The biochemical control over isoprene emission rate

542

III.

General forms of the models used to predict the leaf isoprene emission rate

543

IV.

Modeling the short-term responses to photon flux density

545

V.

Resolving problems with the current Guenther algorithm covering the PPFD-dependence of E_{i}

546

VI.

The temperature dependence of isoprene emission rate

547

VII.

Clarifying issues with the current Guenther algorithm covering the temperature-dependence of E_{i}

549

VIII.

The CO_{2} dependence of the isoprene emission rate

549

IX.

Modeling the relation between isoprene emission and leaf conductance

551

X.

Modeling the longer-term processes that control isoprene emission rate

552

XI.

Conclusions

556

References

556

Summary

The leaves of many plants emit isoprene (2-methyl-1,3-butadiene) to the atmosphere, a process which has important ramifications for global and regional atmospheric chemistry. Quantitation of leaf isoprene emission and its response to environmental variation are described by empirically derived equations that replicate observed patterns, but have been linked only in some cases to known biochemical and physiological processes. Furthermore, models have been proposed from several independent laboratories, providing multiple approaches for prediction of emissions, but with little detail provided as to how contrasting models are related. In this review we provide an analysis as to how the most commonly used models have been validated, or not, on the basis of known biochemical and physiological processes. We also discuss the multiple approaches that have been used for modeling isoprene emission rate with an emphasis on identifying commonalities and contrasts among models, we correct some mathematical errors that have been propagated through the models, and we note previously unrecognized covariances within processes of the models. We come to the conclusion that the state of isoprene emission modeling remains highly empirical. Where possible, we identify gaps in our knowledge that have prevented us from achieving a greater mechanistic foundation for the models, and we discuss the insight and data that must be gained to fill those gaps.

Isoprene (2-methyl-1,3-butadiene) is a highly volatile and reactive hydrocarbon released in large amounts from the leaves of many plants (Harley et al., 1999; Kesselmeier & Staudt, 1999; Loreto & Schnitzler, 2010). Global isoprene emissions are estimated to contribute c. 550 Tg C yr^{−1} to the atmosphere (Arneth et al., 2008), which is of the same magnitude as global methane emissions. Isoprene is highly reactive with atmospheric oxidants and because of this reactivity it contributes to the photochemical production of other atmospheric constituents, including tropospheric ozone, organic nitrates, organic acids, formaldehyde, carbon monoxide and, finally, carbon dioxide (Trainer et al., 1987; Fehsenfeld et al., 1992; Fuentes et al., 2000; Monson & Holland, 2001; Monson, 2002). At the global scale, isoprene emissions have the potential to influence the lifetime of radiatively active, ‘greenhouse’ gases, such as methane (Lelieveld et al., 1998; Poisson et al., 2000). Finally, evidence has been provided that the atmospheric oxidation of isoprene has the potential to affect (both positively and negatively) the formation of secondary organic aerosol particles, which potentially function as cloud condensation nuclei (Pöschl et al., 2010) and which influence the radiation budget of the Earth’s surface (Claeys et al., 2004; Henze & Seinfeld, 2006; Paulot et al., 2009; Kiendler-Scharr et al., 2009, 2012; Kanawade et al., 2011; Forkel et al., 2012). Clearly, our ability to understand and quantify leaf isoprene emissions is important for the prediction of atmospheric chemistry and Earth’s climate.

Given its importance to chemistry and climate there is continuing interest in developing and improving isoprene emission models (Guenther et al., 2006; Arneth et al., 2007; Grote & Niinemets, 2008; Niinemets et al., 2010a). In order to be useful for predicting the rapid photochemical transformations that occur in the regional and global atmosphere, these models need to be of high spatial (1–50 km^{2}) and temporal (c. 1 h) resolution (Logan, 1989; Fiore et al., 2003; Loughner et al., 2007). Isoprene emission from entire forests or landscapes are typically modeled from the leaf scale upwards using our understanding of the dominant environmental factors affecting leaf processes (Monson et al., 1995, Monson et al. 2007; Niinemets et al., 2010a,b). The principal environmental factors influencing isoprene emission rate are: temperature, photosynthetic photon flux density and intercellular CO_{2} concentration, in the short term; and recent weather, soil water availability, atmospheric CO_{2} concentration, position in the canopy and developmental stage of the leaf, in the long term (Harley et al., 1999; Sharkey & Yeh, 2001; Niinemets et al., 2010a,b). These factors influence enzyme activity, substrate availability and gene expression depending on which timescale is considered.

Most of the models that have been produced to date focus on the shape of observed responses of emission rate to environmental variation. A few have been derived from knowledge of biochemical processes, especially with regard to short-term changes in environment, but these derivations have been fewer in number than those based on purely empirical description. This bias toward empirical convenience as a basis for modeling is even more evident in derivations of emission responses to longer-term influences, such as those associated with recent weather, precipitation regimes, canopy environment, atmospheric CO_{2} concentration and leaf ontogeny. Interactions among the shorter- and longer-term influences are present in some models, but once again, these have been entirely derived from observed patterns, not process knowledge.

In this review, we provide a synthesis of our progress to date on the development of leaf isoprene emission models and we discuss those areas where future work is needed. Our review differs from some in the recent past that have considered the topic of emission models (e.g. Grote & Niinemets, 2008; Niinemets et al., 2010a,b) in that we focus on the biochemical underpinnings of the models; the past efforts have focused on the adequacy for such models in predicting observed responses. The principal question we asked ourselves in preparing this review was: where do we stand with regard to understanding how alternate models relate to one another and how do those models relate to our knowledge of the biochemical processes that underlie isoprene biosynthesis and emissions?

II. The biochemical control over isoprene emission rate

Observations of whole-leaf isoprene emission rate have shown a dependence on the absorbed photosynthetic photon flux density, leaf temperature, and atmospheric CO_{2} and O_{2} concentrations (Sharkey & Yeh, 2001); the same environmental variables that dominate control over leaf photosynthesis rates. It has been known for several decades that recently assimilated CO_{2} from photosynthesis is channeled into isoprene biosynthesis (Sanadze et al., 1972; Mgalobilishvili et al., 1978). Discovery of the 2-C-methyl-D-erythritol 4-phosphate (MEP) pathway in the chloroplasts of leaves (Lichtenthaler et al., 1997; Schwender et al., 1997) provided formal understanding of the biochemical connection between photosynthesis and isoprene biosynthesis. Carbon substrates used to construct isoprene originate from recently produced glyceraldehyde 3-P (GAP) and pyruvate (Pyr). GAP is derived through the reductive pentose phosphate (RPP) pathway in the chloroplast, but the origin of chloroplast Pyr is still uncertain. It has been suggested that plastids lack some of the key enzymes of glycolysis such as enolase and phosphoglycerate mutase (Givan, 1999), and are therefore unable to synthesize Pyr. This has led to a hypothesized ‘loop’ whereby carbon is exported as GAP from the chloroplast, rearranged to phosphoenolpyruvate (PEP) in the cytosol, and then imported back into the chloroplast where it is converted to Pyr by pyruvate kinase (Flügge & Gao, 2005; Flügge, 2012). The appearance of recently assimilated ^{13}C in isoprene can be traced to both GAP and Pyr (Karl et al., 2002; Trowbridge et al., 2012), meaning that if cytosolic PEP is transported into the chloroplast, and converted to Pyr, it must occur concurrently with GAP export. Recently, phosphoglycerate mutase and enolase have been detected in plastids (Andriotis et al., 2010; Joyard et al., 2010; Bayer et al., 2011), and it was suggested that an obligatory cytosolic step is not necessary. However, the plastidic concentrations of these enzymes are low, and so their role in generating chloroplast Pyr is yet to be resolved. Older carbon, that is not traceable to recently assimilated ^{13}C, also appears in isoprene, even after several hours of labeling, and this most likely originates from carbon reserves (Funk et al., 2004; Schnitzler et al., 2004; Brilli et al., 2011). This latter carbon, however, is likely to only account for 15–20% of that used for isoprene biosynthesis in most conditions (Kreuzwieser et al., 2002; Brilli et al., 2007).

The MEP pathway is not only dependent on photosynthesis for carbon substrates, but also for NADPH reductant, CTP and ATP, which are required as co-factors for five of the enzymes in the MEP pathway (Lichtenthaler et al., 1997; Phillips et al., 2008). Recently, it has been found that two of the Fe-S-containing enzymes in the MEP pathway can directly accept electrons from ferredoxin, the final electron acceptor in the chloroplast electron transport chain, in the light (Seemann et al., 2006; Seemann & Rohmer, 2007). This direct access to reductant ‘power’ would reduce the dependence of isoprene biosynthesis on NADPH. Work remains to be done to sharpen our understanding of the energetic and reductant requirements for isoprene biosynthesis.

The products of the MEP pathway are IDP and DMADP, which come to equilibrium through enzymatically catalyzed isomerization (Brüggemann & Schnitzler, 2002a; 2002c). DMADP is the immediate precursor to the formation of isoprene, which occurs through catalytic elimination of pyrophosphate by the enzyme isoprene synthase (Silver & Fall, 1991, 1995; Wildermuth & Fall, 1996; Köksal et al., 2010). This reaction will proceed uncatalyzed in the absence of isoprene synthase, but at rates too slow to account for leaf emission rates. The activity of isoprene synthase is regulated in the short term by substrate affinity and turnover in the active site, and by the influence of temperature on catalytic processes (Silver & Fall, 1995; Monson et al., 1992; Lehning et al., 1999). In much of the early research on isoprene emissions it was assumed that control over the isoprene emission rate was almost completely due to the activity of isoprene synthase (see Sharkey & Yeh, 2001). However, more recently a view has emerged in which control by the availability of substrate may be just as important, if not more important, than catalytic affinity and turnover, depending on conditions (Brüggemann & Schnitzler, 2002b; Rosenstiel et al., 2003; Loreto et al., 2006; Wiberley et al., 2008; Rasulov et al., 2009, 2010).

III. General forms of the models used to predict the leaf isoprene emission rate

Working with isoprene emissions from live oak (Quercus virginiana) leaves, Tingey et al. (1979) modeled the responses to both incident photosynthetic photon flux density (PPFD), and leaf temperature (T_{L }) using a general logistic function:

(Eqn 1)

(y, instantaneous isoprene emission rate (E_{i}) expressed as a dependent variable; x, value of PPFD or T_{L} expressed as an independent variable; a, a ‘tunable coefficient’ representing the difference between the minimum and the maximum values of y as it responds to either PPFD or T_{L}; b, a ‘shape parameter’ that determines the slope of the response; c, a ‘location parameter’ that determines the point along the x-axis where the curve is centered on the response; d, the minimum value predicted by the function). The values for a, b, c and d will be different when Eqn 1 is applied to PPFD or T_{L}. Note that in an effort toward clarity in our use of symbols, we have used nonitalicized symbols to describe environmental flux densities or state variables (such as PPFD or leaf temperature) and we have used italicized symbols to describe coefficients, constants or calculated variables. Eqn 1 defines the shape of a general nonlinear response that approaches an asymptote as PPFD or T_{L} increase. The parameter values for Eqn 1 are determined using nonlinear least-squares regression. The only sense of true biochemical mechanism that can be extracted from Eqn 1 is the observation that many physiological processes respond to environmental factors in a nonlinear fashion, with responses often taking the form of a rectangular hyperbola. It is important that we emphasize the focus on ‘form of the response’ at this point, because during the history of model development since 1979, it was the form of these responses that led researchers to alternative forms of the models that have been proposed, not enhanced understanding of biochemical processes and mechanisms. It is also important to point out one additional limitation of the Tingey et al. (1979) model. As applied to the responses to PPFD and T_{L}, two variables that simultaneously affect E_{i}, the modeled responses were disjunct in the sense that they had to be applied separately for each variable.

Working with isoprene emissions from eucalyptus leaves, Guenther et al. (1991) produced a nondisjunct, connected model, in which the responses to PPFD, T_{L}, relative humidity (RH) and atmospheric CO_{2} concentration (C_{ac}) are used together, as fractional scalars, to adjust an emission factor (B_{i}) to instantaneous changes in the environment. The value for B_{i} was defined for a standard set of conditions (PPFD at 1000 μmol m^{−2} s^{−1}, T_{L} at 30°C, RH at 40% and C_{ac} at 330 ppmv):

(Eqn 2)

where L, T, H and C are calculated variables (or coefficients) determined by functions linked to PPFD, T_{L}, RH and C_{ac}, respectively (see Table 1 for a list of abbreviations and symbols for all equations). In Guenther et al. (1991), the RH and C_{ac} were shown to be small when considered across the range of conditions normally encountered by an isoprene-emitting leaf and so in later studies the value of B_{i} was referenced only to standard values for PPFD (1000 μmol m^{−2} s^{−1}) and T_{L} (30°C). A principal breakthrough that can be attributed to the Guenther et al. (1991) model is that it partitioned the environmental control over E_{i} into processes that determine longer-term (hours-to-months) dynamics (which were included in B_{i}) and shorter-term (seconds-to-hours) dynamics (which were included in L, T, H and C ). The value for E_{i} is often referred to as the instantaneous emission rate. The value for B_{i} is often referred to as the basal emission rate. The basal emission rate (B_{i}) is assumed to be under control of longer-term processes that influence gene expression and metabolic acclimation. The controls expressed in L, T, H and C are assumed to represent shorter-term influences on enzyme activity and substrate availability.

Coefficient defining the isoprene synthase formation term (h^{−1})

μ

Coefficient defining the enzyme decay term (h^{−1})

ρ

Relative annual amplitude of E_{imax} (unitless)

τ

Kurtosis of the seasonal amplitude in B_{i} (unitless)

θ

Soil water content (m^{3} m^{−3})

θ_{w}

Soil water content at the leaf wilting point (m^{3} m^{−3})

θ_{1}

Soil moisture scaling factor (m^{3} m^{−3})

Γ*

Photo-compensation point (μmol CO_{2} mol^{−1} dry air)

Θ

Tunable ‘curvature factor’ in electron transport response to PPFD (unitless)

Following publication of the Guenther et al. (1991) study most research focused on explaining and validating the shorter-term processes associated with L and T (Monson et al., 1995). However, as the leaf-scale model was deployed in larger-scale models (e.g. those used to predict global isoprene emissions and their responses to future climate change) the separation into shorter- and longer-term processes was blurred, or ignored. For example, the response to multi-decadal future climate warming was modeled using the Guenther et al. (1991) equation, which was derived for instantaneous changes in T_{L} (Naik et al., 2004; Tao & Jain, 2005). These practices were criticized by Monson et al. (2007) in a call for the modeling community to re-evaluate controls across the most relevant temporal scales, and it has led to the development of community-written papers intended to inform modeling efforts more broadly about the biological controls over leaf isoprene emission (Niinemets et al., 2010a,b, 2011).

IV. Modeling the short-term responses to photon flux density

At the time that the original Guenther et al. (1991) model was developed, evidence had already been provided through the numerous studies of Sanadze (1964), Tingey et al. (1981), Monson & Fall (1989) and Loreto & Sharkey (1990) that a functional linkage exists between photosynthetic CO_{2} assimilation rate (A) and E_{i}. It was assumed that this linkage is carried into the dependencies of A and E_{i} on PPFD, as both processes exhibited similar shapes in their PPFD-response curves. Guenther and co-workers used this inferred linkage as the basis on which to develop an algorithm for the response of E_{i} to PPFD. The original form of the light-dependent component of the Guenther et al. (1991) algorithm is similar to that developed for the photosynthetic electron transport rate (J, μmol m^{−2} s^{−1}), beginning from:

(Eqn 3)

(a, the fraction of incident PPFD absorbed by the leaf; f, the fraction of the incident PPFD diverted to processes other than J ). In this relation it is assumed that two photons of photosynthetically active radiation (PAR) must be used to energize the movement of an electron from H_{2}O to NADP^{+} in the photosynthetic electron transport system. Implicit in Eqn 3 is that none of the leaf cells is saturated by the PPFD that is incident on the upper surface of the leaf (and therefore in zero-order dependence). As the electron transport rate becomes saturated by PPFD in some cells, the dependence of J on PPFD will exhibit progressive influence of an asymptote. Recognizing that J is influenced by an upper limit (J_{max}, μmol m^{−2} s^{−1}), and recognizing that the influence of J_{max} on J increases as PPFD increases, the following quadratic equation can be developed which describes a rectangular hyperbola in which a continuous transition occurs from J =0 at PPFD = 0 to J = J_{max} at saturating PPFD:

(Eqn 4)

(Θ is a tunable ‘curvature factor’ that varies from 0 to 1). Taking the root of Eqn 4 leads to:

The parameters c_{P1} and c_{P2} are tunable coefficients that, in composite, account for: (1) the fact that the molar stoichiometry of electron transport required to synthesize isoprene is different than that required to assimilate CO_{2}, (2) the requirement for a curvature coefficient (Θ), and (3) the requirement for an upper limit defined by J_{max}. Eqn 6 is the first iteration of the ‘Guenther light algorithm’, and it clearly has origins in the fundamental assumption that the dependence of E_{i} on PPFD is due to variation in J.

(α, the initial slope of the response (often called the ‘quantum yield’)). Eqn 8 defines the shape of a rectangular hyperbola that approaches an asymptote at relatively high values for PPFD. Guenther et al. (1993) adopted a modified form of Eqn 8 for the PPFD-dependence of E_{i}:

(Eqn 9)

Here α is analogous to the quantum yield of Eqn 8, and thus should be expressed as mol isoprene mol^{−1} photons incident on the leaf. In reality there are some mathematical errors in making this analogy, which are discussed below. Furthermore, there is a mathematical violation in the denominator of Eqn 9 in that the square root quantity contains a sum that mixes a unitless constant (1.0) with the product of two terms (α and PPFD) both of which are defined with units. Once again, we will confront this violation during further discussion below. If, however, we take Eqn 9 as a valid empirical means of representing the PPFD-dependence of E_{i} (which has been the case for approximately two decades since 1993), and thus we assume that α should carry units mol isoprene mol^{−1} photons, then c_{P3} must be unitless (presumably representing the ratio of B_{i} : B_{imax}, where B_{imax} is the maximum PPFD-saturated emission factor). The value for c_{P3} was set at 1.066 in Guenther et al. (1993). The overall (unintended) effect of shifting the definition of L from Eqn 6 in Guenther et al. (1991) to Eqn 9 in Guenther et al. (1993) was to obscure mechanistic connections to J and even further emphasize the general shape of the dependence of E_{i} on PPFD.

The coefficients for α and c_{P3} in Eqn 9 were assumed to be constant in the Guenther et al. (1993) analysis. Following this analysis, observations revealed that the PPFD response of E_{i} varied depending on leaf position in the canopy. In a later modeling analysis, Guenther et al. (1999) tuned L to canopy position by rendering α and c_{P3} dependent on the cumulative leaf area index (LAI) above the leaf under consideration:

(Eqn 10)

(Eqn 11)

Niinemets et al. (1999) and Martin et al. (2000) moved the model for the dependence of isoprene emission rate on PPFD back toward an explicit connection with J. This treatment began with an expression of the dependence of net CO_{2} assimilation (A, μmol m^{−2} s^{−1}) on J (under conditions of ribulose-1,5-bisphosphate (RuBP) regeneration limitation to A) expressed as:

(Eqn 12)

(C_{ic}, CO_{2} mole fraction in the intercellular air spaces of the leaf; Γ*, photo-compensation point (the CO_{2} compensation point in the presence of only gross photosynthetic CO_{2} uptake and photorespiration, μmol mol^{−1}); R_{d}, the mitochondrial (or ‘dark’) respiration rate (μmol m^{−2} s^{−1})). Using this relation, Niinemets et al. (1999) modeled the relation between E_{i} and J as:

(Eqn 13)

(ε, the fraction of J required to synthesize isoprene). The numbers in the denominator reflect: first, the 6 carbons that are required as substrate in the MEP pathway of the chloroplast, and second, the different stoichiometries for the use of electron transport for isoprene biosynthesis vs CO_{2} assimilation with regard to C_{ic} and Γ^{*}. The dependence of J on PPFD was modeled using Eqn 8 and the resultant value of J was inserted into Eqn 13 to provide E_{i}. In Martin et al. (2000), available ATP was used as the basis for predicting the isoprene emission rate, with availability determined by the balance between production, through J, and consumption, through A. The logic presented in the Martin et al. (2000) model provided a tight connection between E_{i} and the quantum yield for CO_{2} uptake, which in turn reflects interactions among J, A and photorespiration. More experimental work is needed to clarify the factors determining the quantum yield for E_{i}, and whether it is indeed determined by ATP availability, the same factor that determines the quantum yield for net CO_{2} uptake.

Zimmer et al. (2000) modeled E_{i} on the basis of changes in the metabolite pools of the photosynthetic carbon reduction cycle. Their numerical model named ‘Biochemical Isoprene emission Model’ (BIM) is based on reaction rates derived from Michaelis–Menten kinetics. Dynamics in the concentration of Pyr and GAP were linked to dynamics in photosynthesis and then used to determine the chloroplast concentration of DMADP. Ultimately, the response of E_{i} to PPFD was based on the use of J to determine d[Pyr + GAP]/dt. The dependence of Pyr and GAP production rates on PPFD is modeled with a light fleck photosynthesis model (Kirschbaum et al., 1998) that uses the same dependence of J on PPFD reflected in Eqn 5:

(Eqn 14)

(f, ‘function of’; V_{maxIs}, maximum reaction rate catalyzed by isoprene synthase in nmol m^{−2} (leaf area) s^{−1}; K_{mIs}, Michaelis–-Menten constant for isoprene synthase catalysis expressed in molar units). Thus, while the theoretical foundation for modeling the PPFD dependence of E_{i} was now grounded in photosynthetic carbon metabolism, it was ultimately driven by the same dependence of J on PPFD that was reflected in the Guenther et al. (1991) and Niinemets et al. (1999) models. The Zimmer et al. (2000) model has been carried through to future modeling efforts by the same research group (e.g. Grote et al., 2006), though some modifications have been made such as the use of additional photosynthesis models (taken from Farquhar et al., 1980) that infer direct dependence of MEP production on J.

The evolution of the dominant algorithms developed to describe the dependence of isoprene emission rate on PPFD is shown in Fig. 1. One of the principal points to take away from this analysis is that there exists a common ‘quasi-mechanistic’ basis for our current modeling of the PPFD dependence of E_{i}. We have used the term ‘quasi-mechanistic’ because we are not absolutely sure that the PPFD dependence is due to a connection to J; definitive observations establishing this connection have not been made. However, it is clear that daytime production of NADPH (as well as electrons taken directly from ferredoxin) and ATP in the chloroplast is driven by J. It is also clear that NADPH and ATP are required to link isoprene biosynthesis to the assimilation of CO_{2} and to drive the conversion of GAP and Pyr to isoprene in the MEP pathway. These facts provide a firm basis for inferring the PPFD dependence of E_{i} as being due to the PPFD dependence of J.

V. Resolving problems with the current Guenther algorithm covering the PPFD-dependence of E_{i}

Eqn 9 represents the most often used equation for describing the PPFD-dependence of E_{i}. As stated above, it was derived as an analog to Eqn 8, which was originally derived from the so-called ‘Smith Equation’ that was used to describe photosynthetic responses to increasing light intensity. The original Smith Equation (Smith, 1938), using our notation, can be stated as:

(Eqn 15)

(A_{max}, maximum CO_{2} assimilation rate (observed at saturating PPFD); c_{P4}, a coefficient with units m^{2} s μmol^{−1}). Tenhunen et al. (1976a) relied on the Smith Equation to derive Eqn 8 shown above, which was then used to describe the PPFD-dependence of the photosynthetic electron transport rate. In using the Smith Equation and the subsequent derivation of Eqn 8 to derive the Guenther et al. (1993) algorithm for the PPFD-dependence of E_{i} (i.e. Eqn 9), an error was apparently incorporated. Eqn 9 is not valid because the denominator contains a squared term that sums a unitless constant (1.0) with the product of two terms (α and I) defined by units. Proper derivation of Eqn 9 should resolve to:

(Eqn 16)

where α continues to be defined with units mol isoprene mol^{−1} photons absorbed, c_{P3} is now defined with units m^{2} s μmol^{−1}, PPFD continues to be defined with units μmol m^{−2} s^{−1}; an additional coefficient, c_{P5}, is introduced with units μmol m^{−2} s^{−1}. With these adjustments, Eqn 10 will still resolve to the unitless scalar, L, but in this case with proper mathematical relations among all variables and parameters. Eqn 16 will yield the same approximate dependence between L and PPFD as that represented in the original Guenther et al. (1993) algorithm shown in Eqn 9 if the value of α is left at 0.0027, the value of c_{P3} is left as 1.066 (both as defined in the original Guenther et al., 1993 derivation), and if c_{P5} is set to 1.0. Thus, the correction we have provided is more of a ‘housekeeping’ correction intended to clean up the form of the theory, but it will have no significant effect on past predictions generated by the Guenther et al. (1993) algorithm.

More importantly, there exists a ‘hidden’ covariance in the Guenther et al. (1993) model, which has not been previously recognized. Expression of L as a normalized value produces an obscure, but important dependence of α on B_{i}. Using Eqn 16 to define L, which is then used to scale B_{i} and thus calculate E_{i} as a function of PPFD, results in:

(Eqn 17)

Now, taking the first derivative of Eqn 17 yields:

(Eqn 18)

which reveals that as PPFD → 0, the quantum yield, taken as the slope of the E_{i} vs PPFD dependence, is resolved as (α c_{P3}B_{i}). This result does not lead to a clean resolution of α as the quantum yield of E_{i}. If B_{i} is approximated as constant, such as the case for most analyses of the response of E_{i} to PPFD, which occur over the timespan of tens of minutes, then α can also be approximated as constant, and no significant complications from the covariance will emerge. The dependence of α on B_{i}, however, will become problematic when the E_{i} vs PPFD dependence is used to derive α, especially in comparative studies when leaves or species are compared with different values of B_{i}. In those cases, the derived value of α will not solely reflect inherent biochemical and photochemical constraints on dE_{i} /d PPFD, but rather will include an influential component due to dE_{i} /dB_{i}. Thus, caution should be used in interpreting the ‘apparent’ quantum yield for the PPFD-dependence of E_{i}.

VI. The temperature dependence of isoprene emission rate

E_{i} is highly dependent on temperature, increasing in exponential fashion as temperature increases up to a maximum, beyond which it decreases precipitously (e.g. Monson & Fall, 1989; Loreto & Sharkey, 1990; Monson et al., 1992). Guenther et al. (1991) developed an algorithm that resembles the ‘Arrhenius relation’ that is often used to define temperature dependencies in reaction kinetics studies. An appropriate starting point for discussion of the temperature-dependence of E_{i}, therefore, is consideration of the Arrhenius relation. The Arrhenius relation is derived from the Maxwell–Boltzmann statistical distribution of kinetic energies expected in a reaction system at any given temperature. In general form, the model relates the reaction rate coefficient (k) to temperature in a manner that is dependent on the reaction’s activation energy (E_{a}) as:

(Eqn 19)

(k has units s^{−1}; M, a reaction-specific constant that accounts for components of the reaction that do not respond to temperature in exponential fashion (also called the ‘frequency factor’) with units s^{−1}; T, absolute temperature of the reaction system; R, the universal gas constant with units J K^{−1} mol^{−1}).

It is important to note that the Arrhenius relation is derived from observations, not theory, though it was based on expected shifts in the Boltzmann–Maxwell distribution as T changes (Davidson & Janssens, 2006). Following publication of the Arrhenius relation in the late 1800s work was conducted to reconcile the mathematical relation of M and E_{a} with fundamental physics, particularly with knowledge of quantum mechanics and kinetic theory. In the 1930s, Henry Eyring from Princeton University derived these terms within the context of the transition-state complex, rather than the energies of reactant molecules. Eyring proposed a relation that differed from that proposed by Arrhenius (Eyring, 1935):

(Eqn 20)

(κ_{E}, the fractional transmission coefficient which accounts for some transition-state complexes that do not go on to form product; k_{B}, the Boltzmann constant in m^{2} kg s^{−2} K^{−1}; h, Planck’s constant in m^{2} kg s^{−1}; , indicates that the free energy difference (ΔG) is determined between reactants and an intermediate state (the transition-state complex), rather than between reactants and products). The Eyring equation is similar in form to the Arrhenius equation in that the exponential response of k to temperature is present in both. However, the Eyring equation is founded on transition-state mechanics and is therefore considered more precise in terms of the underlying physics. Working from the Eyring equation, Johnson et al. (1942) developed a more general equation to describe the temperature dependence of enzyme-catalyzed reactions:

(Eqn 21)

(H_{a}, enthalpy of activation in J mol^{−1}; H_{d}, enthalpy of de-activation in J mol^{−1}; S, entropy in J K^{−1} mol^{−1}; c, a scaling constant). This form of the equation was first used to model the temperature dependence of partial processes of photosynthesis (Tenhunen et al., 1976a,b) and an analog form of this relation was used by Guenther et al. (1993) for the definition of T, the temperature-dependent scaling coefficient used to modify B_{i} and thus estimate E_{i}, according to Eqn 2:

(Eqn 22)

(c_{T1} (J mol^{−1}), c_{T2} (J mol^{−1}) and T_{M} (K) are ‘tunable’ coefficients; T_{L}, leaf temperature; T_{S}, a standard temperature (typically taken as 303 K)).

In Guenther et al. (1999), the form of Eqn 22 was modified slightly to reduce the number of tunable coefficients, and to reference the temperature scaling function to the temperature optimum of E_{i}, rather than to T_{S}:

(Eqn 23)

where

(Eqn 24)

E_{opt} (nmol m^{−2} s^{−1}) was estimated to be B_{i} multiplied by 1.9 for most plants (Guenther et al., 1999), c_{T3} (J mol^{−1}) and c_{T4} (J mol^{−1}) are tunable coefficients and T_{opt} is the temperature optimum in K for E_{i}. It can be shown through algebraic manipulation that Eqn 23 is equivalent to Eqn 22, with some differences absorbed into the coefficients of Eqn 23; thus changing the point of reference to T_{opt}, rather than T_{S}.

In the model of Niinemets et al. (1999) and Martin et al. (2000), Eqn 21 was used as the basis by which to predict the temperature response of isoprene synthase, which in turn was assumed to reflect the temperature dependence of E_{i}. In this case, the absolute catalyzed rate of isoprene emission was desired, not a rate scaled to B_{i}; so Eqn 21 was more appropriate than Eqn 22. The parameters that determine the temperature dependence expressed in Eqn 21 were derived from observations of the temperature dependence of isoprene synthase activity extracted from crude leaf extracts of poplar (Monson et al., 1992) and oak (Lehning et al., 1999). In the model of Grote et al. (2006), Eqn 21 was used to determine the temperature dependencies of those enzymes in the MEP pathway leading to DMADP synthesis, as well as the temperature dependence of isoprene synthase. In the case of Grote et al. (2006), Eqn 21 was parameterized for isoprene synthase as described in Niinemets et al. (1999) (traceable back to the crude extract studies of Monson et al., 1992 and Silver & Fall, 1995). Furthermore, they used an inverse parameter estimation approach to tune the temperature-dependent parameters from MEP pathway enzymes.

Singsaas & Sharkey (1998, 2000) applied a form of the Arrhenius model to the prediction of isoprene emission from oak leaves exposed to rapid fluctuations in temperature (i.e. non-steady-state). In that case, high-temperature deactivation of isoprene emission was not observed, so that only E_{a} need be considered:

(Eqn 25)

At this point it is important to note that a common thread of logic extends through the history of modeling the temperature dependence of E_{i}; that is, the dependency is determined by a response that can be modeled through the Arrhenius or Eyring approaches, both of which reflect the fundamental energetics and thermodynamics of enzyme–substrate interactions. It has been assumed that the enzyme controlling the temperature dependence of E_{i} is ultimately isoprene synthase, based on the observed similarities in the temperature dependencies of this enzyme and whole-leaf isoprene emissions (e.g. Monson et al., 1992).

Rasulov et al. (2010) validated the modeling framework described to this point by showing that at temperatures up to 30°C, the temperature response of isoprene emission rate was not limited by DMADP substrate availability, but rather by isoprene synthase activity. At temperatures > 30°C, however, isoprene emission rate was influenced by both substrate availability and isoprene synthase activity. This mixed control affects the deactivation term of the models (ΔH_{d}), which likely includes both direct influences on catalytic efficiency and progressive limitation by substrate as temperature is increased above 30°C. In studies by Magel et al. (2006) and Li et al. (2011), DMADP was also observed to limit the temperature dependence of isoprene emission rate, but only at temperatures above 35°C and only after considerable time (c. 1 h). In practical applications, this means that the numerical coefficients used in the existing algorithms may vary depending on temperature range and temperature history. Such temperature dependency in the parameter coefficients themselves is currently not reflected in the models. This is an issue in need of resolution.

VII. Clarifying issues with the current Guenther algorithm covering the temperature dependence of E_{i}

Eqn 22 represents the most often used equation for describing the temperature dependence of E_{i}. Upon close inspection, Eqn 22 differs from the original Johnson et al. (1942) form of the Eyring equation (Eqn 21) in that Eqn 22 contains some unique combinations of terms, such as the product of two temperatures in the denominator of the ‘Guenther algorithm’ (essentially a T^{2} term). Eqn 22 can be reconciled with Eqn 21 if we assume that the relevant T in Eqn 21 is the leaf temperature (T_{L}) and that:

(Eqn 26)

(Eqn 27)

(Eqn 28)

(Eqn 29)

These assumptions are consistent across units for all terms, as both c_{T1} and c_{T2} were originally defined in J mol^{−1} (Guenther et al., 1991), the same units as those for enthalpy, and S is defined in J K^{−1} mol^{−1}, which reconcile to J mol^{−1} in the term ΔS T_{S}. The definition of c, the scaling constant in Eqn 21, in terms of the change in enthalpy (J mol^{−1}) referenced to the ideal gas constant (R) at a standardized temperature (T_{S}) is consistent conceptually with the unitless scaling constant used in Guenther et al. (1993) and intended to establish the thermodynamic state of the system at a standardized temperature (recognizing that RT = PV/n, where P and V define state parameters of the system and n is molar equivalents). The definition of ΔH_{d} as equivalent to ΔS T_{M} is consistent between both models (Johnson et al. and Guenther et al.) with regard to units and within the thermodynamic context of deactivation as a loss of internal energy in the catalytic system through increases in entropy and decreases in enthalpy. This exercise in establishing mathematical and conceptual analogy is important because it clarifies that the Guenther et al. (1993) temperature model is indeed grounded in the mechanistic and thermodynamic theory of the Eyring equation. The advantage of using Eqn 22 as an equivalent to Eqn 21 is that it provides a tractable means of referencing the change in E_{i} to B_{i}, which is determined at T_{S}, a standardized temperature. Thus, the form of Eqn 22 allows the modeling to fit more conveniently into the framework described by Eqn 2.

One might ask, why the derivation of Eqn 22 differed from Eqn 21 in the original treatment of Guenther et al. (1993). The answer lies in the fact that Eqn 22 was derived as the best fit of the Johnson et al. model to observed responses of E_{i} to T_{L}, and in achieving that best fit, some of the mathematical relations in the original model were modified. Thus, Eqn 22 is a derived form of Eqn 21, after achieving a form that produced minimal error between the model and observations.

VIII. The CO_{2} dependence of the isoprene emission rate

Dependence of E_{i} on changes in the atmospheric CO_{2} concentration has been known since the observations reported in Sanadze (1964). In that seminal study, it was shown that E_{i} for poplar leaves decreased as the atmospheric CO_{2} concentration (C_{ac}) increased. Sanadze’s observations were not, at first examination, consistent with evidence that accumulated shortly thereafter (from other experiments by Sanadze and colleagues), showing that the biosynthesis of isoprene was biochemically coupled to photosynthetic CO_{2} assimilation (Sanadze, 1966; Sanadze & Kursanov, 1966; Sanadze & Dzhaiani, 1972). Since those early observations, Sanadze (2004) has developed a biochemical hypothesis to explain his results that depends on the competitive partitioning of chloroplast reductant and ATP between the reductive pentose phosphate pathway and the MEP pathway, which in turn depends on the intercellular CO_{2} concentration (C_{ic}) and the activity of Rubisco. Thus, at low C_{ic} when the demand for reductant and ATP by the reductive pentose phosphate pathway is also low, these compounds will be diverted toward E_{i}; conversely, when C_{ic} is high the reductant and ATP that would otherwise go to E_{i} will be diverted back toward photosynthesis.

The logic proposed by Sanadze in 2004 was foreshadowed in the model of E_{i} developed by Niinemets et al. (1999). Recall from Eqn 13 that the Niinemets et al. (1999) model is based on photosynthetic CO_{2} assimilation with isoprene biosynthesis rate defined by the fraction of J that is partitioned to the MEP pathway. Niinemets et al. (1999) used this connection to explain that as C_{ic} and A decrease, a greater fraction of reductant and ATP in the chloroplast will be channeled to E_{i}; in other words, ε increases as C_{ic} decreases and vice versa as C_{ic} increases. As originally postulated, however, the Niinemets et al. (1999) model did not provide a fundamental relation to define ε as a function of C_{ic}. A more direct connection between ε and C_{ic} was developed as an empirical relation in subsequent work (Arneth et al., 2007).

In the model produced by Martin et al. (2000), the CO_{2} response of E_{i} is represented similarly to that in the Niinemets et al. (1999) model; it is driven by competitive partitioning of ATP between photosynthesis and E_{i}. In this model, as C_{ic} increases, negative feedback is imposed on E_{i} due to the limited turnover of sugar-phosphates and associated limitations by inorganic phosphate (P_{i}) on ATP production. Monson & Fall (1989) showed that E_{i} is sensitive to P_{i}-linked feedback, as evidenced by loss of O_{2} sensitivity of photosynthesis and concomitant reductions in E_{i} in some aspen leaves. Loreto & Sharkey (1993) showed that both isoprene emission rate and leaf ATP concentrations decrease at elevated C_{ic}, and hypothesized that there exists a causal link between these two responses.

More recently, studies by Rosenstiel and others (Rosenstiel et al., 2003, 2004; Loreto et al., 2007) have shown that the CO_{2} sensitivity of E_{i} can be explained by competition for carbon substrate between cytosolic and chloroplastic processes, controlled by the activity of the cytosolic enzyme phosphoenolpyruvate (PEP) carboxylase. The Wilkinson et al. (2009) model is based on this proposed mechanism and it contains the following assumptions: first, at low C_{ic}, the availability of recently produced photosynthate limits E_{i} and it is highly dependent on stored carbohydrate reserves; second, at intermediate C_{ic} the availability of recently produced photosynthate is adequate and the isoprene biosynthesis rate is co-limited by the supply of GAP and Pyr; finally, at high C_{ic} the isoprene biosynthesis rate is progressively more limited by Pyr due to increased activity of PEP carboxylase and concomitant decreased availability of PEP for transport into the chloroplast. In the case of the Wilkinson et al. (2009) paper, the source of Pyr for isoprene biosynthesis is assumed to be cytosolic. The model by Martin et al. (2000) allows for the production of Pyr to limit isoprene biosynthesis at low and intermediate C_{ic}, but in this case the source of the Pyr is assumed to be chloroplastic – as a secondary reaction of Rubisco. The Martin et al. (2000) model includes a shift from limitation by GAP and Pyr availability at low C_{ic} to ATP availability at high C_{ic}. The variable assumptions of these models are in need of more thorough validation, although most of the evidence that has accumulated since the Niinemets et al. (1999) and Martin et al. (2000) models indicates that the source of Pyr for isoprene biosynthesis is extra-chloroplastic, as represented in the Wilkinson et al. (2009) model.

Wilkinson et al. (2009) hypothesized that the three phases of substrate limitation, on which their model is based, are enabled in progressive series as C_{ic} is increased, resulting in an inverse sigmoidal response (Fig. 2). The sigmoidal shape of the CO_{2} response is assumed to reflect a switch among metabolic limitations. Mathematically, a switch in controlling functions can be related to the same independent variable by a Heaviside function H(x):

(Eqn 30)

(d, a critical threshold value of independent variable x that allows the form of the function to be switched between two alternatives (f_{1} and f_{2})). When combined to a single function we can write:

(Eqn 31)

which allows f_{1} to control the value of the dependent variable below the critical switch (designated as u_{x1}), and the sum of f_{1} and f_{2} to control its value at or above the critical switch (designated as u_{x2}). Wilkinson et al. (2009) used the general form of a Heaviside function to derive the relation:

(Eqn 32)

Eqn 32 describes a switch in the CO_{2} response such that below a critical value of C_{ic} (C_{i1}), the maximum value of E_{i} is limited by the rate of mobilization of GAP from older, stored carbohydrate reserves (switch u_{Ci1}, Term I); at or above C_{i1}, the response is driven by the Michaelis–Menten type activity of an enzyme, which we assume to be PEP carboxylase (switch u_{Ci2}, Term II). The coefficient C_{ic50} is analogous to K_{m}, a Michaelis constant that constrains the quasi-first order domain of the response at low C_{ic}. An analytical form of Eqn 32 that does not depend on stepwise triggers, but rather is driven by continuous dependence of E_{i} on C_{ic} is written as:

(Eqn 33)

Here the unitless scaling coefficient c_{C1} forces the right-hand term to be reduced exponentially at low C_{ic} but increased exponentially at high C_{ic}. The net result of c_{C1} is to force the function to produce an inverse sigmoid response as a function of increasing C_{ic}. The term C_{ic50} was re-defined as C* such that C* becomes a more generalized scalar without requiring strict analogy with the Michaelis constant, K_{m}.

Possell & Hewitt (2011) developed a model similar in form to Eqn 33, but rather than basing it on C_{ic}, they used DMADP substrate concentration. Because DMADP decreases as C_{ic} increases, in those cases where a negative CO_{2} response has been observed, the model need not take the form of a mathematical difference as in Eqn 33, and can take the form of the Hill equation for enzymes, which reflects the property of cooperativity:

(Eqn 34)

(V_{max} and K_{m}, Michaelis coefficients for isoprene synthase; c_{C2}, a unitless scaling coefficient, analogous to the Hill coefficient that is used in biochemical models and describes the cooperative nature of the enzyme-substrate interactions). This model was shown to provide good descriptions of the CO_{2} response in numerous species.

The models that have been based on cytosol–chloroplast competition for substrate (e.g., Wilkinson et al., 2009; Possell & Hewitt, 2011) have not been able to explain one aspect of the CO_{2} response – the steep reduction toward zero of the isoprene emission rate at a critically low value of C_{ic} (Loreto & Sharkey, 1990; Rasulov et al., 2009, 2011). Typically, this value is close to the photosynthetic CO_{2} compensation point, and it is rarely reached in leaves in their native environments. Nevertheless, the declining part of CO_{2} response curve below this critical threshold can provide fundamental information of the mechanism(s) responsible for the overall CO_{2} dependence of isoprene emission. This is an issue in need of further study.

Rasulov et al. (2009) used observations of the response of E_{i} and DMADP pool size as a function of C_{ic} to argue that the CO_{2} effect on E_{i} is due to variations in chloroplast ATP supply, not variations in the channeling of PEP from the cytosol to the chloroplast; this brought the focus of the CO_{2} response back to the original processes described in the Niinemets et al. (1999) and Martin et al. (2000) models. This shift in focus was justified on the presumption that carbon availability should be sufficient under most CO_{2} concentrations to support the isoprene biosynthesis rate, but chloroplast ATP availability should not. Both hypotheses rely on the fundamental observation that plastidic DMADP pool size decreases as C_{ic} increases; the debate posed by Rasulov et al. (2009), as a counterpoint to the perspective of Rosenstiel et al. (2004), is focused on the cause of that decrease. Most of the evidence underlying both perspectives is correlative – positive correlations between ATP availability and E_{i} have been observed (Loreto & Sharkey, 1993) and negative correlations between PEP carboxylase activity and E_{i} have been observed (Rosenstiel et al., 2003, 2004; Loreto et al., 2007; Possell & Hewitt, 2011). In a recent study by Trowbridge et al. (2012), proton-transfer mass spectrometry was used to detect the differential kinetics of ^{13}C incorporation into fragments of isoprene presumed to come from cytosolic vs chloroplastic sources. The results during periods of low vs high C_{ic} suggested slower labeling in the fragment purported to come from cytosolic sources, and this fragment was more highly labeled in the presence of low CO_{2}, compared to that derived from GAP directly. These latter results can be interpreted as supporting the Rosenstiel et al. (2003) perspective more than the Rasulov et al. (2009) perspective. Once again, this is an issue that needs more study before a definitive algorithm for C, the factor used for scaling B_{i} according to C_{ic}, can be formulated.

IX. Modeling the relation between isoprene emission and leaf conductance

From some of the earliest studies on isoprene emission from leaves it was recognized that E_{i} is independent of stomatal conductance (g_{s}) in the steady-state condition (Monson & Fall, 1989; Fall & Monson, 1992). Fall & Monson (1992) hypothesized that steady-state reductions in g_{s} were compensated by increases in Δp_{i}, the difference in isoprene partial pressure between the intercellular air spaces of the leaf and the ambient atmosphere in kPa; thus, E_{i} = g_{si} (Δp_{i}/P), where g_{si} is the stomatal conductance to isoprene diffusion in mmol m^{−2} s^{−1} and P is atmospheric pressure in Pa. The theory underlying this relation and its application to a range of emitted volatile organic compounds (VOCs) demonstrated that for compounds such as isoprene, which have relatively high Henry’s Law partitioning coefficients, perturbations to g_{si} should result in rapid (within seconds) establishment of a new diffusion steady-state (Niinemets & Reichstein, 2003). These relations would not be true for VOCs with lower Henry’s Law coefficients (e.g. oxygenated isoprenoids, organic acids or methanol). Niinemets & Reichstein (2003) formalized the theory on these relations by stating:

(Eqn 35)

(H_{i} , Henry’s Law constant for isoprene (Pa m^{3} mol^{−1}); C_{wi}, concentration of isoprene in the liquid (water) phase of the cell (mol m^{−3}); p_{ia}, partial pressure of isoprene in the atmosphere outside the leaf (kPa)).

X. Modeling the longer term processes that control isoprene emission rate

To this point we have discussed reactions and equations to describe the leaf isoprene flux under the assumption that B_{i} is constant. We have access to empirical evidence, however, that B_{i} changes as a function of leaf development, recent dynamics in the temperature and light microenvironments of the leaf, and growth in different atmospheric CO_{2} regimes. Expression of the gene for isoprene synthase, availability of substrate for isoprene synthase, activation of MEP pathway genes and diurnal rhythms of photosynthetic processes are all highly variable and responsive to developmental and environmental cues. Variation in these processes causes variation in B_{i} (Mayrhofer et al., 2005; Wiberley et al., 2005, 2009; Liovamäki et al., 2007; Steinbrecker et al., 2009; Sun et al., 2009, 2012). Capturing that variation in process models has been one of the central challenges in modeling the longer-term dynamics in E_{i}, and its dependence on B_{i}.

1. Seasonal influences on B_{i}

It has long been known that plants of the temperate zone emit different amounts of isoprene depending on day of year but independent of instantaneous light and temperature conditions (Ohta, 1986; Monson et al., 1994). Despite this knowledge, and recognition of its importance for estimating total annual isoprene emissions, modeling strategies that capture the seasonal effect have been slow to develop. Schnitzler et al. (1997) proposed an asymmetric equation to define the seasonal factor, S, which was intended as an additional multiplier of B_{i} in Eqn 2, and was described by an equation analogous to those used for enzyme activity modeling:

(Eqn 36)

(D, day of the year; a, b, c and d, curve fitting coefficients). Pier & McDuffie (1997) used a polynome with three parameters to describe symmetric seasonal variation in the E_{i} observed for white oaks:

(Eqn 37)

Staudt et al. (2000) also proposed a symmetric equation describing a Gaussian (bell-shaped) response with parameters intended to modify B_{i} for both the light-dependent monoterpene and isoprene emissions from leaves:

(Eqn 38)

(ρ, the relative annual amplitude of the maximum possible seasonal emission rate (which is assumed to be between 0 and 1.0 for deciduous species and zero for evergreen species); D and D_{0}, Julian days with D_{0} representing the day on which the emission capacity reaches its maximum; τ, the breadth (kurtosis) of the seasonal amplitude (in days)). A slightly modified version has been also used by Lavoir et al. (2011) to simulate monoterpene emissions of holm oaks (and presumably this is also valid for application to isoprene emissions):

(Eqn 39)

Keenan et al. (2009) used a different asymmetric equation adopted from phenological investigations of photosynthetic processes to approximate monoterpene emission from Pinus pinea and Quercus ilex, which also may applicable to the prediction of leaf isoprene emissions. In this case, they directly adjusted the standardized emission rate, E_{i}, rather than create a multiplier of B_{i}:

(Eqn 40)

(E_{0}, minimum and E_{max} peak standardized emission rates).

One approach to modeling seasonal variation in B_{i} has been to assume that leaf developmental processes, controlled by genetic × environment interactions, underlie expression of the gene for isoprene synthase, and therefore the value of B_{i}. While this mechanistic justification has been given to support such models, the logic used in the model does not actually derive from knowledge of gene expression processes. Rather, the models are once again empirical in nature and are intended to replicate the shape of observed seasonal dynamics in B_{i}. One such model was developed by Lehning et al. (2001). The Seasonal Isoprenoid synthase Model (SIM) is split into a phenological equation that determines the timing of bud break, leaf ontogenesis, and autumn senescence, and an emission activity equation that resolves longer-term environmental influences on B_{i} by calculating daily dynamics in isoprene synthase activity. Evergreen plants are assumed to exhibit no dependence of B_{i} on leaf developmental stage. For deciduous plants seasonal changes in B_{i} are modeled as:

(Eqn 41)

(Eqn 42)

(Eqn 43)

(B_{imax}, seasonal maximum value of B_{i}, [equivalent to B_{i} at D_{0} in Eqn 38]; D, Julian day; D_{e}, Julian day of leaf emergence; D_{m}, Julian day at which maximum Chl a content (per unit leaf area) is reached; D_{1/2m}, Julian day at which half maximum Chl a content is reached; D_{s}, Julian day at which Chl a content starts to decline during leaf senescence; D_{1/2s}, Julian day at which Chl a content has declined to the half maximal value during leaf senescence).

Once again, we note that in all of these models, the parameters and mathematical forms do not reflect specific biological mechanisms known to cause phenological changes in B_{i} or E_{i}. These are truly empirical models, and they are only intended to replicate the overall shape of the observed seasonal variation in B_{i}. We have presented some of the principal symmetric, asymmetric, and weather dependent approaches that have been used for seasonal adjustment of B_{i} in Fig. 3. The general shape of the seasonal responses and their maxima near Day of Year 200 is generally conserved. However, the slopes of the responses for the ascending and descending trajectories on either side of the maximum differs, and this is where model-dependent differences are likely to be greatest.

In the Model of Emissions of Gases and Aerosols from Nature (MEGAN) model (Guenther et al., 2006) a similar approach was used, though in this case, developmental effects were parameterized as fractional adjustments in B_{i}: newly developed foliage was assumed to exhibit B_{i} = 0.05 times the rate of old, fully expanded leaves; young expanding leaves were assumed to exhibit B_{i} at 0.6 times the rate of old, fully expanded leaves; recently matured leaves were assumed to exhibit B_{i} at 1.125 times the rate of old, fully expanded leaves; and old, fully expanded leaves were assumed to exhibit B_{i} at a factor of 1.0. The values of these emission parameters were derived from past empirical studies of leaf development in several species (Guenther et al., 1991; Monson et al., 1994; Goldstein et al., 1998; Petron et al., 2001; Karl et al., 2003). The fraction of each leaf development class within a growing deciduous canopy is modeled as a function of time-dependent increase in LAI and the number of days between leaf emergence and the initiation or maximum expression of B_{i} (D_{i} and D_{m}, respectively).

2. Longer-term effects of temperature,and photosynthetic photon flux density and atmospheric CO_{2} concentration on B_{i}

where is the mean air temperature (K) of the previous time-step interval in the phenology scheme of the model (between 1 wk and 1 month depending on the seasonal phenology database that is used for parameterization of the model). These temperature adjustments to D_{i} and D_{m} are intended to accelerate or decelerate the rate at which leaves move through each seasonal or phenological stage (e.g. newly-developed, young-expanding, fully expanded, etc.), which in turn are linked to specific values of B_{i} as described above.

While not truly mechanistic in their underpinnings, all of the models of seasonal variation in B_{i}, have tried to account for past observations of acclimation due to dynamics in the mean temperature and mean PPFD during the hours-to-weeks preceding an instant in time (see Sharkey et al., 1999; Petron et al., 2001). In the MEGAN model the fundamental form of Eqn 2 was retained as a basis for predicting E_{i} and the fundamental form of Eqn 9 was retained for defining L. However, the parameterization scheme for coefficients α and c_{P3} (described in Eqns 10, 11) was modified. Thus, adjustments to B_{i} were assumed to be reflected as changes in the shape of the curve defining L. Rather than defining α and c_{P3} according to the canopy LAI above target leaves, as described in Guenther et al. (1999), these coefficients were parameterized using the estimated mean values for PPFD during the previous day (24 h) and 10 d (240 h). These researchers reasoned that an expression parameterized by mean daily PPFD would account for the combined effects of LAI and recent weather. Thus:

(Eqn 47)

(Eqn 48)

where and are the mean values for PPFD during the previous 10 and 1 d, respectively, and is a base value intended to differentiate the PPFD incident on sun- vs shade-adapted leaves. The value for was assumed to be 200 μmol m^{−2} s^{−1} and 50 μmol m^{−2} s^{−1} for sun and shade leaves, respectively. The forms of Eqs 47, 48 were derived from empirical analyses reported in Sharkey et al. (1999) and Geron et al. (2000).

We have presented a comparative analysis of the various forms of the ‘Guenther light algorithm’ (defined by L) in Fig. 4. The model runs A–C show results from the algorithms for L described in Guenther et al. (1993, 1999, 2006), respectively. Model Run B shows the simulated increase in the quantum yield for isoprene emissions (the initial slope of the E_{i} vs PPFD relation) in shade-adapted leaves; a response that has been observed in past observations (e.g. Harley et al., 1996; Litvak et al., 1996). This relation disappears in the Guenther et al. (2006) model, wherein shade-adapted leaves are observed to exhibit reduced quantum yields for isoprene emissions when exposed to either ‘sunny’ or ‘cloudy’ weather during the previous 10 d. The higher quantum yield for shade-adapted leaves can be accommodated in the Guenther et al. (2006) model through user-designed modifications to α (see Eqn 10), such that it resolves to a higher value for shade-adapted leaves during either ‘sunny’ or ‘cloudy’ weather, compared with sun-adapted leaves.

In the same way that changes in PPFD cause acclimation in the coefficients that determine L at the scale of hours to days, it has been recognized that changes in the daily mean leaf temperature can influence the coefficients that determine T, the temperature-dependent modifier of B_{i}. In Guenther et al. (2006) equations were developed to estimate E_{opt} and T_{opt} from an assumed ratio of E_{opt} : B_{i} of 2.034, rather than the default value of 1.9 used previously (Guenther et al., 1999), and to take account of recent influences of ambient temperature on the acclimation of both E_{opt} and T_{opt}. Thus:

(Eqn 49)

(Eqn 50)

where and represent the average leaf temperature () over the most recently past 10 and 1 d, respectively. The influence of recent temperature history on E_{opt} and T_{opt} were intended to account for observations in past studies that both B_{i} (see Sharkey et al., 2000; Petron et al., 2001) and T_{opt} (see Monson et al., 1992) change as a function of temperature history.

The growth of isoprene-emitting plants at elevated atmospheric CO_{2} concentrations generally causes a reduction in B_{i}. While this growth-related response may be due to limitations in substrate availability and enzyme activity, as is the case for instantaneous reductions in E_{i}, the ultimate causes of such limitations in the case of growth effects are likely to be different; in this case due to changes in the expression of genes that control activity of the MEP pathway and or isoprene synthase enzyme concentration, or both. Wilkinson et al. (2009) showed that growth at elevated CO_{2} tended to decrease E_{imax}, increase c_{C1} and decrease C*, the coefficients that determine the modeled influence of CO_{2} on B_{i} (see Eqn 33). The modeled result of these changes in parameter values is that the calculated B_{i} decreases as the atmospheric CO_{2} concentration during growth increases. This is consistent with observations (Wilkinson et al., 2009). Thus, the longer-term growth and shorter-term instantaneous effects of atmospheric CO_{2} on E_{i} can be predicted by the same mathematical equation, but with component parameters optimized differentially for the growth CO_{2} condition.

Longer-term influences of PPFD and T_{L} on B_{i} have also been introduced into the SIM model, but in this case within a framework of changes in the activity of isoprene synthase. In the SIM model it is assumed that the amount of isoprene synthase protein on any particular day is set by an initial physiological state and then modified according to the daily PPFD sum and a daily temperature variable (T_{d}) that is dependent on daily (Grote et al., 2010). It is further assumed that isoprene synthase activity and its contributions to B_{i} are decreased due to a continuous degradation (protein turnover) process. Taking both processes together, the difference in enzyme activity, and its influence on B_{i}, from 1 d to the next is described by:

(Eqn 51)

(Eqn 52)

(B_{i0}, the isoprene emission factor (determined by amount of isoprene synthase enzyme) of the previous day (μmol m^{−2} s^{−1}); R, the gas constant (8.3143 J mol^{−1}); φ, a coefficient defining the enzyme formation term (h^{−1}); μ, a coefficient defining the enzyme decay term (h^{−1}); m, a unitless factor for normalizing the Arrhenius term to 1 at 30°C (660.1 × 10^{6}); E_{a}, the activation energy for a doubling of the reaction velocity (51164.8 J mol^{−1})). The two parameters φ and μ are used for species-specific adjustment (which in the case of Quercus robur were assumed to be 0.014 and 0.175, respectively). The term T_{d} is analogous to the Arrhenius-type models used for modifying B_{i} due to changes in the instantaneous leaf temperature, but in this case, referenced to mean daily leaf temperature.

3. Longer-term effects of drought on B_{i}

In addition to the influence of variable temperature and PPFD, as time-integrated, long-term influences on B_{i}, the impact of drought has been studied in several investigations (Fang et al., 1996; Brüggemann & Schnitzler, 2002b; Pegoraro et al., 2004; Brilli et al., 2007). However, until recently, drought has not been considered as a modifier of B_{i} in isoprene emission models. There are three ways that drought influences B_{i}. The first is through reductions in leaf transpiration and concomitant increases in T_{L}, which is accommodated in the modeling through those functions that relate to B_{i} (e.g. Eqns 45, 46). The second is through decreases in C_{ic}, and thus an effect of reducing B_{i} through longer-term growth influences of CO_{2} concentration (see Pegoraro et al., 2005). This CO_{2} effect can be modeled through Eqn 29 when properly parameterized to a reduced CO_{2} growth regime. Finally, there is the direct effect of drought on metabolic processes. Stress and the cellular growth reductions that accompany water stress tend to trigger a cascade of metabolic feedbacks that function to balance metabolism with growth potential. With the MEGAN model, Guenther et al. (2006) introduced a drought scaling factor (D) as a linear relation between relative water availability and B_{i}, which can be used as an additional multiplier in Eqn 2.

(Eqn 53)

(θ, extractable water content (m^{3} m^{−3}); θ_{w}, soil water content that defines the leaf wilting point; , an empirically-determined parameter defined as 0.06 following Pegoraro et al. (2004)). Thresholds for determining the direct influence of θ on B_{i} were set as D =1 for θ > θ_{1}, D determined by Eqn 53 for θ_{w} < θ < θ_{1}, and D =0 for θ < θ_{w}. Associated with those thresholds, it is assumed that θ_{1} = θ_{w} + . One of the difficulties with using this type of model is the determination of θ_{w} and interpretation of its physiological meaning. Guenther et al. (2006) used the wilting point database of Chen & Dudhia (2001) for global mapping of E_{i}. However, there are no studies to date that have established the wilting point as a relevant determinant of B_{i}. Additional possibilities for parameterization of θ_{L} were provided by Müller et al. (2008) who obtained empirical data for the soil water content in the rooting zone of several different types of plants. Without any soil layer stratification but based on local measurements, Grote et al. (2010) and Lavoir et al. (2011) applied the drought impact model for a Mediterranean forest, assuming a linear decrease in D at θ < 0.7. In a separate study, Grote et al. (2009) took advantage of the detailed metabolite characterization permitted by the SIM model to represent drought effects on isoprene and light-dependent monoterpene emissions through the availability of Pyr and GAP, the initial substrates of the MEP pathway. One premise of this approach, however, is that tight coupling exists between leaf carbon balance, as influenced by leaf photosynthesis rate, and isoprenoid emission. This premise has not been corroborated through empirical observations (Teuber et al., 2008). The greatest barrier to progressing in our ability to model drought stress effects on B_{i} is our incomplete understanding of the metabolic connections among drought, expression of isoprene synthase activity, availability of DMADP substrate, and drought-induced changes in the sensitivities of E_{i} to PPFD, temperature and intercellular CO_{2} concentration. Future studies should focus on these connections, which may allow us to integrate drought-stress models more effectively with those on the longer-term effects of T_{L}, PPFD and C_{ic}.

XI. Conclusions

Historically, models of leaf isoprene emission rate have been developed within a highly empirical framework. Lacking details on the biochemical and physiological controls over emission rate, and in the face of interest and support from air quality regulatory agencies in North America and Europe, modelers have searched for the most convenient ways by which to represent leaf-scale processes in regional emission inventory models. This approach relied heavily on searches for mathematical functions that matched the shapes of observed responses of isoprene emission rate to single environmental drivers. The effect of multiple factors varying at once was represented by serial multipliers reflecting single-factor effects that increase or decrease B_{i}. This approach has worked successfully in the sense that landscape-to-global scale emission rates could be predicted, when constrained by observations of atmospheric chemistry (e.g. Guenther et al., 1999; Poisson et al., 2000). However, it also provided a level of comfort and confidence in the assumption that the models were reflecting true biological processes.

As these empirically-based mathematical models emerged, collaborators with experience in biochemical or physiological processes began to justify the form of the models, through a post priori process using knowledge of relevant processes at the leaf or chloroplast scales. This application of first-principles ‘hindsight’ worked effectively in some cases, such as that for the response of steady-state isoprene emissions to T_{L}. In parallel studies, researchers familiar with biochemistry and physiology developed models from the bottom-up. Typically, these efforts began with established models of A or the MEP pathway, and assumptions that activities in these pathways should control the leaf isoprene emission rate (e.g. Niinemets et al., 1999; Martin et al., 2000; Wilkinson et al., 2009). With this approach, however, critical information was missing as to the stoichiometries by which metabolites from the MEP pathway and reductive pentose phosphate (photosynthetic) pathways are channeled into isoprene biosynthesis. Thus, the modeling could be kept close to the first-principles of biochemistry and physiology to a point; but, at some point late in the modeling procedure, an arbitrary partitioning factor had to be introduced to channel some fraction of metabolic potential into E_{i}.

A key limitation to pushing these efforts forward and establishing a more accurate foundation for models that define the instantaneous influences of environment on E_{i} is knowledge of metabolite origins and dynamics in the channeling of Pyr and GAP into the MEP pathway, potential feedbacks of E_{i} on DMADP substrate formation, and the processes that govern the partitioning of reductant, ATP and photochemically-energized electrons to E_{i}. With regard to the longer-term influences on B_{i}, the models are likely to continue to be based on the shapes of seasonal trajectories until we can figure out how to link changes in the seasonal environment, especially with regard to cumulative T_{L} and PPFD, to controls over gene expression for isoprene synthase and MEP pathway enzymes. In addition to improving our ability to predict seasonal dynamics in B_{i} and E_{i}, it is these latter influences that will be especially important to improving our ability to predict changes in isoprene emissions to the slower, even interannual, environmental dynamics that accompany climate change. This convergence of molecular genetics, physiology and biochemistry will have to lie at the core from which predictions emerge as to how plants will affect atmospheric chemistry on the future Earth.