Two parameters account for the flocculated growth of microbes in biodegradation assays

Microbes in activated sludge tanks mostly occur in flocs rather than in cell suspensions. Flocculation results in a limited supply of substrate to the bacteria inside the flocs, which reduces the biodegradation rate of organic compounds by several orders of magnitude. This article presents a simple two-parameter extension of growth models for cell suspensions to account for the ensuing reduction of the degradation rate. The additional parameters represent floc size at division and diffusion length. The biomass of small flocs initially increases exponentially at a rate equal to that of cell suspensions. After this first phase, the growth rate gradually decreases and finally the radius becomes a linear function of time. At this time flocs are large and have a kernel of dead biomass. This kernel arises when the substrate concentration decreases below the threshold level at which cells are just able to pay their maintenance costs. We deduce an explicit approximative expression for the interdivision time of flocs, and thereby for the growth of flocculated microbial biomass at constant substrate concentrations. The model reveals that the effect of stirring on degradation rates occurs through a reduction of the floc size at division. The results can be applied in realistic biodegradation quantifications in activated sludge tanks as long as substrate concentrations change slowly.     

Cell cycle model to describe animal cell size variation and lag between cell number and biomass dynamics

The use of cell numbers rather than mass to quantify the size of the biotic phase in animal cell cultures causes several problems. First, the cell size varies with growth conditions, thus yields expressed in terms of cell numbers cannot be used in the normal mass balance sense. Second, experience from microbial systems shows that cell number dynamics lag behind biomass dynamics. This work demonstrates that this lag phenomenon also occurs in animal cell culture. Both the lag phenomenon and the variation in cell size are explained using a simple model of the cell cycle. The basis for the model is that onset of DNA synthesis requires accumulation of G1 cyclins to a prescribed level. This requirement is translated into a requirement for a cell to reach a critical size before commencement of DNA synthesis. A slower growing cell will spend more time in G1 before reaching the critical mass. In contrast, the period between onset of DNA synthesis and mitosis, tau(B), is fixed. The two parameters in the model, the critical size and tau(B), were determined from eight steady-state measurements of mean cell size in a continuous hybridoma culture. Using these parameters, it was possible to predict with reasonable accuracy the transient behavior in a separate shift-up culture, i.e., a culture where cells were transferred from a lean environment to a rich environment. The implications for analyzing experimental data for animal cell culture are discussed. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 372-379, 1997.     

Competition between two microbial populations in a sequencing fed-batch reactor: theory, experimental verification, and implications for waste treatment applications

Competition between two microbial populations for a single pollutant (phenol) was studied in a sequencing fed-batch reactor (SFBR). A mathematical model describing this system was developed and tested experimentally. It is based on specific growth rate expressions revealed from pure culture batch experiments. The species employed were Pseudomonas putida (ATCC 17514) and Pseudomonas resinovorans (ATCC 14235). It was found that both species biodegrade phenol following inhibitory kinetics which can be described by Andrews' expression. The model predicts that the dynamics of a SFBR, and the kinetics of biodegradation, result in a complex set of operating regimes in which neither species, only one species, or both species can survive at steady cycle. The model also predicts the existence of multiple outcomes, achievable from different start-up conditions, in some domains of the operating parameter space. Experimental results confirmed the model predictions. There was excellent agreement between predicted and measured concentrations of phenol, total biomass, and the biomass of each individual species. This study shows how serious discrepancies can arise in scale-up of biodegradation data if population dynamics are not taken into account. It also further confirms experimentally the theory of microbial competition in periodically forced bioreactors. (c) 1993 John Wiley & Sons, Inc.     

Simulating the effects of N availability, straw particle size and location in soil on C and N mineralization

Predicting the C and N mineralization of straw added to soil is important for forecasting subsequent soil N availability during and between crop growth cycles. The decomposition module of the STICS model, parameterized under optimal conditions, was used to predict straw decomposition in sub-optimal conditions, i.e. when contact between soil and residue was poor (due to large size residues or surface placement) or when mineral N availability was restricted. The data used in the simulations were obtained from published studies of effects of residue size, location and N availability on C and N mineralization from straw under controlled laboratory conditions. We selected studies in which the dynamics of C and N mineralization were measured simultaneously. The dynamics of straw mineralization could be well predicted by the model under optimal conditions with standard parameter values as derived from measured C/N ratios of the residues, but not under sub-optimal conditions which required a new parameterization. A good fit could be obtained on these treatments by a marked reduction in the rate constants of residue and microbial biomass decomposition and a marked increase in the microbial biomass C/N ratio. Our results show the need to include in decomposition models routines for simulating effects of spatial heterogeneity of residue distribution, different particle sizes and limiting N availability.     

Optimization of biomass composition explains microbial growth-stoichiometry relationships

Integrating microbial physiology and biomass stoichiometry opens far-reaching possibilities for linking microbial dynamics to ecosystem processes. For example, the growth-rate hypothesis (GRH) predicts positive correlations among growth rate, RNA content, and biomass phosphorus (P) content. Such relationships have been used to infer patterns of microbial activity, resource availability, and nutrient recycling in ecosystems. However, for microorganisms it is unclear under which resource conditions the GRH applies. We developed a model to test whether the response of microbial biomass stoichiometry to variable resource stoichiometry can be explained by a trade-off among cellular components that maximizes growth. The results show mechanistically why the GRH is valid under P limitation but not under N limitation. We also show why variability of growth rate-biomass stoichiometry relationships is lower under P limitation than under N or C limitation. These theoretical results are supported by experimental data on macromolecular composition (RNA, DNA, and protein) and biomass stoichiometry from two different bacteria. In addition, compared to a model with strictly homeostatic biomass, the optimization mechanism we suggest results in increased microbial N and P mineralization during organic-matter decomposition. Therefore, this mechanism may also have important implications for our understanding of nutrient cycling in ecosystems.     

The effect of dilution rate variation on the performance of continuous fermentation

The effect of dilution rate variation on the performance of bioreactors used for continuous biomass producton is discussed. Calculations based on Powell's model of microbial dynamics show that such a mode of operation does not improve the performance of the process. Remarkable improvement of the process performance reported in literature is caused by incorrectness of the dynamic model of microbial growth which has been used in the calculations.     

Predicting microbial growth dynamics in response to nutrient availability

Developing mathematical models to accurately predict microbial growth dynamics remains a key challenge in ecology, evolution, biotechnology, and public health. To reproduce and grow, microbes need to take up essential nutrients from the environment, and mathematical models classically assume that the nutrient uptake rate is a saturating function of the nutrient concentration. In nature, microbes experience different levels of nutrient availability at all environmental scales, yet parameters shaping the nutrient uptake function are commonly estimated for a single initial nutrient concentration. This hampers the models from accurately capturing microbial dynamics when the environmental conditions change. To address this problem, we conduct growth experiments for a range of micro-organisms, including human fungal pathogens, baker’s yeast, and common coliform bacteria, and uncover the following patterns. We observed that the maximal nutrient uptake rate and biomass yield were both decreasing functions of initial nutrient concentration. While a functional form for the relationship between biomass yield and initial nutrient concentration has been previously derived from first metabolic principles, here we also derive the form of the relationship between maximal nutrient uptake rate and initial nutrient concentration. Incorporating these two functions into a model of microbial growth allows for variable growth parameters and enables us to substantially improve predictions for microbial dynamics in a range of initial nutrient concentrations, compared to keeping growth parameters fixed.     

Representing Density Dependent Consequences of Life History Strategies in Aquatic Ecosystems: EcoSim II

EcoSim II uses results from the Ecopath procedure for trophic mass-balance analysis to define biomass dynamics models for predicting temporal change in exploited ecosystems. Key populations can be represented in further detail by using delay-difference models to account for both biomass and numbers dynamics. A major problem revealed by linking the population and biomass dynamics models is in representation of population responses to changes in food supply; simple proportional growth and reproductive responses lead to unrealistic predictions of changes in mean body size with changes in fishing mortality. EcoSim II allows users to specify life history mechanisms to avoid such unrealistic predictions: animals may translate changes in feeding rate into changes in reproductive rather than growth rates, or they may translate changes in food availability into changes in foraging time that in turn affects predation risk. These options, along with model relationships for limits on prey availability caused by predation avoidance tactics, tend to cause strong compensatory responses in modeled populations. It is likely that such compensatory responses are responsible for our inability to find obvious correlations between interacting trophic components in fisheries time-series data. But Ecosim II does not just predict strong compensatory responses: it also suggests that large piscivores may be vulnerable to delayed recruitment collapses caused by increases in prey species that are in turn competitors/predators of juvenile piscivores. Received 24 February 1999; accepted 3 August 1999.     

Influence of gap size and soil properties on microbial biomass in a subtropical humid forest of north-east India

We examined the effects of treefall gap size and soil properties on microbial biomass dynamics in an undisturbed mature-phase humid subtropical broadleaved forest in north-east India. Canopy gaps had low soil moisture and low microbial biomass suggesting that belowground dynamics accompanied changes in light resources after canopy opening. High rainfall in the region causes excessive erosion/leaching of top soil and eventually soil fertility declines in treefall gaps compared to understorey. Soil microbial population was less during periods when temperature and moisture conditions are low, while it peaked during rainy season when the litter decomposition rate is at its peak on the forest floor. Greater demand for nutrients by plants during rainy season (the peak vegetative growth period) limited the availability of nutrients to soil microbes and, therefore, low microbial C, N and P. Weak correlations were also obtained for the relationships between microbial C, N and P and soil physico–chemical properties. Gap size did influence the microbial nutrients and their contribution to soil organic carbon, total Kjeldhal nitrogen and available-P. Contribution of microbial C to soil organic carbon, microbial N to total nitrogen were similar in both treefall gaps and understorey plots, while the contribution of microbial P to soil available-P was lower in gap compared to the understorey. These results indicate that any fluctuation in microbial biomass related nutrient cycling processes in conjunction with the associated microclimate variation may affect the pattern of regeneration of tree seedlings in the gaps and hence be related with their size. This revised version was published online in June 2006 with corrections to the Cover Date.     

``BACWAVE,' a Spatial–Temporal Model for Traveling Waves of Bacterial Populations in Response to a Moving Carbon Source in Soil

Abstract Previously, we discovered the phenomenon of wavelike spatial distributions of bacterial populations and total organic carbon (TOC) along wheat roots. We hypothesized that the principal mechanism underlying this phenomenon is a cycle of growth, death, autolysis, and regrowth of bacteria in response to a moving substrate source (root tip). The aims of this research were (i) to create a simulation model describing wavelike patterns of microbial populations in the rhizosphere, and (ii) to investigate by simulation the conditions leading to these patterns. After transformation of observed spatial data to presumed temporal data based on root growth rates, a simulation model was constructed with the Runge–Kutta integration method to simulate the dynamics of colony-forming bacterial biomass, with growth and death rates depending on substrate content so that the rate curves crossed over at a substrate concentration within the range of substrate availability in the model. This model was named ``BACWAVE,' standing for ``bacterial waves.' Cyclic dynamics of bacteria were generated by the model that were translated into traveling spatial waves along a moving nutrient source. Parameter values were estimated from calculated initial substrate concentrations and observed microbial distributions along wheat roots by an iterative optimization method. The kinetic parameter estimates fell in the range of values reported in the literature. Calculated microbial biomass values produced spatial fluctuations similar to those obtained for experimental biomass data derived from colony forming units. Concentrations of readily utilizable substrate calculated from biomass dynamics did not mimic measured concentrations of TOC, which consist not only of substrate but also various polymers and humic acids. In conclusion, a moving pulse of nutrients resulting in cycles of growth and death of microorganisms can indeed explain the observed phenomenon of moving microbial waves along roots. This is the first report of wavelike dynamics of microorganisms in soil along a root resulting from the interaction of a single organism group with its substrate. Received: 2 October 1999; Accepted: 9 March 2000; Online Publication: 28 August 2000     

Advantages of the division of labour for the long-term population dynamics of cyanobacteria at different latitudes

A fundamental advancement in the evolution of complexity is division of labour. This implies a partition of tasks among cells, either spatially through cellular differentiation, or temporally via a circadian rhythm. Cyanobacteria often employ either spatial differentiation or a circadian rhythm in order to separate the chemically incompatible processes of nitrogen fixation and photosynthesis. We present a theoretical framework to assess the advantages in terms of biomass production and population size for three species types: terminally differentiated (heterocystous), circadian, and an idealized species in which nitrogen and carbon fixation occur without biochemical constraints. On the basis of real solar irradiance data at different latitudes, we simulate population dynamics in isolation and in competition for light over a period of 40 years. Our results show that in isolation and regardless of latitude, the biomass of heterocystous cyanobacteria that optimally invest resources is comparable to that of the idealized unconstrained species. Hence, spatial division of labour overcomes biochemical constraints and enhances biomass production. In the circadian case, the strict temporal task separation modelled here hinders high biomass production in comparison with the heterocystous species. However, circadian species are found to be successful in competition for light whenever their resource investment prevents a waste of fixed nitrogen more effectively than do heterocystous species. In addition, we show the existence of a trade-off between population size and biomass accumulation, whereby each species can optimally invest resources to be proficient in biomass production or population growth, but not necessarily both. Finally, the model produces chaotic dynamics for population size, which is relevant to the study of cyanobacterial blooms.     

Continuous cultures limited by a gaseous substrate: Development of a simple, unstructured mathematical model and experimental verification with Methanobacterium thermoautotrophicum

This article presents a simple, unstructured mathematical model describing microbial growth in continuous culture limited by a gaseous substrate. The model predicts constant gas conversion rates and a decreasing biomass concentration with increasing dilution rate. It has been found that the parameters influencing growth are primarily the gas transfer rate and the dilution rate. Furthermore, it is shown that, for correct simulation of growth, the influence of gaseous substrate consumption on the effective gas flow through the system has to be taken into account.Continuous cultures of Methanobacterium thermoautotrophicum were performed at three different gassing rates. In addition to the measurement of the rates of biomass production, product formation, and substrate consumption, microbial heat dissipation was assessed using a reaction calorimeter. For the on-line measurement of the concentration of the growth-limiting substrate, H(2), a specially developed probe has been used. Experimental data from continuous cultures were in good agreement with the model simulations. An increase in gassing rate enhanced gaseous substrate consumption and methane production rates. However, the biomass yield as well as the specific conversion rates remained constant, irrespective of the gassing rate. It was found that growth performance in continuous culture limited by a gaseous substrate is substantially different from "classic" continuous culture in which the limiting substrate is provided by the liquid feed. In this report, the differences between both continuous culture systems are discussed.     

Cell cycle modelling

Models able to describe the events of cellular growth and division and the dynamics of cell populations are useful for the understanding of functional control mechanisms and for the theoretical support for automated analysis of flow cytometric data and of cell volume distributions. This paper reports on models that we have developed with this aim for different kinds of cells. The models are composed by two subsystems: one describes the growth dynamics of RNA and protein, and the second accounts for DNA replication and cell division, and describe in a rather unitary frame the cell cycle of eukaryotic cells, like mammalian cells and yeast, and of prokaryotic cells. The model is also used to study the effects of various sources of variability on the statistical properties of cell populations, and we find that in microbial cells the main source of variability appears to be an inaccuracy of the molecular mechanism that monitors cell size. In normal mammalian cells another source of variability, that depends upon the interaction with growth factors which give competence, is apparent. An extended version of the model, which comprises also this additional variability, is presented and used to describe the properties of mammalian cell growth.     

Linking cell division to cell growth in a spatiotemporal model of the cell cycle

Cell division must be tightly coupled to cell growth in order to maintain cell size, yet the mechanisms linking these two processes are unclear. It is known that almost all proteins involved in cell division shuttle between cytoplasm and nucleus during the cell cycle; however, the implications of this process for cell cycle dynamics and its coupling to cell growth remains to be elucidated. We developed mathematical models of the cell cycle which incorporate protein translocation between cytoplasm and nucleus. We show that protein translocation between cytoplasm and nucleus not only modulates temporal cell cycle dynamics, but also provides a natural mechanism coupling cell division to cell growth. This coupling is mediated by the effect of cytoplasmic-to-nuclear size ratio on the activation threshold of critical cell cycle proteins, leading to the size-sensing checkpoint (sizer) and the size-independent clock (timer) observed in many cell cycle experiments.     

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.     

Modeling the impact of the indigenous microbial population on the maximum population density of Salmonella on alfalfa

Within a microbial risk assessment framework, modeling the maximum population density (MPD) of a pathogenic microorganism is important but often not considered. This paper describes a model predicting the MPD of Salmonella on alfalfa as a function of the initial contamination level, the total count of the indigenous microbial population, the maximum pathogen growth rate and the maximum population density of the indigenous microbial population. The model is parameterized by experimental data describing growth of Salmonella on sprouting alfalfa seeds at inoculum size, native microbial load and Pseudomonas fluorescens 2–79. The obtained model fits well to the experimental data, with standard errors less than ten percent of the fitted average values. The results show that the MPD of Salmonella is not only dictated by performance characteristics of Salmonella but depends on the characteristics of the indigenous microbial population like total number of cells and its growth rate. The model can improve the predictions of microbiological growth in quantitative microbial risk assessments. Using this model, the effects of preventive measures to reduce pathogenic load and a concurrent effect on the background population can be better evaluated. If competing microorganisms are more sensitive to a particular decontamination method, a pathogenic microorganism may grow faster and reach a higher level. More knowledge regarding the effect of the indigenous microbial population (size, diversity, composition) of food products on pathogen dynamics is needed in order to make adequate predictions of pathogen dynamics on various food products.     

Individual-based modeling of phytoplankton: evaluating approaches for applying the cell quota model

Present phytoplankton models typically use a population-level (lumped) modeling (PLM) approach that assumes average properties of a population within a control volume. For modern biogeochemical models that formulate growth as a nonlinear function of the internal nutrient (e.g. Droop kinetics), this averaging assumption can introduce a significant error. Individual-based (agent-based) modeling (IBM) does not make the assumption of average properties and therefore constitutes a promising alternative for biogeochemical modeling. This paper explores the hypothesis that the cell quota (Droop) model, which predicts the population-average specific growth or cell division rate, based on the population-average nutrient cell quota, can be applied to individual algal cells and produce the same population-level results. Three models that translate the growth rate calculated using the cell quota model into discrete cell division events are evaluated, including a stochastic model based on the probability of cell division, a deterministic model based on the maturation velocity and fraction of the cell cycle completed (maturity fraction), and a deterministic model based on biomass (carbon) growth and cell size. The division models are integrated into an IBM framework (iAlgae), which combines a lumped system representation of a nutrient with an individual representation of algae. The IBM models are evaluated against a conventional PLM (because that is the traditional approach) and data from a number of steady and unsteady continuous (chemostat) and batch culture laboratory experiments. The stochastic IBM model fails the steady chemostat culture test, because it produces excessive numerical randomness. The deterministic cell cycle IBM model fails the batch culture test, because it has an abrupt drop in cell quota at division, which allows the cell quota to fall below the subsistence quota. The deterministic cell size IBM model reproduces the data and PLM results for all experiments and the model parameters (e.g. maximum specific growth rate, subsistence quota) are the same as those for the PLM. In addition, the model-predicted cell age, size (carbon) and volume distributions are consistent with those derived analytically and compare well to observations. The paper discusses and illustrates scenarios where intra-population variability in natural systems leads to differences between the IBM and PLM models.