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Team:ZJU-China/Modeling
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components, which form the biofilm solid matrix. In the model of single species of bacteria, the | components, which form the biofilm solid matrix. In the model of single species of bacteria, the | ||
particulate components are assumed to be uniform, defined as</p> | particulate components are assumed to be uniform, defined as</p> | ||
| - | <img src="http://ung.igem.org/wiki/images/0/02/Function1.png" width="720" style=" | + | <img src="http://ung.igem.org/wiki/images/0/02/Function1.png" width="720" style="margin-left:10px;" alt="function1" /> |
<p>where<strong> X_M</strong> is the concentration of the attached component, <strong>ρ</strong> is its density , defined as the mass | <p>where<strong> X_M</strong> is the concentration of the attached component, <strong>ρ</strong> is its density , defined as the mass | ||
divided by the volume of the cell or particle, and<strong> ε</strong> is its volume fraction, defined as volume of | divided by the volume of the cell or particle, and<strong> ε</strong> is its volume fraction, defined as volume of | ||
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Project/Modeling
IntroductionCompartment:The biofilm itself is distinguished from the overlying water and the substratum to which it is attached. A mass-transport boundary layer separates the biofilm from the overlying water. Within each compartment are components: include different types of biomass ,substrates , products. biomass is often divided into active microbial species, inert cells, and extracellular polymeric substances(EPS). The components can undergo transformation, transport, and transfer processes. For example, substrate is consumed, and this leads to the synthesis of new active biomass. All process affecting each component in each compartment are mathematically linked together into a mass balance equation that contains rate terms and parameters for each process. Model Selection:Many kinds of Mathematics models have been founded to describe a system of biofilm. Models of different dimensions (1d, 2d, 3d) focus on different properties of a biofilm. Since we care most about the oxygen concentration gradients perpendicular to the substratum, numerical 1-dimensional dynamic model(N1) would be a proper choice for us. CompartmentThe biofilm:A biofilm is a gel-like aggregation of microorganisms and other particles embedded in extracellular polymeric substances. A biofilm contains water inside it, but its main physical characteristic is that it is a solid phase. A biofilm normally is anchored to a solid surface called the substratum on one side and in contact with liquid on its other side. Frequently, a mass-transfer boundary layer is included between the bulk liquid and the biofilm itself. Thus, following figure illustrates a biofilm having four compartments: the substratum, the biofilm itself, the boundary layer, and the bulk liquid outside of the biofilm. While it is complex even for a homogeneous biofilm morphology, we assume the biofilm surface is flat and all material below the maximum biofilm thickness as part of the biofilm components, and they have a constant density. 
the mass-transport boundary layerExperimental observations clearly indicate strong concentration gradients for solutes just outside the biofilm when these solutes are utilized or produced by the microorganisms in the biofilm. Consequently, the solute concentrations at the biofilm surface and in a completely mixed bulk liquid often are significantly different.So we introduce the mass-transport boundary layer,which is a hypothetical layer of liquid above the biofilm and in which all the resistance to mass transport of dissolved components outside the biofilm occurs. The bulk liquid: In our experiments, the bulk liquid is large compared to the biofilm. So the simplest way seems to consider it as a boundary condition of the biofilm compartment and specify the concentrations of dissolved. However, dissolved components can exchange between the biofilm and the bulk liquid, and it has a profound impact on the concentrations in the bulk liquid. Thus we include the bulk liquid not only as a boundary condition, but also as a separate, completely mixed compartment, varying according to the inflow, outflow, and the exchanges with the biofilm. The substratum: In our basic model, the substratum is a separate compartment and impermeable. So it does not have much effect on the biofilm system. However in some bioreactor, the substratum may be permeable, or include organic solids that are biodegraded by attached microorganisms. ComponentDissolved components: There are two kind of dissolved components in our model, one is oxygen, the other is substrates where nutrients are inside. They are expressed by inflow concentration of oxygen, inflow concentration of substrates and monod half saturation constant for substrates, monod half saturation constant for oxygen.Diffusion coefficient of oxygen and diffusion coefficient of substrate are also used to characterize the property of these components. Particulate components: In our model, the particulate components are microbes and EPS. We assume that they are homogeneously mixed in the same proportion in all parts of the biofilm. So can use bacterial density to relate the amount of bacterial and the volume it takes up. And the volume fraction of EPS in Bacterial is specified by a constant. ProcessTransformation processes usually are biochemical reactions that produce or consume one or more components: e.g., consumption of substrate, production of metabolic end-products, microbial growth and decay, and production of EPS. The transport processes that regularly are considered in biofilm models are advection, molecular diffusion, and turbulent dispersion. In special cases, transport of charged components by migration in an electric field created is included. The general, 1d expression to model the specific mass flux of a component is calculuted in the direction z Transfer processes exchange mass of dissolved or particulate components between two compartments. At the interface between the compartments, a continuity condition for the component concentration C and the specific flux j of the exchanged mass must be fulfilled. Continuity means that C and j are the same on both sides of the interface between the compartments. C and j can be calculated at each side of the interface from boundary conditions ModelingAn introduction to N1 model: The core of the N1 model consists of a system of stiff, non-linear partial differential equations. Because of the stiffness of the equation system, integration methods tailored for stiff systems must be used, or the dissolved and the particulate components have to be treated differently. Furthermore, biofilm growth, i.e., the displacement of the interface between biofilm and bulk liquid, creates a so-called moving boundary problem (Kissel et al. 1984; Wanner and Gujer 1986).We performed simulations with the N1 model using the software package AQUASIM (Reichert 1998a, 1998b), since it can handle stiff systems and the moving boundary. The N1 model implemented in AQUASIM is described in next Section. The processes considered in the model include:
Definitions and equations As discussed above, biofilms are multiphase systems. Consequently, in the N1 model three different phases are distinguished. The solid attached phase is made up by the particulate components, which form the biofilm solid matrix. In the model of single species of bacteria, the particulate components are assumed to be uniform, defined as 
where X_M is the concentration of the attached component, ρ is its density , defined as the mass divided by the volume of the cell or particle, and ε is its volume fraction, defined as volume of the component per unit biofilm volume. The porosity or biofilm pore volume fraction θ is 
The pore volume is formed by two phases: the phase of the suspended particulate components with concentrations X_P and the biofilm liquid phase, with the liquid phase volume fraction ε_liquid:
Mass balances in biobilm models Conservation of mass of a component in a dynamic and open system states that: 
SimulationCalculate growth and oxygen concentration gradient of a biofilm which consists of Heterotrophic bacteria. The water inflow at a constant rate contains substrates and Oxygen. Growth occurs with Monod-type rate laws. Respiration occurs with specific rate. A program variable LF referring to Biofilm Thickness and 3 dynamic volume state variables O_2, S, X referring to oxygen, substrates and the Heterotrophic bacteria are defined. All necessary constants and initial conditions are defined in following table. Some of their values are based on average results provided by IWA. 
Definition of Process: 
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