Exploring the mismatch between the theory and application of photosynthetic quotients in aquatic ecosystems

Estimates of primary productivity in aquatic ecosystems are commonly based on variation in O2 , rather than CO2 . The photosynthetic quotient (PQ) is used to convert primary production estimates from units of O2 to C. However, there is a mismatch between the theory and application of the PQ. Aquatic ecologists use PQ = 1–1.4. Meanwhile, PQ estimates from the literature support PQ = 0.1–4.2. Here, we describe the theory on why PQ may vary in aquatic ecosystems. We synthesize the current understanding of how processes such as NO3− assimilation and photorespiration can affect the PQ. We test these ideas with a case study of the Clark Fork River, Montana, where theory predicts that PQ could vary in space and time due to variation in environmental conditions. Finally, we highlight research needs to improve our understanding of the PQ. We suggest departing from fixed PQ values and instead use literature‐based sensitivity analyses to infer C dynamics from primary production estimated using O2 .

Primary production in aquatic ecosystems fixes dissolved carbon dioxide (CO 2 ) and subsequently releases dissolved oxygen (O 2 ) via photosynthesis. Primary production can control the carbon (C) cycling in aquatic ecosystems, which has implications for the availability of energy to higher trophic levels, dissolved nutrient availability and cycling, and global climate dynamics due to the effects on atmospheric CO 2 concentrations (McKinley et al. 2017). Primary production can be measured either via the rate of gaseous product (O 2 ) generation or as the rate of reactant (CO 2 ) consumption following the generic photosynthesis reaction (Eq. 1) or by measuring the C assimilated into biota using isotopic approaches measured in microcosms (Nielsen 1951;Peterson 1999) to whole ecosystems (Bower et al. 1987).
Most studies measure primary production as the change in O 2 , rather than CO 2 assimilated in aquatic ecosystems, because O 2 is easier to continuously and accurately measure using sensors. Indeed, much of the early research on aquatic ecosystem metabolic processes is based on models measuring the change of O 2 with either diel variation (Odum 1956) or experimental manipulation of light (Nielsen 1951). Furthermore, methods to measure CO 2 lagged that of O 2 , and also require knowing pH or alkalinity along with more difficult computation to account for the overall dissolved inorganic carbon (DIC) pool (Barnes 1983;Gattuso et al. 1999;Stets et al. 2017).
Because we are often more interested in C cycling we convert primary production measured with O 2 data to C using the photosynthetic quotient (PQ), that is, the molar ratio of the flux of O 2 released to CO 2 assimilated during photosynthesis (Eqs. 2, 3; Note that this equation naturally is negative due to the decline of DIC during photosynthesis, although we follow the common convention of referencing PQ as a positive value).
Despite its evident importance to translating from one chemical species to another, the PQ term has been dogmatically employed without appreciating the existing degree of uncertainty and its implications. A mismatch between theory and application of the PQ exists within and across aquatic ecosystem subfields. For example, it is common to assume a PQ of 1 or 1.2 for freshwater ecosystems, even though these values are not necessarily backed by measurements or theory. Marine researchers commonly use a value of 1.1-1.4, which is in part based on theory, but under specific assumptions, such as whether production is new or recycled (Williams et al. 1979;Laws 1991;Hedges et al. 2002). Fixed and fieldspecific PQ values are used despite PQ measurements from the literature ranging from 0.1 to 4.2 (Table 1).
Physiological mechanisms that co-occur with oxygenic carbon fixation alter the rate of net O 2 production because they occur at rates and scales commensurate with that employed by our methods even as those methods fail to distinguish them. For example, photorespiration and nitrate (NO À 3 ) assimilation are net sinks and sources of O 2 (respectively) that are rarely accounted for in models that estimate primary production (Fig. 1). These oversights may have extensive and unappreciated consequences for our understanding of C fluxes associated with primary production across ecosystems.
Here, we explore the current status of PQ values in aquatic ecosystems and their historical application, with the goal to enable better estimates of C transformations based on O 2 production. Specifically, we will: 1. summarize measurements of PQ within and between marine and freshwater aquatic ecosystems;  Laws (1991), Williams and Robertson (1991), Raine (1983), Williams et al. (1979), Iriarte (1999), Carvalho (2014) DIC 4 0.4-1.6 Burris (1981), Smith et al. (2012), Du et al. (2018), Bolden et al. (2019), Mercado et al. (2003), Iriarte (1999), Rosenberg et al. (1995) Freshwater 14 C 2 0.3-2.0 Bell and Kuparinen (1984), Sakamoto et al. (1984) DIC 3 0.13-1.4 Pokorny et al. (1989), Hanson et al. (2003), Sakamoto et al. (1984), Kliphuis et al. (2011) 2. describe the state of chemical and biological theory addressing how and why PQ varies across species and environmental conditions; 3. provide a case study of a river where theory predicts that PQ should vary in space and time based on environmental conditions and algal succession; and 4. highlight research needs to improve our understanding of how we translate between C and O metabolism.
Overall, we suggest departing from minimally informed fixed PQ values and instead use literature-based sensitivity analyses to inform how PQ choice affects interpreting C dynamics from primary production estimated using O 2 . Furthermore, we encourage simultaneous measurements of O 2 and CO 2 when possible, to provide empirical evidence of environmental PQ variation and accurate elemental translation.
We recognize from the outset that PQ is difficult to assess because we cannot directly measure photosynthesis. Measurements of photosynthesis will always co-occur with respiration and parsing the two measurements is problematic; it is common to measure respiration (in the dark), then measure net ecosystem production (in the light), and subtract respiration from net ecosystem production. The light/dark approach assumes that respiration is constant throughout the day and ignores other abiotic processes that are related to diel cycles of light (e.g., photo-oxidation), and it is becoming clear that these assumptions are often not valid (Hotchkiss and Hall 2014;Schindler et al. 2017). Thus, variation in respiration, and in particular, respiration quotients (RQ) will affect our assessment of PQ. In this work, we are not considering the effects of RQ, since a summary of the RQ variation with environmental conditions already exists (Berggren et al. 2012). Even so, RQ and PQ should both be considered when interpreting O 2 and CO 2 metabolism data.

History of PQ measurements and values used across aquatic ecosystems
Most of our knowledge of the PQ comes from experiments and measurements in marine ecosystems on planktonic algae. Although our purpose is not to exhaustively review the literature, the papers we encountered were mostly marine with few from freshwater, including only one study with measurements of PQ on freshwater benthic algae (Table 1). The bias of PQ measurements towards phytoplankton and marine ecosystems may be due to historic method constraints on the accurate measurement of CO 2 (Williams and Robertson 1991 and CO 2 are affected by photosynthesis and respiration, while O 2 is uniquely affected by NO À 3 assimilation and photorespiration. O 2 concentrations are much more likely to be affected by captive bubbles or ebullition since O 2 has low solubility above saturation. In contrast, CO 2 equilibration with DIC reduces the potential for bubble formation. Note that the magnitude of the effect of any given process on dissolved gas concentrations depends on a variety of environmental conditions specific to that process. (Strickland 1960) to measure primary productivity (referred to here as 14 C productivity). Given the complexities of using radioactive materials in the natural environment, this method is much more favorable at small scales (e.g., bottles that capture microscopic primary producers) and in the laboratory, thus the initial bias towards phytoplankton and marine ecosystems. However, the 14 C approach is not directly comparable to measures of O 2 evolution because the incubation results in something between gross primary production and net primary production (Peterson 1980) as a result of the potential respiration of 14 C-labeled photosynthetic products and/or the recycling of 12 CO 2 into the photosynthetic pool. These uncertainties in the measurement of C fixation bias estimates of the PQ.
A review by Williams and Robertson (1991) showed PQ ranged between 0.5 and 3.5, with most measurements made based on 14 C productivity. They attributed the broad range in PQ to the 14 C productivity problem (Peterson 1980), and suggested that a more realistic range of values for phytoplankton is between 1.1 and 1.4 (Williams and Robertson 1991). This range was confirmed by Laws (1991), who used chemical equations of photosynthetic products, and reported estimates of phytoplankton chemical composition to show that PQ ranged from 1.1 to 1.4. These values of PQ represent broad averages, and more contemporary site-specific studies using more accurate methods to measure primary production (i.e., direct measurement of CO 2 or DIC compared to 14 C productivity) have resulted in PQ values that differ from 1.1 to 1.4 based on specific species and environmental conditions ( Table 1). Using a fixed value of 1.35-1.45 is still common in marine ecology (Bolden et al. 2019); however, in general, marine ecologists tend to be more aware of the inherent variability of the PQ.
Freshwater ecologists also commonly use fixed PQs but usually employ values of 1 or 1.2, based more on tradition rather than on theory or data. We know of no direct source suggesting the use of PQ = 1 although it is probably safe to assume that this value reflects only the photosynthetic creation of glucose (C 6 H 12 O 6 ), which results in molar equivalence between O 2 released and CO 2 fixed. The assumption of glucose as the only photosynthetic product oversimplifies organic matter synthesis and ignores other processes that can affect dissolved O 2 concentrations. The PQ value of 1.2 similarly does not have a common source, although some attribute this value to Wetzel (2001) or Bott et al. (1978), despite neither source providing convincing evidence for a fixed value. Although the range of reported PQ measurements in fresh waters is between 0.13 and 2.0 (Table 1), we have not found an article that applies a fixed value other than 1 or 1.2 when the PQ is not directly measured. Moreover, none have accounted for the implications associated with the uncertainty of the PQ values employed.
We suggest that aquatic ecologists need to use greater scrutiny when applying PQ values to translate O 2 -based metabolism to rates of C fixation. Given the need for an accurate PQ, we summarize knowledge of why the PQ may vary across species and environmental conditions, and use this knowledge to generate hypotheses about the biological and environmental conditions causing the PQ to vary. Although most of this knowledge is based on studies of phytoplankton from marine ecosystems, the same concepts often hold in freshwater ecosystems.

PQ theory
PQ can be separated into multiple components, with each dictated by the characteristics of the species of primary producer and the environmental conditions surrounding the organism (Williams and Robertson 1991). The appropriate PQ value that represents biological effects on O 2 and CO 2 (PQ Bio ) can be represented by Eq. 4: where PQ C refers to the change in O 2 and CO 2 associated with photosynthesis, PQ NO 3 refers to the release of O 2 associated with the reduction of nitrate (NO À 3 ) for nitrogen (N) assimilation, and PQ pr is the reduction of O 2 associated with photorespiration. The ellipsis (…) represents the myriad of other biological processes that may affect O 2 or CO 2 availability (e.g., nitrification, sulfate reduction). We focus on the three described components identified by considerable evidence to measurably affect the PQ in most aquatic ecosystems.
Carbon PQ component PQ C ð Þ The PQ C represents the change in O 2 and CO 2 associated with photosynthesis. The PQ C can be broken down into the various products of photosynthesis dictated by the state of reduction of the products for the general compound C H ð Þ Y O ð Þ Z (Eq. 5) (Ryther 1956;Laws 1991): where Y and Z represent the number of H and O atoms in the photosynthetic product, respectively. Thus, the net change in O 2 and CO 2 associated with the sum of C H ð Þ Y O ð Þ Z products equals PQ C . The most common products intimately coupled with photosynthesis are glucose (C 6 H 12 O 6 ), glycolic acid (C 2 H 3 O 3 ), saturated fatty acids (e.g., C 20 H 30 O 2 ), nucleic acids (e.g., C 38 H 47 O 28 N 15 P 4 ), and protein (e.g., C 18 H 24 O 6 N 5 ). The proportion of C in each of these compounds relative to the total C in the algal cell will dictate the value of the PQ C . The range of plausible PQ C H ð Þ Y O ð Þ Z values is 0.625-1.5, with the lowest value assuming the production of glycolic acid only, and the highest value assuming the sole production of saturated fatty acids. In reality, these extremes are not likely, and the PQ C will fall somewhere in the middle of this range. Laws (1991) estimated the 'typical' planktonic cell to contain 40% each of proteins and carbohydrates, 15% lipids, and 5% nucleic acids, which equates to a PQ C of 1.08. This value is specific to phytoplankton, and there is likely species-specific variation in PQ C based on algal physiology and life history. Notably, benthic algae may have a different PQ C than phytoplankton given the need for making structures to anchor to the substratum (among other physiological differences); however, there is little information available to calculate the PQ C of benthic algae.
Few papers have measured the effect that specific photosynthetic end products have on the PQ, although speculation abounds that environmentally driven changes in lipid production increase the PQ. These changes in the PQ are likely associated with variation in light and energy availability occurring in environments replete with other possible limiting nutrients as the algal cell uses the opportunity to generate lipids that require more energy and C, thus increasing the PQ (Norici et al. 2011;Palmucci et al. 2011). Indeed, many studies have observed increased PQ with diel increases in light (Carvalho 2014), and experimental light manipulation (Du et al. 2018). The mechanistic link (or lack thereof) between environmental variables and lipid production is a knowledge gap relevant to estimates of PQ in freshwater and marine ecosystems.
The biosynthesis associated with primary production also requires N, and the concurrent reduction of NO À 3 by the cell for assimilation will release O 2 , thus increasing the PQ. Specifically, 2 mol of O 2 are released for every mole of NO À 3 reduced to NH 3 .
Organismal demand for N controls PQ NO À 3 ; C : N ratios in organismal biomass are a good indicator of this demand (Redfield et al. 1963) with lower values reflecting greater demand for N. Finally, it is energetically favorable for organisms of all sizes to assimilate N as ammonium (NH þ 4 ), which does not produce O 2 , relative to NO À 3 . Therefore, the relative amount of NO À 3 in the total dissolved inorganic nitrogen pool (NO À 3 þ NH þ 4 ; DIN) also controls the amount of O 2 released as a part of N assimilation (Smith et al. 2012).
Lab experiments demonstrate that O 2 release due to NO À 3 assimilation increases PQ values (Raine 1983;Bell and Kuparinen 1984). Raine (1983) manipulated NO À 3 concentrations of salt water containing phytoplankton in the lab, while keeping all other environmental conditions constant. PQ varied from 1.0 to 2.25 and the measured change in PQ was similar to the theoretical change in PQ based on chemical stoichiometry (e.g., Eq. 7; Raine 1983). Field measurements tell the same story; Smith et al. (2012) observed an increase in the PQ from 1.24 to 1.42 over $ 30-yr in a Rhode Island estuary. The observed increase in the PQ coincided with increased proportion of DIN present as NO À 3 associated with a new wastewater treatment plant upstream of the estuary. The change in relative N availability (i.e., Eq. 7) suggests that the PQ should have increased by 0.15-0.23, which includes the observed increase of 0.18 (Smith et al. 2012). Note that we do have some hesitation about the validity of Eq. 7 as it is not immediately clear that the units are balanced and Smith et al. (2012) do not show their derivation; however, the match between expected values derived from chemical stoichiometry and empirical measurements enhances confidence in estimates of PQ NO À 3 estimated from Eq. 7. Nonetheless, the lowest confidence is assigned to estimates of PQ pr , which addresses the complicated and poorly addressed influences of photorespiration.

Photorespiration PQ component PQ pr À Á
Photorespiration is the light-dependent O 2 reduction and CO 2 release by photoautotrophs. Unlike mitochondrial respiration, photorespiration only occurs in light, does not conserve energy in ATP, and does not use substrates from the tricarboxylic acid cycle (Tolbert 1974). Instead, photorespiration reduces O 2 via the RuBisCO mediated reaction with ribulose 1,5-bisphosphate (RuBp) that generates C-2 compounds (e.g., 2-phosphoglycolate) further processing of which consumes more O 2 and liberates CO 2 .
Although the end result of this process includes the release of CO 2 , some or all of this CO 2 is "recycled," or immediately fixed by photosynthesis, rather than being released from the cell (Søndergaard and Wetzel 1980). Photorespiration was initially referred to as a "wasteful process" given that glycolate is not needed and is often excreted when in excess (Tolbert and Zill 1956). Later studies discovered that glycolate is eventually funneled through multiple reactions to contribute to the Calvin-Benson cycle, although at a higher energy cost than via photosynthesis (Moroney et al. 2013). During photorespiration O 2 occupies the RuBisCO binding site, limiting CO 2 fixation and reducing rates of net photosynthesis. Photosynthesis inhibition at high O 2 concentrations is a common phenomenon explained by photorespiration. Most algal species have developed C concentrating mechanisms to build up DIC around the RuBisCO enzyme, limiting photorespiration (Giordano et al. 2005). Aquatic photoautotrophs need C concentrating mechanisms because they commonly face an environment with low CO 2 concentrations, and need to build up CO 2 near RuBisCO for photosynthesis (Moroney et al. 2013).
Despite the evolution of C concentrating mechanisms to limit its effects, photorespiration is still necessary. For example, mutant cyanobacteria missing some photorespiration pathways could not to survive outside of abnormally high CO 2 environments (Eisenhut et al. 2008), evidence that photorespiration is necessary despite C concentrating mechanisms.
Although the purpose and potential benefits of photorespiration are not yet fully understood (Osmond and Grace 1995;Shi and Bloom 2021), generally 3 moles of O 2 are reduced for every mole of CO 2 produced during photorespiration, which ultimately lowers PQ (Stewart 1974). Multiple experiments have shown that photorespiration constitutes 20-40% of net photosynthesis across a variety of aquatic ecosystems and measurement scales (Burris 1981;Parkhill and Gulliver 1998;Kliphuis et al. 2011;Buapet et al. 2013), particularly when O 2 is readily available and CO 2 is relatively low (Vance and Spalding 2005). Photorespiration rates are often $ 0 below 100% O 2 saturation, low at or below 100% O 2 saturation, and highest when O 2 is super-saturated. Although, low CO 2 (and indirectly elevated pH) can often increase photorespiration rates as long as some O 2 is present (Kliphuis et al. 2011). Multiple studies have measured PQ < 1, which is often attributed to photorespiration (Hough 1974;Burris 1981;Pokorny et al. 1989). Furthermore, some studies have experimentally manipulated light (Iriarte 1999) or O 2 concentrations (Burris 1981) and found PQ < 1 matching the theoretical effect of photorespiration. We note that photorespiration has not been observed (when it otherwise should have been) under some environmental conditions and with some specific algal species (Lloyd et al. 1977;Birmingham et al. 1982).
Toward predicting when and where PQ may vary based on environmental conditions PQ estimates from the literature and direct experimentation provide a theoretical basis for hypothesizing when and where PQ may vary with environmental conditions. Here, we synthesize the relevant information to aid researchers in addressing variation in the PQ. Literature supports the assertion that PQ NO 3 and PQ pr are likely to have the strongest effect on the PQ, and these effects can be predicted based on the proportion of NO À 3 that is DIN, the C:N ratio of the algal biomass, and the % O 2 saturation. We used data from the literature to simulate the effect of environmental conditions on the PQ using Eq. 7 (for effects of PQ NO3 ) and data from Burris (1981) (for effects of the PQ pr ; Fig. 2). We conducted the simulations across the plausible range of each variable in the environment, where the environmental conditions ranged for the proportion of DIN that is NO 3 from 0 and 1, for O 2 saturation from 40% and 180%, and with an algal C : N composition of 5 or 20. This simulation provides a broad hypothetical framework for predicting when and where the PQ may vary. Results from the simulation (Fig. 2) illustrate the potential for PQ pr to increase the PQ Bio by as much as 0.8 when dissolved O 2 is low, and decrease by as much as 1.2 when O 2 is well above saturation. Changes associated with PQ NO 3 are less dramatic, with changes in the PQ Bio of 0.2 when N demand is high (low biomass C : N, Fig. 2A) and less when N demand is low (high biomass C : N, Fig. 2B). We note that the simulated changes in PQ Bio are relative to an assumed PQ C of 1.1, an assumption that carries its own degree of uncertainty. While the literature supports the possibility of variation in the PQ C , we do not believe there is enough information to adequately predict anything other than a fixed PQ C .
Primary productivity varies with light intensity, and thus PQ can change with the diel progression of light (Hough 1974; Fig. 2. The simulated change in PQ associated with the PQ NO À 3 (y-axis) and the PQ pr (x-axis) components. The top panel assumes a fixed C : N of 5, while the bottom panel assumes a fixed C : N of 20. Both simulations assume PQ C ¼ 1:1 and the dashed line shows the combination of environmental conditions where PQ does not vary from 1.1. The PQ will increase (positive change relative to PQ C ) when NO À 3 is a high proportion of DIN and the aquatic ecosystem is below 100% O 2 saturation. In contrast, the PQ will decline the most when NO À 3 is a low proportion of DIN and waters are supersaturated with O 2 . Finally, when the demand for N is low (indicated by high C : N ratio), the effect of PQ NO À 3 on the PQ change is less than when N demand is high. Iriarte 1999;Mercado et al. 2003). Furthermore, the degree to which the PQ will vary during a day will depend on the rate of primary productivity, given that ecosystems with low productivity are unlikely to supersaturate with O 2 , limiting the potential effect PQ pr may have on PQ Bio . The proportion of NO À 3 that is DIN can vary seasonally or spatially, particularly in lotic ecosystems. Accordingly, we simulated how different fixed rates of gross primary production (GPP), in units of moles O 2 released (GPP O ), interact with environmental conditions to control the PQ Bio . Specifically, we simulated three scenarios: (1) low GPP rates (i.e., O 2 saturation never exceeds 100%) and low NO À 3 assimilation rates (assuming mineralization of NH þ 4 as is likely at low productivity rates); (2) high GPP rates (i.e., O 2 saturation exceeds 100% during most of the photic period) and low NO À 3 assimilation rates (assuming a low proportion of NO À 3 that is DIN); and (3) high GPP rates and high NO À 3 assimilation rates (assuming a high proportion of NO À 3 that is DIN). We again assume that PQ C is fixed at 1.1. We simulated the diel O 2 saturation curves assuming low and high fixed values for GPP (19 and 250 mmol m À3 d À1 , respectively) and ecosystem respiration scaled with GPP, while gas-exchange, average depth, temperature, and pressure were held constant (SI 1, SI Table 2 in Supporting Information). We used the theory presented above to predict PQ C , PQ NO À 3 , PQ pr , and PQ Bio in each scenario. We then used the PQ Bio derived from each scenario to calculate GPP, in units of fixed C (GPP C ). Finally, we compared the GPP C predicted with the PQ Bio from each scenario with the GPP C estimated using fixed PQ values   (Eq. 4). The mean PQ x for each scenario was fixed based on our current knowledge of the effect of environmental conditions on each compartment. In the low GPP scenario (D), PQ Bio ≈ PQ C since the effects of PQ Pr and PQ NO À 3 are negligible when O 2 saturation is < 100% and NO À 3 concentration is low. For the high productivity and low NO À 3 scenario (E), the PQ Bio is < PQ C due to photorespiration occurring when O 2 saturation is ≥ 100%. For the high productivity and high NO À 3 scenario (F), PQ Bio > PQ C primarily controlled by the large effect of PQ NO À 3 . The final row shows how the fixed GPP value in units of O 2 would be transformed to C units using: the mean PQ Bio from (D) to (F), the two most common fixed freshwater (FW) values (1, 1.2), the most common marine value (1.4), and a conservative range of PQ values observed in the literature (0.5 and 3.5, respectively). The error associated with using the conventional fixed PQ values is low when GPP rates are low, and vice versa when rates are high. The highest error when converting GPP from units of O 2 to C occurs when GPP rates are high, NO À 3 is low, and photorespiration is occurring.
commonly used in marine and freshwater ecosystems, and from the range observed in the literature. In low productivity ecosystems, N assimilation and photorespiration are likely low, thus the PQ Bio ≈ PQ C (Fig. 3). Based on our current understanding of PQ C , when PQ Bio ≈ PQ C the choice of traditional fixed PQ values will minimally bias estimates of GPP C from measured GPP O . However, if future research found evidence that the PQ C varies with environmental variables or by species-specific characteristics, then PQ Bio may diverge from canonical values. In contrast, highly productive ecosystems with low NO À 3 will have a PQ Bio that is < 1. This scenario causes the largest bias of GPP C estimates from GPP O . PQ values < 1 will have a higher bias than values > 1 of the same magnitude (i.e., 0.8 compared to 1.2) (Fig. SI  1 Supporting Information). In the final scenario, the O 2 consumed through photorespiration is negated by the addition of O 2 from NO À 3 assimilation, resulting in a PQ Bio > PQ C . In this scenario, the PQ Bio is similar in magnitude with the traditional fixed values, with minimal bias based on the choice of PQ Bio .

Considerations of observation error and measurement bias
The above theory provides testable hypotheses that can be confronted with measurements of the PQ (PQ Meas ) across environmental conditions. In an ideal world, PQ Meas would match PQ Bio ; however, this scenario is unlikely ever true. It is necessary to consider the observation error PQ Obs error ð Þand measurement bias (PQ Meas bias ).

PQ Meas ¼ PQ Bio þ PQ Obs error þ PQ Meas bias ð8Þ
PQ Obs error is the error imposed by imperfectly measuring O 2 or DIC. For example, an improperly calibrated or drifting sensor may add observation error to the PQ. In general, it is possible to limit PQ Obserror with careful calibration and lab work. Even so, there are statistical approaches to account for measurement error (e.g., state-space models, Pedersen et al. 2011;Appling et al. 2018). We define PQ Meas bias as the error imposed when a model for dissolved O 2 or DIC does not reflect the process of interest. PQ Meas bias is process error and can occur for a variety of reasons. For example, abiotic processes other than the biological processes associated with primary production can bias estimates of GPP (e.g., bubble formation in chambers). PQ Meas bias will be most prevalent when comparing photosynthesis based on O 2 or CO 2 at the ecosystem level and provides a substantial challenge for discerning the biological signal within PQ Meas . Although PQ Meas bias can take many forms, we highlight a few sources of PQ Meas bias that deserve further consideration and research.
For ecosystems with attached benthic primary producers (Fig. 1A), captive bubbles (i.e., bubbles from primary production trapped in biofilms) are common, but understudied in productive ecosystems and are more likely to affect O 2 than CO 2 . Dissolved CO 2 readily equilibrates with the DIC pool in many circumneutral pH environments, while the CO 2 remaining as dissolved gas has a much higher solubility than O 2 . Bolden et al. (2019) attribute observed captive bubbles on marine coral reefs to abnormally low PQ measurements after ruling out the contribution of PQ pr and PQ NO À 3 . The bias in measured O 2 was large enough that the authors suggested that O 2 -based estimates of metabolism are unreliable in highly productive benthic environments (Bolden et al. 2019). Ebullition occurs when captive bubbles are released into the water column, and eventually traverse the air-water divide. The effect of ebullition on primary production estimates has been documented in many lakes, where emissions of bubbles can be similar in magnitude to diffusive O 2 exchange (Koschorreck et al. 2017), and contain up to 20% of daily net oxygen production (Howard et al. 2018). Ultimately, captive and ebullated bubbles can affect estimates of PQ Bio in variable ways depending on the degree to which gases in bubbles equilibrate with the water around them (Hall and Ulseth 2020).
Groundwater inputs can affect O 2 and CO 2 concentrations in receiving freshwater ecosystems (Sear et al. 1999;Duvert et al. 2018). Generally groundwater in local flow is replete with CO 2 and low in O 2 due to soil respiration (Tank et al. 2018); low O 2 groundwater can bias estimates of GPP (Hall and Tank 2005). The contribution of groundwater CO 2 can generally be predicted by stream hydrology, with higher groundwater CO 2 contributions in small headwater streams compared to larger rivers (Hotchkiss et al. 2015;Horgby et al. 2019b); however, knowledge of local land use and geology, among other factors, are needed to adequately predict groundwater O 2 and CO 2 contributions (Tank et al. 2018;Horgby et al. 2019a;Hutchins et al. 2021). Furthermore, groundwater contributions are not static and will likely vary with environmental conditions (Ulseth et al. 2018). Overall, groundwater inputs of CO 2 are likely to be high in many freshwater ecosystems, which may lead to an underestimation of PQ Bio .

Case study: The Upper Clark Fork River, Montana
The literature clearly supports that the PQ Bio can vary from the canonical fixed values based on environmental conditions with measures further complicated by a mismatch between PQ Bio and PQ Meas . Next, we provide a case study of a river where changing environmental conditions predict that the PQ Bio varies in space and time, and differs from PQ Meas . This case study reveals the inability to reconcile the application of canonical PQs, PQ theory, and measurements of PQ.
The Upper Clark Fork River in western Montana is a midorder river that is highly productive during the summer growing season due to ample light, warm water, and plenty of nutrients (both N and P). The NO À 3 concentration is high near the headwaters of the river, due to a combination of natural and anthropogenic inputs, and generally declines moving downstream. Phosphorus (as soluble reactive phosphorus) is high in the middle section of the Clark Fork due to weathering of P-rich rocks in this region (Carey 1991). Variability in nutrients and productivity make the Clark Fork a useful location to explore how environmental conditions may translate to variation of the PQ Bio in space and time. Specifically, we explore how variation in NO À 3 , and the maximum daily O 2 saturation might control spatial and temporal variation in the PQ Bio during the summer growing season. Data, metadata, and code are publicly available (Trentman et al. 2023).
We present data collected in the Clark Fork from summer 2020 (1 August-30 October) at two sites, an upstream site near Warm Springs, Montana (Site 1) and a downstream site near Clinton, Montana (Site 2). Data presented are: O 2 concentrations collected using continuous sensors (PME) at 10-min. intervals, and DIN concentrations (as NO À 3 þ NH 4 ) collected as bi-weekly filtered grab samples and analyzed using standard methods. The O 2 data were used to calculate maximum O 2 saturation for each day, and estimate GPP O using a single-station approach. More details on data collection and analyses are provided in SI 3 in Supporting Information. We use these data and the theory presented above to predict PQ Bio at each site over the algal growing season. Data presented here are unpublished and associated with long-term efforts to characterize the relationship between nutrients, algal biomass, and metabolism in the Clark Fork (H.M. Valett unpubl. data).
Diel O 2 reflected a highly productive river with large daily excursions. Maximum daily O 2 saturation exceeded 100% at both sites for most of the time series (Fig. 4). Maximum daily O 2 saturation dipped below 100% at the downstream site for the final 2 weeks, and at the upstream site for the final 3 d of data. Our current understanding of the link between O 2 saturation above 100% and photorespiration suggests that photorespiration was likely affecting PQ Bio . The effect of photorespiration likely declined in the later portion of the sampling period as max O 2 saturation dipped below 100%, with a longer dip at the downstream site. The proportion of DIN that was NO À 3 was generally higher (between 0.6 and 0.95) at the upstream site, compared to the downstream site (between 0.2 and 0.9; Fig. 4). At both sites, the proportion declined by 0.4 during the growing season and then rebounded toward the end of the growing season. The predicted effect of O 2 release from NO À 3 assimilation on the PQ was greater at the upstream site compared to the downstream site, with similar amounts of within-site variation over the growing season (Fig. 4).
We applied the predicted PQ Bio reflecting the influences of the components addressed herein to the GPP O measurements derived at the upstream and downstream sites to estimate GPP C (Fig. 5). We then compared GPP C rates estimated with the predicted PQ Bio to the GPP C estimated with the canonical PQ value of 1.2, as well as a conservative range of PQ values from the literature (0.4-3.5). At both sites, GPP O rates declined by 50-60% during the measurement period. During the early part of the measurement period, GPP C using the predicted PQ Bio was $ 30% greater than the GPP C estimated using the traditional fixed PQ of 1.2. The difference between estimates declined as the growing season progressed. At Site 1, the two estimates converged for the final 2 weeks of measurements as max O 2 saturation declined to less than 100% saturation and PQ Bio increased to near 1. At Site 2, the two estimates of GPP C nearly converged for the final few days of measurements, again matching patterns in maximum O 2 saturation. Assuming our predicted PQ Bio is accurate, using a fixed value of 1.2 may lead to an error in GPP C estimates of 0-33% based on the site and time of the year.
To capture the potential uncertainty imposed by the various PQs used here, we also compared the GPP C estimated from the canonical value with the range of PQs found in the literature. The GPP C using the range of literature PQ values was AE 63% different than the GPP C estimates using PQ = 1.2 (Fig. 5), which is a wide range that has implications for how we interpret and analyze GPP C estimates.
Finally, to employ a more controlled assessment of PQ Bio , we compared the PQ Bio predicted from environmental conditions with a set of measured PQ values using instruments to directly quantify CO 2 and O 2 . We used sealed recirculating acrylic chambers (Dodds and Brock 1998) to measure the PQ from substrate collected from the Clark Fork River in summer 2021. The PQ was estimated using the change in O 2 (measured from continuous sensors) and DIC (estimated from CO 2 measured using in situ sensors, DeGrandpre 1993, and ex situ measurements of alkalinity). Six PQ measurements were made based on independent substrata and at varying levels of productivity, provided by altering the available light using a shade cloth. SI 3 in Supporting Information contains a detailed description of the methods. Based on the linear regression of molar rates among the six experiments, the PQ Meas from the sealed chambers was 1.6 (AE 0.5 SE of the regression slope; Fig. 6).
The difference between the mean predicted PQ Bio (0.7) and the mean PQ Meas (1.6) in the Clark Fork River shows that there were either substantial bias and error in our measures of C and O 2 metabolism (i.e., PQ Meas ) or poor predictions based on gaps in the current theory (i.e., PQ Bio ). These data cannot adjudicate the true value of the PQ and rather exposes our extensive knowledge gaps in theory and measurement of PQ. It is necessary to resolve these problems in PQ theory and measurements in order to generate accurate assessments of GPP C .

Photorespiration
Photorespiration may be more prevalent when estimating aquatic primary production than we know. Studies that estimate photorespiration at the ecosystem level are uncommon (but see Parkhill and Gulliver 1998), despite ample evidence that photorespiration is occurring in highly productive and pH-neutral waters. Thus, more ecosystem-level estimates of the prevalence and magnitude of photorespiration are necessary. Although some evidence shows a relationship between photorespiration and O 2 saturation or the ratio of O 2 to CO 2 , the data representing these empirical relationships are lacking. Furthermore, it is not clear if empirical relationships are species-specific. For example, some species of red and brown algae suppress photorespiration more than many green algae species (Reiskind et al. 1989). In general, it is not clear how strongly species differ in the magnitude and prevalence of photorespiration, and if the potential variation is relevant to ecosystem-level estimates of primary production.
NO À 3 assimilation Although multiple field and lab studies have characterized the effect of O 2 release during NO À 3 assimilation on the PQ, our ability to predict this effect is more complicated than our current knowledge (i.e., Eq. 7). Equation 7 does not account for the scenario where NH þ 4 and NO À 3 are both saturated in respect to nutrient demand (Dodds et al. 2002;e.g., algae are limited by their maximum nutrient uptake rate rather than nutrient availability). For example, a realistic scenario in many agricultural streams is that both NH þ 4 and NO À 3 are present in high concentrations (e.g., 10 and 110 μmol L À1 , respectively; M. T. Trentman pers. observ.). In this scenario, using Eq. 7 would lead to an overestimate of the effect of NO À 3 assimilation on the PQ because the form of N uptake is dominated by NH þ 4 , which is preferentially assimilated by organisms. Furthermore, in marine environments, urea can be a substantial proportion of DIN (Ignatiades 1986;Antia et al. 1991) and is more readily assimilated than NH þ 4 . In short, Eq. 7 should account for the abundance of NH þ 4 and urea in relation to demand as well as the relative availability of NO 3 .  Fig. 6. The relationship between O 2 and DIC-based gross primary production measured simultaneously in a chamber using substrate from the Clark Fork River. The slope of the blue line was calculated using a reduced major axis approach since the axes are symmetrical and represents PQ Meas ¼ 1:6 (AE 0.5 SE of the slope), which is higher than the predicted PQ Bio from the same river.

Photosynthetic products
Information on the production of different C products during photosynthesis are uncommon, specifically for freshwater algae and species that have no commercial interest, making it difficult to calculate the theoretical PQ C H ð Þ Y O ð Þ Z . Thus, our understanding of how PQ C may vary by species and with environmental conditions (e.g., light) is limited. However, most photosynthetic products, including proteins, nucleic acids, and glucose, have a PQ C H ð Þ Y O ð Þ Z of approximately 1. That leaves glycolic acid and saturated fatty acids as two possible sources of variation, which could decrease and increase the PQ C H ð Þ Y O ð Þ Z , respectively, based on their content in an algal cell. It is unlikely that the production of glycolic acid is substantial enough to shift the PQ given the relatively low production rate (Cheng et al. 1972). It is more likely that the species-specific variability in saturated fatty acid production (Harwood 2019) Guschina and Harwood 2006;Li-Beisson et al. 2019), but there are little data on how this variation in lipid chemistry affects the proportion of the total C of algal biomass. Research is needed that quantifies the rate of saturated fatty acid production across algae species, and how this production varies with environmental variability.

PQ measurements across scales and species
More measurements of PQ, particularly on freshwater benthic algae and using methods other than 14 C, would improve our understanding of how the PQ Bio varies among species and with environmental variables. The PQ can be measured through a variety of approaches that can be generally separated by the location of the measurements, whether controlled (e.g., in bottles or chambers) or in the natural environment. Measuring the PQ requires simultaneous monitoring of the C that is fixed and the O 2 that is released during the process of primary production (Although, see the above text that describes the measurement challenges imposed by respiration). The measurement of O 2 can be accurately obtained with low-cost optical sensors (assuming no captive bubbles). Estimating the change in DIC is more involved and includes either direct measurements of DIC or is accomplished by estimating the DIC pool via measures of dissolved CO 2 concentrations and either pH or alkalinity (Dickson et al. 2007). In highly buffered waters, the relative decline in DIC will be small, and this decline is difficult to accurately quantify, but the decline in CO 2 will be large. Given this fact, we briefly highlight the pros, cons, and limitations of the best-known approaches that use accurate measures of O 2 and CO 2 .
Using sealed containers to measure algal productivity allows for control of environmental conditions. Controlled measurements of PQ will promote equivalence between PQ Meas and PQ Bio , since it will be easier to limit PQ Meas bias .
The bottle approach is useful for phytoplankton, while more complex chambers should be used for benthic algae, especially from rivers. Unidirectional flow in rivers is one of many controls on primary productivity, thus chambers with continuous, circulating flow are needed to measure the PQ for riverine benthic algae (Dodds and Brock 1998;Rüegg et al. 2015). PQ measurements sealed from the atmosphere are best because, if used properly, the chambers allow the researcher to rule out gas-exchange between the water and the atmosphere, which can be challenging to measure in low productivity and turbulent ecosystems (among other conditions; Hall and Ulseth 2020). The benefits of bottles or chambers should be balanced with the potential bias of measuring ex situ processes. When working with chambers, it is common to not let O 2 exceed 100% saturation in order to limit bubble formation in the chamber. Thus, chamber experiments may be biased in that the effects of PQ Pr are limited. Finally, large chambers can be expensive and cumbersome to use (M.T. Trentman pers. observ.).
Measuring diel variation in O 2 and DIC to estimate metabolic processes is another approach to estimate the PQ, although in most scenarios PQ Meas ≠ PQ Bio due to PQ Meas bias . The bias of PQ measured with diel sensors will be particularly problematic given that there is a range of environmental controls on reach scale or whole lake O 2 and DIC budgets. Bubbles, groundwater inputs, and the complexities of DIC chemistry (e.g., calcite precipitation), among many other factors, may impose higher variation on PQ Meas than the biological signal. That said, simultaneous monitoring of diel O 2 and CO 2 concentrations is a relatively new approach made possible by the increased availability of accurate CO 2 sensors (Vachon et al. 2020). However, at the time of writing this article, the models used to estimate metabolism from DIC are still in development despite their prior use (Wright and Mills 1967). A degree of caution is necessary to interpret reach-level estimates of PQ given that measurements will likely not reflect PQ Bio . Future methods and models should be developed with the specific goal of parsing PQ Bio from PQ Meas at all levels of measurement. Isotopic approaches at either scale (Ferr on et al. 2016, e.g., with labeled 18 O water or 13 C) may be useful in parsing PQ Bio from PQ Meas or specific process (e.g., photorespiration); however, to our knowledge, specific methods and approaches do not yet exist.

Conclusions
An accurate PQ Bio is needed for understanding C metabolism derived from O 2 data. The current evidence suggests that the PQ Bio can vary with environmental conditions. Thus, the use of fixed canonical PQ Bio values is likely causing inaccurate estimates of C-based metabolism from O 2 data. Although the current theory provides a basic understanding of the processes that may control the PQ Bio , much research will be necessary to use environmental conditions to predict PQ Bio values more accurately and precisely. Furthermore, consideration needs to be given to observation error and measurement bias that will be common when measuring the PQ in the natural environment. Until we improve PQ theory and our ability to reconcile error and bias from PQ Meas , we suggest we should employ more scrutiny when applying PQ values to convert O 2 -based metabolism to C. Specifically, researchers should move beyond fixed values of the PQ. Instead, we suggest matching our current knowledge of how the PQ varies with the environmental conditions at any given measurement site to generate a range of plausible PQ values (e.g., Fig. 5) from the literature. Such an approach will better account for the uncertainty of the PQ when estimating C-based metabolism from O 2 data.