A Modeling and Simulation Study of the Role of Suspended Microbial Populations in Nitrification in a Biofilm Reactor

Many biological wastewater treatment processes are based on bacterial biofilms, i.e. layered aggregates of microbial populations deposited on surfaces. Detachment and (re-)attachment leads to an exchange of biomass between the biofilm and the surrounding aqueous phase. Traditionally, mathematical models of biofilm processes do not take the contribution of the suspended, non-attached bacteria into account, implicitly assuming that these are negligible due to the relatively small amount of suspended biomass compared to biofilm biomass. In this paper, we present a model for a nitrifying biofilm reactor that explicitly includes both types of biomass. The model is derived by coupling a reactor mass balance for suspended populations and substrates with a full one-dimensional Wanner–Gujer type biofilm model. The complexity of this model, both with respect to mathematical structure and number of parameters, prevents a rigorous analysis of its dynamics, wherefore we study the model numerically. Our investigations show that suspended biomass needs to be considered explicitly in the model if the interests of the study are the details of the nitrification process and its intermediate steps and compounds. However, suspended biomass may be neglected if the primary interests are the overall reactor performance criteria, such as removal rates. Furthermore, it can be expected that changes in the biofilm area, attachment, detachment, and dilution rates are more likely to affect the variables primarily associated with the second step of nitrification, while the variables associated with the first step tend to be more robust.     

Modelling of biofilms

A mixed-culture biofilm (MCB) model is available which describes the progression of biofilm thickness and the spatial distribution and development in time of dissolved and paniculate components in the biofilm. The MCB model is able to predict the physico-chemical conditions at the interface between the biofilm and the solid surface, on which the biofilm grows, as a function of the conditions in the bulk fluid, the microbial composition of the biofilm, and the transport and transformation processes which take place in the biofilm. The mass balance equations of the MCB model are generally valid and can be applied to almost any microbial system if its kinetics and stoichiometry can be provided. AQUASIM is a new computer program for the identification and simulation of aquatic systems. The program solves the equations of the MCB model. It has a window-type user interface and includes routines for simulation, sensitivity analysis, automatic parameter estimation and data fitting. The MCB model has been developed and is primarily used in the field of waste water treatment. However, under certain conditions and with some additional simplifications this model can also be used for the investigation of biofouling and biocorrosion problems. The possibilities and limitations of the application of the MCB model and of AQUASIM to this type of problem are briefly discussed.     

A structured-population model of Proteus mirabilis swarm-colony development

In this paper we present continuous age- and space-structured models and numerical computations of Proteus mirabilis swarm-colony development. We base the mathematical representation of the cell-cycle dynamics of Proteus mirabilis on those developed by Esipov and Shapiro, which are the best understood aspects of the system, and we make minimum assumptions about less-understood mechanisms, such as precise forms of the spatial diffusion. The models in this paper have explicit age-structure and, when solved numerically, display both the temporal and spatial regularity seen in experiments, whereas the Esipov and Shapiro model, when solved accurately, shows only the temporal regularity. The composite hyperbolic-parabolic partial differential equations used to model Proteus mirabilis swarm-colony development are relevant to other biological systems where the spatial dynamics depend on local physiological structure. We use computational methods designed for such systems, with known convergence properties, to obtain the numerical results presented in this paper.     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

Bifurcations in Turing systems of the second kind may explain blastula cleavage plane orientation

Spontaneous pattern formation (emergence of Turing structures) may take place in biological systems as primary and secondary bifurcations to nonlinear parabolic partial differential equations describing biochemical reaction-diffusion systems. Bipolarity in mitosis and cleavage planes in cytokinesis may be related to this formation of prepatterns. Cleavage planes in early blastulas have an apparently well controlled spatial relationship to the polarity known as the animal-vegetal (A-V) axis: the mitotic spindles form perpendicular to this axis in the first two division stages, with cleavage planes going strictly through the A-V poles. The third-stage spindles are parallel to the A-V axis, and cleavage is roughly in the equatorial plane, thus separating the A-V poles. The reason for these phenomena are poorly understood with current mitosis/cytokinesis models based on intrinsic spindle properties. It is shown here by numerical simulation that a simple modification to the usual Turing equations yields selection rules which lead directly to these orientations of the prepatterns, without any further ad hoc assumptions. These results strongly support the prepattern model for mitosis and cytokinesis and the viewpoint that prepatterns play a fundamental role in nature.     

Two-Fluid Model of Biofilm Disinfection

We consider a dynamic model of biofilm disinfection in two dimensions. The biofilm is treated as a viscous fluid immersed in a fluid of less viscosity. The bulk fluid moves due to an imposed external parabolic flow. The motion of the fluid is coupled to the biofilm inducing motion of the biofilm. Both the biofilm and the bulk fluid are dominated by viscous forces, hence the Reynolds number is negligible and the appropriate equations are Stokes equations. The governing partial differential equations are recast as boundary integral equations using a version of the Lorenz reciprocal relationship. This allows for robust treatment of the simplified fluid/biofilm motion. The transport of nutrients and antimicrobials, which depends directly on the velocities of the fluid and biofilm, is also included. Disinfection of the bacteria is considered under the assumption that the biofilm growth is overwhelmed by disinfection. Supported by NSF award DMS-0612467.     

In vitro model systems for exploring oral biofilms: From single-species populations to complex multi-species communities

Numerous in vitro biofilm model systems are available to study oral biofilms. Over the past several decades, increased understanding of oral biology and advances in technology have facilitated more accurate simulation of intraoral conditions and have allowed for the increased generalizability of in vitro oral biofilm studies. The integration of contemporary systems with confocal microscopy and 16S rRNA community profiling has enhanced the capabilities of in vitro biofilm model systems to quantify biofilm architecture and analyse microbial community composition. In this review, we describe several model systems relevant to modern in vitro oral biofilm studies: the constant depth film fermenter, Sorbarod perfusion system, drip–flow reactor, modified Robbins device, flowcells and microfluidic systems. We highlight how combining these systems with confocal microscopy and community composition analysis tools aids exploration of oral biofilm development under different conditions and in response to antimicrobial/anti-biofilm agents. The review closes with a discussion of future directions for the field of in vitro oral biofilm imaging and analysis.     

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.     

Simplified models for packed-bed biofilm reactors

For a packed-bed biofilm reactor two reactor models are proposed. One model is for the limiting case of a biofilm with a constant biofilm thickness in which diffusion within the biofilm is shown to be negligible. The second model assumes that the thickness of the biofilm is limited by the concentration of substrate within the biofilm. The analytical solutions for these reactor models are shown to agree very well with the numerical solutions to the exact differential equations.     

Simulation of citric acid production by rotating disk contactor

A simple model was presented to describe the time courses of citric acid production by a rotating disc contactor (RDC) using Aspergillus niger. The model is expressed by Monod-type cell growth, Luedeking-Piret-type citric acid production rate equations, and the diffusion equation for oxygen in the biofilm. The model contains five parameters which were determined by the nonlinear least squares method by fitting the numerical solution to the experimental data. In solving the equations, the cell density of the biofilm was estimated from the value of cellular mass per unit of biofilm area using an empirical equation. The experimental time courses in citric acid production period were well simulated with this model. The relation between the specific biofilm surface area and the rate of citric acid production was also explained by the simulation using the average values of five parameters of twelve runs. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 689-696, 1997.     

Hypercycles versus parasites in the origin of life: model dependence in spatial hypercycle systems

Spatial hypercycle systems can be modelled by means of cellular automata or partial differential equations. In either model, two dimensional spirals or clusters can be formed. Different models give rise to slightly different spatial structures, but the response to parasites is fundamentally different: In cellular automata the hypercycle is resistant to parasites that are fatal in a partial differential equations model. In three dimensions scroll rings correspond to the two dimensional spirals. Numerical simulations on a partial differential equations model indicate that the scroll rings are unstable: The contract by a power law and disappear. Therefore, in three dimensions clusters seem to be the best candidate for the hypercycle resistant to parasites.     

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.     

A spatial mechanism with higher pairs for modelling the human knee joint

By generalizing a previous model proposed in the literature, a new spatial kinematic model of the knee joint passive motion is presented. The model is based on an equivalent spatial parallel mechanism which relies upon the assumption that fibers within the anterior cruciate ligament (ACL), the medial collateral ligament (MCL) and the posterior cruciate ligament (PCL) can be considered as isometric during the knee flexion in passive motion (virtually unloaded motion). The articular surfaces of femoral and tibial condyles are modelled as 3-D surfaces of general shapes. In particular, the paper presents the closure equations of the new mechanism both for surfaces represented by means of scalar equations that have the Cartesian coordinates of the points of the surface as variables and for surfaces represented in parametric form. An example of simulation is presented in the case both femoral condyles are modelled as ellipsoidal surfaces and tibial condyles as spherical surfaces. The results of the simulation are compared to those of the previous models and to measurements. The comparison confirms the expectation that a better approximation of the tibiofemoral condyle surfaces leads to a more accurate model of the knee passive motion.     

A fluid dynamics model of the growth of phototrophic biofilms

A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing.     

Analysis of three- and four-compartment models forin vivo radioligand-neuroreceptor interaction

Currently applied three-copartment models for analyzing kinetic data derived fromin vivo positron emission tomographic (PET) studies of radioligand-neuroreceptor interactions require assumptions which may not be strictly valid. Such assumptions include very rapid kinetics for nonspecific binding and the absence of multiple specific receptors or subtypes. Computer simulations, based on an exact analytical solution of the relevant differential equations, indicate the numerical errors that can arise when the assumptions are invalid. We propose a fourcompartment model which requires fewer assumptions. A simple relationship is derived for expressing the microscopic rate constants of either the three- or four-compartment model as explicit functions of the experimentally-observed macroscopic rate constants. This could eliminate the need for time-consuming, iterative, non-linear, curve-fitting approaches and numerical integration. The usefulness of the four-compartment model is limited, however, by the sensitivity and temporal resolution of current PET imaging devices.     

Calculation of effective diffusivities for biofilms and tissues.

In this study we describe a scheme for numerically calculating the effective diffusivity of cellular systems such as biofilms and tissues. This work extends previous studies in which we developed the macroscale representations of the transport equations for cellular systems based on the subcellular-scale transport and reaction processes. A finite-difference model is used to predict the effective diffusivity of a cellular system on the basis of the subcellular-scale geometry and transport parameters. The effective diffusivity is predicted for a complex three-dimensional structure that is based on laboratory observations of a biofilm, and these numerical predictions are compared with predictions from a simple analytical solution and with experimental data. Our results indicate that, under many practical circumstances, the simple analytical solution can be used to provide reasonable estimates of the effective diffusivity.     

Global Kinetic Explorer: A new computer program for dynamic simulation and fitting of kinetic data

We describe a new dynamic kinetic simulation program that allows multiple data sets to be fit simultaneously to a single model based on numerical integration of the rate equations describing the reaction mechanism. Unlike other programs that allow fitting based on numerical integration of rate equations, in the dynamic simulation rate constants, output factors, and starting concentrations of reactants can be scrolled while observing the change in the shape of the simulated reaction curves. Fast dynamic simulation facilitates the exploration of initial parameters that serve as the starting point for nonlinear regression in fitting data and facilitates exploration of the relationships between individual constants and observable reactions. The exploration of parameter space by dynamic simulation provides a powerful tool for learning kinetics and for evaluating the extent to which parameters are constrained by the data. This feature is critical to avoid overly complex models that are not supported by the data.     

Modeling and simulation of competition between two microorganisms for a single inhibitory substrate in a biofilm reactor

A simple biofilm model was developed to simulate the competition between two microorganisms for a common inhibitory substrate. The following assumptions were made for the simulations: (1) the biofilm has a uniform thickness and is composed of 5 segments, (2) growth of two microorganisms A and B which utilize the common substrate is expressed by the Haldane kinetics with a spatial limitation term and is independent of the other microorganism in the biofilm reactor, and (3) diffusion of the substrate, movement of the microorganisms, and continuous loss of the biomass by shearing are expressed by Fick's Law-type equations. The qualitative behavior of the biofilm reactor is characterized by five regions, I-V, depending on the operation conditions, the substrate concentration in feed, and the dilution rate. In region I, both microorganisms are washed out of the biofilm reactor. In region II, microorganism B is washed out, and in region III, microorganism A is washed out of the biofilm. In region IV, both microorganisms coexist with one another. In region V, both microorganisms coexist with a sustained oscillatory behavior. Convergence to regions I-V depends on the initial conditions. In regions II-V, washout of either or both microorganisms is also observed with initial conditions too far away.     

Resolving discrepancies between deterministic population models and individual-based simulations

ABSTRACT This work ties together two distinct modeling frameworks for population dynamics: an individual-based simulation and a set of coupled integrodifferential equations involving population densities. The simulation model represents an idealized predator-prey system formulated at the scale of discrete individuals, explicitly incorporating their mutual interactions, whereas the population-level framework is a generalized version of reaction-diffusion models that incorporate population densities coupled to one another by interaction rates. Here I use various combinations of long-range dispersal for both the offspring and adult stages of both prey and predator species, providing a broad range of spatial and temporal dynamics, to compare and contrast the two model frameworks. Taking the individual-based modeling results as given, two examinations of the reaction-dispersal model are made: linear stability analysis of the deterministic equations and direct numerical solution of the model equations. I also modify the numerical solution in two ways to account for the stochastic nature of individual-based processes, which include independent, local perturbations in population density and a minimum population density within integration cells, below which the population is set to zero. These modifications introduce new parameters into the population-level model, which I adjust to reproduce the individual-based model results. The individual-based model is then modified to minimize the effects of stochasticity, producing a match of the predictions from the numerical integration of the population-level model without stochasticity.