygen concentration, F is the inlet flow rate, V is the reactor hold-up, h is the removal factor (0 £ h £ 1), q is the cycle-length function, D is the death probability density function (PDF), K is the division PDF, and the hyperbolic terms in Eq. 3 are the S-dependent and DO-dependent terms of the single-cell substrate consumption rate, model equations are as follows:

In this work, the information used to guide the formulation of the statistical terms and substrate uptake rate of the model, as well as to adjust its parameters, is macrokinetic data on batch cultures of the yeast Saccharomyces cerevisiae (biomass and substrate concentrations).

Death and division PDF's are products of individual (and statistically independent) influence factors of aging and environmental conditions on these processes. With one remarkable exception, these individual terms are based upon the sigmoid function

This yeast divides asymmetrically into a larger mother cell and a smaller daughter, and the completion of each cell cycle leaves a scar on the surface of the mother. The mother should grow to a given mass before it divides in two, making impossible subsequent divisions within a given period (i.e., there is a critical – minimum – mass that should be achieved before a cell starts replicating all its constituents to produce a newborn). Since just after a division, the mass of the mother is close to this critical mass and the mass of the newborn is much farther from it, the cell cycle of the newborn is longer. Finally, the environmental conditions have two basic influences on division: (i) low S concentrations may prevent arrest of the division, as mother cells need it to synthesize all the constituents of the daughter, and (ii) low DO levels may reduce some synthesis rates (as fermentative metabolism yields less energy than respiratory metabolism), but do not prevent cell cycle arrest. Thus, the division PDF is

Note that the exponential term in Eq. 5 accounts for the necessary lag between two successive divisions and is calculated as normalized variable z (T is the truncation function).

Concerning death, we assume that the longer a cell lived, the greater is its chance to die, and also that under poor nutritional conditions and at low DO levels, the probability of death increases. However, total lack of substrate and/or dissolved oxygen does not imply death (i.e., D(x,0,0) < 1). Finally, there is a "random mortality factor," a residual death probability for cells of all ages in any environment. The death PDF is

Both PDF's are depicted in Fig. 2. For the sake of brevity and visibility, here we only show plots in (x, DO) space and the age span was limited to 10 cycles.

Application of the Galerkin formalism to Eqs. (1) to (3) yields a large set of ordinary differential equations (only time-dependent). The basis function used was the unit pulse. This set was integrated using the Runge-Kutta method (Duarte, 2000).

RESULTS

Simulating the set of ODE's resulting from application of the Galerkin formalism to Eqs. (1) to (3) requires knowledge of the cycle-length funtion q (x) for this yeast. Also, a mass-distribution function M(x) to compute cell mass as a function of age is necessary. Functions q (x) and M(x) (and their parameters) and M₀ were taken from the literature (Chen et al., 2000). Here we assumed here M₀ to be the value of the critical mass (for beginning of budding). The shape of M(x) is depicted in Fig. 3a, clearly showing the asymmetrical nature of S. cerevisiae cell division. Note that all peaks (values for a given cell just before division) of the M(x)/M₀ ratio are around 1.7, while values for newborn cells are around 0.7 and values for mothers just after division are around 1.

As a straightforward consequence of (i) this asymmetrical division and (ii) the mass checkpoint for beginning of budding, duration of cell cycles is a function of age, as newborns are farther from entering the budding phase than mothers. We consider that this function q (x) is dependent only on genealogical age, in the following fashion: the cell-cycle duration of any mother has the same value, which is shorter than that of a newborn. A histogram of this function is shown in Fig. 3b.

Finally, the initial age distributions of living and dead cells were chosen to agree with Lord and Wheals's work (1980). We assumed these distributions to have the same shape, with the value of the dead cell distribution arbitrarily picked as 10% of the value of the living cell density.

Some model parameters were qualitatively adjusted (based upon previous knowledge and several simulations) and others were estimated from data on batch cultures. These values as well as experimental methodologies and optimization procedures are reported elsewhere (Duarte, 2000).

The model was used to study the behavior of continuous yeast cultures. It is known that this kind of system may show periodic behavior in the absence of a forcing function (for example, a periodic time-varying inlet flowrate). These autonomous oscillations are due to occur under conditions of a constantly high V/F ratio.

As Figs. 4 and 5 show, the model reproduces the experimentally observed periodic behavior. Note that these oscillations are not caused by fluctuations in inlet flow rate or substrate concentration (respectively, F and Sⁱⁿ) and that they are maintained for a long time (for over a hundred periods). Figure 4 is shown on a smaller time scale (100 hours, about 3 periods), just to enhance visibility.

CONCLUSIONS

Despite the nonsegregated experimental data used to estimate the parameters of the ASPBM, it shows features that a broad variety of other models fail to capture, such as the capability to predict spontaneous oscillations in continuous cultures of yeast. In fact, this is a major advantage of this ASPBM, as other PBM's found in the literature need segregated data (which is expensive and technically harder to obtain) for their parameters to be estimated. Computation of the density of dead cells is another major feature of the model, as it is unusual to find an approach to bioprocess modeling that accounts for them. This formulation illustrates the capability of PBM's to incorporate cell-cycle information, as in the case of death and division PDF's, even though this information is implicit.

This ASPBM is an easily extensible framework, which can be expanded to account for other types of death and division PDF's and different influences of the variables already included in it (for example, a segregated substrate uptake function or a DO mass balance) or to incorporate new variables (e.g. ethanol). Another important advantage of this formulation is the relatively simple method of solution and numerical simulation.

ACKNOWLEDGEMENTS

The scholarship and grant from CNPq - National Council for Scientific and Technological Development - are gratefully acknowledged.

Received: March 5, 2002

Accepted: August 30, 2002

• Chen, K. C., Csikasz-Nagy, A., Gyorffy, A., Val, J., Novak, B. and Tyson, J. (2000), Kinetic Analysis of a Molecular Model of the Budding Yeast Cell Cycle, Mol. Biol. Cell, 11, 369-391.
• Duarte, M. V. E. (2000), Uma modelagem de dinâmica populacional microbiana estruturada em idade, M.Sc. Thesis, Escola de Química/UFRJ, Rio de Janeiro, Brazil.
• Lord, P. G. and Wheals, A. E. (1980), Asymmetrical Division of Saccharomyces cerevisiae, J. Bacteriol., 142, 808-818.
• Villadsen, J. and Nielsen, J. (1992), Modelling of Microbial Kinetics", Chem. Eng. Sci., 47, 4225.

Publication Dates

• Publication in this collection
  19 Mar 2003
• Date of issue
  Mar 2003

History

• Received
  05 Mar 2002
• Accepted
  30 Aug 2002