Estimation of true growth and product yields in aerobic cultures

In the microbial production of useful products, it is important to understand the allocation of substrate energy for maintanance, growth, and product formation. Methods are presented to obtain point and 95% confidence interval estimates for the true growth yield parameter, true product yield parameter, and the maintenance parameter. Methods are presented which allow all data to be used simultaneously for those cases where more than the minimum number of measurements are made at each specific growth rate (or dilution rate). Three estimation methods and two forms of the energy allocation equations are investigated. Point estimates are similar for the three methods, but interval estimates are considerably larger for one of the three methods. The results depend on the form of the equations.     

Simultaneous versus sequential optimal experiment design for the identification of multi-parameter microbial growth kinetics as a function of temperature

Optimal experiment design for parameter estimation (OED/PE) has become a popular tool for efficient and accurate estimation of kinetic model parameters. When the kinetic model under study encloses multiple parameters, different optimization strategies can be constructed. The most straightforward approach is to estimate all parameters simultaneously from one optimal experiment (single OED/PE strategy). However, due to the complexity of the optimization problem or the stringent limitations on the system's dynamics, the experimental information can be limited and parameter estimation convergence problems can arise. As an alternative, we propose to reduce the optimization problem to a series of two-parameter estimation problems, i.e., an optimal experiment is designed for a combination of two parameters while presuming the other parameters known. Two different approaches can be followed: (i) all two-parameter optimal experiments are designed based on identical initial parameter estimates and parameters are estimated simultaneously from all resulting experimental data (global OED/PE strategy), and (ii) optimal experiments are calculated and implemented sequentially whereby the parameter values are updated intermediately (sequential OED/PE strategy).This work exploits OED/PE for the identification of the Cardinal Temperature Model with Inflection (CTMI) (Rosso et al., 1993). This kinetic model describes the effect of temperature on the microbial growth rate and encloses four parameters. The three OED/PE strategies are considered and the impact of the OED/PE design strategy on the accuracy of the CTMI parameter estimation is evaluated. Based on a simulation study, it is observed that the parameter values derived from the sequential approach deviate more from the true parameters than the single and global strategy estimates. The single and global OED/PE strategies are further compared based on experimental data obtained from design implementation in a bioreactor. Comparable estimates are obtained, but global OED/PE estimates are, in general, more accurate and reliable.     

Expanded thermodynamic true yield prediction model: adjustments and limitations

Bacterial yield prediction is critical for bioprocess optimization and modeling of natural biological systems. In previous work, an expanded thermodynamic true yield prediction model was developed through incorporating carbon balance and nitrogen balance along with electron balance and energy balance. In the present work, the application of the expanded model is demonstrated in multiple growth situations (aerobic heterotrophs, anoxic, anaerobic heterotrophs, and autolithotrophs). Two adjustments are presented that enable improved prediction when additional information regarding the environmental conditions (pH) or degradation pathway (requirement for oxygenase- or oxidase-catalyzed reactions) is known. A large data set of reported yields is presented and considered for suitability in model validation. Significant uncertainties of literature-reported yield values are described. Evaluation of the model with experimental yield values shows good predictive ability. However, the wide range in reported yields and the variability introduced into the prediction by uncertainty in model parameters, limits comprehensive validation. Our results suggest that the uncertainty of the experimental data used for validation limits further improvement of thermodynamic prediction models.     

Estimation of yield, maintenance, and product formation kinetic parameters in anaerobic fermentations

In many anaerobic fermentation processes, high energy bonds in adenosine triphosphate (ATP) are produced when available electrons are converted from organic substrate into extracellular organic products such as ethanol. The true growth yield and maintenance parameters are directly related to the product formation kinetic parameters for these anaerobic processes. Methods are presented which allow all of the experimental measurements to be used simultaneously to estimate these parameters. Results are presented for several different anaerobic fermentations.     

Analysis of exponential growth data for yoghurt cultures

Obtaining accurate estimates of maximum specific growth rate, growth yield, and product yield is important for many fermentation processes. A systematic procedure is presented to select the exponential growth region and estimate the maximum specific growth rate using the covariate adjustment method with all the available measured variables (i.e. biomass, substrate, and product). The procedure is applied to data collected during growth of pure and mixed cultures of Lactobacillus bulgaricus and Streptococcus thermophilus on 3% dry milk under anaerobic conditions. The estimation procedure gives good estimates with relatively narrow confidence intervals even though biomass concentration is measured by an indirect method. The estimated values of maximum specific growth rate range from 0.2805 h(-1) for S. thermophilus (ATCC-19258) to 0.4672 h(-1) for S. thermophilus (Microlife). Growth and product yields are estimated using regression analysis and the data for the exponential growth region. The growth yields are compared to their theoretical maximum values.     

Estimation of the true growth yield and maintenance coefficient for yoghurt cultures

Point and 95% confidence interval estimates of the true growth yield and maintenance coefficient are reported for four pure cultures andtwo mixed cultures of Lactobacillus bulgaricus and Streptococcus thermophilus grown anaerobically on 3% nonfat dry milk using batch followed by fed batch culture. An improved method of data analysis, which includes separate analysis of the exponential growth region and a new method of selecting smoothing spline functions, is utilized. The parameter estimation results are compared with the theoretical maximum yields for one and two moles of ATP produced per mole of glucose and to the growth yield estimated from the exponential growth region analysis.     

Utilization of statistics and experimental design in data collection and analysis

Application of experimental design techniques to Pirt's yield model shows that it is important to collect data at the lowest and highest specific growth rates. In the fed-batch fermentation process, values of specific growth rate can be varied from the maximum value at the start of the process to very low values near the end of the experiment. Candida utilis was cultivated using batch followed by fed-batch culture with glucose as the main source of carbon and energy. Values of substrate concentration, oxygen consumption, carbon dioxide evolution, liquid volume, flow rate cell concentration, and nitrogen concentration, which was an indirect measure of biomass, were measured. Least-squares estimates of the true biomass energetic yield and maintenance coefficient were obtained using a multivariate statistical analysis procedure referred to as the covariate adjustment procedure. Methods of selecting the best estimates using covariate adjustment are illustrated. The results show that useful parameter estimates with relatively short confidence intervals can be obtained using these statistical methods.     

Expanded thermodynamic model for microbial true yield prediction

Thermodynamic methods to predict true yield and stoichiometry of bacterial reactions have been widely used in biotechnology and environmental engineering. However, yield predictions are often inaccurate for certain simple organic compounds. This work evaluates an existing method and identifies the cause of prediction errors for compounds with low degree of reductance of carbon. For these compounds, carbon, not energy or reducing equivalents, constrains growth. Existing thermodynamically-based models do not account for the potential of carbon-limited growth. The improved method described here consists of four balances: carbon balance, nitrogen balance, electron balance, and energy balance. Two efficiency terms, K1 and K2 are defined and estimated from a priori analysis. The results show that K1 and K2 are nearly the same in value so that only one coefficient, K = 0.41 is used in the modified model. Comparisons with observed yields show that use of the new model and parameters results in significantly improved yield estimation based on inclusion of the carbon balance. The average estimation error is less than 6% for the data set presented.     

Assessment of macroenergetic parameters for an anaerobic upflow biomass bed and filter (UBF) reactor

This article reports the steady-state performance of two hybrid anaerobic digesters treating soluble synthetic sugar wastes of 1 and 0.5% strength and the assessment of the associated macroenergetic parameters (growth yield, so-called maintenance coefficient). A theoretical development shows a "nongrowth" parameter concept to be more appropriate than maintenance or decay. Combined energy and mass balances are used to develop a model for growth rate which compares well with experimental data. The COD removal efficiency had no significant effect on growth yield and the maintenance parameter, although a dual combined balance indicated the possibility of such an effect. Macroenergetic parameters did not vary significantly with the specific feeding rate of the system. We thus conclude that a single model may be used over a broad range of feeding and performance conditions.     

Identifiability and retrievability of unique parameters describing intrinsic Andrews kinetics

A key factor contributing to the variability in the microbial kinetic parameters reported from batch assays is parameter identifiability, i.e., the ability of the mathematical routine used for parameter estimation to provide unique estimates of the individual parameter values. This work encompassed a three-part evaluation of the parameter identifiability of intrinsic kinetic parameters describing the Andrews growth model that are obtained from batch assays. First, a parameter identifiability analysis was conducted by visually inspecting the sensitivity equations for the Andrews growth model. Second, the practical retrievability of the parameters in the presence of experimental error was evaluated for the parameter estimation routine used. Third, the results of these analyses were tested using an example data set from the literature for a self-inhibitory substrate. The general trends from these analyses were consistent and indicated that it is very difficult, if not impossible, to simultaneously obtain a unique set of estimates of intrinsic kinetic parameters for the Andrews growth model using data from a single batch experiment.     

Efficient parameter estimation for RNA secondary structure prediction

MOTIVATION: Accurate prediction of RNA secondary structure from the base sequence is an unsolved computational challenge. The accuracy of predictions made by free energy minimization is limited by the quality of the energy parameters in the underlying free energy model. The most widely used model, the Turner99 model, has hundreds of parameters, and so a robust parameter estimation scheme should efficiently handle large data sets with thousands of structures. Moreover, the estimation scheme should also be trained using available experimental free energy data in addition to structural data. RESULTS: In this work, we present constraint generation (CG), the first computational approach to RNA free energy parameter estimation that can be efficiently trained on large sets of structural as well as thermodynamic data. Our CG approach employs a novel iterative scheme, whereby the energy values are first computed as the solution to a constrained optimization problem. Then the newly computed energy parameters are used to update the constraints on the optimization function, so as to better optimize the energy parameters in the next iteration. Using our method on biologically sound data, we obtain revised parameters for the Turner99 energy model. We show that by using our new parameters, we obtain significant improvements in prediction accuracy over current state of-the-art methods. AVAILABILITY: Our CG implementation is available at http://www.rnasoft.ca/CG/.     

The stoichiometry and energetics of oxygenic phototrophic growth

The values of gross metabolic flows in cells are essentially interconnected due to conservation laws of chemical elements and interrelations of biochemical coupling. Therefore, the overall stoichiometry of cellular metabolism, such as the biomass quantum yield, the ratio between linear and circular flows via the electron transport chain, etc., can be calculated using balances of metabolic flows in the network branching points and coupling ratios related to ATP formation and expenditures. This work has studied the energetic stoichiometry of photosynthetic cells by considering the transfer of reductivity in the course of biochemical reactions. This approach yielded rigorous mathematical expressions for biomass quantum yield and other integral bioenergetic indices of cellular growth as functions of ATP balance parameters. The effect of cellular substance turnover has been taken into account. The obtained theoretical estimation of biomass quantum yield is rather close to experimental data which confirms the predictive capacity of this approach.     

Quantifying nonlinear anisotropic elastic material properties of biological tissue by use of membrane inflation

Determination of material parameters for soft tissue frequently involves regression of material parameters for nonlinear, anisotropic constitutive models against experimental data from heterogeneous tests. Here, parameter estimation based on membrane inflation is considered. A four parameter nonlinear, anisotropic hyperelastic strain energy function was used to model the material, in which the parameters are cast in terms of key response features. The experiment was simulated using finite element (FE) analysis in order to predict the experimental measurements of pressure versus profile strain. Material parameter regression was automated using inverse FE analysis; parameter values were updated by use of both local and global techniques, and the ability of these techniques to efficiently converge to a best case was examined. This approach provides a framework in which additional experimental data, including surface strain measurements or local structural information, may be incorporated in order to quantify heterogeneous nonlinear material properties.     

FEDMA--a simple algorithm for theoretical modeling of linear metabolic pathways: from fuzzy data sets to prediction and experiment

A theoretical model of a chain of irreversible Michaelis-Menten reactions proceeding inside a living cell, taking cell growth, division and subcellular compartmentation into account, was proposed. It became a basis for the construction of a "fuzzy" enzymatic data-modeling algorithm (FEDMA) - a procedure allowing the estimation of missing parameter values for the modeled system, in accordance both with the derived theoretical rules and the available experimental data. The obtained tool was tested to model the heme biosynthesis pathway in Saccharomyces cerevisiae, where about 40% of parameters remain unknown. The missing parameters estimated by means of FEDMA fall in the range of expected values.     

Sensitivity analysis and parameter estimation in a model of anaerobic waste water treatment processes with substrate inhibition

Anaerobic waste water treatment processes are commonly presented by the fifth order Hill and Barth non-linear model, describing three main stages of anaerobic digestion. The model investigated in the present work is a modified version of the Hill and Barth model, which includes substrate inhibition of growth of methanogenic bacteria. Parameter estimation of this model is a difficult problem because of the high number of parameters to be estimated and the rather restricted information concerning the variables. The aim of the present work is to use sensitivity theory to find a method for selecting the most significant parameters. In particular, we develop a sensitivity model for anaerobic digestion using relative sensitivity functions. Simulation results from the sensitivity model show that the number of parameters to be estimated can be reduced from 12 to 4. We suggest a 2-step procedure for parameter estimation, which is also based on sensitivity analysis. This procedure gives results, which allow for off-line determination of parameter values if the experimental data for biogas production rate are known.     

Determination of parameter identifiability in nonlinear biophysical models: A Bayesian approach

A major goal of biophysics is to understand the physical mechanisms of biological molecules and systems. Mechanistic models are evaluated based on their ability to explain carefully controlled experiments. By fitting models to data, biophysical parameters that cannot be measured directly can be estimated from experimentation. However, it might be the case that many different combinations of model parameters can explain the observations equally well. In these cases, the model parameters are not identifiable: the experimentation has not provided sufficient constraining power to enable unique estimation of their true values. We demonstrate that this pitfall is present even in simple biophysical models. We investigate the underlying causes of parameter non-identifiability and discuss straightforward methods for determining when parameters of simple models can be inferred accurately. However, for models of even modest complexity, more general tools are required to diagnose parameter non-identifiability. We present a method based in Bayesian inference that can be used to establish the reliability of parameter estimates, as well as yield accurate quantification of parameter confidence.     

Development of a phenomenological modeling approach for prediction of growth and xanthan gum production using Xanthomonas campestris

An unstructured kinetic model for xanthan production is described and fitted to experimental data obtained in a stirred batch reactor. The culture medium was composed of several nitrogen sources (soybean hydrolysates, ammonium and nitrate salts) consumed sequentially. The model proposed is able to describe this sequential consumption of nitrogen sources, the consumption of inorganic phosphate and carbon, the evolution of biomass, and production of xanthan. The parameter estimation has been performed by fitting the kinetic model in differential form to experimental data. Runs of the model for simulating xanthan gum production as a function of the initial concentration of inorganic phosphate have shown the positive effect of phosphate limitation on xanthan yield, though diminishing rates of production. The model was used to predict the kinetic parameters for a medium containing a 2-fold lower initial phosphate concentration. When tested experimentally, the measured fermentation parameters were in close agreement with the predicted model values, demonstrating the validity of the model.     

In search of a thermodynamic description of biomass yields for the chemotrophic growth of microorganisms

Correlations for the prediction of biomass yields are valuable, and many proposals based on a number of parameters (Y(ATP), Y(Ave), eta(o), Y(c), Gibbs energy efficiencies, and enthalpy efficiencies) have been published. This article critically examines the properties of the proposed parameters with respect to the general applicability to chemotrophic growth systems, a clear relation to the Second Law of Thermodynamics, the absence of intrinsic problems, and a requirement of only black box information. It appears that none of the proposed parameters satisfies all these requirements. Particularly, the various energetic efficiency parameters suffer from major intrinsic problems. However, this article will show that the Gibbs energy dissipation per amount of produced biomass (kJ/C-mod) is a parameter which satisfies the requirements without having intrinsic problems. A simple correlation is found which provides the Gibbs energy dissipation/C-mol biomass as a function of the nature of the C-source (expressed as the carbon chain length and the degree of reduction). This dissipation appears to be nearly independent of the nature of the electron acceptor (e.g., O(2), No(3) (-), fermentation). Hence, a single correlation can describe a very wide range of microbial growth systems. In this respect, Gibbs energy dissipation is much more useful than heat production/C-mol biomass, which is strongly dependent on the electron acceptor used. Evidence is presented that even a net heat-uptake can occur in certain growth systems.The correlation of Gibbs energy dissipation thus obtained shows that dissipation/C-mol biomass increases for C-sources with smaller chain length (C(6) --> C(1)), and increases for both higher and lower degrees of reduction than 4. It appears that the dissipation/C-mol biomass can be regarded as a simple thermodynamic measure of the amount of biochemical "work" required to convert the carbon source into biomass by the proper irreversible carbon-carbon coupling and oxidation/reduction reactions. This is supported by the good correlation between the theoretical ATP requirement for biomass formation on different C-sources and the dissipation values (kJ/C-mol biomass) found. The established correlation for the Gibbs energy dissipation allows the prediction of the chemotrophic biomass yield on substrate with an error of 13% in the yield range 0.01 to 0.80 C-mol biomass/(C)-mol substrate for aerobic/anaerobic/denitrifying growth systems.