Use of Stochastic Models To Assess the Effect of Environmental Factors on Microbial Growth

Abstract

We present a novel application of a stochastic ecological model to the study and analysis of microbial growth dynamics as influenced by environmental conditions in an extensive experimental data set. The model proved to be useful in bridging the gap between theoretical ideas in ecology and an applied problem in microbiology. The data consisted of recorded growth curves of Escherichia coli grown in triplicate in a base medium with all 32 possible combinations of five supplements: glucose, NH 4 Cl, HCl, EDTA, and NaCl. The potential complexity of 2 5 experimental treatments and their effects was reduced to 2 2 as just the metal chelator EDTA, the presumed osmotic pressure imposed by NaCl, and the interaction between these two factors were enough to explain the variability seen in the data. The statistical analysis showed that the positive and negative effects of the five chemical supplements and their combinations were directly translated into an increase or decrease in time required to attain stationary phase and the population size at which the stationary phase started. The stochastic ecological model proved to be useful, as it effectively explained and summarized the uncertainty seen in the recorded growth curves. Our findings have broad implications for both basic and applied research and illustrate how stochastic mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding basic processes in microbial ecology.Mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding the interaction of organisms with their physical and biological environment (8,47,48,50,52,56). Studies in the field of predictive microbiology have shown that successful modeling requires both adequate models and thorough data sets (57; for extensive reviews, see references 10, 42, and 55). In predictive microbiology a two-step modeling approach is used (reference 55 and citations therein). Primary models describe the basic rules of how microbial numbers change over time (52). Next, these simple models are used to derive secondary models that account for the effect of a set of factors in microbial growth.The forces of the environment and the events of reproduction and growth are themselves stochastic in nature (3,21,24,32,36,39,44,45,51,53), yet simple ecological models, such as the Verhulst logistic equation, result in deterministic predictions. However, smooth convergence to asymptotic results is not what is usually seen, even in rigorous experimental settings (17, 31). Hence, a more realistic alternative to population growth modeling is to confront stochastic equations with the data at hand (references 5, 14, and 17 and citations therein and references 41 and 47).In this paper, we describe a novel application of a stochastic population model to analyze how environmental conditions influence microbial growth dynamics using an extensive experimental data set. The primary model used was the stochastic Ricker (SR) equation (27, 51). Our secondary model...

Evolution of Diversity in Spatially Structured Escherichia coli Populations

“…The stochastic Ricker model, which accounts for departures from the deterministic predictions that occur in a given time series of population growth (11), was used to fit the time series data from the growth of each clone. The model used incorporated environmental stochasticity (11,40), which accounts for variability in the population growth rate that arises from external factors (such as pH and temperature) that equally affect all the individuals in the population. It expresses the predicted population size as a function of the current population size and accounts for density dependence.…”

“…The MLEs were found by minimization of the negative log-likelihood function using the NelderMead simplex algorithm (37) in MATLAB (The MathWorks, Natick, MA). As with a two-way analysis of variance, the sources of variation were divided into two factors: biofilm depth and growth media (BMG1, BMG1ϩStr, and BMG1ϩAmp) (40). The 64 time series obtained for each of these 6 ϫ 3 experimental conditions were used as replicates to assess the variability between clones.…”

“…The purpose of this analysis was to determine whether the growth dynamics of clones varied within and between depths in the biofilm. Data from 1,152 time series (6 depths ϫ 3 media ϫ 64 clones) were analyzed and then compared to the 64 ϫ 3 time series of clones from the parental population using the stochastic Ricker model of population dynamics (11,40). The stochastic Ricker model, which accounts for departures from the deterministic predictions that occur in a given time series of population growth (11), was used to fit the time series data from the growth of each clone.…”

“…The growth kinetics of clones from six different biofilm depths were measured and modeled using an analysis-of-variance formulation of the stochastic Ricker model of population dynamics with environmental noise (11,40). Rigorous statistical methods were used to show that after 1 month of cultivation, the extant diversity in E. coli biofilms was extraordinarily high and varied with depth.…”

Visualization of Growth Curve Data from Phenotype Microarray Experiments

“…In general, two approaches are used to characterize growth curves: growth curve models [14,16] and calculation of growth indices [17]. We will evaluate the latter method first and will link the display of the growth curves with the calculated growth indices in order to be able to easily evaluate how well the growth indices reflect the actual growth.…”