A new method to solve a non-steady-state multispecies biofilm model

A transient multispecies model for quantifying microbial space competition in biofilm is derived from existing models, introducing a new approach to biomass detachment modelling. This model includes inert biomass, substrate diffusion and utilization rate within the biofilm and diffusional layers. It predicts the evolution of biofilm thickness, bulk substrate concentration, species distribution and substrate concentration within the biofilm. A zero-dimensional transient model is described. Its steady-state solution is used to set up initial conditions of the one-dimensional model and case computation towards steady-state solution. Some numerical tools have been developed, enabling fast computation on microcomputers. Simulations show the validity of a zero-dimensional model and perturbated systems are also simulated. Simulations with experimental data give acceptable results.     

Activity of Pseudomonas aeruginosa in biofilms: Steady state

Aerobic glucose metabolism by Pseudomonas aeruginosa in steady-state biofilms at various substrate loading rates and reactor dilution rates was investigated. Variables monitored were substrate (glucose), biofilm cellular density, biofilm extracellular polymeric substance (EPS) density, and suspended cellular and EPS concentrations. A mathematical model developed to describe the system was compared to experimental data. Intrinsic yield and rate coefficients included in the model were obtained from suspended continuous culture studies of glucose metabolism by P. aeruginosa. Experimental data compared well with the mathematical model, suggesting that P. aeruginosa does not behave differently in steady-state biofilm cultures, where diffusional resistance is negligible, than in suspended cultures. This implies that kinetic and stoichiometric coefficients for P. aeruginosa derived in suspended continuous culture can be used to describe steady-state biofilm processes.     

Model of steady-state-biofilm kinetics

A steady-state biofilm is defined as one that has neither net growth nor decay over time. The model, developed for steady-state-biofilm kinetics with a single substrate, couples the flux of substrate into a biofilm to the mass (or thickness) of biofilm that would exist at steady-state for a given bulk substrate concentration. Based on kinetic and energetic constraints, this model predicts for a single substrate that a steady-state bulk concentration, S_min, exists below which a steady-state biofilm cannot exist. Thus, in the absence of adsorption of bacteria from the bulk water and for substrate concentration below S_min, substrate flux and biofilm thickness are zero. Equations are provided for calculating the steady-state substrate flux and biofilm thickness for S greater than S_min. An example is provided to demonstrate the use of the steadystate model.     

Steady-state solution of a two-species biofilm problem

Through a thorough investigation of the boundary conditions for a general two-species biofilm model, a simple and fast method for solving the steady-state case is developed and presented. The methods used may be extended to biofilm models in which more than two species are considered. Four different sets of boundary conditions are possible for the two-species biofilm model. Each set is shown to be asymptotically stable. A biofilm model describing the competition between autotrophic and heterotrophic bacteria and a biofilm model considering only Nitrosomonas and Nitrobacter are used for illustration. A parameter L(crit), critical film thickness for bacterial coexistence, is introduced from which criteria on the bulk concentrations for coexistence are derived. From these criteria it is seen that the thinner the biofilm, the more restrictive the conditions are for steady-state coexistence. For thin biofilms there may, in many cases, be no point in considering more than one species in the biofilm model. Furthermore, the gradients of the bacterial concentrations are in many cases negligible in thin biofilms, and the biofilm may then be assumed to be homogeneous. The criteria on the bulk concentrations together with the four sets of boundary conditions provide the necessary information for a direct solution of the steady-state two-species biofilm model by means of an ordinary differential and algebraic equation solver. (c) 1996 John Wiley & Sons, Inc.     

Biofilm model calibration and microbial diversity study using Monte Carlo simulations

Mathematical models are useful tools for studying and exploring biological conversion processes as well as microbial competition in biological treatment processes. A single‐species biofilm model was used to describe biofilm reactor operation at three different hydraulic retention times (HRT). The single‐species biofilm model was calibrated with sparse experimental data using the Monte Carlo filtering method. This calibrated single‐species biofilm model was then extended to a multi‐species model considering 10 different heterotrophic bacteria. The aim was to study microbial diversity in bulk phase biomass and biofilm, as well as the competition between suspended and attached biomass. At steady state and independently of the HRT, Monte Carlo simulations resulted only in one unique dominating bacterial species for suspended and attached biomass. The dominating bacterial species was determined by the highest specific substrate affinity (ratio of µ/K_S). At a short HRT of 20 min, the structure of the microbial community in the bulk liquid was determined by biomass detached from the biofilm. At a long HRT of 8 h, both biofilm detachment and microbial growth in the bulk liquid influenced the microbial community distribution. Biotechnol. Bioeng. 2013; 110: 1323–1332. © 2012 Wiley Periodicals, Inc.     

Stratified mixed-culture biofilm model for anaerobic digestion

Development of a novel two-layer anaerobic biofilm model is based on substrate utilization kinetics and mass transport. The model is applied to steady-state conditions for a fixed-film anaerobic reactor. The microbial film is considered to consist of two distinct biofilm layers, one adjacent to the second, with an acidogenic bacteria biofilm forming the outer layer and a methanogenic film the inner one. The model assumes that sugars are only metabolized by the first layer and converted into volatile fatty acids (VFA), while fatty acids are taken up only by the inner layer. The model is able to predict both substrate flux net uptake and methane production for steady-state conditions. The results of modeling agree with methane production experimental data published elsewhere. Further, the model shows why layered fixed-film reactors can withstand high and inhibitory concentrations of volatile fatty acids as well as severe overloading without failure.     

Accurate pseudoanalytical solution for steady-state biofilms

An extremely accurate pseudoanalytical solution for the flux of substrate into a steady-state biofilm is developed. The standard deviations between the substrate fluxes computed from the pseudoanalytical solution and the numerical solution were less than 2.6%. Additional advantages of the pseudoanalytical solution are that it has no inaccuracies around S(min) (*) = 1 and it is composed of single continuous functions applicable to the whole S(min) (*) region.     

Evaluation of steady-state-biofilm kinetics

Laboratory-scale biofilm reactors were used to evaluate a model of the kinetics of steady-state biofilm and the concept that there is a minimum concentration, S_min, below which no steady-state activity can occur. With acetate as the ratelimiting substrate, the steady-state concept of S_min was verified for naturally grown biofilms. Substrate removal and biofilm thickness declined rapidly as the substrate concentration approached S_min, which was 0.66 mg/liter for acetate. Using independently derived kinetic parameters, the model of steady-state-biofilm kinetics successfully predicted substrate utilization and biofilm thickness without the need for fitting factors. The results imply that organic materials may persist in water and wastewater, in part, because they are too low in concentration to supply sufficient energy to sustain the microorganisms.     

Modeling of the evolution of a water-remedying biofilm with consideration for its erosion

A model is proposed that describes the growth and destruction of a biofilm due to the consumption of contaminants dissolved in the water being purified. The mathematical solution involves equations describing the balance of biomass, the delivery and uptake of substrate, and the dynamics of biofilm thickness. The effect of erosion on the characteristics of the steady-state regime was shown. The concentrations of the substrate and biomass, the flow of the substrate into the film, and changes in biofilm thickness were calculated.     

A multispecies biofilm model

Using a continuum approach and observing conservation principles, an analytical mathematical model of microbial interaction in biofilms was developed. The model predicts changes in biofilm thickness and describes the dynamics and spatial distribution of microbial species and substrates in the film. It allows for biomass detachment due to shear stress and sloughing, external mass transfer limitations, as well as variations in substrate concentrations in the bulk liquid. A computer implementation of the model is provided using an example of heterotrophicautotrophic competition to illustrate how the observed phenomena can be numerically reproduced and indicating how they might affect overall biofilm performance.     

Analytical effectiveness calculations concerning the degradation of an inhibitive substrate by a steady-state biofilm

A reaction engineering model for the degradation of an inhibitory substrate by a steady-state biofilm is presented. The model describes both the metabolic rate controlling behavior of this substrate in the biofilm and the effect of diffusion limitation caused by an arbitrary substrate on the active biofilm thickness. An analytical expression for the biocatalyst effectiveness factor is presented on the basis of Pirt kinetics for cell maintenance, first order substrate inhibition kinetics, and zero order substrate consumption kinetics. The proposed expression for the biocatalyst effectiveness factor is much more convenient to incorporate into a macroreactor model than the numerical alternatives. Simple criteria are presented to check the applicability of the model in case of true Monod kinetics. The analytical solution is expected to be particularly applicable to processes where a low soluble organic substrate controls the biomass growth, a situation which is often met in wastewater purification processes of industrial importance. The degradation of phenol by Pseudomonas sp. is treated as an example. (c) 1993 John Wiley & Sons, Inc.     

Prediction of substrate consumption rate, average biofilm density and active thickness for a thin spherical biofilm at pseudo-steady state

In this work, a new approach is proposed to evaluate substrate consumption rate, average biofilm density and active thickness of a spherical bioparticle in a completely mixed fluidized bed system. The substrate consumption rate and average biofilm density are predicted for a given biofilm surface substrate concentration and operational biofilm thickness. A diffusion and reaction model is developed with an effective diffusion coefficient that depends on the average biofilm density. This approach, a first in the literature, predicts the optimum average density of a biofilm to yield the maximum substrate consumption rate within the biofilm. A reasonable correlation was observed between the model prediction and experimental results for substrate consumption rate and average biofilm density for thin and fully active biofilms.     

A model-based approach to detect interspecific interactions during biofilm development

A model-based approach was developed to detect interspecific interactions during biofilm development. This approach relied on the comparison of experimental data with a simple null model of biofilm growth dynamics where individual species grew independently of one another, except that they competed for space. Such a model was directly parameterized with a 4D confocal image series of biofilms and then used as a null model to detect interspecific interactions between pairs of bacterial species. This approach was tested in two bispecific competitive trials. In the first trial, the progressive exclusion of Pseudomonas fluorescens by Pseudomonas putida appeared to be due solely to the different intrinsic growth rates of the two strains. In contrast, modelling results suggested the presence of interference competition between Pseudomonas aeruginosa and P. putida in mixed biofilms. The authors’ approach enables the detection of ecologically relevant interactions which constitute a prerequisite to building a comprehensive view of the dynamics and functioning of spatially structured bacterial communities.     

The Role of the Liquid Phase in the Degradation of Toluene and m‐Cresol within a Biofilm Trickle‐Bed Reactor

The degradation of toluene and m‐cresol in a biofilm trickle‐bed reactor was experimentally and theoretically investigated. Degradation is the result of the cooperation between suspended and immobilized microorganisms in the trickling film and the biofilm. The role of the trickling film is that of a barrier for mass transfer to the biofilm or that of an additional reaction space. This is the result of physical availability of pollutants to the liquid phase as well as co‐substrate degradation of inherent biomass. An instationary reactor balance model is presented. In addition to this the change in wetting behavior of carrier surface due to biofilm formation is discussed. A partial wetting of biofilm surface by rivulets of the trickling film is proposed. The model was verified by experimental data. The different reactor operation modes denoted as biofilm regime versus trickling film regime for the chosen pollutant system were expressed in terms of dimensionless reactions and transfer numbers. It is shown that the volumetric reaction rates for toluene in a trickling film regime reaches values twice as high as that of a biofilm regime due to the presence of the second substrate m‐cresol. The limiting step in both cases is the mass transfer of oxygen to bacteria in the biofilm or trickling film.     

Film analysis of activated sludge microbial discs by the Taguchi method and grey relational analysis

A biofilm model with substrate inhibition is proposed for the activated sludge growing discs of rotating biological contactor (RBC); this model is different from the steady-state biofilm model based on the Monod assumption. Both deep and shallow types of biofilms are examined and discussed. The biofilm models based on both Monod and substrate inhibition (Haldane) assumptions are compared. In addition, the relationships between substrate utilization rate, biofilm thickness, and liquid phase substrate concentration are discussed. The influence order of the factors that affect the biofilm thickness is studied and discussed by combining the Taguchi method and grey relational analysis. In this work, a Taguchi orthogonal table is used to construct the series that is needed for grey relational analysis to determine the influence priority of the four parameters S _B , kX _f , K _s, and K _i .     

Improved pseudoanalytical solution for steady-state biofilm kinetics

Simple algebraic expressions for the flux of substrate into a steady-state biofilm are developed. This pseudoanalytical solution, which eliminates the need for repetitiously solving numerically a set of nonlinear differential equations, is based on an analysis of the numerical results from the numerical solution of the differential equations. The critical advantage of this new pseudoanalytical solution is that it is highly accurate for the entire range of substrate concentrations and kinetic parameters. The article also illustrates that previous pseudoanalytical solutions for steady-state biofilm kinetics are seriously inaccurate for certain ranges of substrate concentration and kinetic parameters.     

Modeling how soluble microbial products (SMP) support heterotrophic bacteria in autotroph-based biofilms

Multi-species biofilm modeling has been used for many years to understand the interactions between species in different biofilm systems, but the complex symbiotic relationship between species is sometimes overlooked, because models do not always include all relevant species and components. In this paper, we develop and use a mathematical model to describe a model biofilm system that includes autotrophic and heterotrophic bacteria and the key products produced by the bacteria. The model combines the methods of earlier multi-species models with a multi-component biofilm model in order to explore the interaction between species via exchange of soluble microbial products (SMP). We show that multiple parameter sets are able to describe the findings of experimental studies, and that heterotrophs growing on autotrophically produced SMP may pursue either r- or K-strategies to sustain themselves when SMP is their only substrate. We also show that heterotrophs can colonize some distance from the autotrophs and still be sustained by autotrophically produced SMP. This work defines the feasible range of parameters for utilization of SMP by heterotrophs and the nature of the interactions between autotrophs and heterotrophs in multi-species, multi-component biofilms.     

Consecutive reaction kinetics involving distributed fraction of methanogens in fluidized-bed bioreactors

A kinetic model involving the distributed fractions of acidogens and methanogens is proposed. To determine the fluxes and biochemical reaction rates of the substrate sucrose and its intermediates, volatile fatty acids (VFAs) in bulk liquid and within the biofilm, a kinetic model was developed by combining the solid-phase model with the liquid-phase model. The predicted substrate removal efficiencies of the conventional and tapered fluidized-bed bioreactors (CFB, TFBs) are in good agreement with the experimental results. The biofilm thickness in TFBs are thicker than that in CFB, resulting in performance enhancement with TFBs. The simulated results obtained from the kinetic model show that methanogenesis is the rate-limiting step of degradation of the simple organic compound (sucrose), and the chemical oxygen demand (COD) concentration in the effluent is mainly contributed by the intermediates VFAs. The distributed fractions of acidogens and methanogens determined experimentally are 0.4 and 0.6, respectively.