Daniel López

Individual-based Modelling: An Essential Tool for Microbiology

Micro-organisms play a central role in every ecosystem and in the global biomass cycle. They are strongly involved in many fields of human interest, from medicine to the food industry and waste control. Nevertheless, most micro-organisms remain almost unknown, and nearly 99% of them have not yet been successfully cultured in vitro. Therefore, new approaches and new tools must be developed in order to understand the collective behaviour of microbial communities in any natural or artificial setting. In particular, theoretical and practical methodologies to deal with such systems at a mesoscopic level of description (covering the range from 100 to 108 cells) are required. Individual-based modelling (IBM) has become a widely used tool for describing complex systems made up of autonomous entities, such as ecosystems and social networks. Individual-based models (IBMs) provide some advantages over the traditional whole-population models: (a) they are bottom-up approaches, so they describe the behaviour of a system as a whole by establishing procedural rules for the individuals and for their interactions, and thus allow more realistic assumptions for the model of the individuals than population models do; (b) they permit the introduction of randomness and individual variability, so they can reproduce the diversity found in real systems; and (c) they can account for individual adaptive behaviour to their environmental conditions, so the evolution of the whole system arises from the dynamics that govern individuals in their pursuit of optimal fitness. However, they also present some drawbacks: they lack the clarity of continuous models and may easily become rambling, which makes them difficult to analyse and communicate. All in all, IBMs supply a holistic description of microbial systems and their emerging properties. They are specifically appropriate to deal with microbial communities in non-steady states, and spatially explicit IBMs are particularly appropriate to study laboratory and natural microbiological systems with spatial heterogeneity. In this paper, we review IBM methodology applied to microbiology. We also present some results obtained from the application of Individual Discrete Simulations, an IBM of ours, to the study of bacterial communities, yeast cultures and Plasmodium falciparum-infected erythrocytes in vitro cultures of Plasmodium falciparum-infected erythrocytes.

Mathematical modelling methodologies in predictive food microbiology: A SWOT analysis

Predictive microbiology is the area of food microbiology that attempts to forecast the quantitative evolution of microbial populations over time. This is achieved to a great extent through models that include the mechanisms governing population dynamics. Traditionally, the models used in predictive microbiology are whole-system continuous models that describe population dynamics by means of equations applied to extensive or averaged variables of the whole system. Many existing models can be classified by specific criteria. We can distinguish between survival and growth models by seeing whether they tackle mortality or cell duplication. We can distinguish between empirical (phenomenological) models, which mathematically describe specific behaviour, and theoretical (mechanistic) models with a biological basis, which search for the underlying mechanisms driving already observed phenomena. We can also distinguish between primary, secondary and tertiary models, by examining their treatment of the effects of external factors and constraints on the microbial community. Recently, the use of spatially explicit Individual-based Models (IbMs) has spread through predictive microbiology, due to the current technological capacity of performing measurements on single individual cells and thanks to the consolidation of computational modelling. Spatially explicit IbMs are bottom-up approaches to microbial communities that build bridges between the description of micro-organisms at the cell level and macroscopic observations at the population level. They provide greater insight into the mesoscale phenomena that link unicellular and population levels. Every model is built in response to a particular question and with different aims. Even so, in this research we conducted a SWOT (Strength, Weaknesses, Opportunities and Threats) analysis of the different approaches (population continuous modelling and Individual-based Modelling), which we hope will be helpful for current and future researchers.

Simulation modelling of bacterial growth in yoghurt.

INDISIM, an individual-based simulator, was used to specifically study the influence of the shape and size of Streptococcus salivarius subsp. thermophilus and Lactobacillus delbrueckii subsp. bulgaricus in yoghurt processing. To this effect, two different sets of simulations were carried out. In the first set of simulations, it was assumed that the initial acidity of the medium has the same value as the acidity of the cytoplasm of the microorganisms. Hence, the differences in bacterial growth by the two species are only attributable to differences in their geometry. It was found that, in an optimum culture, the growth in biomass of S. salivarius subsp. thermophilus is larger than that of L. delbrueckii subsp. bulgaricus. An important factor for understanding this difference might be the larger mass-to-surface ratio of the former. In the second set of simulations, a simplified model of yoghurt, the parametrisation differs both in the geometry and the metabolism of the two species. The results of these simulations are in very good qualitative agreement with the experimental data of [Lait 69 (1989) 519]. Finally, by inhibiting the uptake of amino acids by S. salivarius subsp. thermophilus, the large relative importance of lactic acid in yoghurt processing was highlighted.

Analysis and IbM simulation of the stages in bacterial lag phase: Basis for an updated definition

The lag phase is the initial phase of a culture that precedes exponential growth and occurs when the conditions of the culture medium differ from the pre-inoculation conditions. It is usually defined by means of cell density because the number of individuals remains approximately constant or slowly increases, and it is quantified with the lag parameter lambda. The lag phase has been studied through mathematical modelling and by means of specific experiments. In recent years, Individual-based Modelling (IbM) has provided helpful insights into lag phase studies. In this paper, the definition of lag phase is thoroughly examined. Evolution of the total biomass and the total number of bacteria during lag phase is tackled separately. The lag phase lasts until the culture reaches a maximum growth rate both in biomass and cell density. Once in the exponential phase, both rates are constant over time and equal to each other. Both evolutions are split into an initial phase and a transition phase, according to their growth rates. A population-level mathematical model is presented to describe the transitional phase in cell density. INDividual DIScrete SIMulation (INDISIM) is used to check the outcomes of this analysis. Simulations allow the separate study of the evolution of cell density and total biomass in a batch culture, they provide a depiction of different observed cases in lag evolution at the individual-cell level, and are used to test the population-level model. The results show that the geometrical lag parameter lambda is not appropriate as a universal definition for the lag phase. Moreover, the lag phase cannot be characterized by a single parameter. For the studied cases, the lag phases of both the total biomass and the population are required to fully characterize the evolution of bacterial cultures. The results presented prove once more that the lag phase is a complex process that requires a more complete definition. This will be possible only after the phenomena governing the population dynamics at an individual level of description, and occurring during the lag and exponential growth phases, are well understood.

Individual based simulations of bacterial growth on agar plates

The individual based simulator, INDividual DIScrete SIMulations (INDISIM) has been used to study the behaviour of the growth of bacterial colonies on a finite dish. The simulations reproduce the qualitative trends of pattern formation that appear during the growth of Bacillus subtilis on an agar plate under different initial conditions of nutrient peptone concentration, the amount of agar on the plate, and the temperature.

Statistical aspects of biological organization

Abstract It is shown that some fundamental aspects of biological organization can be derived and understood in the light of time-honoured principles and techniques of physics. In particular, we claim that an appropriate Maximum Entropy Formalism (MAXENT) and an adequate version of the Monte Carlo simulation programs provide a powerful insight into the biomass distributions of the populations of living systems.

To achieve an earlier IFN-γ response is not sufficient to control Mycobacterium tuberculosis infection in mice.

The temporo-spatial relationship between the three organs (lung, spleen and lymph node) involved during the initial stages of Mycobacterium tuberculosis infection has been poorly studied. As such, we performed an experimental study to evaluate the bacillary load in each organ after aerosol or intravenous infection and developed a mathematical approach using the data obtained in order to extract conclusions. The results showed that higher bacillary doses result in an earlier IFN-γ response, that a certain bacillary load (BL) needs to be reached to trigger the IFN-γ response, and that control of the BL is not immediate after onset of the IFN-γ response, which might be a consequence of the spatial dimension. This study may have an important impact when it comes to designing new vaccine candidates as it suggests that triggering an earlier IFN-γ response might not guarantee good infection control, and therefore that additional properties should be considered for these candidates.

Self-organized criticality in Monte Carlo simulated ecosystems

Abstract A self-organized critical state is observed in a simple ecosystem model based on a Monte Carlo simulation approach. A 1 ⨍ Fourier spectrum is obtained, related with a dynamical process with fluctuations on a wide range of time scales, showing a well defined power law. The system shows a fractal spatial organization generated from a spatially homogeneous energy flow. Indeed our studies provide evidence for self-organized critically in realistic biological models, showing the reliability of a previous conjecture about the application of this approach to real living systems. Some general implications are also discussed.

Low Dose Aerosol Fitness at the Innate Phase of Murine Infection Better Predicts Virulence amongst Clinical Strains of Mycobacterium tuberculosis

Background Evaluation of a quick and easy model to determine the intrinsic ability of clinical strains to generate active TB has been set by assuming that this is linked to the fitness of Mycobacterium tuberculosis strain at the innate phase of the infection. Thus, the higher the bacillary load, the greater the possibility of inducting liquefaction, and thus active TB, once the adaptive response is set. Methodology/Principal Findings The virulence of seven clinical Mycobacterium tuberculosis strains isolated in Spain was tested by determining the bacillary concentration in the spleen and lung of mice at weeks 0, 1 and 2 after intravenous (IV) inoculation of 104 CFU, and by determining the growth in vitro until the stationary phase had been reached. Cord distribution automated analysis showed two clear patterns related to the high and low fitness in the lung after IV infection. This pattern was not seen in the in vitro fitness tests, which clearly favored the reference strain (H37Rv). Subsequent determination using a more physiological low-dose aerosol (AER) inoculation with 102 CFU showed a third pattern in which the three best values coincided with the highest dissemination capacity according to epidemiological data. Conclusions/Significance The fitness obtained after low dose aerosol administration in the presence of the innate immune response is the most predictive factor for determining the virulence of clinical strains. This gives support to a mechanism of the induction of active TB derived from the dynamic hypothesis of latent tuberculosis infection.

Local Inflammation, Dissemination and Coalescence of Lesions Are Key for the Progression toward Active Tuberculosis: The Bubble Model

The evolution of a tuberculosis (TB) infection toward active disease is driven by a combination of factors mostly related to the host response. The equilibrium between control of the bacillary load and the pathology generated is crucial as regards preventing the growth and proliferation of TB lesions. In addition, some experimental evidence suggests an important role of both local endogenous reinfection and the coalescence of neighboring lesions. Herein we propose a mathematical model that captures the essence of these factors by defining three hypotheses: (i) lesions grow logistically due to the inflammatory reaction; (ii) new lesions can appear as a result of extracellular bacilli or infected macrophages that escape from older lesions; and (iii) lesions can merge when they are close enough. This model was implemented in Matlab to simulate the dynamics of several lesions in a 3D space. It was also fitted to available microscopy data from infected C3HeB/FeJ mice, an animal model of active TB that reacts against Mycobacterium tuberculosis with an exaggerated inflammatory response. The results of the simulations show the dynamics observed experimentally, namely an initial increase in the number of lesions followed by fluctuations, and an exponential increase in the mean area of the lesions. In addition, further analysis of experimental and simulation results show a strong coincidence of the area distributions of lesions at day 21, thereby highlighting the consistency of the model. Three simulation series removing each one of the hypothesis corroborate their essential role in the dynamics observed. These results demonstrate that three local factors, namely an exaggerated inflammatory response, an endogenous reinfection, and a coalescence of lesions, are needed in order to progress toward active TB. The failure of one of these factors stops induction of the disease. This mathematical model may be used as a basis for developing strategies to stop the progression of infection toward disease in human lungs.