Environ. Sci. Technol. 2007, 41, 2855-2860
Accounting for Intrapopulation Variability in Biogeochemical Models Using Agent-Based Methods FERDI L. HELLWEGER* AND EHSAN KIANIRAD Department of Civil & Environmental Engineering, Northeastern University, Boston, Massachusetts 02115
Present biogeochemical models typically use a lumpedsystem (population-level) modeling (LSM) approach that assumes average properties of a population within a control volume. For modern models that formulate phytoplankton growth as a nonlinear function of the internal nutrient (e.g., Droop kinetics), this averaging assumption can introduce a significant error. Agent-based (individual-based) modeling (ABM) is an alternative approach that does not make the assumption of average properties. This paper presents a new agent-based phytoplankton model called iAlgae. The model is contrasted to a conventional lumped-system model, constructed based on identical underlying sub-models of nutrient uptake (including luxury uptake) and growth (cell quota, Droop model). The two models are validated against laboratory data and applied to a realistic scenario, consisting of a point source nutrient discharge into a river. For the realistic scenario, the ABM-predicted phytoplankton bloom is significantly lower than the LSM-predicted one, which is due to the intrapopulation distribution in cell quotas (due to different life histories of individuals) and nonlinearity of the growth rate model. In the ABM, a fraction of the population accumulates nutrients in excess of their immediate growth requirement (luxury uptake), leaving less for the remainder. Because the model is nonlinear, this results in a suboptimal (from a population perspective) utilization of nutrient and a lower population-level growth rate, compared to the case of no intrapopulation variability assumed by the LSM model. In general, the ABM and LSM approaches can produce significantly different results when incompletely mixed conditions lead to intrapopulation variability in cell properties (i.e., cell quota) and the model equations are nonlinear.
Introduction Biogeochemical (a.k.a. eutrophication, water quality) models are important environmental science and engineering tools. They allow scientists to quantitatively evaluate process models against field data. Engineers use biogeochemical models to relate nutrient discharges from point and nonpoint sources to receiving water conditions, which is an essential component of a total maximum daily load (TMDL) analysis. The present biogeochemical modeling approach, as described in textbooks (1, 2) and implemented in operational models (e.g., WASP, 3; AQUATOX, 4), can be * Corresponding author phone: (617) 373-3992; fax: (617) 3734419; e-mail:
[email protected]. 10.1021/es062046j CCC: $37.00 Published on Web 03/16/2007
2007 American Chemical Society
characterized as a lumped-system (a.k.a. population-level) modeling (LSM) approach. Phytoplankton cells are essentially considered to be chemical molecules and are quantified as a chemical concentration (e.g., mmol C L-1). Nutrient uptake and growth is considered to be a chemical reaction between the phytoplankton and nutrient molecules, and rates are formulated based on the average properties (e.g., concentration) of the population within a waterbody or mass balance segment. This “chemistry approach” to biogeochemical modeling is adequate when phytoplankton growth is simulated as a function of the external nutrient concentration (i.e., Monod kinetics). However, it is recognized for some time now that formulating growth based on the internal nutrient is more mechanistically correct (5, 6), and biogeochemical models that decouple nutrient uptake and growth are being developed (7, 8). As illustrated in this paper, for these models, the assumption of average population properties, which is solidly embedded in the LSM approach, can introduce a significant error. Agent-based (a.k.a. individual-based) modeling (ABM) is an alternative approach that does not make the assumption of average properties. In ecology, where organisms (e.g., predator fish) are more functionally complex, this modeling approach is well established (9, 10). Rather than simulating population-level properties (fish biomass per volume) using differential equations (e.g., Lotka-Volterra), agent-based models simulate individual members of the population, including their states (e.g., body weight) and behavior (e.g., predation). As a result of the cumulative behavior of individuals, a population-level response emerges. Due to the advantages of the ABM approach and increasing computational resources, ABMs are now beginning to be applied to lower trophic levels, like zooplankton (11, 12), phytoplankton (13, 14) and bacteria (15, 16). The following simple example quantitatively illustrates the difference between the LSM and ABM approaches. Consider a water body or mass balance segment containing a population of phytoplankton cells with varying amounts of internal nutrient (q, mmol mol C-1, cell quota), due to different life histories. The growth rate (µ, day-1) is calculated using the nonlinear cell quota model (5; µ ) µMAX (1 - q0/q); µMAX, day-1, max. µ; q0, mmol mol C-1, subsistence quota). To simplify the calculations, we will assume there are two sub-populations of equal numbers. Sub-population A has q ) q0 and sub-population B has q ) 3 q0. In the LSM approach, the population-average cell quota is used to calculate the population-average growth rate, which is µ ) µMAX (1 - q0/ [2 q0]) ) 0.5 µMAX. In the ABM approach, the growth rate for each individual cell is calculated separately based on the individual cell quotas. The individuals in sub-population A have µ ) µMAX (1 - q0/q0) ) 0.0, and the individuals in subpopulation B have µ ) µMAX (1 - q0 / [3 q0]) ) 0.67 µMAX. The population-average growth rate is µ ) 0.33 µMAX, which is significantly lower than that computed using the LSM approach. In mathematics, this is known as Jensen’s inequality. The ABM approach constitutes a potential alternative for biogeochemical models. But will the ABM and LSM approaches produce different results for a realistic scenario? If yes, will that difference be significant in an engineering context? For what model types and/or hydrodynamic conditions can we expect a significant difference? The purpose of this paper is to introduce the ABM approach for biogeochemical modeling and contrast it to the conventional LSM approach. We conducted the following steps. First, we constructed LSM and ABM models based on identical VOL. 41, NO. 8, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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phosphorus (PP). The reverse reaction is excretion of phosphate. Upon death, PP becomes particulate organic phosphorus (POP) and dissolved organic phosphorus (DOP), based on a user-specified fraction. POP hydrolyzes to DOP, and DOP mineralizes to PO4, closing the cycle. All state variables in the LSM are on a mass per volume water basis (e.g., mmol P L-1), with one variable for each mass balance segment. The ABM has a similar construct, except that the phytoplankton nutrient contents are now a property of individuals. The corresponding individual state variables are size (m, pmol C cell-1), nitrogen quota (qN, mmol N mol C-1) and phosphorus quota (qP, mmol P mol C-1). Also, since cells are transported (advection, dispersion, settling), each individual has a position (x, m; y, m; z, m). A representative number (SR) is included to scale from individuals to super-individuals (discussed further later in the paper). As in the LSM, extracelluar constituents (e.g., POP, DOP, PO4) are modeled as lumped-system state variables. The model, therefore, uses a hybrid Eulerian-Lagrangian approach, as was done previously for nutrient-phytoplankton (13) and plankton-fish (17) systems. Mass Balances. The mass balance equation for phytoplankton C in the LSM is
dPC ) (µP - µR - µD)PC dt
FIGURE 1. State variables for (a) population-level and (b) individualbased models. State variables: DO is dissolved oxygen, POC is particulate organic carbon, DOC is dissolved organic carbon, PC is phytoplankton carbon, PON is particulate organic nitrogen, DON is dissolved organic nitrogen, NH4 is dissolved ammonia, NO3 is dissolved nitrate, PN is phytoplankton nitrogen, POP is particulate organic phosphorus, DOP is dissolved organic phosphorus, PO4 is dissolved phosphate, PP is phytoplankton phosphorus, m is cell size, qN is nitrogen cell quota, qP is phosphorus cell quota, x, y, z are position, SR is representative number. underlying sub-models of nutrient uptake and growth. Second, we validated them against laboratory data. Third, we confirmed that the models produce the same results for linear uptake and growth sub-models. Fourth, we applied them to a realistic field scenario and compared the results.
Model Description This section presents the key details of the ABM and LSM models, and highlights the differences between the two approaches. The underlying sub-models of nutrient uptake and growth are based on the OldLace model (8). For simplicity, a number of components of that model are omitted, including Si, multiple phytoplankton species, multiple reactivity classes for particulate and dissolved organic matter, explicit simulation of zooplankton, and the sediment bed. State Variables. The state variables and their interaction are illustrated in Figure 1. The LSM state variables are similar to those of present operational biogeochemical models, except that separate variables are included for phytoplankton carbon (PC), nitrogen (PN) and phosphorus (PP), allowing for a variable cell stoichiometry (luxury uptake). Zooplankton are not explicitly simulated, but their effect is included implicitly in the death reaction (e.g., PC f POC). The phosphorus cycle is as follows: Phosphate (PO4) is taken up as a nutrient by the phytoplankton to become phytoplankton 2856
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(1)
where µP (day-1) is the specific photosynthesis rate, µR (day-1) is the specific respiration rate, and µD (day-1) is the specific death rate. The specific growth rate is defined as µG ) µP µR. Note that transport terms (advection, dispersion, settling) are omitted from this and subsequent equations for clarity. Transport is discussed further later in the paper. The mass balance equation for phytoplankton P is
dPP ) (VPO4 - WPO4)PC - µDPP dt
(2)
where VPO4 (mol P mol C-1 day-1) is the specific PO4 uptake rate, and WPO4 (mol P mol C-1 day-1) is the specific PO4 excretion rate. The first term on the RHS of eq 2 is the uptake and excretion of PO4, and the second term is the production of POP and DOP by death due to zooplankton grazing or some other factor (e.g., viral lysis). Growth dilution is not included in the mass balance equation, because the intracellular concentration is on a per volume water basis (18). The mass balance equation for PO4 is
dPO4 ) (WPO4 - VPO4)PC + kDOPDOP dt
(3)
where kDOP (day-1) is the DOP mineralization rate constant. The first term on the RHS of eq 3 is again the excretion and uptake of PO4, and the second term is the production of PO4 due to mineralization of DOP. In the ABM, there is no differential mass balance equation for the phytoplankton population, rather a discrete accounting routine tracks the individuals. The size of individuals changes as a result of photosynthesis and respiration:
dm ) mµG dt
(4)
The mass balance equation for intracellular P for each individual is
dqP ) VPO4 - WPO4 - µG qP dt
(5)
where qP (mmol P mol C-1) is the P cell quota, VPO4 (mol P
mol C-1 day-1) is the specific PO4 uptake rate, and WPO4 (mol P mol C-1 day-1) is the specific PO4 excretion rate. The first and second term on the RHS of eq 5 are the uptake and excretion of PO4. The last term accounts for growth dilution, since the cell quota is defined on a carbon-basis. For extracellular PO4, the mass balance equation is
dPO4 dt
)
1
∑
V s agents
[(WPO4 - VPO4)m] + kDOPDOP
(6)
where s V (m3) is the segment volume. The first term on the RHS of eq 6 includes a summation term for all the individuals in the control volume and accounts for the excretion and uptake of phosphate, and the second term accounts for the production of PO4 by mineralization of DOP. Note that the parameters describing nutrient uptake and growth (VPO4, WPO4, µG) are the same for the LSM and ABM models and are computed using the same sub-models, as described subsequently. Photosynthesis and Respiration. Photosynthesis is simulated using the cell quota model, modified to explicitly define limited and replete conditions for each nutrient, which is needed for the dual uptake model (discussed in the next section) (18-20):
[(
µP ) µP,MAX min 1 -
) ]
)(
q0,N q0,P , 1, LH qN qP
(7)
where µP,MAX (day-1) is the maximum specific photosynthesis rate, q0,P (mmol P mol C-1) is the P subsistence quota below which the photosynthesis rate is zero, qP (mmol P mol C-1) is the P quota, and LH is the hypothetical limitation term, which accounts for limitation by a nutrient not explicitly considered. Therefore, each term in the minimum operator on the RHS of eq 7 represents limitation by a specific nutrient. Note that the maximum attainable µP is µP,MAX LH. For limitation by a single nutrient, the LH constant can be related to a cell quota below which the nutrient is limiting, the limitation quota qL ) q0/(1 - LH). Using the limitation quota, eq 7 can be rewritten:
(
µP ) µP,MAX 1 -
q0
)
min[q,qL]
(8)
which more clearly highlights the three different zones of photosynthesis 1: none (q < q0, µP ) 0), 2: nutrient limited (q0 < q < qL, µP ) f(q)) and 3: nutrient replete (qL < q, µP ) µP,MAX LH). Limitation by temperature and light is not considered in this paper and, therefore, omitted from the equations for clarity. Respiration is accounted for using a constant first-order rate constant. Reproduction and Death. During the cell cycle, the cell size (m) increases from a minimum (m0) to twice that (2 m0), a requirement for balanced growth in the case of binary fission. The m0 parameter can be related to the population average cell size by ave(m) ) m0 2 ln(2) (see Section S1, Supporting Information). Note that in this paper, model results are presented on a mmol C L-1 basis, which makes the choice of m0 irrelevant. In other words, even though the ABM has m0 as an additional parameter, the value of m0 effectively cancels out in the model (unless individual cell parameters are considered). Once the cell size reaches 2 m0, the cell divides into two daughter cells. To prevent numerical or artificial synchronization (14) and to account for real biological randomization, the size (m) of daughter cells is varied somewhat from the ideal 50/50 split. The split fraction SF is defined as the size of the first daughter cell, with (1 SF) being the size of the second daughter cell. For each cell division event, SF is drawn randomly from a normal distri-
bution with a mean of 0.5, coefficient of variation (RCV) of 0.1, truncated to (2 standard deviation and >0 (15). See Section S2 in the Supporting Information for further discussion on stochastic variability. Cell division results in the creation of an additional super-individual, rather than a doubling of the representative number. Since the cell quotas (qN, qP) are biomass-based, the daughter cells will have the same cell quota as the mother cell. Cell death in the ABM is simulated as a stochastic process. The probability of a cell dying in a given time step is µD ∆t. For each individual, a random number between 0 and 1 is drawn, and if that number is below the probability of dying, the cell dies. Nutrient Uptake. Nutrient uptake is simulated using a two-part uptake model for nutrient-limited and -replete (luxury uptake) conditions. The luxury uptake model is (18, 21)
{
PO4 VMAX,PO4 q < qL KM,PO4 + PO4 VPO4 ) PO4 µG q*,PO4 q > qL KM*,PO4 + PO4
(9)
where VMAX,PO4 (mol P mol C-1 day-1) is the maximum specific PO4 uptake rate, KM,PO4 is the PO4 uptake half-saturation constant for limited conditions, q*,PO4 (mmol P mol C-1) is the steady-state luxury uptake quota for PO4 uptake, and KM*,PO4 is the PO4 uptake half-saturation constant for replete conditions. Therefore, for P-limited conditions, the top term on the RHS of eq 9 is used, and for P-replete conditions the bottom term is used. Transport. Transport of lumped Eulerian parameters (e.g., PO4) is modeled using conventional advection and dispersion formulations. Transport of individual Lagrangian particles occurs as a result of advective and random walk components. The particle transport algorithm is designed to produce the same population-level transport as for lumped parameters (22). A constant settling velocity is applied to phytoplankton and particulate organic matter. Implementation. Since the number of phytoplankton cells can be very large, simulating each individual is not feasible given present computing technology. The approach used here is to simulate a number of representative superindividuals, each identical in behavior to individuals of the population and representative of a number of population individuals (e.g., refs 14, 23). The model framework, iAlgae, is based on the hydrodynamic model ECOMSED (22, 24, 25) and the biogeochemical models RCA (26) and OldLace (8). More details on the model heritage and code are provided in Supporting Information Section S5.
Model Application Model Validation Against Laboratory Data. As described in the introduction, any differences between the ABM and LSM models will likely be related to intrapopulation variability in cell properties (i.e., quota), which will be mainly due to different life histories of individuals. Therefore, the two approaches should produce the same results for well-mixed laboratory reactors, where the life history of individuals is controlled and intrapopulation variability is minimized. To verify this and also validate the models, they were applied to the steady chemostat and batch culture laboratory experiments of Droop (19, 20). The chemostat results (Figure 2a) show how the growth rate increases as a hyperbolic function of the cell quota. The batch results (Figure 2b) show very high cell quotas during the exponential phase due to the luxury uptake mechanisms, and lower quotas at the substance level during the stationary phase. The LSM and ABM model results are practically identical and consistent with the data. This exercise demonstrates that the two VOL. 41, NO. 8, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. LSM and ABM application to (a) continuous and (b) batch culture. Data are from Droop (19, 20). Cell quotas were converted from cell basis (fmol P cell-1) to carbon basis (mmol P mol C-1) using an average cell size of 1.04 pmol cell-1 (27). Parameter values are as in refs 18-20.
FIGURE 3. LSM and ABM predicted phytoplankton concentration downstream of a phosphate point source. Implementation validation using the simplified linear model. approaches produce the same results for well-mixed systems, which can apply to laboratory reactors as well as lakes and reservoirs. Description of Test Case. The purpose of this paper is to illustrate the differences between the LSM and ABM approaches, and for that reason, a simplified scenario is simulated. The scenario consists of a continuous phosphate point source into a straight channel river (width ) 1 km, depth ) 10 m, grid spacing ) 1 km). Light, temperature and nitrogen are assumed to be at optimal conditions, and only limitation by P is considered. The upstream inflow rate was set at 1000 m3 s-1, which results in a velocity of 0.1 m s-1. The longitudinal dispersion in the model is 150 m2 s-1 (28). The point source has a discharge rate of 0.10 mol P sec-1 (0.27 t P day-1), which translates into 0.10 µmol L-1, once diluted over the cross section. The model coefficients are listed and compared to literature values in Section S3, Supporting Information. All simulations are run to a steadystate. Implementation Validation Using Simplified Linear Model. The ABM and LSM models are implemented in different computer codes and, to eliminate bugs or other mistakes as potential source of any difference between the models, a docking (29) exercise is performed. Since the root cause of any model difference will be the nonlinearity of the model equations, the two models should produce the same results if the equations were linear. To test this we (temporarily) linearize the model by using a linear growth equation (µP ) µMAX PO4/KM,PO4, i.e., linear portion of Monod), and a linear uptake equation where uptake is proportional to growth assuming a fixed cell quota (V ) µPq0, i.e., Redfield stoichiometry). Figure 3 illustrates that the two modeling approaches produce the same results, except for slight differences due to stochastic variability in the ABM. Therefore, for the ABM and LSM approaches to produce different results, the model equations need to be nonlinear. Application of Full Nonlinear Model. The results for the full nonlinear versions of the LSM and ABM models are 2858
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FIGURE 4. LSM and ABM predicted (a) phytoplankton concentration, (b) phosphate concentration, and (c) cell quota downstream of a phosphate point source. Full nonlinear model. presented in Figure 4. The general pattern predicted by the two models is similar. A phytoplankton bloom occurs some distance downstream of the source (Figure 4a). The phytoplankton concentration in mmol C L-1 can be converted to more common units by assuming 30 µg Chl.a mg C-1 (2). Using this conversion, 0.1 mmol C L-1 corresponds to 36 µg Chla L-1, which is characteristics of eutrophic conditions (30). The PO4 concentration jumps up at the source location, and then decreases due to uptake by phytoplankton (Figure 4b). The cell quota is highest (close to q*,PO4 ) 40 mmol P mol C-1) just downstream from the source and then decreases to the subsistence quota (q0,P ) 1.0 mmol P mol C-1) with distance from the source (Figure 4c). Quantitatively, the results of the two models are significantly different. The ABM bloom is lower and occurs farther downstream than the LSM bloom. It is important to realize that the two models are based on the same sub-models of nutrient uptake and growth, are parametrized using the same values, and that they produce the same results in a laboratory
FIGURE 5. Intrapopulation distribution of (a) photosynthesis rates and (b) cell quotas predicted by the individual-based model, 20 km ((0.5 km) downstream of the discharge point (n ) 1465). setting (Figure 2). However, under realistic environmental conditions, including advection and dispersion, the LSM and ABM approaches produce significantly different results. It is interesting to note that similar comparisons of LSM vs ABM (31, 32) model approaches for the activated sludge wastewater treatment process produced analogous results. When the intrapopulation variability in microorganism properties is accounted for, the population-average growth rate decreases. The difference is significant in a management context. The average Chlorophyll a concentration (converted as discussed above) for the stretch of the river shown (-10 to 70 km) for the LSM model is 14 µg L-1, which is clearly in the eutrophic stage (>10 µg L-1, 30). The average for the ABM is 10 µg L-1, which would be considered borderline between mesotrophic and eutrophic. To understand the cause for the lower phytoplankton concentration in the ABM model, the intrapopulation distribution of cell quotas and corresponding photosynthesis rates, 20 km downstream of the source are examined in Figure 5. Due to dispersion (random walk), the population at any given location will be composed of individuals with different life histories (see also Supporting Information Section S4). As a result of the different life histories of individuals, the cell quotas within the population are quite distributed (Figure 5b), resulting in a distribution of photosynthesis rates (Figure 5a). Even though, the average cell quota (qP) is greater than the limiting quota (qL,P), the average photosynthesis rate is below the attainable maximum (µP,MAXLH). The underlying reason for this apparent discrepancy lies in the nonlinearity of the photosynthesis equation (eq 8). Specifically, the discrepancy can be explained by the limitation threshold, which states that when the cell quota is above the threshold, the photosynthesis rate is at the attainable maximum. For the whole population, the average cell quota is above the threshold, and the LSM approach would predict a population photosynthesis rate at the attainable maximum. For many of the individuals the cell quota is also above the threshold, and their photosynthesis rate is at the attainable maximum. However, a significant number of individuals have cell quotas below the threshold, and their photosynthesis rate is below the attainable maximum. As a result, the population average photosynthesis rate is also below the attainable maximum. The population-average photosynthesis rate would increase if some of the individuals with very high cell quota would
“give” some of their nutrient to the individuals with lower cell quota. This would not reduce the photosynthesis rate of the individuals with high cell quota because they would still be above the limitation quota. However, it would increase the photosynthesis rate of the individuals with low cell quota because their photosynthesis rate is limited by P. One way of looking at the situation is that some individuals are not “thinking as a team”, taking more nutrient than they (presently) need, causing the team performance to be lower. It is reasonable to consider this because intercellular communication (quorum sensing) and coordination of activities is increasingly being recognized as important in microorganism populations (33). Additional nonlinearities are in the hyperbolic shape and the subsistence quota cutoff of the nutrient limitation term, as well as in the uptake model (eq 9), which also contribute to the difference between the LSM and ABM models. Outlook. This paper demonstrated that the ABM and LSM approaches can produce significantly different results for a biogeochemical model applied to a realistic scenario. However, it was also demonstrated that when the intrapopulation variability is minimized due to well-mixed conditions, in a laboratory reactor, lake or reservoir, the two approaches produce the same results. Also, when the model equations are linear, or effectively linear (e.g., conditions remain in the linear portion of the Monod growth curve), the results are the same. Those two conditions (incompletely mixed conditions and nonlinear model equations) can be used to gauge the need for using the ABM approach. The advantage of ABM over LSM approaches increases with the complexity and nonlinearity of microorganisms. As our knowledge of microorganisms increases, so does our realization of their complexity and desire to model it, which means that agentbased techniques should become increasingly important in environmental science and engineering. In application, the main disadvantage of ABMs may be the high computational cost. This will increase significantly when direct agent-agent interactions (e.g., zooplankton grazing) are included, although simplifying assumptions can reduce the computational burden (34). There are many exciting research areas where the averaging assumption of LSMs introduces problems and ABMs may be useful, including nutrient deficiency and cell cycle dynamics (35), trace metal transformations (18), diel growth and division patterns (36), circadian rhythms (37), intracellular biochemistry (38), species competition/ interaction, and complex environmental system dynamics (biocomplexity). ABMs should also be useful when adaptive behavior or complex life cycles (e.g., the formation of resting stage cells, 39) are important.
Supporting Information Available Details on relating the minimum cell-size to the population average cell-size, stochastic variability in the full model, model parameters, life history of individuals, and model heritage and code. This material is available free of charge via the Internet at http://pubs.acs.org.
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Received for review August 25, 2006. Revised manuscript received January 5, 2007. Accepted February 9, 2007. ES062046J
