A kinetic model for product formation of microbial and mammalian cells

Growth of microbial and mammalian cells can be classified into substrate-limited and substrate-sufficient growth according to the relative availability of the substrate (carbon and energy source) and other nutrients. It has been observed for a number of microbial and mammalian cells that the consumption rate of substrate and energy (ATP) is generally higher under substratesufficient conditions than under substrate limitation. Accordingly, the product formation under substrate excess often exhibits different patterns from those under substrate limitation. The extent of increase or decrease in product formation may depend not only on the nature of limitation and cell growth rate but also on the residual substrate concentration in a relatively wide range. The product formation kinetic models existing in literature cannot describe these effects. In this study, the Luedeking-Piret kinetic is extended to include a term describing the effect of residual substrate concentration. The extended model has a similar structure to the kinetic model for substrate and energy consumption rate recently proposed by Zeng and Deckwer. The applicability of the extended model is demonstrated with three microbial cultures for the production of primary metabolites and three hybridoma cell cultures for the production of ammonia and lactic acid over a wide range of substrate concentration. The model describes the product formation in all these cultures satisfactorily. Using this model, the range of residual substrate concentration, in which the product formation is affected, can be quantitatively assessed. (c) 1995 John Wiley & Sons, Inc.     

The kinetics of cometabolism

Experimental observations indicate that the rates of cometabolic transformation are linked to the consumption of growth substrate during growth and to the consumption of cell mass and/or energy substrate in the absence of growth substrate. Three previously proposed models (models 1 through 3) describing the kinetics of cometabolism by resting cells are compared, and the interrelationships and underlying assumptions for these models are explored. Models 1 to 3 are shown to converge at high concentrations of the nongrowth substrate. An expression describing nongrowth substrate transformation in the presence of growth substrate is proposed, and this expression is integrated with an expression for cell growth to give a single unstructured model (model 4) that encompasses models 1 to 3 and describes cometabolism by both resting and growing cells. Model 4 couples transformation of nongrowth substrate to consumption of growth substrate and biomass, and predicts that cometabolism will result, and decreased specific growth rates for a cometabolizing population. Competitive inhibition can also be incorporated in the model. Experimental aspects of model calibration and verification are discussed. The need for models that distinguish between the exhaustion of cell activity and cell death is emphasized. (c) 1993 Wiley & Sons, Inc.     

A mathematical model for microbial growth under limitation by conservative substrates

A mathematical model is suggested for growth of microorganisms under limitation by “conservative” substrates such as inorganic ions or vitamins that are not broken down after uptake into the cells, but that wholely or partly remain available for production of biomass. The specific growth rate is expressed here as a function of the intracellular “concentration” of the limiting substrate, defined as the amount of substrate within the cells per unit of cell dry weight. In the model, the intracellular substrate is divided into two parts. One part is a “structural” substrate not available for further growth. The other part is an “excess” or “functional” substrate that is used for biomass production and is assumed to be converted into structural substrate proportionally to growth. The rate of growth is believed to be controlled by the intracellular concentration of excess substrate.     

Analytic solution to steady-state radial diffusion of a substrate with first-order reaction kinetics in the tissue of a Krogh's cylinder

It is often useful to calculate the concentration profile for a substrate undergoing reaction in the tissue surrounding a capillary. In this paper, we consider a model geometry consisting of a long straight cylinder of tissue surrounding a capillary. Substrate diffuses radially out of the capillary through the tissue, with consumption of substrate in the tissue directly proportional to substrate concentration (i.e., first-order reaction kinetics). The model is extended to include the case where a cylinder of necrotic tissue surrounds a metabolically active inner tissue cylinder. A simple analytic solution is derived, and concentration profiles are generated for various combinations of parameters. Compared to the case where substrate consumption is independent of concentration, this model predicts much more rapid depletion of substrate near the capillary interface. This can have significant implications for the calculation of the hypoxic fraction (e.g., tissue with pO(2)<0.5-5 mmHg) when tumor oxygenation is modeled. The model also permits calculation of the limiting substrate concentration for cell viability when the reaction rate constant is known and vice versa.     

Factor Xa active site substrate specificity with substrate phage display and computational molecular modeling

Structural origin of substrate-enzyme recognition remains incompletely understood. In the model enzyme system of serine protease, canonical anti-parallel beta-structure substrate-enzyme complex is the predominant hypothesis for the substrate-enzyme interaction at the atomic level. We used factor Xa (fXa), a key serine protease of the coagulation system, as a model enzyme to test the canonical conformation hypothesis. More than 160 fXa-cleavable substrate phage variants were experimentally selected from three designed substrate phage display libraries. These substrate phage variants were sequenced and their specificities to the model enzyme were quantified with quantitative enzyme-linked immunosorbent assay for substrate phage-enzyme reaction kinetics. At least three substrate-enzyme recognition modes emerged from the experimental data as necessary to account for the sequence-dependent specificity of the model enzyme. Computational molecular models were constructed, with both energetics and pharmacophore criteria, for the substrate-enzyme complexes of several of the representative substrate peptide sequences. In contrast to the canonical conformation hypothesis, the binding modes of the substrates to the model enzyme varied according to the substrate peptide sequence, indicating that an ensemble of binding modes underlay the observed specificity of the model serine protease.     

A kinetic study of lipase-catalyzed reversible kinetic resolution involving verification at miniplant-scale

Lipase-catalyzed kinetic resolution of racemates is a popular method for synthesis of chiral synthons. Most of these resolutions are reversible equilibrium limited reactions. For the first time, an extensive kinetic model is proposed for kinetic resolution reactions, which takes into account the full reversibility of the reaction, substrate inhibition by an acyl donor and an acyl acceptor as well as alternative substrate inhibition by each enantiomer. For this purpose, the reversible enantioselective transesterification of (R/S)-1-methoxy-2-propanol with ethyl acetate catalyzed by Candida antarctica lipase B (CAL-B) is investigated. The detailed model presented here is valid for a wide range of substrate and product concentrations. Following model discrimination and the application of Haldane equations to reduce the degree of freedom in parameter estimation, the 11 free parameters are successfully identified. All parameters are fitted to the complete data set simultaneously. Six types of independent initial rate studies provide a solid data basis for the model. The effect of changes in substrate and product concentration on reaction kinetics is discussed. The developed model is used for simulations to study the behavior of reaction kinetics in a fixed bed reactor. The typical plot of enantiomeric excess versus conversion of substrate and product is evaluated at various initial substrate mixtures. The model is validated by comparison with experimental results obtained with a fixed bed reactor, which is part of a fully automated state-of-the-art miniplant.     

Substrate deformation determines actin cytoskeleton reorganization: A mathematical modeling and experimental study

A mathematical model has been developed to define the relationship between the actin cytoskeleton reorganization of a cell and substrate deformation acting on the cell. The model is based on the following major assumptions: (a) normal substrate strain, not the shear substrate strain, determines the actin cytoskeleton reorganization; (b) the normal substrate strain is transmitted to individual actin filaments; (c) each actin filament has a basal strain energy (BSE) when the cell adheres to the substrate without stretching; and (d) the actin filaments undergo disassembly when their strain energies are decreased to zero or increased to twice their BSEs. The resulting model predicts that the actin filaments are formed in the direction where their BSEs are minimally altered. This direction is therefore the one without normal substrate strain. The prediction was confirmed by experiments conducted on both fibroblasts and endothelial cells. The present model may be relevant for understanding better the effects of mechanical stimuli on the cells.     

Model of steady-state-biofilm kinetics

A steady-state biofilm is defined as one that has neither net growth nor decay over time. The model, developed for steady-state-biofilm kinetics with a single substrate, couples the flux of substrate into a biofilm to the mass (or thickness) of biofilm that would exist at steady-state for a given bulk substrate concentration. Based on kinetic and energetic constraints, this model predicts for a single substrate that a steady-state bulk concentration, S_min, exists below which a steady-state biofilm cannot exist. Thus, in the absence of adsorption of bacteria from the bulk water and for substrate concentration below S_min, substrate flux and biofilm thickness are zero. Equations are provided for calculating the steady-state substrate flux and biofilm thickness for S greater than S_min. An example is provided to demonstrate the use of the steadystate model.     

A Unified Kinetic Hypothesis of Carrier-Mediated Transport: Its Applications

A model analysis of the process of carrier mediated membrane transport is presented, wherein the carrier is present in two forms of differing affinity for substrate. The two forms of carrier undergo interconversion by asymmetric metabolic reactions on each side of the membrane. From this model system expressions are derived for the steady-state distribution ratio for substrate, for the unidirectional fluxes of substrate and hence for the initial velocity of uptake of substrate, and for the effect of preloading cells upon the initial velocity of uptake of labeled substrate. These expressions are applied to published data for glycine transport in Ehrlich ascites tumor cells to obtain numerical values for the parameters of a concentrative membrane carrier system. Concentrative uptake is shown to be consequent to the differing affinities of the two forms of carrier. When the affinities of the two forms are equal, equilibrative uptake occurs. The model analysis is applied to the phenomena of metabolic and competitive inhibition.     

Analytical effectiveness calculations concerning the degradation of an inhibitive substrate by a steady-state biofilm

A reaction engineering model for the degradation of an inhibitory substrate by a steady-state biofilm is presented. The model describes both the metabolic rate controlling behavior of this substrate in the biofilm and the effect of diffusion limitation caused by an arbitrary substrate on the active biofilm thickness. An analytical expression for the biocatalyst effectiveness factor is presented on the basis of Pirt kinetics for cell maintenance, first order substrate inhibition kinetics, and zero order substrate consumption kinetics. The proposed expression for the biocatalyst effectiveness factor is much more convenient to incorporate into a macroreactor model than the numerical alternatives. Simple criteria are presented to check the applicability of the model in case of true Monod kinetics. The analytical solution is expected to be particularly applicable to processes where a low soluble organic substrate controls the biomass growth, a situation which is often met in wastewater purification processes of industrial importance. The degradation of phenol by Pseudomonas sp. is treated as an example. (c) 1993 John Wiley & Sons, Inc.     

Relating damped oscillations to sustained limit cycles describing real and ideal batch fermentation processes

A batch fermentation model is presented in which the specific growth rate and yield functions are chosen such that sustained oscillations in both the cell and substrate concentration occur. This phenomenon is shown to be a Hopf bifurcation in the underlying system of non-linear ordinary differential equations which comprises the model. It is shown that for oscillations in the substrate concentration to occur it is necessary for the yield term to depend on both the cell and substrate levels.     

Prediction of substrate consumption rate, average biofilm density and active thickness for a thin spherical biofilm at pseudo-steady state

In this work, a new approach is proposed to evaluate substrate consumption rate, average biofilm density and active thickness of a spherical bioparticle in a completely mixed fluidized bed system. The substrate consumption rate and average biofilm density are predicted for a given biofilm surface substrate concentration and operational biofilm thickness. A diffusion and reaction model is developed with an effective diffusion coefficient that depends on the average biofilm density. This approach, a first in the literature, predicts the optimum average density of a biofilm to yield the maximum substrate consumption rate within the biofilm. A reasonable correlation was observed between the model prediction and experimental results for substrate consumption rate and average biofilm density for thin and fully active biofilms.     

A model for cell motility on soft bio-adhesive substrates

Mechanical stiffness of bio-adhesive substrates has been recognized as a major regulator of cell motility. We present a simple physical model to study the crawling locomotion of a contractile cell on a soft elastic substrate. The mechanism of rigidity sensing is accounted for using Schwarz's two-spring model Schwarz et al. (2006). The predicted dependency between the speed of motility and substrate stiffness is qualitatively consistent with experimental observations. The model demonstrates that the rigidity dependent motility of cells is rooted in the regulation of actomyosin contractile forces by substrate deformation at each anchorage point. On stiffer substrates, the traction forces required for cell translocation acquire larger magnitude but show weaker asymmetry which leads to slower cell motility. On very soft substrates, the model predicts a biphasic relationship between the substrate rigidity and the speed of locomotion, over a narrow stiffness range, which has been observed experimentally for some cell types.     

A new modeling technique and computer simulation of bacterial growth

A mathematical model that describes substrate utilization and cell growth in terms of two potentially rate-limiting enzyme systems has been developed. Consideration of substrate inhibition and enzyme repression have been incorporated. The model provides a rational approach for characterizing non-steady-state phenomena. The model has been used to analyze batch test data to illustrate the effects of inhibition, repression, and concurrent substrate utilization. Its utility lies in the fact that it provides a quantitative framework for describing changes in the activity levels of cells that result from changes in substrate concentration and/or substrate type. The lag phase resulting from exposure to a new substrate can be modeled.     

Unstructured model for L-lysine fermentation under controlled dissolved oxygen

An unstructured model was developed for batch cultivation of Corynebacterium lactofermentum (ATCC 21799) under controlled dissolved oxygen. The model is capable of predicting batch experiments performed at various initial substrate concentrations. By extending the batch culture model to a fed-batch model and using a heuristic approach to optimize the fed-batch cultivation, it is shown that fed-batch cultivation is superior to batch operation due to increased productivity at high substrate concentrations.     

Single-Occupancy Binding in Simple Bounded and Unbounded Systems

The number of substrate molecules that can bind to the active site of an enzyme at one time is constrained. This paper develops boundary conditions that correspond to the constraint of single-occupancy binding. Two simple models of substrate molecules diffusing to a single-occupancy site are considered. In the interval model, a fixed number of substrate molecules diffuse in a bounded domain. In the spherical model, a varying number of molecules diffuse in a domain with boundary conditions that model contact with a reservoir containing a large number of substrate molecules. When the diffusive time scale is much shorter than the time scale for entering the single-occupancy site, the dynamics of binding are accurately described by simple approximations.     

Steady state model for evaluation of external and internal mass transfer effects in an immobilized biofilm

A steady model for the evaluation of external liquid film diffusion and internal pore diffusion effects in an immobilized biofilm system under continuous mode of operation was developed. The model takes into account, substrate diffusion through external liquid film and biofilm. Average rate of substrate consumption in the biofilm was considered. The overall efficiency of the biofilm was mathematically represented by considering the combined effects of substrate penetration and substrate utilization in the biofilm. The model was illustrated using a case study of pyridine biodegradation in a rotating biological contactor immobilized with pyridine degrading microbial film. The model is able to effectively predict both internal and external mass transfer effects in an immobilized biofilm system.     

The ideal free distribution and bacterial growth on two substrates

A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.     

A new method to solve a non-steady-state multispecies biofilm model

A transient multispecies model for quantifying microbial space competition in biofilm is derived from existing models, introducing a new approach to biomass detachment modelling. This model includes inert biomass, substrate diffusion and utilization rate within the biofilm and diffusional layers. It predicts the evolution of biofilm thickness, bulk substrate concentration, species distribution and substrate concentration within the biofilm. A zero-dimensional transient model is described. Its steady-state solution is used to set up initial conditions of the one-dimensional model and case computation towards steady-state solution. Some numerical tools have been developed, enabling fast computation on microcomputers. Simulations show the validity of a zero-dimensional model and perturbated systems are also simulated. Simulations with experimental data give acceptable results.     

Role of substrate inhibition kinetics in enzymatic chemical oscillations.

Two chemical kinetic models are investigated using standard nonlinear dynamics techniques to determine the conditions under which substrate inhibition kinetics can lead to oscillations. The first model is a classical substrate inhibition scheme based on Michaelis-Menten kinetics and involves a single substrate. Only when this reaction takes place in a flow reactor (i.e., both substrate and product are taken to follow reversible flow terms) are oscillations observed; however, the range of parameter values over which such oscillations occur is so narrow it is experimentally unobservable. A second model based on a general mechanism applied to the kinetics of many pH-dependent enzymes is also studied. This second model includes both substrate inhibition kinetics as well as autocatalysis through the activation of the enzyme by hydrogen ion. We find that it is the autocatalysis that is always responsible for oscillatory behavior in this scheme. The substrate inhibition terms affect the steady-state behavior but do not lead to oscillations unless product inhibition or multiple substrates are present; this is a general conclusion we can draw from our studies of both the classical substrate inhibition scheme and the pH-dependent enzyme mechanism. Finally, an analysis of the nullclines for these two models allows us to prove that the nullcline slopes must have a negative value for oscillatory behavior to exist; this proof can explain our results. From our analysis, we conclude with a brief discussion of other enzymes that might be expected to produce oscillatory behavior based on a pH-dependent substrate inhibition mechanism.