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Using national-scale data to develop nutrient-microcystin relationships that guide management decisions Lester L Yuan, and Amina I Pollard Environ. Sci. Technol., Just Accepted Manuscript • Publication Date (Web): 31 May 2017 Downloaded from http://pubs.acs.org on June 6, 2017

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Environmental Science & Technology

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Using national-scale data to develop nutrient-

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microcystin relationships that guide management

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decisions

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Lester L. Yuan*

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Amina I. Pollard

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Office of Water, U.S. Environmental Protection Agency, 1200 Pennsylvania Ave., Washington

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DC 20460

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*email: [email protected], phone: 202-566-0908, fax: 202-566-0441

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ABSTRACT. Quantitative models that predict cyanotoxin concentrations in lakes and reservoirs

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from nutrient concentrations would facilitate management of these resources for recreation and

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as sources of drinking water. Development of these models from field data has been hampered

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by the high proportion of samples in which cyanotoxin concentrations are below detection limits

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and by the high variability of cyanotoxin concentrations within individual lakes. Here, we

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describe a national-scale hierarchical Bayesian model that addresses these issues and that

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predicts microcystin concentrations from summer mean total nitrogen and total phosphorus

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concentrations. This model accounts for 69% of the variance in mean microcystin concentrations

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in lakes and reservoirs of the conterminous United States. Mean microcystin concentrations were

1 ACS Paragon Plus Environment


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more strongly associated with differences in total nitrogen than total phosphorus. A general

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approach for assessing this and similar types of models for their utility for guiding management

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decisions is also described.

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TABLE OF CONTENTS/ABSTRACT ART

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KEYWORDS. Cyanotoxin, microcystin, hierarchical Bayesian model, lake, reservoir,

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environmental management

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INTRODUCTION. High concentrations of cyanotoxins can restrict the use of lakes and

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reservoirs for recreation and as sources of drinking water1. Cyanobacterial blooms can have

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negative impacts on fish and other biological communities2. Exposure to low concentrations of

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cyanotoxins such as microcystin can cause liver cancer3, while exposure to very high doses can

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cause other immediate health effects4,5. Recognizing these hazards, health and environmental

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agencies have specified concentrations of different cyanotoxins below which exposure in

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drinking water or during recreation are considered acceptable3,6,7. For example, a microcystin

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concentration of 0.3 µg/L has been published as an acceptable concentration in drinking water3.

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Cyanotoxins are produced by different species of cyanobacteria (e.g, Microcystis,

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Planktothrix), and while the physiological or ecological reasons for their production are still 2 ACS Paragon Plus Environment

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under debate8,9, the environmental factors associated with increased concentrations of cyanotoxin

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in a waterbody are generally known. First, increased concentrations of nitrogen and phosphorus

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stimulates the growth of cyanobacteria, increasing their abundance, and increasing the likelihood

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of high cyanotoxin concentrations10. Other environmental factors, such as high temperatures11

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and strongly stratified conditions12, may also be particularly hospitable to the growth of certain

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species of cyanobacteria, which can then produce cyanotoxins. Recent experimental and

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observational work has also identified conditions under which certain cyanobacteria species are

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more likely to actually produce toxins. Some examples of these conditions include temporary

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nitrogen limitation followed by an imbalance in cellular ratios between nitrogen and carbon13,

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and iron limitation14.

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Among the environmental factors that are associated with increased cyanotoxin concentrations,

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nitrogen and phosphorus concentrations are most amenable to control by environmental

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managers seeking to reduce the likelihood of high cyanotoxins. To this end, accurate and precise

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relationships between observed cyanotoxins and nutrient concentrations would facilitate

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management decisions by quantifying the environmental benefits (i.e., lower cyanotoxin

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concentrations) that would be expected for a given reduction in nutrient loading. Criterion values

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for nitrogen and phosphorus based on these relationships would also provide numeric values that

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would facilitate compliance monitoring, setting allowable discharge concentrations, and

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assessing existing conditions.

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Available analyses of relationships between cyanotoxins and nutrient concentrations have thus

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far yielded relatively imprecise relationships or are based on data summaries that limit their

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utility for informing management decisions. For example, analysis of data collected in lakes and

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reservoirs of the Midwest U.S. found that TN and TP concentrations accounted for only a small



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proportion of the observed variance in microcystin concentrations (one of the most commonly

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observed cyanotoxins)15. Other analyses of field data have summarized microcystin observations

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into categories (e.g., greater than or less than 1 µg/L) before relating to environmental

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factors16,17, or examined the maximum observed microcystin concentrations at different nutrient

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concentrations15,18.

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approaches applied by researchers arise from two complications that are inherent to field

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observations of cyanotoxins: (1) a high proportion of samples are observed with cyanotoxin

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concentrations below detection limits, and (2) a very high level of temporal and spatial

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variability is observed in cyanotoxin concentrations.

The relatively imprecise relationships and the variety of analytical

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Here, we describe a hierarchical Bayesian model relating summer average concentrations of

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total nitrogen (TN) and total phosphorus (TP) to microcystin concentrations in lakes and

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reservoirs of the conterminous United States. The model described here accounts for detection

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limits in microcystin and explicitly models different components of variability in observed

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microcystin concentrations. The final model can be used to link nutrient concentrations to

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probabilities of exceedance of different microcystin thresholds and in doing so, provides a tool

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for managing lakes to prevent undesirable microcystin concentrations. We argue that this model

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is sufficiently precise to effectively direct management actions to minimize the deleterious

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effects of high microcystin concentrations, and further propose an approach for quantifying the

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effects of model uncertainty on management decisions.

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MATERIALS AND METHODS

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Data. Data from lakes and reservoirs in the United States were collected by the U.S.

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Environmental Protection Agency’s National Lakes Assessment (NLA) in the summers (May-

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September) of 2007 and 2012

19,20

. The NLA consisted of a random sample of lakes from the


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conterminous United States. In 2007, lakes with surface areas greater than 4 ha, and in 2012,

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lakes greater than 1 ha were selected from the United States using a stratified randomized

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sampling design21. The final data set was supplemented by a small number of hand-picked lakes

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and reservoirs that were a priori identified as being less disturbed by human activities22.

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Between these two assessment periods, a total of 3690 microcystin samples were analyzed.

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These data were collected from 1869 different lakes. Of these, 403 lakes were randomly selected

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and visited in both 2007 and 2012. An additional 193 lakes were randomly selected and

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resampled on a different day in the same year after the initial visit. The timing of the second visit

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varied among lakes, but on average, the second sample was collected approximately 46 days

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after the first.

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During each visit to a selected lake, an extensive suite of abiotic and biological variables was

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measured. Only brief details on sampling protocols are provided here regarding the parameters

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used in this analysis; more extensive descriptions of sampling methodologies are available19,20.

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At each lake, a sampling location was established in open water at the deepest point of each lake

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(up to a maximum depth of 50 m) or in the mid-point of reservoirs. In 2012, an additional

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sampling location for the collection of a second microcystin sample was established in the littoral

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zone approximately 10 m out from a randomly selected point on the shoreline.

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At the open water site, a water sample was collected using a vertical, depth-integrated

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methodology that collected water from the photic zone of the lake (to a maximum depth of 2 m).

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Multiple sample draws were combined in a rinsed, 4 liter (L) cubitainer. When full, the

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cubitainer was gently inverted to mix the water, and an aliquot was taken as the water chemistry

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sample. This subsample was placed on ice and shipped overnight to the Willamette Research

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Station in Corvallis, Oregon, where total nitrogen (TN), total phosphorus (TP) were measured.



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Another aliquot from the depth-integrated water sample was taken as a microcystin sample. At

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the littoral zone site, an additional water sample was collected using a 2 L brown bottle 0.3 m

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below the surface at a near-shoreline location with a total depth of at least 1 m. Both of these

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samples were placed on ice and shipped to the US Geological Survey (in 2007) and BSA

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Environmental Services, Inc. (in 2012), where microcystin concentrations were measured as

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described below.

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TN and TP were measured in the lab with persulfate digestion and colorimetric analysis.

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Microcystin sample processing began with three sequential freeze/ thaw cycles to lyse

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cyanobacteria23. Processed samples were filtered using 0.45 micron polyvinylidene difluoride

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membrane syringe filters and stored frozen until analysis. The concentration of microcystin in

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the filtered water sample was measured with a polyclonal Enzyme-Linked Immuno-Sorbent

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Assay (ELISA) using an Abraxis kit for Microcystin-ADDA, which refers to an amino acid that

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is unique to microcystin and other similar compounds. The binding mechanism of the

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Microcystin-ADDA assay is specific to the microcystins, nodularins, and their congeners;

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therefore, results from this assay may include contributions from any compound within the

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ADDA functional group24,25. The minimum reporting level (i.e., the detection limit) for the assay

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was 0.1 µg/L as microcystin-LR.

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Statistical analysis. As noted earlier, two aspects of field observations of microcystin

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complicate statistical analysis: (1) the high proportion of samples in which microcystin

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concentration is below detection limits, and (2) the high temporal and spatial variability of

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microcystin concentrations. To address the issue of below-detection-limit, or censored samples,

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we compared two approaches for modeling the statistical distribution of microcystin

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measurements. First, we modeled microcystin concentration as a log-normally distributed


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variable that was left-censored at the detection limit. This approach is commonly applied to

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water quality measurements subject to a detection limit26. Second, we assigned a concentration

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of zero to non-detects and modeled microcystin as an over-dispersed Poisson distribution.

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Poisson distributions are typically used when observed values of the variable of interest have true

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zeroes in the distribution (e.g., the abundance of a particular species27). So, with this second

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approach, non-detects are modeled as if no microcystin was present in the sample.

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To address the high temporal and spatial variability of microcystin concentrations, we

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identified four different components of this variability: (1) sampling variability, which includes

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variability due to measurement error and in samples collected at different locations in the same

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lake on the same day, (2) intra-annual variability, or variability in concentrations collected in the

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same lake on different days in the same year, (3) inter-annual variability, or variability in

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concentrations collected in the same lake in different years, and (4) among-lake variability, or

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variability in concentrations among different lakes. We then fit two “intercept-only” models that

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partitioned observed variance in microcystin concentrations into different components, using the

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two probability distributions. The censored log-normal model can be written as follows: log = + [] + [] + [] +

(1)

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where each log-transformed observation of microcystin (MCi) is modeled as the sum of an

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overall mean value a and four different components of variance: (1) among-lake variability, bj[i],

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where the index j indicates different sites, corresponding to sample i; (2) inter-annual variability,

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ck[i], where the index k indicates different site-year combinations, (3) intra-annual variability,

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dm[i], where the index m indicates different sampling days for a given site-year combination, and

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(4) sampling variability, ui. Each component of variance is modeled as a normal distribution with

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a mean value of zero and a standard deviation (sb, sc, sd, and su) estimated from the available



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data. Each observation recorded as being below the detection limit was included as another

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parameter estimated by the model, subject to the same mean value and variance components as

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the other observations, but limited to values below the detection limit28. In essence, each

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observation below the detection limit is represented as a distribution of possible values with a

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maximum value equal to the detection limit.

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The over-dispersed Poisson model29 can be expressed in an identical form, log = ′ + ′[] + ′[] + ′[] + ′

(2)

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with log-transformed mean microcystin concentrations at each site modeled as the sum of an

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overall mean value and different variance components. Then, each observation of microcystin

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(MCi) is assumed to be a drawn from a Poisson distribution with the corresponding mean value

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(λi). Because the Poisson distribution is specified only for non-negative integers, before fitting

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the model, we transformed observed microcystin values by rounding to the nearest 0.1 µg/L and

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multiplying by 10. As noted earlier, for this model, microcystin concentrations below the

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detection limit were assumed to be zero. The predicted distributions of microcystin

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concentrations resulting from the two models were compared with observed distributions with

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quantile-quantile plots. Based on this initial analysis, we selected an over-dispersed Poisson

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distribution for subsequent models (see Results).

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Using the over-dispersed Poisson distribution, we next estimated the effects of summer mean

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concentrations of TN and TP on observed microcystin. Two model components were required to

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represent the relationship between TN and TP and microcystin concentration: a model to

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estimate summer mean TN and TP concentrations from sample measurements and a model to

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estimate the relationship between summer mean TN and TP and microcystin concentration.

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Because TN and TP concentrations were available as 1 – 2 grab samples from each lake during


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the summer sampling season, we could model the summer mean TN and TP concentration, using

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data from lakes with repeat visits within the same year to quantify the variability of TN and TP

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concentrations during the sampling season30,31. TN and TP were first log-transformed and then

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scaled by subtracting their overall mean values and dividing by their standard deviation. This

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standardization of the nutrient measurements improves the convergence rate of the Bayesian

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models and facilitates comparison of the relative effects of TN and TP. Models for summer mean

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TN and TP can then be written as follows, log = log [] + , log = log [] +

(3)

,

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Where TNi and TPi are standardized grab sample measurements, and log [] and

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log [] are the summer means of log(TN) and log(TP) at each lake-year combination, j. The

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distributions of summer mean concentrations for TN and TP among all lake-year combinations

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in the survey were assumed to be log-normally distributed. That is, log ~" , #,$%& log ~" , #

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,$%&

(4)

Where µTN and µTP are the overall mean log(TN) and log(TP) concentrations among all lake-

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years and sTN,

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concentrations among all lake-years. Each grab sample measurement of TN or TP is modeled as

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the lake summer mean concentration plus a random effect (pTN,i and pTP,i) attributed to sampling

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and temporal variability. These random effects were also assumed to be log-normally distributed

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with , ~0, #,$()*& and , ~0, #

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site

sTP,site are the standard deviations of the distributions of summer mean

,$()*& .

The second component of the model estimates the effects of differences in summer mean nutrient concentrations on microcystin:



log = ′ + +, log [] + +- log [] + ′[] + ′[] + ′[] + ′

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(5)

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where summer mean nutrient concentrations for each site were provided by the first model

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component, and the regression coefficients, f1 and f2, quantify the effects of changes in summer

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mean TN and TP on the expected mean microcystin concentration. Variables describing

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contributions from different components of variability are the same as described earlier. Here

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again, the observed microcystin concentration for sample i was assumed to follow a Poisson

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distribution with a mean value of µi.

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All statistical analyses were performed with R32. Hierarchical Bayesian models were fit using the R library Rstan33.

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RESULTS AND DISCUSSION

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Non-detects. The distribution of observed microcystin concentrations was strongly skewed,

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with 60% of samples recorded as being below the detection limit of 0.1 µg/L (Table 1, grey bars

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in Figure 1). Modeling possible values of microcystin below the detection limit yielded a

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combined distribution that was approximately log-normal (Figure 1). However, the modeled log-

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normal distribution under-predicted observed microcystin concentrations at the upper quantiles

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of the distribution (Figure 2), and observed values of microcystin exceeding 5 µg/L were

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generally under-predicted by the censored log-normal model. In contrast, the predicted

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distribution from the over-dispersed Poisson model more closely matched observations, and only

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slightly over-predicted observed microcystin concentrations greater than 40 µg/L.

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Figure 1. Distribution of observed and modeled microcystin concentrations. White bars:

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distribution of observed microcystin values. Grey bars: one realization of modeled microcystin

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values below the detection limit, using the censored log-normal model.

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Figure 2. Quantile-quantile plot comparing predicted distribution of microcystin with observed

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microcystin concentrations above the detection limit. Open grey circles: over-dispersed Poisson

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distribution, filled black circles: censored log-normal distribution. Solid line shows the 1:1

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relationship.



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The superior performance of the over-dispersed Poisson model in predicting microcystin

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concentrations was somewhat surprising because many water quality parameters are effectively

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modeled as log-normal distributions34. One possible explanation for this behavior is the

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biological origin of microcystin. Microcystin is a secondary metabolite of particular

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cyanobacteria, and its concentration is therefore associated with the abundance of cyanobacteria.

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Indeed, microcystin concentrations were measured after first lysing cyanobacterial cells, and so

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observed concentrations were directly related to the abundance of cyanotoxin producing species.

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Hence, the Poisson distributions that are typically used to model the abundances of different

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biological organisms35 may also apply to modeling microcystin concentrations.

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One additional statistical reason for the superior performance stems from the treatment of non-

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detects. When using the over-dispersed Poisson, we assumed that microcystin concentrations that

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were less than the detection limit indicated that no microcystin was present (i.e., a zero

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concentration). Hence, samples with non-detects contributed more strongly to the model fitting

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compared to non-detect samples in the censored log-normal model. This subtle increase in

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sample size may have improved the overall model fit. However, the assumption that non-detects

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are zero is a simplification of the true concentration, as some small amount of microcystin likely

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existed in a proportion of the samples in which it was not detected. For the purposes of

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developing models that can be used for management decisions, these very low concentrations are

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of much less interest than concentrations that exceed known thresholds for human health effects,

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and therefore the over-dispersed Poisson model provides a more useful tool for guiding

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management decisions.

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Variance components. In the intercept-only models, among-lake differences accounted for the

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largest proportion of variance in microcystin concentrations. The standard deviation of among-


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lake variability was 2.24 (in units of log-transformed mean microcystin concentration), and this

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component accounted for 62% of the total variance in mean microcystin observations. The

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standard deviation of inter-annual variability within the same lake was 1.36, which accounted for

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23% of the total variance. The standard deviations of inter-annual temporal variability within a

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given year at a given lake and sampling variability among samples collected on the same day in a

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given lake were 0.91 and 0.54, respectively, and accounted for the remaining 11% and 4% of

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total variability. These variance components define the variance of a log-normal distribution that

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provides mean values for a Poisson distribution, and so the subsequent sampling from the

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Poisson distribution contributes additional variability to the final predicted distribution. In the

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present model, though, the variance of the underlying log-normal distribution is so large that the

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additional variance contributed by the Poisson distribution is negligible.

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The model for summer mean TN estimated the standard deviation of within-lake year

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variability as 0.28 (in units of standardized, log-transformed TN), while the standard deviation of

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among-lake year variability was 0.97. So, variability in TN measurements within a particular

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lake-year accounted for 8% of the overall variance in TN. Similarly, standard deviation of

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within- and among-lake variability of TP were 0.27 and 1.09, respectively. The inclusion of these

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models for TN and TP not only provides a more accurate estimate of the summer mean nutrient

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concentrations in each lake, but also allows the uncertainty in these estimates to propagate to the

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predictions of microcystin concentration36. TN and TP measurements were correlated with a

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Pearson coefficient of 0.76, a correlation which, given the number of samples, was not strong

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enough to influence the estimates of separate effects of TN and TP 37.

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Including summer mean TN and TP concentrations to account for among-lake differences in

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microcystin concentration substantially reduced the residual among-lake variance. The standard



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deviation of among-lakes variability in microcystin decreased to 1.24, so differences in nutrient

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concentrations accounted for 69% of the variance in microcystin concentrations among different

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lakes. As expected, same day sampling variability and within-year temporal variability were

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unchanged after including of nutrient concentration in the model. Surprisingly, between-year

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temporal variability was also unchanged after including nutrient concentrations, as one might

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expect that some of the changes in microcystin concentration between the 2007 and 2012 surveys

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would have been correlated with changes in summer mean TN and TP, thus reducing the residual

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inter-annual variance. Evidently, any changes in seasonal TN and TP between 2007 and 2012

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among the 403 lakes sampled in both surveys were not associated with changes in microcystin

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concentrations. Other analyses of these same data have observed an increase in phosphorus

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concentrations in the population of lakes and a notable reduction in the proportion of

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oligotrophic lakes between 2007 and 201238.. A comparable analysis of microcystin

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concentrations did not show population level changes between the assessment periods, which

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suggests that these phosphorus changes were not manifested as increased concentrations of

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microcystins. In lakes with higher nutrient concentrations, smaller changes were observed

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between 2007 and 2012 (Table 1).

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Estimating different components of variability provides a means of more effectively

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characterizing model performance in terms of the component of variability in microcystin that

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seasonally averaged changes in nutrients would influence. That is, changes in summer mean TN

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and TP concentrations would not be expected to influence microcystin measurements in a given

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lake on the same day or in the same season, and therefore, interpretation of the model precision

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should not include these components. The focus of the present analysis on summer mean TN and

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TP differs from other analyses of temporally-resolved field data that have observed that


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microcystin production varies with intra-annual changes in concentrations of different N and P

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species13,39,40. This difference stems partly from the regional scale of the present analysis, in

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which the temporal resolution of the data was not fine enough to explore these within-lake

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temporal trends, and partly from the fact that among-lake patterns in the relationship between N,

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P, and microcystin were likely stronger than within-lake relationships. Also, the nutrient

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measurements used here are expressed in terms of total N and total P, and while these integrative

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measures of nutrient concentrations often best represent overall nutrient loading41, they obscure

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the dynamics associated with transformation among the different N and P species.

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Beyond the potential effects of nutrient speciation, differences in microcystin concentrations

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associated with different days in the same season could also be associated with the specific

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environmental conditions that favor the formation of cyanobacteria blooms such as high

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temperatures and a stable water column42,43, while sampling variability could arise mainly from

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local factors such as a wind direction that concentrates cyanobacteria on the shore of a lake

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where the littoral sample is collected44, or laboratory measurement error. These potentially

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important environmental variables might improve predictive capacity, but require fine-scale

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temporal or local information that is outside of the scope of the current large-scale analysis.



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Figure 3. Observed total nitrogen (TN), total phosphorus (TP), and microcystin (MC). Size of

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each symbol is proportional to the number of samples with the indicated combination of TN and

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TP concentrations. Symbol shading indicates the mean microcystin concentration computed from

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samples at the indicated TN and TP. Contour lines show the modeled mean microcystin

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concentration.

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Nutrient-microcystin relationship. The standardized coefficient for TN was 1.69 (90%

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credible intervals: 1.51 – 1.87). For TP, the standardized coefficient was 0.19 (0.03 – 0.35), so

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the majority of the variations in microcystin concentration was associated with changes in TN.

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These relative effects are evident in the nearly vertical contour lines showing predicted mean

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concentrations (Figure 3). The patterns in the observed microcystin concentrations qualitatively

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agree with modeled relationships between TN, TP, and microcystin concentration.

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The strength of the association between TN and microcystin is consistent with an analysis of

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similar continental scale data in which the relative effects of TP and TN were considered 16 ACS Paragon Plus Environment

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simultaneously16, and with another analysis of relationships between nutrients and microcystin

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conducted over a similar geographic extent45. However, consensus on a mechanistic explanation

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for this result is not presently available. Emerging literature based on analysis of gene expression

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data suggests that microcystin is produced under high light and during iron and nitrogen limiting

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conditions8. Beyond triggering microcystin production, though, the more relevant environmental

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factor to consider may be conditions that are conducive to rapid growth of microcystin-

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producing cyanobacterial species. In this regard, both increased N and increased P have been

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observed in laboratory studies to increase cyanobacterial cell growth and the associated

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concentration of microcystin46. Additional analyses of these patterns at large spatial scales will

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help further refine these relationships, but the consistency of our present results with other

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available analyses and laboratory evidence suggests that these relationships are appropriate for

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informing management decisions.

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The model described here provides a predictive relationship between TN, TP, and microcystin,

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but for any given lake, additional information regarding the magnitudes of N and P loadings

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would help guide specific management actions. In some lakes, high P loading and cyanobacterial

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fixation of atmospheric N provide the nutrients necessary for cyanobacterial growth39, while in

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other lakes, high loads of inorganic N and P obviate the need for fixation47. The present model

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provides a means of linking ambient concentrations of TN and TP to the probability of different

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outcomes regarding microcystin concentration, but the different mechanisms by which TN and

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TP can be increased (or decreased) in a lake may require consideration of lake-specific

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characteristics when selecting management options that will effectively change observed

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concentrations of TN and TP.



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Management applications. One approach for quantifying the effects of model uncertainty on

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management decisions is to estimate the range of outcomes among different lakes in the study

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area, given a single value of TP and a single value of TN. These TP and TN values (i.e., criterion

343

values) are specified such that maintaining lake and reservoir concentrations at these levels will

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protect the uses that have been designated for the waterbody (e.g., recreation). As discussed

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above, observed microcystin concentrations in a lake vary in time and vary depending on where

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samples are collected on the lake, and the magnitudes of these sources of variability were

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quantified in the model. Average microcystin concentrations also differ among different lakes,

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even in lakes with the same concentrations of TN and TP. Hence, managing all lakes to a single

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TN and a single TP concentration (e.g., enforcing one criterion value for TN and one for TP)

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would result in a range of average microcystin concentrations in lakes in the study area.

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Visualizing the range of outcomes associated with different criteria may provide useful

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information for managers using the models to guide decisions.

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To further expand on this idea, we first derive candidate numeric nutrient criteria that account

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for management decisions regarding the frequency with which microcystin might be allowed to

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exceed a critical target concentration. Illustrative values are selected here for each of the

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different “management decisions”. For example, we first select 4 µg/L as a critical target

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concentration for microcystin, a threshold that has been proposed for incidental ingestion during

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recreation48. To simplify the discussion, we next combine all temporal and sampling variance

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components into a single distribution, characterized by a standard deviation, swithin, as follows, #.%/0 = 1#2- + #3- + #4- = 1.74

(6)

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This simplification yields two components of variance: one that quantifies variation in

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microcystin concentration within each lake, and one that quantifies variation in microcystin 18 ACS Paragon Plus Environment

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among different lakes, conditioned on TN and TP concentrations. To derive criteria, two

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additional management decisions are needed. First, we specify that a 5% exceedance probability

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is allowed for all samples collected within a particular lake (pwithin). Second, we specify that

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criteria are computed based on the 25th percentile of the among-lake distribution (pamong). These

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initial choices are provided for illustrative purposes, and the effects of these choices are

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discussed below.

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To calculate candidate criteria, we return to the model equation, and replace the within-lake

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variance components with a single value that expresses the allowable frequency of exceedance of

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any sample collected from a particular lake. Similarly, we replace the among-lake variance

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component with a single value that expresses the contribution from the selected percentile of the

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among-lake distribution: 29% + +- log 29% + Φ

Using National-Scale Data To Develop Nutrient ... - ACS Publications

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