On the lag phase and initial decline of microbial growth curves

The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of non-duplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves.     

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.     

Cell growth and division: a deterministic/probabilistic model of the cell cycle

A model of the cell cycle, incorporating a deterministic cell-size monitor and a probabilistic component, is investigated. Steady-state distributions for cell size and generation time are calculated and shown to be globally asymptotically stable. These distributions are used to calculate various statistical quantities, which are then compared to known experimental data. Finally, the results are compared to distributions calculated from a Monte-Carlo simulation of the model.     

Density functions of residence times for deterministic and stochastic compartmental systems

A significant consideration in modeling systems with stages is to obtain models for the individual stages that have probability density functions (pdfs) of residence times that are close to those of the real system. Consequently, the theory of residence time distributions is important for modeling. Here I show first that linear deterministic compartmental systems with constant coefficients and their corresponding stochastic analogs (stochastic compartmental systems with linear rate laws) have the same pdfs of residence times for the same initial distributions of inputs. Furthermore, these are independent of inflows. Then I show that does not hold for non-linear deterministic systems and their stochastic analogs (stochastic compartmental systems with non-linear rate laws). In fact, for given initial distributions of inputs, the pdfs of non-linear determistic systems without inflows and of their stochastic analogs, are functions of the initial amounts injected. For systems with inflows, the pdfs change as the inflows influence the occupancies of the compartments of the system; they are state-dependent pdfs.     

The integration of cell proliferation and growth in leaf morphogenesis

A number of recent publications have assessed the outcome on leaf development of targeted manipulation of cell proliferation. The results of these investigations have awakened interest in the long-standing debate in plant biology on the precise role of cell division in morphogenesis. Does cell proliferation drive morphogenesis (cell theory) or is it subservient to a mechanism which acts at the whole organ level to regulate morphogenesis (organismal theory)? In this review, the central role of growth processes (distinct from cell proliferation) in morphogenesis is highlighted and the limitations in our understanding of the basic mechanisms of plant growth control are highlighted. Finally, an attempt is made to demonstrate how sequential local co-ordination of growth might provide an interpretation of some of the recent observations on cell proliferation and leaf morphogenesis.     

Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion

We propose a mathematical modelling system to investigate the dynamic process of tumour cell proliferation, death and tumour angiogenesis by fully coupling the vessel growth, tumour growth and blood perfusion. Tumour growth and angiogenesis are coupled by the chemical microenvironment and the cell-matrix interaction. The haemodynamic calculation is carried out on the updated vasculature. The domains of intravascular, transcapillary and interstitial fluid flow were coupled in the model to provide a comprehensive solution of blood perfusion variables. An estimation of vessel collapse is made according to the wall shear stress criterion to provide feedback on vasculature remodelling. The simulation can show the process of tumour angiogenesis and the spatial distribution of tumour cells for periods of up to 24 days. It can show the major features of tumour and tumour microvasculature during the period such as the formation of a large necrotic core in the tumour centre with few functional vessels passing through, and a well circulated tumour periphery regions in which the microvascular density is high and associated with more aggressive proliferating cells of the growing tumour which are all consistent with physiological observations. The study also demonstrated that the simulation results are not dependent on the initial tumour and networks, which further confirms the application of the coupled model feedback mechanisms. The model enables us to examine the interactions between angiogenesis and tumour growth, and to study the dynamic response of a solid tumour to the changes in the microenvironment. This simulation framework can be a foundation for further applications such as drug delivery and anti-angiogenic therapies.     

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.     

Stability of the steady-state size distribution in a model of cell growth and division

The approach to steady-state size distribution is studied for a growing population of cells. The model incorporates cell growth at a linear rate and division into two equal daughters after a random time composed of an exponentially distributed phase and a constant deterministic phase.This work was supported by the National Science Foundation under Grant No. MCS 8300559This work was supported by the National Science Foundation under Grant No. MCS 8301104     

Intelligent modelling of bioprocesses: a comparison of structured and unstructured approaches

This contribution moves in the direction of answering some general questions about the most effective and useful ways of modelling bioprocesses. We investigate the characteristics of models that are good at extrapolating. We trained three fully predictive models with different representational structures (differential equations, differential equations with inheritance of rates and a network of reactions) on Saccharopolyspora erythraea shake flask fermentation data using genetic programming. The models were then tested on unseen data outside the range of the training data and the resulting performances were compared. It was found that constrained models with mathematical forms analogous to internal mass balancing and stoichiometric relations were superior to flexible unconstrained models, even though no a priori knowledge of this fermentation was used.Paper presented at the international conference on trends in monitoring and control of life science applications, 7–8 October 2002, Lyngby, Denmark.     

Individual-based modelling of bacterial cultures to study the microscopic causes of the lag phase

The lag phase has been widely studied for years in an effort to contribute to the improvement of food safety. Many analytical models have been built and tested by several authors. The use of Individual-based Modelling (IbM) allows us to probe deeper into the behaviour of individual cells; it is a bridge between theories and experiments when needed. INDividual DIScrete SIMulation (INDISIM) has been developed and coded by our group as an IbM simulator and used to study bacterial growth, including the microscopic causes of the lag phase. First of all, the evolution of cellular masses, specifically the mean mass and biomass distribution, is shown to be a determining factor in the beginning of the exponential phase. Secondly, whenever there is a need for an enzyme synthesis, its rate has a direct effect on the lag duration. The variability of the lag phase with different factors is also studied. The known decrease of the lag phase with an increase in the temperature is also observed in the simulations. An initial study of the relationship between individual and collective lag phases is presented, as a complement to the studies already published. One important result is the variability of the individual lag times and generation times. It has also been found that the mean of the individual lags is greater than the population lag. This is the first in a series of studies of the lag phase that we are carrying out. Therefore, the present work addresses a generic system by making a simple set of assumptions.     

The balance between deterministic and stochastic processes in structuring lake bacterioplankton community over time

One major goal in microbial ecology is to establish the importance of deterministic and stochastic processes for community assembly. This is relevant to explain and predict how diversity changes at different temporal scales. However, understanding of the relative quantitative contribution of these processes and particularly of how they may change over time is limited. Here, we assessed the importance of deterministic and stochastic processes based on the analysis of the bacterial microbiome in one alpine oligotrophic and in one subalpine mesotrophic lake, which were sampled over two consecutive years at different time scales. We found that in both lakes, homogeneous selection (i.e., a deterministic process) was the main assembly process at the annual scale and explained 66.7% of the bacterial community turnover, despite differences in diversity and temporal variability patterns between ecosystems. However, in the alpine lake, homogenizing dispersal (i.e., a stochastic process) was the most important assembly process at the short‐term (daily and weekly) sampling scale and explained 55% of the community turnover. Alpha diversity differed between lakes, and seasonal stability of the bacterial community was more evident in the oligotrophic lake than in the mesotrophic one. Our results demonstrate how important forces that govern temporal changes in bacterial communities act at different time scales. Overall, our study validates on a quantitative basis, the importance and dominance of deterministic processes in structuring bacterial communities in freshwater environments over long time scales.     

Stochastic and deterministic simulations of heterogeneous cell population dynamics

A Monte Carlo algorithm, which can accurately simulate the dynamics of entire heterogeneous cell populations, was developed. The algorithm takes into account the random nature of cell division as well as unequal partitioning of cellular material at cell division. Moreover, it is general in the sense that it can accommodate a variety of single-cell, deterministic reaction kinetics as well as various stochastic division and partitioning mechanisms. The validity of the algorithm was assessed through comparison of its results with those of the corresponding deterministic cell population balance model in cases where stochastic behavior is expected to be quantitatively negligible. Both algorithms were applied to study: (a) linear intracellular kinetics and (b) the expression dynamics of a genetic network with positive feedback architecture, such as the lac operon. The effects of stochastic division as well as those of different division and partitioning mechanisms were assessed in these systems, while the comparison of the stochastic model with a continuum model elucidated the significance of cell population heterogeneity even in cases where only the prediction of average properties is of primary interest.     

A thermodynamic theory of microbial growth

Our ability to model the growth of microbes only relies on empirical laws, fundamentally restricting our understanding and predictive capacity in many environmental systems. In particular, the link between energy balances and growth dynamics is still not understood. Here we demonstrate a microbial growth equation relying on an explicit theoretical ground sustained by Boltzmann statistics, thus establishing a relationship between microbial growth rate and available energy. The validity of our equation was then questioned by analyzing the microbial isotopic fractionation phenomenon, which can be viewed as a kinetic consequence of the differences in energy contents of isotopic isomers used for growth. We illustrate how the associated theoretical predictions are actually consistent with recent experimental evidences. Our work links microbial population dynamics to the thermodynamic driving forces of the ecosystem, which opens the door to many biotechnological and ecological developments.     

Changes of Escherichia coli cell cycle parameters during fast growth and throughout growth with limiting amounts of thymine

It is generally accepted that during fast growth of Escherichia coli, the time (D) between the end of a round of DNA replication and cell division is constant. This concept is not consistent with the fact that average cell mass of a culture is an exponential function of the growth rate, if it is also accepted that average cell mass per origin of DNA replication (M_i) changes with growth rate and negative exponential cell age distribution is taken into account. Data obtained from cell composition analysis of E. coli OV-2 have shown that not only (M_i) but also D varied with growth rate at generation times () between 54 and 30 min. E. coli OV-2 is a thymine auxotroph in which the replication time (C) can be lengthened, without inducing changes in , by growth with limiting amounts of thymine. This property has been used to study the relationship between cell size and division from cell composition measurements during growth with different amounts of thymine. When C increased, average cell mass at the end of a round of DNA replication also increased while D decreased, but only the time lapse (d) between the end of a replication round and cell constriction initiation appeared to be affected because the constriction period remained fairly constant. We propose that the rate at which cells proceed to constriction initiation from the end of replication is regulated by cell mass at this event, big cells having shorter d times than small cells.Abbreviations OD₄₅₀ and OD₆₃₀ Optical density at a given wavelength in nm Dedicated to Dr. John Ingraham to honor him for his many contributions to Science     

Estimating the mode of growth of individual microbial cells from cell volume distributions

Two new methods are derived for inferring the mode of growth of individual microbial cells from measurements made of the volume distributions of populations. One is based on statistics of the observed distribution and has the particular advantage that it is very easy to use. The second, which requires gradient centrifugation, yields the mode of growth directly, rather than by comparison with theoretically derived distributions. Both methods have been found to be more sensitive than those previously suggested.     

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).     

The potential of mining slag as a substrate for microbial growth and the microbiological analysis of slag and slag seepage

The potential of a Cu/Ni mining slag to act as a substrate for the growth of the bacteria Thiobacillus ferrooxidans, Thiobacillus thiooxidants, and Thiobacillus thioparus was examined. As well, slag and seepage samples were screened for the presence of the Thiobacillus species. For the 28 samples employed in the environmental recovery studies, T. ferrooxidans was recovered in 25 samples, T. thiooxidans in 19 samples, and T. thioparus in 27 samples. For T. ferrooxidans, the development of a colour change in the medium corresponded with the presence of motile bacilli as detected microscopically. For T. thiooxidans and T. thioparus, a decrease in culture pH of greater than 0.2 units usually corresponded with the presence of motile bacilli. The potential for growth on slag was determined by adding slag samples to media (devoid of an electron donor) appropriate for the growth of the three Thiobacillus species. All pulverized slag samples supported the growth of the three species.     

Transition phenomena in bacterial growth between logarithmic and stationary phases

Summary A method to calculate the age distribution of the cells in the transition phase starting from that of the cells in logarithmic phase is described. It is clarified that two transition phenomena (decrease in the growth rate of cell number and partial synchronization) in the transition phase come, mathematically, from the fact that da_g/dt > 0 (a_g = generation time).The cell age at which septum becomes observable is estimated from the age distribution of the cells and the ratio of septated cells at each time in the transition phase. The result suggests that the cell age at which septum synthesis starts increases in the transition phase and that the mode of septum synthesis changes during that phase.