Two mathematical models for the development of a single microbial pellet

A mathematical model for pellet development of filamentous microorganisms is presented, which simulates in detail location and growth of single hyphal elements. The basic model for growth, septation and branching of discrete hyphae is adopted from Yang et al. [2, 23]. Exact solutions to the intracellular mass-balance equations of a growth-limiting key component is given for two types of either branched or unbranched cellular compartments. Furthermore, the growth model was extended in regard to the external mass-balance equations of limiting substrates (oxygen, glucose) under the assumption that the substrates can enter the denser regions of the pellet only diffusively. Penetration of the substrates into the more porous outer regions of the pellet occurs more easily due to microeddies in the surrounding fluid. Chipping of hyphae from the pellet surface by shear forces was included in the model as well. The application of shear forces leads to a marked smoothing of the simulated pellet surface. The development of pellets from spore germination up to late stages with cell-lysis due to shortage of substrates in the pellet centre can be described. The effects of various model parameters are discussed.List of Symbols A _i algebraic coefficient (i = 1, 2,..., 6) - B _i algebraic coefficient (i = 1, 2,..., 6) - C _i mass-concentration of component i (i = O₂, S) (gl^–1) - C _i,crit concentration of substance i critical for lysis (i=O₂, S) (gl^–1) - C _i,stop concentration of substance i below which cells are inactivated (gl^–1) - C(l _i,t) intracellular concentration of the key component at site l _i and time t (gl^–1) - C _m maximal intracellular concentration of the key component (gl^–1) - C _X Concentration of dry biomass (gl^–1) - D intracellular diffusion coefficient of the key component (m² h^–1) - D _max,i maximal molecular diffusion coefficient of substrate i (i = O₂, S) (m² h^–1) - D _eff,i effective diffusion coefficient of component i (i = O₂) (m² h^–1) - d _h cross-sectional diameter of hyphae (m) - k production coefficient for the key component (h^–1) - K _s Monod coefficient for glucose (gl^–1) - k ₀ Monod coefficient for oxygen (gl^–1) - L _c total length of a compartment (m) - L _i total length of branch i (i=1, 2, 3) (m) - l _i position on branch i (i=1, 2, 3) - L _m maximal length of a segment (m) - m _i maintenance coefficient of substrate i (h^–1) - N _m maximal number of segments in a compartment - n _iR number of tips of type i in layer R, i=1, 2 - p auxiliary variable (see Eq. (7)) - P _Br probability that a hypha is chipped off (%h^–1) - pO ₂ partial pressure of oxygen in the liquid phase (%) - Q auxiliary variable (see Eq. (8)) - Q _i uptake rate of substrate i (i = O₂, S) (gl^–1 h^–1) - q auxiliary variable (see Eq. (7)) - R index of radial layer (R=1, 2, 3,..., R _max) - r radius (m) - r _crit critical radius, Eq. (15) (m) - r _max pellet radius (m) - r _tip distance from the pellet centre to the tip position (m) - r _thr threshold radius (m) - s auxiliary variable (see Eq. (7)) - S index for glucose - t time (h) - v _R volume of layer R (1) - Y _Mi observable yield coefficient of biomass on substrate i (gg^–1) - Y _Xi yield coefficient of biomass on substrate i (gg^–1) Greek Letters _i actual tip expansion rate (m h^–1) - _i,m actual maximal extension rate of tip i (i=1, 2) (m h^–1) - _1y lysis rate (h^–1) - _m maximal tip extension rate (m h^–1) - auxiliary variable in Eq. (2) - auxiliary variable in Eq. (3) - auxiliary variable defined in Eq. (4) (m^–1) - _shear shear force parameter - _R overall specific growth rate in layer R (h^–1) - _m maximal specific hyphal growth rate (h^–1) - cell volume density (l cell volume per 1) - _crit critical cell volume density in Eq. (15) - _S shear force parameter - _X cell mass density (g dry weight per 1 wet cells) - (C _i) growth kinetics on substrate i - proportional factor in Eq. (34) (l g^–1) We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.     

Modification and verification of the dynamic cell cycling model for baker's yeast

The cell cycling model (CCM) for S. cerevisiae proposed earlier is modified and tested with our own experimental data. Although the original CCM was well verified in steady states and exponential growth with data available in literature, some discrepancies between model predictions and experiments were found for the dynamics of fed-batch culture. The redistribution pattern of the age distribution of daughter cells is suggested as cause of the model error. With an exponential type of redistribution, instead of the original linear one, the model behaviour in transients is improved. The modified model was verified with data of fraction of budding cells and cell number for five fed-batch cultivations. The model agreed well with the experimental data. The simulation results suggest that the cell cycling process indeed is essentially in a pseudo-steady state during fed-batch cultivation, as was assumed in the model. Due to the strong correlation between the quality of baker's yeast and the state of the population in the cell cycling process, the model was applied to optimize the feeding rate of a fedbatch process with consideration of final product quality. The optimal feeding was used succesfully in a laboratory experiment, which demonstrates the validity of the model.List of Symbols B h length of budding phase - C _b1, C _d1, C _p1 parameters in cycling phase equations - C _b2, C _d2, C _p2 h parameters in cycling phase equations - d(i) number of cells in ith cycling interval inU _d - E kg m^–3 ethanol concentration - F m³ h^–1 substrate feeding rate - F _max and F _min m³ h^–1 upper and lower limit of F - FBC, FDC, FPC % fraction of budding cells, unbudded daughter cells and unbudded parent cells - K _B1, K _B2, K _B3, K _EG, K _Ad parameters in the metabolic model - m _ATP mol(gh)^–1 maintenance coefficient for ATP - n _b, n _d, n _p number of age intervals in the budding phase, daughter phase and parent phase - PO _min, PO _max minimal and maximal effectiveness of oxidative phosphorylation - r _Acmax mol(gh)^–1 saturation value of the specific acetyl-CoA-reaction rate - S kg m^–3 concentration of total reduceable sugars - S _R kg m^–3 substrate concentration in the feed - T cell number doubling time - T _fh fermentation period - U _dh length of unbudded daughter cell cycling phase - U _ph length of unbudded parent cell cycling phase - V _cell m³ average volume of yeast cells - V _L m³ liquid volume of the reactor - X kg m^–3 cell mass concentration - X _N cm^–3 cell number concentration - Y _ATP g mol_ATP ^–1 yield coefficient of ATP - parameter in the exponential redistribution function - h^–1 specific growth rate - h length of the discrete age interval of cell cycle phases - suffix old and new denote the value before and after increasing of      

A dynamic cell cycling model for growth of baker's yeast and its application in profit optimization

A cell cycling model for unequal budding yeast Saccharomyces cerevisiae is proposed and verified by steady state data from experiments available in the literature. This model can be used to determine the relative fraction of the cells in any cycling phase or with any genealogical age during fermentation. As the quality of yeast is strongly influenced by the cycling process, the model could therefore be used to control the quality of the harvested yeast cells. The input of the cell cycling model is the specific growth rate , which is obtained from a metabolic model for S. cerevisiae proposed earlier. With this extended model system not only the quality control, but also the whole economical profit optimization can be carried out. Simulations were done to optimize the profit of a commercial scale baker's yeast production process by manipulating substrate feeding rate and substrate concentration under different aeration rates, fermentation periods and other conditions applied in industry.List of Symbols B h budding phase - C _d1, C _d2, C _p1' parameters in cycling phase equations - C _p2, C _b1, C _b2, d _s m Sauter-diameter - E kg/m³ ethanol concentration - E1, E2 state variables in the metabolic model - E _G mean relative gas hold-up - f parameter vector of the regulation model - F system matrix of the regulation model - F or F(t) m³/h substrate feeding rate - Fr Froude number - FBC, FDC, FPC % fraction of daughter cells, unbudded daughter cells and unbudded parent cells - g m/s² acceleration of gravity - K _B1–3, K _EG parameters in metabolic model - K ₃, K _Ad , L ₃ K ₃ E, K_O, K_S limitation constants for ethanol, oxygen and substrate - k _La h^–1 volumetric mas transfer coefficient - m _ATP mol(gh)^–1 maintenance coefficient - nb, nd, np number of cycling age intervals in budding cycling phase, unbudded daughter cycling phase and unbudded parent cycling phase - Nt number of total cells - OF mg/dm³ concentration of dissolved oxygen - P kg total yeast product in dry weight - P/O effectiveness of oxidative phosphorylation - q _O20 mol(gh)^–1 minimum specific oxygen uptake ability - q _O2 mol(gh)^–1 specific oxygen uptake rate - q _O2max mol(gh)^–1 maximum q _O2 given by metabolic regulation - q _s mol(gh)^–1 specific glucose uptake rate - q _Smax mol(gh)^–1 maximum q _S - R(·) switch function - r _Ac mol(gh)^–1 specific acetyl-CoA-consumption rate - r _Acmax saturation rate of r _Ac - r _E1 mol(gh)^–1 specific ethanol production rate - r _E2 mol(gh)^–1 specific ethanol uptake rate - r _SO mol(gh)^–1 minimum value of r _Smax - r _s mol(gh)^–1 specific rate of glycolysis - r _Smax mol(gh)^–1 maximum specific rate of gluconeogenesis given by metabolic regulation - S kg/m³ total reduced sugar concentration - S _R kg/m³ substrate concentration in feed - T h cell number doubling time - T _f h fermentation period - Ud h unbudded daughter phase - Up h unbudded parent phase - V _F m³ volume of liquid phase in fermentor - V _G m³/h aeration rate - w _sg m/s superficial gas velocity - X kg/m³ dried cell concentration - Y _ATP g(molATP)^–1 yield coefficient of ATP - z state vector in regulation model - the factor of fermentative activity decrease caused by budding cells - or(t) h^–1 specific growth rate - h discrete unit of cycling age     

Macrokinetic model for methylotrophic Pichia pastoris based on stoichiometric balance

A macrokinetic model for Pichia pastoris expressing recombinant human serum albumin is proposed. The model describes the balances of some key metabolites, ATP and NADH, during glycerol and methanol metabolism. In the glycerol growth phase, the metabolic pathways mainly include phosphorylation, glycolysis, tricarboxylic acid cycle, and respiratory chain. In the methanol growth phase, methanol is oxidized to formaldehyde at first. Then, while a part of formaldehyde is oxidized to formate, the rest is condensed with xylulose-5-monophosphate to form glyceraldehyde-3-phosphate, and further assimilated to form cell constituents. The metabolic pathways following glyceraldehyde-3-phosphate were assumed to be similar to those in the glycerol growth phase. Based on the model, the macrokinetic bioreaction rates such as the specific substrate consumption rate, the specific growth rate, the specific acetyl-CoA formation rate as well as the specific oxygen uptake rate are obtained. The specific substrate consumption rate and the specific growth rate are then coupled into a bioreactor model such that the relationship between substrate feeding rates and the main state variables, i.e., the medium volume, the concentrations of the biomass, the substrate, and the product, is set up. Experimental results demonstrate that the model can describe the cell growth and the protein production with reasonable accuracy.     

Dynamics of growth in batch and continuous cultures of Saccharomyces cerevisiae during shifts from aerobiosis to anaerobiosis and reverse

Summary Saccharomyces cerevisiae H 1022 was cultivated in batch and continuous culture on a glucose substrate. The yeast was subjected to a sudden change from aerobic to anaerobic growth conditions by switching the inlet-gas stream from air to dinitrogen. The dynamics of growth and product formation during the periods of adaptation were studied. A structured growth model based on Monod-Blackman-kinetics was applied to simulate these shift-experiments. The immediate switch of the yeast to maximum anaerobic growth and ethanol production predicted by this model was confirmed very well in the chemostat-experiments. However, a slow adaptation was evident for the switch from anaerobiosis back to aerobiosis.Gesellschaft für Biotechnologische Forschung mbH, Mascheroder Weg 1, D-3300 Braunschweig-Stöckheim, Federal Republic of Germany     

The production of 2,3-butanediol by fermentation of high test molasses

Summary Klebsiella oxytoca fermented 199 g·l^–1 high test or invert molasses using batch fermentation with substrate shift to produce 95.2–98.6 g 2,3-butanediol·l^–1 and 2,4–4.3 g acetoin·l^–1 with a diol yield of 96–100% of the theoretical value and a diol productivity of 1.0–1.1 g·l^–1·h^–1. Fermentation was performed numerous times with molasses in repeated batch culture with cell recovery. Such repeated batch fermentation, in addition to a high product yield, also showed a very high product concentration. For example, 118 g 2,3-butanediol·l^–1 and 2.3 g acetoin·l^–1 were produced from 280 g·l^–1 of high test molasses. The diol productivity in this fermentation amounted to 2.4 g·l^–1·h^–1 and can undoubtedly be further increased by increasing the cell concentration. Because the Klebsiella cultures ferment 2,3-butanediol at an extremely high rate once the sugar has been consumed, the culture was inhibited completely by the addition of 15 g ethanol·l^–1 and switching off aeration. Offprint requests to: A. S. Afschar     

Influence of age and structure of Pencillium chrysogenum pellets on the internal concentration profiles

Pellets of Penicillium chrysogenum which were spontaneously formed after a certain stage of a batch fermentation, displayed a considerable structural change in course of their lifetime. Microelectrode studies showed the internal mass transport properties of these pellets (diameter 1–3 mm) to be highly effected by their morphological structure. Relatively young pellets, in an early stage of the batch fermentation, possessed a homogeneous and dense structure. These pellets were only partly penetrated by oxygen (ca. 70 μm) at air saturated bulk conditions. Older pellets, in a final stage of the batch fermentation, were stratified and fluffy. They were completely penetrated by oxygen due to a decreased activity and a higher diffusivity. Investigations with glucose microelectrodes revealed that glucose consumption inside pellets of all lifetimes exclusively occurred in the periphery, indicating that growth was restricted to these regions only.