Minimax D-optimal designs for the logistic model

We propose an algorithm for constructing minimax D-optimal designs for the logistic model when only the ranges of the values for both parameters are assumed known. Properties of these designs are studied and compared with optimal Bayesian designs and Sitter's (1992, Biometrics, 48, 1145-1155) minimax D-optimal kk-designs. Examples of minimax D-optimal designs are presented for the logistic and power logistic models, including a dose-response design for rheumatoid arthritis patients.     

A single mutation in the NAD-specific formate dehydrogenase from Candida methylica allows the enzyme to use NADP

An homology model of Candida methylica formate dehydrogenase (cmFDH) was constructed based on the Pseudomonas sp. 101 formate dehydrogenase (psFDH) structure. An aspartic acid residue in the model was predicted to interact with the adenine ribose of the NAD cofactor, in common with many NAD-dependent oxoreductases. Replacement of this aspartic acid residue by serine in cmFDH removed the absolute requirement for NAD over NADP shown by the wild type enzyme. Taken with similar results shown by d- and l-lactate dehydrogenases, this suggests that an aspartic acid in this position is a major determinant of coenzyme specificity in NAD/NADP-dependent dehydrogenases.     

Estimating the energetic contribution of hydrogen bonding to the stability of Candida methylica formate dehydrogenase by using double mutant cycle

An homology model of Candida methylica formate dehydrogenase (cm FDH) was constructed based on the Pseudomonas sp. 101 formate dehydrogenase (ps FDH) structure. In wild type cm FDH, Thr169 and Thr226 can form hydrogen bonds with each other. We measured the interaction energy between the two threonines independent of other interactions in the proteins by using a so-called double mutant cycle and assessing the protein stability from the concentration of guanidine hydrochloride needed to denature 50% of the molecules. We conclude that the hydrogen bonds stabilize the wild type protein by -4 kcal mol(-1).     

Macrokinetic model for Gluconobacter oxydans in 2-keto-L-gulonic acid mixed culture

A set of kinetic models have been developed for the production of 2-keto-L-gulonic acid from L-sorbose by a mixed culture of Gluconobacter oxydans and Bacillus megaterium. A metabolic pathway is proposed for Gluconobacter oxydans, and a macrokinetic model has been developed for Gluconobacter oxydans, where the balances of some key metabolites, ATP and NADH are taken into account. An unstructured model is proposed for concomitant bacterium Bacillus megaterium. In the macrokinetic model and unstructured model, the mechanism of interaction between Gluconobacter oxydans and Bacillus megaterium is investigated and modeled. The specific substrate uptake rate and the specific growth rate obtained from the macrokinetic model are then coupled into a bioreactor model such that the relationship between the substrate feeding rate and the main state variables, such as the medium volume, the biomass concentrations, the substrate, and the is set up. A closed loop regulator model is introduced to approximate the induction of enzyme pool during lag phase after inoculation. Experimental results demonstrate that the model is able to describe 2-keto-L-gulonic acid fermentation process with reasonable accuracy.     

A new method for the adaptive determination of optimum pH and temperature

An adaptive control algorithm for the on-line determination of optimal temperature or pH for biomass production in a continuous fermentor is presented. The algorithm requires no prior information and uses a dynamic Hammerstein model to identify parameters and to estimate an optimal steady-state control value. A check of the estimated performance measure second derivative is included to ensure that the target extremum is an optimum. The process is driven towards this optimum with a variable step size that depends on the quality of the on-line identified model. Numerical simulations are performed on a dynamic chemostat model that incorporates a metabolic time delay. The algorithm successfully finds the optimum temperature or pH values and maintains the reactor at the optimum steady state.     

Optimal dynamic experiments for bioreactor model discrimination

 This paper describes a general approach fordynamic model discrimination for continuous cultures and presents dynamic models for pure cultures of E. coli and C. utilis obtained using the method. For each pure culture system, four candidate models representing various levels of structure were considered. All models reduce to Monod growth kinetics at steady state. An optimized set of multivariable step inputs in selected manipulative variables was used to discriminate between candidate models. The models that best predicted the dynamic behavior were selected by comparison of model predictions with experimental data. Two discrimination functions were compared in terms of their ability to determine the optimal set of multivariable step inputs to discriminate between candidate models. Results indicate that model discrimination based on maximizing the minimum absolute difference between any two models for a given set of inputs possessed good potential for discrimination between candidate models. Models selected for E. coli andC. utilis from the model discrimination work arepresented and compared with experimental data. Received: 24 May 1994/Received revision: 28 September 1994/Accepted: 5 December 1994     

Solid-state production of polygalacturonase by Aspergillus sojae ATCC 20235

The effect of solid substrates, inoculum and incubation time were studied using response surface methodology (RSM) for the production of polygalacturonase enzyme and spores in solid-state fermentation using Aspergillus sojae ATCC 20235. Two-stage optimization procedure was applied using D-optimal and face-centered central composite design (CCD). Crushed maize was chosen as the solid substrate, for maximum polygalacturonase enzyme activity based on D-optimal design. Inoculum and incubation time were determined to have significant effect on enzyme activity and total spore (p<0.01) based on the results of CCD. A second order polynomial regression model was fitted and was found adequate for individual responses. All two models provided an adequate R(2) of 0.9963 (polygalacturonase) and 0.9806 (spores) (p<0.001). The individual optimum values of inoculum and incubation time for maximum production of the two responses were 2 x 10(7) total spores and 5-6 days. The predicted enzyme activity (30.55 U/g solid) and spore count (2.23 x 10(7)spore/ml) were very close to the actual values obtained experimentally (29.093 U/g solid and 2.31 x 10(7)spore/ml, respectively). The overall optimum region considering the two responses together, overlayed with the individual optima. Solid-state fermentation provided 48% more polygalacturonase activity compared to submerged fermentation under individually optimized conditions.     

Modeling of glycerol production by fermentation in different reactor states

A kinetic model of glycerol production by fermentation with the osmophilic yeast Candida krusei was studied firstly by analogies to published works. Considering that the glycerol produced competes with glucose, as a second carbon source for energy maintenance, mathematical models of glucose utilization and glycerol accumulation were modified further. By adjusting only two variable macrokinetic parameters, K_S and β, the model simulations could fit experimental data well when the reactor was changed from Airlift Loop Reactor in different scale or airlift mode to Stirred Vessel. To avoid a significant reduction in glycerol production in the latter fermentation stage, the final condition of the fermentation, determined by the concentration ratio of glycerol to glucose, was also investigated in four different Reactor States. The kinetic models and simulation results can provide certain reference for scale up of glycerol production by fermentation.     

Maximin D-optimal designs for longitudinal mixed effects models

In this article, the optimal selection and allocation of time points in repeated measures experiments is considered. D-optimal cohort designs are computed numerically for the first- and second-degree polynomial models with random intercept, random slope, and first-order autoregressive serial correlations. Because the optimal designs are locally optimal, it is proposed to use a maximin criterion. It is shown that, for a large class of symmetric designs, the smallest relative efficiency over the model parameter space is substantial.     

Generalized linear latent variable models for repeated measures of spatially correlated multivariate data

Observations of multiple-response variables across space and over time occur often in environmental and ecological studies. Compared to purely spatial models for a single response variable in the exponential family of distributions, fewer statistical tools are available for multiple-response variables that are not necessarily Gaussian. An exception is a common-factor model developed for multivariate spatial data by Wang and Wall (2003, Biostatistics 4, 569-582). The purpose of this article is to extend this multivariate space-only model and develop a flexible class of generalized linear latent variable models for multivariate spatial-temporal data. For statistical inference, maximum likelihood estimates and their standard deviations are obtained using a Monte Carlo EM algorithm. We also use a novel way to automatically adjust the Monte Carlo sample size, which facilitates the convergence of the Monte Carlo EM algorithm. The methodology is illustrated by an ecological study of red pine trees in response to bark beetle challenges in a forest stand of Wisconsin.     

DESIGN: computerized optimization of experimental design for estimating Kd and Bmax in ligand binding experiments. I. Homologous and heterologous binding to one or two classes of sites

We have developed a versatile computer program for optimization of ligand binding experiments (e.g., radioreceptor assay system for hormones, drugs, etc.). This optimization algorithm is based on an overall measure of precision of the parameter estimates (D-optimality). The program DESIGN uses an exact mathematical model of the equilibrium ligand binding system with up to two ligands binding to any number of classes of binding sites. The program produces a minimal list of the optimal ligand concentrations for use in the binding experiment. This potentially reduces the time and cost necessary to perform a binding experiment. The program allows comparison of any proposed experimental design with the D-optimal design or with assay protocols in current use. The level of nonspecific binding is regarded as an unknown parameter of the system, along with the affinity constant (Kd) and binding capacity (Bmax). Selected parameters can be fixed at constant values and thereby excluded from the optimization algorithm. Emphasis may be placed on improving the precision of a single parameter or on improving the precision of all the parameters simultaneously. We present optimal designs for several of the more commonly used assay protocols (saturation binding with a single labeled ligand, competition or displacement curve, one or two classes of binding sites), and evaluate the robustness of these designs to changes in parameter values of the underlying models. We also derive the theoretical D-optimal design for the saturation binding experiment with a homogeneous receptor class.     

Modeling isotopomer distributions in biochemical networks using isotopomer mapping matrices

Within the last decades NMR spectroscopy has undergone tremendous development and has become a powerful analytical tool for the investigation of intracellular flux distributions in biochemical networks using (13)C-labeled substrates. Not only are the experiments much easier to conduct than experiments employing radioactive tracer elements, but NMR spectroscopy also provides additional information on the labeling pattern of the metabolites. Whereas the maximum amount of information obtainable with (14)C-labeled substrates is the fractional enrichment in the individual carbon atom positions, NMR spectroscopy can also provide information on the degree of labeling at neighboring carbon atom positions by analyzing multiplet patterns in NMR spectra or using 2-dimensional NMR spectra. It is possible to quantify the mole fractions of molecules that show a specific labeling pattern, i.e., information of the isotopomer distribution in metabolite pools can be obtained. The isotopomer distribution is the maximum amount of information that in theory can be obtained from (13)C-tracer studies. The wealth of information contained in NMR spectra frequently leads to overdetermined algebraic systems. Consequently, fluxes must be estimated by nonlinear least squares analysis, in which experimental labeling data is compared with simulated steady state isotopomer distributions. Hence, mathematical models are required to compute the steady state isotopomer distribution as a function of a given set of steady state fluxes. Because 2(n) possible labeling patterns exist in a molecule of n carbon atoms, and each pattern corresponds to a separate state in the isotopomer model, these models are inherently complex. Model complexity, so far, has restricted usage of isotopomer information to relatively small metabolic networks. A general methodology for the formulation of isotopomer models is described. The model complexity of isotopomer models is reduced to that of classical metabolic models by expressing the 2(n) isotopomer mass balances of a metabolite pool in a single matrix equation. Using this approach an isotopomer model has been implemented that describes label distribution in primary carbon metabolism, i.e., in a metabolic network including the Embden-Meyerhof-Parnas and pentose phosphate pathway, the tricarboxylic acid cycle, and selected anaplerotic reaction sequences. The model calculates the steady state label distribution in all metabolite pools as a function of the steady state fluxes and is applied to demonstrate the effect of selected anaplerotic fluxes on the labeling pattern of the pathway intermediates. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55:831-840, 1997.     

Modular organization of FDH: Exploring the basis of hydrolase catalysis

An abundant enzyme of liver cytosol, 10-formyltetrahydrofolate dehydrogenase (FDH), is an interesting example of a multidomain protein. It consists of two functionally unrelated domains, an aldehyde dehydrogenase-homologous domain and a folate-binding hydrolase domain, which are connected by an approximately 100-residue linker. The amino-terminal hydrolase domain of FDH (Nt-FDH) is a homolog of formyl transferase enzymes that utilize 10-formyl-THF as a formyl donor. Interestingly, the concerted action of all three domains of FDH produces a new catalytic activity, NADP+-dependent oxidation of 10-formyltetrahydrofolate (10-formyl-THF) to THF and CO2. The present studies had two objectives: First, to explore the modular organization of FDH through the production of hybrid enzymes by domain replacement with methionyl-tRNA formyltransferase (FMT), an enzyme homologous to the hydrolase domain of FDH. The second was to explore the molecular basis for the distinct catalytic mechanisms of Nt-FDH and related 10-formyl-THF utilizing enzymes. Our studies revealed that FMT cannot substitute for the hydrolase domain of FDH in order to catalyze the dehydrogenase reaction. It is apparently due to inability of FMT to catalyze the hydrolysis of 10-formyl-THF in the absence of the cosubstrate of the transferase reaction despite the high similarity of the catalytic centers of the two enzymes. Our results further imply that Ile in place of Asn in the FDH hydrolase catalytic center is an important determinant for hydrolase catalysis as opposed to transferase catalysis.     

Synthesis of allysine ethylene acetal using phenylalanine dehydrogenase from Thermoactinomyces intermedius

Allysine ethylene acetal [(S)-2-amino-5-(1,3-dioxolan-2-yl)-pentanoic acid (2)] was prepared from the corresponding keto acid by reductive amination using phenylalanine dehydrogenase (PDH) from Thermoactinomyces intermedius ATCC 33205. Glutamate, alanine, and leucine dehydrogenases, and PDH from Sporosarcina species (listed in order of increasing effectiveness) also gave the desired amino acid but were less effective. The reaction requires ammonia and NADH. NAD produced during the reaction was recyled to NADH by the oxidation of formate to CO(2) using formate dehydrogenase (FDH). PDH was produced by growth of T. intermedius ATCC 33205 or by growth of recombinant Escherichia coli or Pichia pastoris expressing the Thermoactinomyces enzyme. Using heat-dried T. intermedius as a source of PDH and heat-dried Candida boidinii SC13822 as a source of FDH,98%, but production of T. intermedius could not be scaled up. Using heat-dried recombinant E. coli as a source of PDH and heat-dried Candida boidinii 98%. In a third generation process, heat-dried methanol-grown P. pastoris expressing endogenous FDH and recombinant Thermoactinomyces98% ee.     

Ensemble Modeling for Aromatic Production in Escherichia coli

Ensemble Modeling (EM) is a recently developed method for metabolic modeling, particularly for utilizing the effect of enzyme tuning data on the production of a specific compound to refine the model. This approach is used here to investigate the production of aromatic products in Escherichia coli. Instead of using dynamic metabolite data to fit a model, the EM approach uses phenotypic data (effects of enzyme overexpression or knockouts on the steady state production rate) to screen possible models. These data are routinely generated during strain design. An ensemble of models is constructed that all reach the same steady state and are based on the same mechanistic framework at the elementary reaction level. The behavior of the models spans the kinetics allowable by thermodynamics. Then by using existing data from the literature for the overexpression of genes coding for transketolase (Tkt), transaldolase (Tal), and phosphoenolpyruvate synthase (Pps) to screen the ensemble, we arrive at a set of models that properly describes the known enzyme overexpression phenotypes. This subset of models becomes more predictive as additional data are used to refine the models. The final ensemble of models demonstrates the characteristic of the cell that Tkt is the first rate controlling step, and correctly predicts that only after Tkt is overexpressed does an increase in Pps increase the production rate of aromatics. This work demonstrates that EM is able to capture the result of enzyme overexpression on aromatic producing bacteria by successfully utilizing routinely generated enzyme tuning data to guide model learning.     

Scale-up and application of a cyclone reactor for fermentation processes

A cyclone reactor for microbial fermentation processes was developed with high oxygen transfer capabilities. Three geometrically similar cyclone reactors with 0.5?l, 2.5?l and 15?l liquid volume, respectively, were characterized with respect to oxygen mass transfer, mixing time and residence time distribution. Semi-empirically correlations for prediction of oxygen mass transfer and mixing times were identified for scale-up of cyclone reactors. A volumetric oxygen mass transfer coefficient k _L a of 1.0?s^?1 (available oxygen transfer rate with air: 29?kg?m^?3?h^?1) was achieved with the cyclone reactor at a volumetric power input of 40?kW?m^?3 and an aeration gas flow rate of 0.2?s^?1. Continuous methanol controlled production of formate dehydrogenase (FDH) with Candida boidinii in a 15?l cyclone reactor resulted in more than 100% improvement in dry cell mass concentration (64.5?g?l^?1) and in about 100% improvement in FDH space-time yield (300?U?l^?1?h^?1) compared to steady state results of a continuous stirred tank reactor.     

D-optimal design applied to binding saturation curves of an enkephalin analog in rat brain

The D-optimal design, a minimal sample design that minimizes the volume of the joint confidence region for the parameters, was used to evaluate binding parameters in a saturation curve with a view to reducing the number of experimental points without loosing accuracy in binding parameter estimates. Binding saturation experiments were performed in rat brain crude membrane preparations with the opioid mu-selective ligand [3H]-[D-Ala2,MePhe4,Gly-ol5]enkephalin (DAGO), using a sequential procedure. The first experiment consisted of a wide-range saturation curve, which confirmed that [3H]-DAGO binds only one class of specific sites and non-specific sites, and gave information on the experimental range and a first estimate of binding affinity (Ka), capacity (Bmax) and non-specific constant (k). On this basis the D-optimal design was computed and sequential experiments were performed each covering a wide-range traditional saturation curve, the D-optimal design and a splitting of the D-optimal design with the addition of 2 points (+/- 15% of the central point). No appreciable differences were obtained with these designs in parameter estimates and their accuracy. Thus sequential experiments based on D-optimal design seem a valid method for accurate determination of binding parameters, using far fewer points with no loss in parameter estimation accuracy.     

Three algorithms for interpreting models consisting of ordinary differential equations: Sensitivity coefficients,sensitivity functions,global optimization

This paper reports the development and application of three powerful algorithms for the analysis and simulation of mathematical models consisting of ordinary differential equations. First, we describe an extended parameter sensitivity analysis: we measure the relative sensitivities of many dynamical behaviors of the model to perturbations of each parameter. We check sensitivities to parameter variation over both small and large ranges. These two extensions of a common technique have applications in parameter estimation and in experimental design. Second, we compute sensitivity functions, using an efficient algorithm requiring just one model simulation to obtain all sensitivities of state variables to all parameters as functions of time. We extend the analysis to a behavior which is not a state variable. Third, we present an unconstrained global optimization algorithm, and apply it in a novel way: we determine the input to the model, given an optimality criterion and typical outputs. The algorithm itself is an efficient one for high-order problems, and does not get stuck at local extrema. We apply the sensitivity analysis, sensitivity functions, and optimization algorithm to a sixth-order nonlinear ordinary differential equation model for human eye movements. This application shows that the algorithms are not only practicable for high-order models, but also useful as conceptual tools.     

Growth models of cultures with two liquid phases. VI. Parameter estimation and statistical analysis

Parameter estimation studies have been conducted employing mathematical models developed previously by the investigators and experimental data collected by the last author. A batch fermentation process in which Candida lipolytica were cultured on n-hexadecane dissolved in dewaxed gas oil was employed to obtain the experimental data. The kinetic data from a number of batch experiments conducted at different initial substrate concentrations and different dispersed phase volume fractions were analyzed assuming that, the basic model parameters (maximum specific growth rate, saturation constant, substrate phase equilibrium constant, adsorption constant, desorption constant, etc.) did not change from experiment to experiment. The Gauss-Newton method with modification by Greenstadt, Eisenpress, Bard, and Carroll was used to minimize the conventional sum of squares criterion on the IBM 300/50 computer. The individual confidence intervals were obtained for each individual parameter. Tin- models were compared employing the F-test for equality of variances and an analysis of residuals. For the two best models, the estimated parameter values were compared with available experimental information. The results showed good agreement between the experimental data and the values predicted by the mathematical models. The results presented in this work did suggest that growth on small segregated drops may be more important than continuous phase growth on dissolved substrate.     

Effect of surface electrostatic interactions on the stability and folding of formate dehydrogenase from Candida methylica

NAD⁺-dependent formate dehydrogenase (FDH-EC 1.2.1.2) is an important enzyme to regenerate valuable NADH required by NAD⁺-dependent oxidoreductases in enzyme catalysis. The limitation in the thermostability of FDH enzyme is a crucial problem for development of biotechnological and industrial processes, despite of its advantages. In this study, to investigate the contribution of surface electrostatic interaction to the thermostability of FDH from Candida methylica (cmFDH) N187E, H13E, Q105R, N300E, N147R N300E/N147R, N187E/Q105R, N187E/N147R,Y160R, Y302R, Y160E and Y302E mutants were designed using a homology model of cmFDH based on Candida boidinii (cb) by considering electrostatic interactions on the protein surface. The effects of site-specific engineering on the stability of this molecule was analyzed according to minimal model of folding and assembly reaction and deduced equilibrium properties of the native system with respect to its thermal and denaturant sensitivities. It was observed that mutations did not change the unfolding pattern of native cmFDH and increased numbers of electrostatic interactions can cause either stabilizing or destabilizing effect on the thermostability of this protein. The thermodynamic and kinetic results suggested that except relatively improved mutants, three out of the nine single mutations increased the melting temperature of cmFDH enzyme.