Evaluation of data transformations used with the square root and schoolfield models for predicting bacterial growth rate.

A comparison was made between mathematical variations of the square root and Schoolfield models for predicting growth rate as a function of temperature. The statistical consequences of square root and natural logarithm transformations of growth rate use in several variations of the Schoolfield and square root models were examined. Growth rate variances of Yersinia enterocolitica in brain heart infusion broth increased as a function of temperature. The ability of the two data transformations to correct for the heterogeneity of variance was evaluated. A natural logarithm transformation of growth rate was more effective than a square root transformation at correcting for the heterogeneity of variance. The square root model was more accurate than the Schoolfield model when both models used natural logarithm transformation.     

Comparison of a quadratic response surface model and a square root model for predicting the growth rate of Yersinia enterocolitic

Polynomial equations, relating the growth rate of Yersinia enterocolitica to temperature (0–25°C) and pH (4.5–6-5) in a liquid medium were constructed for four different acidulants. The logarithm of the time for a 100-fold increase in bacterial numbers could be represented by a quadratic response surface function of pH and temperature. The interactions between pH and temperature on growth rate were found to be additive. Values for a 2 log cycle increase in growth derived from the model were in good agreement with experimental values. Predictions from the quadratic model and from a square root model were compared with experimental values in laboratory media and UHT milk. The mean square error (MSE) for the quadratic response surface model was smaller than that for the square root model in 81% of cases. In UHT milk the square root model increasingly underestimated growth rate, as the temperature decreased and would 'fail dangerous' if used for predictive purposes. This indicated that the response surface model is more reliable for predicting the growth of Y. enterocolitica under conditions of sub-optimal temperature and pH.     

New modified Square Root and Schoolfield models for predicting bacterial growth rate as a function of temperature

Summary A new modified Square Root model and two new modified Schoolfield models were evaluated for their ability to predict the growth rate ofYersinia enterocolitica as a function of temperature. The new Square Root model fits the data better than both the original Square Root model and the Zwietering Square Root model. Both new Schoolfield models, a six-and a four-parameter equation, fit the data better than the original Schoolfield model. The new four-parameter Schoolfield model was developed by removing the term describing low temperature inactivation from the new six-parameter Schoolfield model. Inclusion of the two extra parameters in the new six-parameter Schoolfield model (F=318) did not significantly improve the fit compared to the new fourparameter Schoolfield model (F=488).     

Variation of ribosomal proteins with bacterial growth rate.

The composition of ribosomal proteins has been examined as a function of the growth rate of Escherichia coli cells. Seven sets of cultural conditions, utilizing different combinations of carbon and nitrogen sources, were employed to provide a 36-fold spread in growth rate. The cellular content of most of the ribosomal proteins in ribosomes decreased to a similar extent in the very slow-growing cultures. Major exceptions were proteins S6 and L12, which exhibited a much more pronounced decrease , and S21, which exhibited an increase. None of the proteins remained invariant with growth rate.     

An evaluation of models for the effect of plasmid copy number on bacterial growth rate

The rates of growth of recombinant bacteria depend on their plasmid content. This is modelled by expressing the specific growth rate in terms of the number of copies of the plasmid per cell. Three models in common use have been tested with different Escherichia coli strains and one strain of Bacillus stearothermophilus containing different plasmids. While no particular model was decisively better than others for all data, that of Bentley & Quiroga (Biotechnol. Bioeng. 1993, 42: 222–234) was the best for specific growth rates which vary inversely with the plasmid copy number, and a modified form of the model of Satyagal & Agarwal (Biotechnol. Bioeng. 1989, 33: 1135–1144) was the best for growth rates which increase with the copy number. The differences appear to be linked to the plasmid replication mechanisms. Contrary to some claims, no model portrayed the experimentally observed inflection points.     

Effective rate models for the analysis of transport-dependent biosensor data.

Optical biosensors, including the BIACORE, provide an increasingly popular method for determining reaction rates of biomolecules. In a flow chamber, with one reactant immobilized on a chip on the sensor surface, a solution containing the other reactant (the analyte) flows through the chamber. The time course of binding of the reactants is monitored. Scientists using the BIACORE to understand biomolecular reactions need to be able to separate intrinsic reaction rates from the effects of transport in the biosensor. For a model to provide a useful basis for such an analysis, it must reflect transport accurately, while remaining simple enough to couple with a routine for estimating reaction rates from BIACORE data. Models have been proposed previously for this purpose, consisting of an ordinary differential equation with 'effective rate coefficients' incorporating reaction and transport parameters. In this paper we investigate both the theoretical basis and numerical accuracy of these and related models.     

Revised enzyme synthesis rate expression in cybernetic models of bacterial growth

A revised enzyme synthesis rate expression for cybernetic models of bacterial growth is presented. The rate expression, which is comprised of inducible and constitutive contributions, provides for a basal enzyme level that is necessary to predict certain types of commonly observed continuous culture transients. The response of a continuous culture to a step change in feed stream composition is simulated using both the old and new formulations, and the ramifications for the "matching-law" formulation are discussed.     

Hidden Markov models used for the offline classification of EEG data.

Hidden Markov models (HMM) are introduced for the offline classification of single-trail EEG data in a brain-computer-interface (BCI). The HMMs are used to classify Hjorth parameters calculated from bipolar EEG data, recorded during the imagination of a left or right hand movement. The effects of different types of HMMs on the recognition rate are discussed. Furthermore a comparison of the results achieved with the linear discriminant (LD) and the HMM, is presented.     

A model for predicting the transmission rate of malaria from serological data

A model is developed to estimate the duration for which malaria antibody levels in the blood remain high in a closed population. This estimate can be used to calculate the transmission rate within a region, in conjunction with the serological information contained in the population. The model is used on data obtained from a study of malaria in the Philippines and shows excellent agreement. It is subsequently utilised for predictions and seems to be an appropriate vehicle for this purpose.     

Medium-dependent control of the bacterial growth rate

By combining results from previous studies of nutritional up-shifts we here re-investigate how bacteria adapt to different nutritional environments by adjusting their macromolecular composition for optimal growth. We demonstrate that, in contrast to a commonly held view the macromolecular composition of bacteria does not depend on the growth rate as an independent variable, but on three factors: (i) the genetic background (i.e. the strain used), (ii) the physiological history of the bacteria used for inoculation of a given growth medium, and (iii) the kind of nutrients in the growth medium. These factors determine the ribosome concentration and the average rate of protein synthesis per ribosome, and thus the growth rate. Immediately after a nutritional up-shift, the average number of ribosomes in the bacterial population increases exponentially with time at a rate which eventually is attained as the final post-shift growth rate of all cell components. After a nutritional up-shift from one minimal medium to another minimal medium of higher nutritional quality, ribosome and RNA polymerase syntheses are co-regulated and immediately increase by the same factor equal to the increase in the final growth rate. However, after an up-shift from a minimal medium to a medium containing all 20 amino acids, RNA polymerase and ribosome syntheses are no longer coregulated; a smaller rate of synthesis of RNA polymerase is compensated by a gradual increase in the fraction of free RNA polymerase, possibly due to a gradual saturation of mRNA promoters. We have also analyzed data from a recent publication, in which it was concluded that the macromolecular composition in terms of RNA/protein and RNA/DNA ratios is solely determined by the effector molecule ppGpp. Our analysis indicates that this is true only in special cases and that, in general, medium adaptation also depends on factors other than ppGpp.     

Model-based clustering and data transformations for gene expression data.

MOTIVATION: Clustering is a useful exploratory technique for the analysis of gene expression data. Many different heuristic clustering algorithms have been proposed in this context. Clustering algorithms based on probability models offer a principled alternative to heuristic algorithms. In particular, model-based clustering assumes that the data is generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The issues of selecting a 'good' clustering method and determining the 'correct' number of clusters are reduced to model selection problems in the probability framework. Gaussian mixture models have been shown to be a powerful tool for clustering in many applications. RESULTS: We benchmarked the performance of model-based clustering on several synthetic and real gene expression data sets for which external evaluation criteria were available. The model-based approach has superior performance on our synthetic data sets, consistently selecting the correct model and the number of clusters. On real expression data, the model-based approach produced clusters of quality comparable to a leading heuristic clustering algorithm, but with the key advantage of suggesting the number of clusters and an appropriate model. We also explored the validity of the Gaussian mixture assumption on different transformations of real data. We also assessed the degree to which these real gene expression data sets fit multivariate Gaussian distributions both before and after subjecting them to commonly used data transformations. Suitably chosen transformations seem to result in reasonable fits. AVAILABILITY: MCLUST is available at http://www.stat.washington.edu/fraley/mclust. The software for the diagonal model is under development. CONTACT: kayee@cs.washington.edu. SUPPLEMENTARY INFORMATION: http://www.cs.washington.edu/homes/kayee/model.     

Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Model for bacterial culture growth rate throughout the entire biokinetic temperature range.

The "square-root" relationship proposed by Ratkowsky et al. (J. Bacteriol. 149:1-5, 1982) for modeling the growth rate of bacteria below the optimum growth temperature was extended to cover the full biokinetic temperature range. Two of the four parameters of this new nonlinear regression model represent minimum and maximum temperature bounds, respectively, for the predicted growth of the culture. The new model is easy to fit and has other desirable statistical properties. For example, the least-squares estimators of the parameters of the model were almost unbiased and normally distributed. The model applied without exception to all bacterial cultures for which we were able to obtain data. Results for 30 strains are reported.     

Efficient inference for random-coefficient growth curve models with unbalanced data

Growth and dose-response curve studies often result in incomplete or unbalanced data. Random-effects models together with a variety of computer-intensive iterative techniques have been suggested for the analysis of such data. This paper is concerned with a noniterative method for estimating and comparing location parameters in random-coefficient growth curve models. Consistent and asymptotically efficient estimators of the location parameters are obtained using estimated generalized least squares. Two criteria for testing multivariate general linear hypotheses are introduced and their asymptotic properties are investigated. The results are applied to clinical data obtained on the blood ultrafiltration performance of hemodialyzers used in the treatment of patients with end-stage renal disease.     

Linear-Arrhenius models for bacterial growth and death and vitamin denaturations

Summary The development of the quantitative, linear-Arrhenius model of Davey for predicting bacterial growth and death (inactivation) is reviewed. The applicability of the model to published data from independent researchers for both the growth phase and lag phase, involving combined environmental factors (T, a_w) is illustrated. Also illustrated is its applicability to thermal inactivation kinetics and vitamin denaturation (with combinedT, pH). Integration of the model to produce complex models describing the thermal sterilization of liquid is demonstrated. Advantages of the model, including its simplicity and the fact that the coefficients to build the model can be obtained easily by relatively unsophisticated users, are highlighted in a comparison with other models.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Ultrasound increases the rate of bacterial cell growth

Ultrasound was employed to increase the growth rate of bacterial cells attached to surfaces. Staphylococcus epidermidis, Pseudomonas aeruginosa, and Escherichia coli cells adhered to and grew on a polyethylene surface in the presence of ultrasound. It was found that low-frequency ultrasound (70 kHz) of low acoustic intensity (<2 W/cm(2)) increased the growth rate of the cells compared to growth without ultrasound. However, at high intensity levels, cells were partially removed from the surface. Ultrasound also enhanced planktonic growth of S. epidermidis and other planktonic bacteria. It is hypothesized that ultrasound increases the rate of transport of oxygen and nutrients to the cells and increases the rate of transport of waste products away from the cells, thus enhancing their growth.     

Reconciling root plasticity and architectural ground rules in tree root growth models with voxel automata

Dynamic models of tree root growth and function have to reconcile the architectural rules for coarse root topology with the dynamics of fine root growth (and decay) in order to predict the strategic plus opportunistic behaviour of a tree root system in a heterogeneous soil. We present an algorithm for a 3D model based on both local (soil voxel level) and global (tree level) controls of root growth, with development of structural roots as a consequence of fine root function, rather than as driver. The suggested allocation rules of carbon to fine root growth in each rooted voxel depend on the success in water uptake in this voxel during the previous day, relative to overall supply and demand at plant level. The allocated C in each voxel is then split into proliferation (within voxel growth) and extension into neighbouring voxels (colonisation), with scale-dependent thresholds and transfer coefficients. The fine root colonisation process defines a dynamic and spatially explicit demand for transport functions. C allocation to development of a coarse root infrastructure linking all rooted voxels depends on the apparent need for adjustment of root diameter to meet the topologically defined sap flow through this voxel during the previous day. The allometric properties of the coarse root system are maintained to be in line with fractal branching theory. The model can predict the dynamics of the shape and structure (fine root density, coarse root topology and biomass) of the root system either independently of soil conditions (purely genetically-driven) or including both the genetic and environmental effects of roots interacting with soil water supply and its external replenishment, linking in with existing water balance models. Sensitivity of the initial model to voxel dimensions was addressed through explicit scaling rules resulting in scale-independent parameters. The model was parameterised for two tree species: hybrid walnut (Juglans nigra × regia) and wild cherry (Prunus avium L.) using results of a pot experiment. The model satisfactorily predicted the root growth behaviour of the two species. The model is sparse in parameters and yet applicable to heterogeneous soils, and could easily be upgraded to include additional local influences on root growth (and decay) such as local success in nutrient uptake or dynamic soil physical properties.     

Evaluation of diverse soybean germplasm for root growth and architecture

Root characteristics of soybean (Glycine max (L.) Merr.) improve drought avoidance by increasing water uptake from the soil profile. Screening genotypes for improved root architecture without breaking the taproots or losing lateral roots is a challenge. Due to difficulty in separating roots from field or potting soil, a rapid and effective screening method with a suitable growth medium to assess root characteristics under controlled conditions needs to be established. We describe two screening techniques “the cone system” and “the tube system” using turface:sand medium. In the cone system thirty four soybean lines including cultivars and exotic plant introduction (PI) lines were evaluated for tap root length and root biomass, 12 days after sowing. Eight replications per line were grown in a growth chamber. Significant differences among genotypes for tap root length were detected by the cone system. Validity of results from the cone system was tested by evaluating root growth 21 days after planting for eight lines in the tube system. A coefficient of determination of 0.72 indicated good agreement between the two screening systems for evaluating genotypes for rooting depth. The cone system will be a useful method to easily and rapidly assess soybean genotypes for root growth.     

Evaluation of regression models in metabolic physiology: predicting fluxes from isotopic data without knowledge of the pathway

This study explores the ability of regression models, with no knowledge of the underlying physiology, to estimate physiological parameters relevant for metabolism and endocrinology. Four regression models were compared: multiple linear regression (MLR), principal component regression (PCR), partial least-squares regression (PLS) and regression using artificial neural networks (ANN). The pathway of mammalian gluconeogenesis was analyzed using [U−¹³C]glucose as tracer. A set of data was simulated by randomly selecting physiologically appropriate metabolic fluxes for the 9 steps of this pathway as independent variables. The isotope labeling patterns of key intermediates in the pathway were then calculated for each set of fluxes, yielding 29 dependent variables. Two thousand sets were created, allowing independent training and test data. Regression models were asked to predict the nine fluxes, given only the 29 isotopomers. For large training sets (>50) the artificial neural network model was superior, capturing 95% of the variability in the gluconeogenic flux, whereas the three linear models captured only 75%. This reflects the ability of neural networks to capture the inherent non-linearities of the metabolic system. The effect of error in the variables and the addition of random variables to the data set was considered. Model sensitivities were used to find the isotopomers that most influenced the predicted flux values. These studies provide the first test of multivariate regression models for the analysis of isotopomer flux data. They provide insight for metabolomics and the future of isotopic tracers in metabolic research where the underlying physiology is complex or unknown.We acknowledge the support of NIH Grant DK58533 and the DuPont-MIT Alliance.