Modeling Approaches

Conceptual and Descriptive Models

The simplest type of model, and therefore often a starting point in a biogeochemical modeling study, is a qualitative, conceptual or descriptive model. This type of model can be effectively communicated using diagrams, like that shown for mercury methylation in Figure 1. The purpose of this model is to describe the requirements for mercury methylation, and the model is simplified for that purpose. Water, for example, is also part of the overall reaction, but it is omitted from the model because it typically does not affect the process. Also, no information on what occurs inside of the bacteria, the biochemical reaction(s), is included in the model. Despite (and maybe because of) the simplifications, the model conveys important aspects of the process for this particular purpose. Another way to communicate a qualitative model is in the form of a narrative, as shown for dissimilatory iron reduction in Box 1.

Chemistry-Type Models

Qualitative models are useful, but often quantitative predictions are needed. One common approach to quantitative biogeochemical modeling is to apply the concepts of chemistry. That is, organisms are quantified as concentrations and their effect on chemistry is considered a reaction. There are a number of ways organisms can be incorporated into chemical models, as described in this section.

Hg(II) Sulfate-

Figure 1 Conceptual model of mercury methylation by sulfate-reducing bacteria.

MeHg

Organic carbon

Sulfide

Figure 1 Conceptual model of mercury methylation by sulfate-reducing bacteria.

Box 1 Narrative presentation of a model for dissimilatory Fe(m) reduction

The oxidation of detritus to CO2 by dissimilatory Fe(iii) reduction is a multistep process. First, the complex organic compound is hydrolyzed to smaller soluble compounds (e.g., amino acids, fatty acids). Then, those compounds are metabolized to acetate by fermentative microorganisms. Finally, the acetate is oxidized to CO2 and the Fe(i i i) is reduced to Fe(i i) by iron-reducing bacteria.

Such organism molecules can also be included in kinetic rate expressions. For growth on a substrate or nutrient, it is often observed that the specific growth rate is proportional to the substrate concentration at low substrate concentrations, meaning the substrate is rate limiting. However, at higher substrate concentrations the growth rate is limited by other factors, like the rate of processing the substrate. This type of behavior can be simulated using the well-known Monod model:

Organism is ignored

The simplest way to model the effect of organisms on chemistry is to simply ignore them, or to not explicitly recognize them or their action in the model. Nitrification, for example, is a two-step process carried out by nitrifying bacteria. First, Nitrosomonas converts ammonia (NH3) to nitrite (NO7) and then Nitrobacter converts nitrite to nitrate (NOf):

As shown in the equation, the reaction is mediated by and requires the two organisms. However, in the natural environment, nitrifying bacteria are often present in sufficient quantities, and therefore they are typically not included in rate expressions:

d[NH3 dt

where k (d_1) is the first-order reaction rate constant for the nitrification process. Sometimes the effect of O2 is included in the rate expression by modifying the rate constant k as a function of [O2].

Organism is included as reactant or product

Another way to model the effect of biological organisms on chemistry is to consider them a chemical molecule that participates in a reaction as a reactant or product. Photosynthesis, for example, is often represented using the following reaction:

106CO2 + 122H2O + 16HNO3 + H3PO4 ! (CH2O)i06(NH3 )i6 (H3PO4) +138O2

where the molecule A is a simplified chemical-type representation of algae, also called Redfield molecule. This type of representation is useful for composition analysis. It says, for example, that the N:P ratio of phyto-plankton is 16:1, which can be used to determine which one of these nutrients will run out first and end up limiting primary production.

where [A] (mmol is the phytoplankton concentration, MMAX (d_1) is the maximum specific growth rate (when [HNO3] >> Km), and Km (mmol l_1) is the half-saturation constant. The Monod model has a hyperbolic shape as illustrated in Figure 2. The amount of nutrient consumed per biomass synthesized can be calculated using a yield coefficient d[HNO3

Y dt where Y (1 mol A/16mol HNO3) is the yield coefficient.

Organism is included as catalyst

Another method is to consider organisms as a mediator in a chemical reaction without being a reactant or product, which is called catalyst in chemistry and enzyme or biological catalyst in biology. The nitrifying bacteria discussed above can be considered catalysts for the nitrification reaction, although the rate expression does not explicitly recognize that. The general sequence of an enzyme-mediated reaction is

TO o

M MAX

/

' I

Substrate concentration (mmol l-1)

Figure 2 Monod growth model. ^MAX — 1.0 d~1, Km = 0.5 mmol l_1.

First the substrate S combines with the enzyme E in a reversible reaction to form the complex SE. Then, the SE reacts to form one or more products P and E in an irreversible reaction. The mechanistic rate expression for enzyme kinetics is the Michaelis-Menten equation:

where VMAX (mmol l-1 d-1) is the maximum reaction velocity, and KM (mmol l-1) is half-saturation constant. The parameters VMAX and KM are related to the enzyme concentration and the rate constants of the individual reactions. The Michaelis-Menten equation has the same hyperbolic shape as the Monod equation illustrated in Figure 2.

Biology-Type Models

In some cases treating organisms as chemical molecules is overly simplistic and introduces excessive error into the model. Then, a more explicit representation of their effect on chemistry is needed. Organisms only directly affect the chemistry of their environment by removing (uptake) or adding (excretion) chemicals from or to their environment. These processes can be passive or active, as discussed in this section.

Passive uptake and excretion

Substances continuously diffuse in and out of organisms across the cell membrane, with a net transport in the direction of decreasing concentration. Movement through cell membranes can be complicated by multiple layers and binding sites, and therefore models often assume simple diffusion across one layer. For that case, the transport rate is proportional to the concentration gradient across the cell membrane:

where V(fmol cell-1 d-1) is the transport rate, Pm (m d-1) is the membrane permeability coefficient, A (m2 cell-1) is the cell surface area, and SIN and Sout (fmol m- ) are the chemical concentrations inside and outside of the cell, respectively. The intracellular concentration can be related to the cell quota (fmol cell-1) using the cell volume (m3 cell-1). For chemicals that speciate, the model applies to the same species, and care should be taken when the speciation chemistry is significantly different inside and outside of the cell.

Active uptake and excretion

Organisms can also actively take up and excrete substances using transport sites (enzymes) on the cell membrane. This process can move chemicals against a

Outside

Inside

ES S Figure 3 Enzyme-mediated uptake.

concentration gradient, in which case it requires an external input of energy. The nutrient phosphate, for example, is taken up by an active uptake process. Cadmium is an example of a toxic chemical that is excreted from cells using an active excretion process. The sequence of events is similar to that of enzyme kinetics described above, and consists of a two-step process, illustrated in Figure 3 for uptake. First, the chemical binds reversibly to the transport site. Then, it is transported into the cell in an irreversible reaction. This is typically modeled using the Michaelis-Menten equation:

V — Vmax km + s where V (fmol cell 1 d 1) is the transport rate, VMAX (fmol cell-1d-1) is the maximum transport rate, S (mmoll-1) is the chemical concentration, and KM (mmoll-1) is half-saturation constant. Active uptake systems are typically designed for and specific to a chemical. However, under certain circumstances other chemicals can be transported by the uptake system by mistake. The equation can be modified for the case where different substances (e.g., phosphate and arsenate) are taken up by the same transport system (competitive inhibition), and for the case where an internal compound slows the reaction (noncom-petitive inhibition), using methods from enzymology. In addition, kinetic data sometimes reveal multiple uptake systems for one compound, and the ability of organisms to switch systems on/off.

Empirical Models

The chemistry- and biology-type models discussed above are based on theoretical or known relationships, like molecular diffusion in the case of passive excretion, and those models are therefore classified as mechanistic. Empirical modeling is an alternative approach, based entirely on data. That is, the model is constructed with the objective of reproducing an observed pattern, and little or no attention is paid to the mechanistic correctness of the model equation(s). If a mechanistic understanding of every detail of a process is known, there is really no reason to adopt the empirical approach. However, when a process is too complex and unknown, the empirical

Figure 4 Temperature dependence of the endogenous respiration rate of Anabaena circinalis akinetes. Data from Fay P (1988) Viability of akinetes of the planktonic cyanobacterium Anabaena circinalis. Proceedings of the Royal Society of London B 234: 283-301.

approach may be the only viable alternative. Unfortunately, this is the rule rather than the exception in biogeochemistry and many operational models therefore contain at least some empirical components. Consider, for example, the temperature dependence of the endogenous respiration rate of akinetes (resting stage cells) of the cyanobacterium Anabaena circinalis shown in Figure 4. If we do not know the mechanism(s) responsible for the observed pattern, we may simply accept that fact and construct an equation that fits the data. The line in Figure 4 corresponds to a simple two-part equation with a slope of 0.0055 below 30 °C and -0.0035 above that. Although this modeling exercise did not further our mechanistic understanding of the endogenous respiration process, it did provide us with a simple means of predicting the rate, which may be useful and needed as a component of a larger mechanistic ecological model or for management purposes.

Since we have already admitted that our model has no mechanistic basis, the form of the equation (e.g., linear, exponential) is not important, and other empirical approaches are available (e.g., neural networks) that are not based on equations at all. However, it is generally accepted that models should be as simple and with as few parameters as possible. Also, since empirical modeling is based entirely on data, it is generally considered to be less transferable to conditions outside ofthose used to develop the model. So we should not apply our empirical model for akinete respiration to temperatures higher than 45 °C, other experimental conditions (e.g., higher/lower nutrient concentration), or other species (e.g., Anabaena flos-aquae).

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