A Bayesian network model for estimating stoichiometric ratios of lake seston components

Abstract

The elemental composition of seston provide insights into the functioning of lake food webs and how nutrients cycle through the environment. Here, we describe a Bayesian network model that simultaneously estimates relationships between dissolved and particulate nutrients, suspended volatile and non-volatile sediments, and algal chlorophyll. The model provides direct estimates of the phosphorus and nitrogen content of phytoplankton, suspended non-living organic matter, and suspended inorganic sediment. We apply this model to data collected from reservoirs in Missouri, USA to test the validity of our assumed relationships. The results indicate that, on average among all samples, the ratio of nitrogen to phosphorus (N:P) in phytoplankton and non-living organic matter in these reservoirs were similar, although under nutrient replete conditions, N:P in phytoplankton decreased. Phosphorus content of inorganic sediment was lower than in phytoplankton and non-living organic matter. The analysis also provided a means of tracking changes in the composition of whole seston over time. In addition to informing questions regarding seston stoichiometry, this modeling approach may inform efforts to manage lake eutrophication because it can improve traditional models of relationships between nutrients and chlorophyll in lakes.

Introduction

Since Redfield’s seminal work (Redfield 1958), aquatic ecologists have measured the elemental composition of seston and sought to explain its relationship to processes such as nutrient cycling (Elser and Urabe 1999), nutrient limitation for both phytoplankton (Klausmeier, Litchman, and Levin 2004; Ptacnik et al. 2010; Hillebrand et al. 2013) and herbivores (Sterner and Hessen 1994) and food web stability (Perhar and Arhonditsis 2009). In general, elemental composition of seston is thought to reflect the basic processes that cycle carbon, phosphorus, and nitrogen in the environment (Sterner et al. 2008). Because human activities have significantly altered nutrient concentrations in aquatic ecosystems, a better understanding of the causes of changes to the elemental composition of seston is critical for managing and mitigating the effects of excess nutrients in the environment (Dodds 2006; Glibert et al. 2011).

Partitioning seston into its components, which include living phytoplankton, non-living organic matter, and inorganic matter, is difficult using data routinely collected from field studies. Many studies rely on a series of filters to separate particles of different size ranges, and then assume each size range is composed primarily of unique aspects of seston such as phytoplankton (Elser and Hassett 1994) or bacteria (Cotner et al. 2010). In some studies, post hoc tests are used to assess accuracy of these initial assumptions (Hassett et al. 1997). Particle size is, however, imperfectly associated with seston type. Consequently, analyses of elemental composition of bulk seston, even in restricted size ranges, provide an incomplete picture of nutrient allocation among various components of the seston. For example, the range of particle sizes associated with inorganic matter overlaps with particle sizes associated with organic matter (Walling and Moorehead 1989).

Laboratory methods for separating different components of seston are in use. For example, centrifuging samples combined with colloidal silica separates living from non-living organic matter (Hamilton et al. 2005). Alternatively, measurements of inorganic particle density via electron microscopy and x-ray analyses can partition total phosphorus measurements into organic and inorganic components (Effler et al. 2014). These methods are promising, but not routinely applied.

Other workers have proposed statistical approaches for separating the components of seston (Hessen et al. 2003), and we expand on this approach here. More specifically, we ask whether the elemental composition of living phytoplankton, non-living organic matter, and inorganic matter can be inferred by invoking basic assumptions and analyzing measurements of total phosphorus (TP), total nitrogen (TN), suspended sediment, and chlorophyll (Chl). We use a Bayesian network model for this analysis that provides specific advantages (described below) over more commonly applied frequentist statistical methods. We then demonstrate that partitioning seston components in this way yields new insights into mechanisms driving changes in seston stoichiometry and potentially improve our ability to predict Chl from nutrient concentrations.

Methods

Data

We illustrate these models with data collected in 15 Missouri (MO) reservoirs. Study reservoirs were constructed in the Glacial Till Plains region of northern Missouri between 1950 and 1992. The range in size (10 – 408 ha), mean depth (1.6 – 5.7 m) and flushing rate (0.1 – 3.4 times per year) represent regional reservoir conditions (Jones et al. 2008). Integrated photic zone samples were collected weekly near the dam of each reservoir between May and August 2004. Total suspended solids were determined by filtering a known volume of lake water through Whatman934-AH filters (nominal filter size: 1.5 μm) that were pre-rinsed, dried, ashed, and tared. Non-volatile suspended sediment (NVSS) was determined by weight after ashing TSS samples, and volatile suspended sediment (VSS) was determined by difference (TSS – NVSS). Samples were analyzed for Chl (uncorrected for degradation products), TP, dissolved P, TN, dissolved inorganic N (DIN), and dissolved N. Chl was measured from material retained on a 1 μm Gelman AE filter. Dissolved nutrient concentrations were estimated in filtrate through the Whatman934-AH filters. A total of 241 samples were available for analysis.

Data collected by EPA’s National Lakes Assessment (NLA) in the summers (May-September) of 2007 and 2012 were used to qualitatively compare with trends estimated from MO reservoirs. The NLA consisted of a random sample of approximately 1800 lakes from the continental United States. During each visit to a selected lake, a sampling location was established in open water at the deepest point of each lake (up to a maximum depth of 50 m) or in the mid-point of reservoirs. A water sample was collected using a vertical, depth-integrated methodology that collected water from the photic zone of the lake (to a maximum depth of 2 m). Multiple sample draws were combined in a rinsed, 4 liter (L) cubitainer. When full, the cubitainer was gently inverted to mix the water, and an aliquot was taken as the water chemistry sample. This subsample was placed on ice and shipped overnight to the Willamette Research Station in Corvallis, Oregon. TN and TP were measured in the laboratory from the open water sample at pre-specified levels of precision and accuracy (US EPA 2012). Typical laboratory methods included persulfate digestion with colorimetric analysis for TN and TP. To measure Chl concentration, 250 mL of lake water was pumped through a glass fiber filter in the field and quantified in the lab to pre-specified levels of precision and accuracy.

Mathematical model

For TP, we assume that three seston components in addition to dissolved P contribute to TP: P sorbed onto inorganic, or non-volatile suspended sediment (NVSS), P associated with living phytoplankton, and P associated with non-living organic, suspended sediment (VSS_NP) (Figure 1).

Figure 1.

Schematic representation of model for Chl-TP relationship. Observed TP originates from phytoplankton, which is associated with chlorophyll (Chl), dissolved P (P_diss), sediment bound P (NVSS), and P associated with non-living organic matter (VSS_NP). The shaded box for VSS_NP indicates that direct measurements of this parameter were not available, and its value is inferred statistically.

Direct measurements of VSS_NP were not available, but we estimated its concentration by assuming that the concentration of all volatile suspended sediments is composed of two main contributors: phytoplankton biomass and non-living organic matter that is not associated with living phytoplankton (i.e., VSS_NP). Hence, we modeled observed concentrations of VSS as follows:

$$VSS=kChl+VS{S}_{NP}$$

Where volatile suspended sediment concentration (VSS) is expressed as the sum of a phytoplankton component, modeled as the product of a coefficient, k, multiplied by Chl concentration, and a non-phytoplankton component, VSS_NP. The distribution of k is assumed to be log-normal to constrain it to positive values. To fit this relationship to available data, we log-transformed both sides of the equation to address skewness in the observed distributions of the different variables, and we introduced a term, ε₁, that allows for residual variability in observed log(VSS):

$$\log \left(VS{S}_i\right)=\log \left(kCh{l}_i+VS{S}_{NP,i}\right)+{\epsilon }_{1,i}$$

Where the subscript, i, refers to individual samples in the data set, and the error term, ε_1,i, is a random variable drawn from a normal distribution with a mean value of zero and a standard deviation of σ₁. Exploratory analysis suggested that mean values for VSS_NP varied among lakes, so values of VSS_NP,i were assumed to be drawn from a log-normal distributions characterized by a mean, μ_VSS-NP, unique to each lake and a standard deviation, σ_VSS-NP:

$$\log \left(VS{S}_{NP,i}\right)=N\left({\mu }_{VSS-NP,j\left[i\right]},{\sigma }_{VSS-NP}\right)$$

Where the subscript j[i] refers to different lakes that correspond with each sample, i. Mean values for each lake (μ_VSS-NP,j) were assumed to be drawn from a single common normal distribution, characterized by an overall mean value and standard deviation.

Observed TP is modeled as the sum of directly observed dissolved P, phosphorus incorporated in phytoplankton, which is assumed to be directly proportional to Chl, phosphorus that is associated with phosphorus associated with non-living organic matter (VSS_NP) and nonvolatile suspended sediment (NVSS).

$$E[T{P}_i]=P_{diss,i}+d_{1}Ch{l}_i+d_{2}VS{S}_{NP,i}+d_{3}NVS{S}_i$$

The three coefficients, d₁, d₂, and d₃, quantify the mass of P in phytoplankton per unit of Chl, and the P content of VSS_NP and NVSS, respectively. Like k, these coefficients are assumed to be log-normally distributed to restrict them to positive values. The subscript, i, again refers to individual samples, and TP_i is enclosed by E[.] to indicate that the equation applies to the expected value of TP for sample i. In contrast to the model for VSS, we do not log-transform the equation. Instead, the skewed distribution of observed values for TP_i is taken into account by modeling the residual distribution of these observed values as a Gamma distribution with a shape parameter of α₁ and a rate parameter of α₁/E[TP_i]. The mean value of this distribution is E[TP_i] and the variance is E[TP_i]²/α₁, so the residual variance is proportional to the square of the expected value. The parameter α₁ adjusts the rate at which residual variance increases with the mean expected value and is fit to the observed data.

We specified an identical model for TN, but initial model fits indicated that the contribution of NVSS to TN was negligibly small, and so we excluded this term in the final model:

$$E\left[T{N}_i\right]=N_{diss,i}+f_{1}Ch{l}_i+f_{2}VS{S}_{NP,i}$$

Here again, the parameters, f₁, and f₂ are specified as being log-normally distributed to constrain them to positive values and observed values of TN are assumed to be drawn from a Gamma distribution with a shape parameter of α₂.

To further explore the potential for this type of analysis to provide insights regarding the mechanisms that determine stoichiometric ratios, we identified MO samples for which DIN > 50 μg/L. In these samples, sufficient N was likely available for phytoplankton growth, and therefore, we hypothesized that stoichiometric ratios associated with phytoplankton would differ in these samples compared to samples in which DIN was near 0 (Klausmeier, Litchman, Daufresne, et al. 2004). We refit the models but estimated different values for the coefficients that quantified N- and P-content of phytoplankton and VSS_NP. More specifically, we expressed our model equations as follows:

$$E[T{P}_i]=P_{diss,i}+d_{1,k}Ch{l}_i+d_{2,k}VS{S}_{NP,i}+d_{3}NVS{S}_i$$

$$E\left[T{N}_i\right]=N_{diss,i}+f_{1,k}Ch{l}_i+f_{2,k}VS{S}_{NP,i}$$

Where the index, k, indexes samples with low (k=1) and high concentrations of DIN (k=2).

Prior to analysis, all measurements were scaled by dividing by their maximum observed value. This scaling improves the efficiency of the Monte-Carlo algorithm used to fit the equation. We then specified weakly informative priors for all the parameters. As discussed earlier, all model coefficients, k, f₁, f₂, d₁, d₂, and d₃ were assumed to be log-normally distributed to restrict then to positive values. Prior distributions for these parameters were specified as log-normal distributions with mean values of 0 and standard deviations of 8. The value of 8 was chosen to be several times larger than the expected standard deviation of any of the coefficients, so that the function of the prior distribution was mainly to steer randomly drawn samples away from extreme values. Similarly, the prior distribution of σ₁ was specified as a half-Cauchy distribution with a scale of 2. Initial estimates of the parameters, α₁ and α₂, were large, and so, to improve the efficiency of the Monte-Carlo sampling, we scaled α₁ and α₂ by a factor of 100 and assigned them weakly informative half-Cauchy priors with a scale of 2. The prior distribution for the mean values of VSS_NP (i.e., μ_VSS-NP) was a normal distribution with a mean value of 0 and a standard deviation of 8. All models are then fit simultaneously using the Bayesian modelling software, RStan (Stan Development Team 2016) and the statistical modeling software, R Version 3.4.2 (R Core Team 2017).

Because the relationships estimated in the current models quantify inherent mean stoichiometric relationships regarding P and N content in phytoplankton cells, these relationships may be similar among different data sets. To qualitatively test this hypothesis, we compared base relationships estimated in MO to TP, TN, and Chl data collected from lakes across the United States (US EPA 2010; US EPA 2017).

Results

The TP and TN models provided accurate predictions of observed TP and TN (Figure 2). Because residual variance increased with the mean predicted values of TP and TN, root mean square prediction error (RMSE) exhibited similar trends, ranging from 4 to approximately 20 μg/L for the TP model, and from 50 to 200 μg/L for (TN), mirroring the predicted increase in residual standard deviation for the Gamma distribution (solid lines in lower panels of Figure 2). We also computed 90% posterior prediction intervals for each sample and confirmed that < 10% of observations of TP and TN deviated from these intervals.

Figure 2.

Predicted TP and TN versus observed TP and TN (upper panels). Root mean square error (RMSE) as a function of predicted values (lower two panels). Dashed lines (upper panels) show 1:1 relationships. Solid line in lower two panels shows the increase in residual standard deviation expected from Gamma distribution. Open circles in lower panels show estimated RMSE computed from approximately 20 samples in a bin around the indicated predicted TP or TN.

Coefficient values for the base model indicated that on average, NVSS was 0.099% phosphorus by weight, whereas nitrogen content was negligible (Table 1). Mean P and N content in non-phytoplankton VSS were 0.67% and 6.9%, respectively, which yielded a molar ratio of N:P in VSS_NP of 23:1. 90% credible intervals on this ratio extended from 19:1 to 27:1.

Table 1.

Model coefficients. See model equations for coefficient definitions.

Model	Coefficient	Mean value (90% confidence limits)
VSS	k	0.092 (0.084, 0.10)
TP	d1	0.61 (0.54, 0.68)
	d2	6.7 (5.9, 7.7)
	d3	0.99 (0.70, 1.28)
TN	f1	5.4 (4.7, 6.1)
	f2	69 (61, 78)

Coefficients for Chl in the TN and TP models quantified the ratio between the mass of N and P in phytoplankton and the mass of Chl. So, we estimated the mean mass ratio between N and P within phytoplankton cells as 5.4/0.61 = 8.8. This mass ratio converts to a molar ratio of 19 in phytoplankton, with 90% credible intervals extending from 17 to 23.

The coefficient k in the model for VSS measures the proportion of the observed VSS mass attributable to phytoplankton (as quantified by Chl). If we assume that volatile suspended sediment concentration can be converted to an estimate of carbon by multiplying by a factor of 0.45 (Findlay et al. 1991; Pribyl 2010), we estimate the molar C:P ratio in phytoplankton as 170:1, with 90% credible intervals of 160:1 and 200:1.

Distributions of TP and TN measurements are highly skewed, and so these variables are frequently log-transformed prior to analysis, as are measurements of different predictor variables (e.g., Chl). In the present model, the relationships are expressed in terms of the variables in their original units, and so the log-transformation would apply to the entire sum of model components. For example, a log-transform applied to the model for TN would be:

$$\text{log}\left(E\left[T{N}_i\right]\right)=\text{log}\left(N_{diss,i}+f_{1}Ch{l}_i+f_{2}VS{S}_{NP,i}\right)$$

Hence, modeled relationships between a predictor variable (e.g., Chl) and E[TN] are non-linear when plotted in log-log space. Examples of this non-linear relationship are shown in Figure 3, where Chl is plotted against particulate TN (defined as the difference between TN and dissolved N). We can compute a limiting relationship between Chl and TN by considering the case in which VSS_NP is negligibly small, in which case, we assume that the term f₂VSS_NP is zero and write the log-transformed model equation as follows,

$$\log \left(E\left[T{N}_p\right]\right)=\log \left(f_1\right)+\text{log}\left(Chl\right)$$

Where TN_p is particulate TN. So, in samples in which VSS_NP is small relative to Chl, the relationship between log(Chl) and log(TN) is a straight line, with a slope of 1, and an intercept value of log(f₁). This limiting relationship is shown as a solid line in Figure 3.

Figure 3.

Predicted relationships between Chl and particulate TN for different levels of VSS_NP. Solid line: relationships between Chl and TN_p when VSS_NP is negligibly small. Dashed lines: relationship between Chl and TN_p for different quantiles of VSS_NP (as indicated).

As the contribution of N associated with VSS_NP increases, the relationship between Chl and TN_p deviates from the limiting relationship. For example, if we fix VSS_NP at the 5^th percentile of its observed distribution of values, the resulting relationship between Chl and TN_p approaches the limiting relationship quickly as Chl increases, but at low concentrations of Chl, the effect of VSS_NP is strong enough to reduce the slope between Chl and TN_p to nearly zero (dashed line labeled “5%” in Figure 3). For relationships computed using higher concentrations of VSS_NP, the curves start to deviate more substantially from the limiting relationship, but always approach the linear limit at high concentrations of Chl. Overall, the distribution of curves matches the distribution of the observed samples, lending support to the appropriateness of the model formulation. The same limiting relationships exist for the relationships between Chl and VSS and between Chl and TP (Figure 4).

Figure 4.

Relationships between Chl and VSS and TP_p.. Solid lines show estimated limiting relationship between Chl and VSS (left panel), between Chl and particulate TP (right panel) in the absence of contributions from any other factors.

Seston contributions to TP and TN varied with time and by lake (Figure 5). In the example oligo/mesotrophic lake (left panel), dissolved N accounted for approximately 80% of observed TN throughout the summer sampling season. Non-living organic seston accounted for the majority of the remaining TN, with only a small increase in phytoplankton-associated N in weeks 24 and 25. Dissolved P accounted for approximately 50% of observed TP throughout the summer in the same lake, and the proportional contribution of phytoplankton P was nearly 20% in week 24. The contribution of inorganic seston to total P was initially nearly 20% and decreased to lower proportions over the summer.

Figure 5.

Allocation of P and N to different compartments for a mesotrophic lake (left column) and eutrophic lake (right column). Dark gray: proportion of nutrient attributed to phytoplankton biomass, medium gray: proportion of nutrient attributed to NVSS (TP plots only), and light grey: proportion of nutrient attributed to non-phytoplankton VSS. Dissolved proportions of N and P account for remaining proportions for each stack of bars.

In the eutrophic lake (right panels, Figure 5), phytoplankton N and P accounted for much greater proportions of TN and TP. The phytoplankton proportion of TN and TP increased and decreased with variations in Chl concentration, indicating that the concentration of other seston components were much less variable than the phytoplankton component. As observed with the oligo/mesotrophic lake, dissolved N still accounted for a large proportion of observed TN, and a smaller proportion of observed TP.

DIN in 56 out of 241 samples was greater than 50 μg/L. Estimates of N and P concentrations per unit of Chl differed substantially between low and high DIN samples, but because a smaller sample size yielded a greater uncertainty for estimates for the high DIN coefficients, none of the differences were statistically significant (Table 1). Mean P content associated with phytoplankton (d₁) increased from low to high DIN samples, whereas N content decreased (f₁). Both P and N content in VSS_NP decreased slightly from low to high DIN samples (d₂ and f₂). These changes translated into a shift in N:P in phytoplankton from 21 (90% credible intervals: 18 – 25) to 12 (6 – 19) from low DIN to high DIN samples, while N:P in VSS_NP exhibited a negligible change from 22 (19 – 27) to 21 (12 – 33).

Whole seston estimates of N:P (quantified as TN_p:TP_p) clustered around a value N:P = 19 estimated for phytoplankton (Figure 6). N:P for VSS_NP was slightly higher, and therefore an increase in the proportion of VSS_NP in a sample would be expected to shift observations slightly above the solid line. Some of the large deviations of values below N:P = 19 can likely be attributed to samples in which nutrients are replete, as phytoplankton in these samples may have much lower N:P (see above).

Figure 6.

Relationship between TP_p and TN_p. Solid line: N:P = 19 ratio expected in phytoplankton dominated sample. Filled circles: samples with DIN > 50 μg/L.

The limiting relationships between TP, TN, and Chl estimated using MO data were similar to the qualitative limits observed in the national data set (Figure 7). The deviation of the cloud of observations for TN from the limiting relationships can likely be attributed to the fact that only TN (rather than TN_p) was available from the NLA data set, and so, observed TN is likely higher than TN_P for all samples due to the contribution from dissolved organic and inorganic N. The limiting relationship for TP more closely corresponded to the lower limits of the distribution of points, suggesting that the dissolved component of P accounted for a smaller proportion of TP. Overall, these results suggest that broadly applicable limiting relationships between TP, TN, and Chl can be estimated.

Figure 7.

National lakes assessment observations of TP, TN, and Chl. Solid lines are relationships for TP_p and TN_p estimated from MO lakes.

Discussion

We have introduced a new approach for estimating stoichiometric ratios of different components of seston in lakes and reservoirs. The model accurately predicts variations in TP and TN among different samples and yields coefficients that are directly interpretable in terms of the nutrient content associated with different components of seston. Application of this modeling approach can help refine our understanding of the mechanisms that determine lake seston stoichiometry and inform the development of models that are used to predict Chl from nutrient inputs.

Bayesian network models provided a means of modeling relationships between measurements of different parameters in a single, internally-consistent framework. The Bayesian model described here builds on statistical approaches described in earlier studies. In particular, our current formulation is similar to a statistical approach for separating seston into living and non-living components described by Hessen et al. (2003). The current approach expands that model to estimate separate contributions from NVSS and VSS_NP and uses Gamma distributions to account for the increase in the variance of observations with mean values. Our approach is also similar to a technique for decomposing the contributions from organic and inorganic seston components described by Effler et al. (2014). In their work, they estimated the nutrient content of different components by analyzing a small number of samples that were dominated by only one constituent. Our estimates of limiting behavior are more precise than this approach because we use all the data and a model formulation that explicitly includes the limiting relationships that occur when one component of seston is the dominant contributor (Figure 3).

Seston stoichiometry

Our analysis highlights how the amounts of N and P associated with different components of seston (phytoplankton, non-living organic material, and inorganic suspended sediment) contribute to variability in whole seston measurements of N:P. We found that N:P in organic, non-living seston was only slightly higher than that estimated in phytoplankton. In contrast, N content in inorganic seston was negligible (Vitousek and Howarth 1991), while P was present. As a proportion of total mass, though, the P content of inorganic seston was much lower than in organic seston. Still, in samples with large concentrations of inorganic sediment, inorganic seston can contribute to variations in whole seston P (Effler et al. 2014). Over the summer sampling season, the contributions of different seston components to TN and TP varied considerably (Figure 5), and these variations would manifest as variations in whole seston N:P.

Our findings that N:P = 19 and C:P = 170 in phytoplankton in our base model were strikingly similar to values estimated from analysis of whole seston from a global data set of lakes (Hecky et al. 1993). In this earlier work, data were confined to lakes with residence times exceeding 6 months to limit the contributions of allochthonous seston, and the final values were estimated by regression of mean N and C content on mean P content across 44 lakes. Our present data were collected from reservoirs with retention times as low as 0.06 years, and we relied on modeling to control for the effects of inorganic and non-living organic matter. The similarity of the ratio estimates from these two contrasting datasets and analyses may suggest that mean nutrient content in the phytoplankton assemblages of different lakes converges to a single value, but analysis of data collected from more systems is required to solidify this conclusion.

Several mechanisms have been proposed that cause variations in whole seston N:P in addition to the differences among seston components on which our analysis is focused. Examples of these mechanisms include nutrient availability (Hecky et al. 1993), light to nutrient ratio (Sterner et al. 1997), temperature, water residence time (They et al. 2017), variations in nutrient content within different phytoplankton species, and variations in phytoplankton species composition (Martiny et al. 2013). Our model does not exclude consideration of these other mechanisms. Instead, we suggest that examining the effects of these different mechanisms on particular components of seston via analyses similar to what we have described here can deepen our understanding of the reasons why N:P varies.

Our exploratory analysis of the effects of nutrient availability on N:P provides one example of how partitioning P and N among seston components may help clarify how environmental conditions affect seston stoichiometry. We found that N:P shifted dramatically from 21 for low DIN samples to 12 for high DIN samples. This change is qualitatively consistent with predictions of changes in N:P from a state of competitive equilibrium (i.e., low levels of biologically available nutrients) to a state of exponential growth (i.e., nutrient replete environments) (Klausmeier, Litchman, Daufresne, et al. 2004). Our exploratory analysis is only illustrative, though, and not intended as an exhaustive test of the effects of excess nutrients. Coupling our analysis of field data with manipulative experiments would provide more robust inferences, and then, the type of analysis we describe here may help bridge the gap between small-scale experiments and observations gleaned from large-scale monitoring data by providing a means of controlling for some of covariates inherent in field data (Elser and Hamilton 2007). The structure of our Bayesian model allows one to introduce variations in coefficient values, as we have in our exploratory analysis. In cases in which several groups of samples can be defined, coefficients can be expressed as a distribution of values characterized by a mean and variance (Gelman and Hill 2007), and a hierarchical structure imposed on coefficient values. We used this approach in our base model to estimate lake-specific mean values of inferred VSS_NP, which improved our model for VSS. Ultimately, modeling the variations in nutrient content for different seston components would improve understanding of which variations in whole seston N:P can be attributed to differences in the composition of the seston versus differences in the nutrient content of different components.

Analyzing differences in the composition of seston components could also improve our understanding of the effects of variations in external loading of organic material and nutrients. For example, differences in watershed land use (Arbuckle and Downing 2001) and in the sources of nutrient loading (Downing and McCauley 2003) have been associated with changes in N:P observed in lakes. Similarly, atmospheric loading (Elser et al. 2009) has altered lake N:P, and particulate N:P varies with residence time (They et al. 2017). In these examples, examination of the allocation of nutrients to different seston components (Figure 5) would likely advance understanding the effects of different types of nutrient loading on seston components.

Predicting Chl from nutrient concentrations

Statistical estimates of relationships between concentrations of TP and Chl and between TN and Chl in lakes have been fundamental to understanding and managing lake eutrophication (Jones and Bachmann 1976; Jones and Knowlton 2005), and our current models provide relationships between this same set of variables. Our model differs, however, from traditional nutrient/Chl models in two important ways that reflect our focus on the elemental composition of seston (Lewis and Wurtsbaugh 2008). First, we used covariates (NVSS and VSS_NP) that accounted for variations in TP and TN, rather than in Chl. In contrast, most other analyses of nutrient/Chl relationships have sought to explain deviations in Chl concentrations from predictions by TP and/or TN. This interpretation of the nutrient/Chl relationship has led to investigations of the effects of covariates such as TN:TP (Guildford and Hecky 2000), zooplankton abundance (Mazumder 1994), lake morphology (Hoyer and Jones 1983; Phillips et al. 2008), and water transparency (Jones and Knowlton 2005; Webster et al. 2008).

We believe our model more accurately represents the processes that govern relationship between simultaneous observations of TP, TN, and Chl, and it provides the means to identify more specific hypotheses regarding the effects of other factors on nutrient/Chl relationships. For example, instead of simply hypothesizing that the presence of a high abundance of zooplankton grazers reduces Chl relative to TP, we can identify particular conditions in which zooplankton would be expected to have the strongest effects on the ratio of Chl to nutrients. In general, as zooplankton reduce phytoplankton abundance, the decrease in phytoplankton (as measured by Chl) should be accompanied by a decrease in the N and P associated with the phytoplankton component of seston. Other components of seston that contribute to TN and TP should be unchanged, so the effects of zooplankton should track lines associated with nearly constant values of the other seston components (as are plotted in Figure 3, and plotted with the x and y variables reversed in Figure 8). Then, in samples in which phytoplankton is the dominant source of particulate N and P, we would expect zooplankton grazing to reduce Chl, TN_p, and TP_p in the nearly the same proportions (e.g., a change from A to B in Figure 8), and the observed ratio between Chl and TN_p and between Chl and TP_p would be nearly constant. In samples in which phytoplankton is not the dominant source of particulate N and P, however, we would expect that with higher levels of grazing the ratio between Chl and nutrients would decrease (a change from B to C1). This effect may be magnified if zooplankton excretion increases the concentration of non-living organic seston (a change from B to C2) (Urabe et al. 2003). Hence, high abundance of zooplankton would be expected to reduce Chl concentration relative to nutrient concentrations, but this effect would be most pronounced in lakes in which phytoplankton is not the dominant component of seston. Hypotheses for the effect of other factors on TP-Chl and TN-Chl relationships could be developed in a similar manner.

Figure 8.

Hypothesized effect of zooplankton grazing on relationship between TNp and Chl. Solid line: predicted relationship between TN_p and Chl assuming a constant value of VSS_np at the median value of VSS_np. Dashed line: predicted relationship between TN_p and Chl, assuming VSS_np equal to the 75^th percentile of observed values.

The second way our current model differs from traditional investigations of nutrient/Chl relationships is that its mathematical formulation is based on untransformed measurements. In contrast, traditional nutrient/Chl models have been based on log-transformed variables. For example, if we apply the conventional approach to the model for TP, the following model equation results:

$$\log \left(TP\right)=b_1+P_{diss}+b_2\log \left(chl\right)+b_3\log \left(NVSS\right)+b_4\text{log}\left(VS{S}_{NP}\right)+{\epsilon }_4$$

In this more commonly used model equation, the model coefficients become exponents of the predictor variables when the equation is expressed in original units, and TP is modeled as the product, rather than the sum, of the different predictors. In this latter formulation, the model coefficients are not immediately interpretable in terms of nutrient content. Furthermore, coefficients are linear, and multiple linear regression can be used to estimate their values, but the limiting behavior discussed above when a particular seston component dominates does not occur. Identifying these limits in the relationships between TP, TN, and Chl may be particularly important in broadening the applicability of models predicting Chl from nutrients. Indeed, our finding that the limiting relationships estimated in MO were qualitatively similar to the limits of observations collected in a continental scale data set suggests that at least at large spatial scales, assuming constant values for phytoplankton P and N content provides a useful approximation.

Uncertainties and conclusions

A few aspects of the current analysis may have introduced uncertainty and error into the results. We used Chl as a proxy for phytoplankton biomass, so any seston components that covaried with Chl would be treated in the analysis as if they were live phytoplankton. For example, nutrients associated with autotrophic organic detritus that increases with phytoplankton biomass would be indistinguishable from phytoplankton biomass. Further comparisons between the present analytical approach and direct measurements of different seston components would help quantify the magnitude of this error. Chl content in phytoplankton also can vary among species and with changes in environmental conditions (Hecky et al. 1993; Kasprzak et al. 2008), and estimating nutrient content in different types of lakes and different conditions will help quantify the magnitude of this variability.

Estimates of carbon content introduce another source of uncertainty to the current estimates of seston stoichiometry. We estimated carbon content as a fixed fraction of volatile suspended sediment, but this fraction can vary with conditions (Grove and Bilotta 2014). Others have employed different approaches for measuring organic carbon, including the use of a relationship between phytoplankton cell volume and carbon content (Hessen et al. 2003), and direct measurements of elemental carbon (Urbansky 2001). Concurrent measurements of VSS and direct measurements of elemental carbon would provide the most robust test of the method used here to estimate carbon.

Measuring dissolved and particulate fractions of N and P requires an arbitrary decision regarding the pore size of the filters, as the dissolved fraction is defined as any material that passes through the filter (Solórzano and Sharp 2003). Similarly, suspended sediment and Chl are measured from material retained by the filters. Hence, the current findings with regard to the nutrient content of different seston components may change with data collected with finer or coarser filters, or with more involved measurements of dissolved nutrients (Karl and Björkman 2015). Indeed, nephelometric measurements of the turbidity of filtrate in the present data suggest that a substantial amount of suspended sediment passes through the 1.5 μm filter (Knowlton and Jones 2000). Because data used here were internally consistent, such that dissolved and particulate nutrients, suspended sediment, and Chl were all measured using filters with comparable pore sizes, the estimates of nutrient content of various seston components are robust. Comparisons of the present results with other studies, however, should take into account the possible effect of different filter pore sizes.

The current statistical approach can be applied to measurements that are easily acquired in routine monitoring, a feature that allows broad use in analyses of both field observations and data from experimental manipulations. Collecting auxiliary information regarding the stoichiometry of different seston components could also enhance interpretation of the model results. For example, laboratory methods for separating different components (Hamilton et al. 2005; Effler et al. 2014) yield complementary data that, if conducted simultaneously with the current analysis approach, would provide data to further validate the approach.

In conclusion, we have shown that we can accurately model TP and TN concentrations in lakes as a function of Chl, suspended sediment, and dissolved nutrients. These models provide insight into the nutrient content of different components of seston, which in turn, can be interpreted with regard to the values of different environmental factors. The present models also inform efforts to develop eutrophication models that predict Chl as a function of nutrient loads.

Table 2.

Estimates of coefficients for exploratory model for low and high DIN samples. Mean coefficient values shown with 90% credible intervals in parentheses.

Model	Coefficient	Low DIN	High DIN
TP	d1	0.58 (0.51, 0.65)	0.73 (0.58, 0.89)
	d2	7.1 (6.1, 8.0)	6.4 (4.4, 8.5)
TN	f1	5.6 (4.8, 6.4)	4.0 (2.0, 6.0)
	f2	71 (63, 81)	61 (39, 84)