Annual cycle of Scots pine photosynthesis

Photosynthesis, i.e. the assimilation of atmospheric carbon to organic molecules with the help of solar energy, is a fundamental and well-understood process. Here, we connect theoretically the fundamental concepts affecting C3 photosynthesis with the main environmental drivers (ambient temperature and solar light intensity), using six axioms based on physiological and physical knowledge, and yield straightforward and simple mathematical equations. The light and carbon reactions in photosynthesis are based on the coherent operation of the photosynthetic machinery, which is formed of a complicated chain of enzymes, membrane pumps and pigments. A powerful biochemical regulation system has emerged through evolution to match photosynthesis with the annual cycle of solar light and temperature. The action of the biochemical regulation system generates the annual cycle of photosynthesis and emergent properties, the state of the photosynthetic machinery and the efficiency of photosynthesis. The state and the efficiency of the photosynthetic machinery is dynamically changing due to biosynthesis and decomposition of the molecules. The mathematical analysis of the system, defined by the very fundamental concepts and axioms, resulted in exact predictions of the behaviour of daily and annual patterns in photosynthesis. We tested the predictions with extensive field measurements of Scots pine (Pinus sylvestris L.) photosynthesis on a branch scale in northern Finland. Our theory gained strong support through rigorous testing.


Introduction
The movement of the globe around the sun generates a conspicuous annual cycle of the solar radiation on the earth, and this cycle is especially strong at high latitudes.Ambient temperatures respond to the cycle of solar energy input and therefore a strong annual cycle exists also in temperature, although a bit delayed.These large variations in light and temperature are greatly influencing the distribution of plant species, especially in the northern regions.As an example, Scots pines (Pinus sylvestris L.), while abundant all over Europe, have also adapted especially well to the annual cycle of radiation and temperature in the northern climate, forming even the treeline in many regions (Juntunen et al., 2002).
As a consequence of the seasonal variation in light and temperature, many perennials including deciduous trees have a strong metabolic annual cycle, as they grow new leaves every spring that then become senescent in the autumn.Temperature affects the timing of many phenological events, i.e. bud burst and flowering (Hänninen and Kramer, 2007;Hari and Häkkinen, 1991;Linkosalo, 2000;Sarvas, 1972).However, the annual cycle is less clear in coniferous trees, although they also have a period of intensive new foliage growth in the spring and a specific time frame when old needles are senescing in the fall.
The annual cycle of light and temperature is manifested in plant metabolism in many ways.Actively metabolising cells are very sensitive to low temperatures, and, as a consequence, they need to inactivate many processes in order to avoid damage during winter in a process called winter hardening (Hänninen, 2016).This means that the metabolism of, for P. Hari et al.: Scots pine photosynthesis example, evergreen Scots pine needles also needs to follow a clear annual cycle.For example, when sufficiently hardened, pine needles tolerate temperatures well below −30 • C in winter; however, they are very sensitive to temperatures below −10 • C during summer (Sakai and Larcher, 1987).The metabolism of photosynthesis recovers gradually from the winter hardened state during spring, and the ambient temperature has an important role in this recovery (Pelkonen and Hari, 1980).
Biochemically, photosynthesis can be defined as a long chain of action of pigments, membrane pumps and enzymes, which use light as a source for energy and atmospheric CO 2 as source for carbon (see e.g.von Caemmerer and Farquhar, 1981).Changes in the concentrations and activities of this photosynthetic machinery generate the annual metabolic cycle of photosynthesis.The physiological basis of the annual cycle at the level of the synchronised action of the photosynthetic machinery is poorly known, especially when it comes to the role of temperature in the synthesis, activation, decomposition and deactivation of the machinery.
Sugars formed in photosynthesis are the source of energy for all cellular metabolic activity and raw materials for growth.The length of the photosynthetically active period is a key factor determining the annual amounts of sugars formed in photosynthesis (Hari et al., 2013) and it plays a very important role in the metabolism and growth of vegetation.Thus, a theoretical understanding of the dynamics of the photosynthetic annual cycle is key to understanding and explaining the growth of the trees growing at high latitudes.
Physiological and biochemical research has provided useful knowledge of the photosynthetic reaction chains, and the details of this machinery at the leaf, organ and tissue levels have been intensively explored over decades, mostly in controlled laboratory conditions (Farquhar and von Caemmerer, 1982;Farquhar et al., 1980;Kirschbaum et al., 1998;Laisk and Oja, 1998).However, field measurements in mature trees are difficult to perform, and the results are not easy to interpret.Therefore, the detailed physiological knowledge that has mostly been obtained from laboratory experiments needs to be translated into the field conditions, i.e. into trees living in their natural environment to increase our understanding of the annual cycle of photosynthesis under field conditions.This was our motivation in developing a conceptual approach to the relationship of photosynthesis and the annual variations in light and temperature.
The field of physics was facing a similar situation in the seventeenth century as field studies on photosynthesis are encountering now.There were plenty of single and scattered experiments and observations, but the unifying theory was missing.Isaac Newton presented an approach to construct theories in his book Principia Mathematica and unified the physical knowledge.He proceeded in four steps when developing his theories, starting from the definition of concepts and followed by the introduction of axioms.The mathematical analysis of the behaviour of the system defined by the concepts and axioms dominated his theory development.Finally, he derived predictions and tested them.The new translation of Newton's famous book Principia Mathematica (Newton, 1999) clearly presents these four steps.
In our previous analysis of photosynthesis taking place during midsummer, we strictly followed Newton's example by introducing the concepts and axioms, by analysing the behaviour of the system defined by these concepts and axioms, and finally by deriving predictions and testing them (Hari et al., 2014).However, it was evident that our theory omits the annual cycle of metabolism and therefore it crucially fails to predict the photosynthesis in the transitional times such as spring and autumn.The daily patterns of measured and predicted CO 2 exchange were quite similar, but the level of predicted photosynthesis was too low, especially in early spring and late autumn.We thus concluded that we have to introduce the annual cycle of metabolism into our theory.Our aim is to develop our theory of photosynthesis to cover the whole growing season and to explain and predict the annual cycle of Scots pine photosynthesis in field conditions.

Theory development
The strong annual cycle of the solar light intensity and ambient temperature is characteristic of the growing area of Scots pine: for example, Finland has mostly a subarctic climate according to the Köppen-Geiger climate classification (Peel et al., 2007), meaning that summers are quite mild, with daily maximum temperatures being around 20 • C, whereas winters are rather cold, with minimum temperatures often below −20 • C. A regulation system has emerged through evolution to match the metabolism and cold tolerance with the annual cycle of the solar radiation and temperature.
The process of photosynthesis consists of a large number of steps that form the light and carbon reactions of photosynthesis.Each step is based on actions of a specific molecule, with the most important being pigments (e.g.chlorophylls and carotenoids), transmembrane proteins and membrane pumps (e.g.ATPases), and Calvin cycle enzymes (e.g.ribulose-1,5-bisphosphate carboxylase/oxygenase, Rubisco; Taiz et al., 2015).A proper functioning of the reaction chain in photosynthesis requires that no single step is blocking the chain of interlinked energy capture, membrane transport or synthesis of new compounds.The core of pigment complexes as well as the membrane pumps and enzymes are all proteins that have a tendency to decay (Araujo et al., 2011;Hinkson and Elias, 2011;Huffaker and Peterson, 1974;Nelson et al., 2014).Proteins are nitrogen-rich macromolecules (many contain 15-16 wt %N; Nelson et al., 2014) and they are costly to produce and maintain.Therefore, it is natural that plants need to be able to use the limited N reserves in an effective way.Since nitrogen has several competing usages in metabolism, maintaining excess proteins is a "waste" of nitrogen.The synthesis and decomposition of active pro-tein molecules balance the concentrations of active protein molecules in the photosynthetic chain.Evidently, maintaining the proper balance of these molecules is a crucial and demanding task for the metabolism of trees.
Large changes in the photosynthetic performance characterise the annual cycle of photosynthesis, generated by changes in the concentrations of the photosynthetic machinery.The maintenance of the proper concentrations of the components in this machinery is taken care of by a very powerful biological regulation system that has emerged through evolution to match the cellular metabolism with the regular annual cycle of solar light and temperature and is capable of modifying the processes within the normal range of conditions but also provides sufficient resilience under sudden (short-term) extreme conditions during the transition from winter to spring (Ensminger et al., 2004a;Zarter et al., 2006).This system synthesises, activates, decomposes and deactivates the critical photosynthetic machinery over timescales of days (Nelson et al., 2014), and it is an acclimation system affecting the activation and deactivation of transcriptional modules responsive to light and temperature cues (e.g.Cazzonelli and Pogson, 2010;Toledo-Ortiz et al., 2014).The changes in the machinery, in turn, generate changes in the relationship between photosynthesis and light.This forms the metabolic basis for our theory of the dynamics of the annual cycle of photosynthesis.

Definitions and axioms
We start our formulation with definitions as Newton did centuries ago.We utilise physiological and physical knowledge in the formulation of the axioms needed for the mathematical formulation.
Definition 1.The photosynthetic machinery is the complex web of pigments, membrane pumps and enzymes forming the biochemical structure underlying photosynthesis.
Plants are able to change the concentrations of active components in the photosynthetic machinery.
Definition 2. Plants have a biochemical regulation system that synthesises, activates, decomposes and deactivates the photosynthetic machinery.
The action of the biochemical regulation system generates the annual cycle of photosynthesis and maintains the balance between the different steps in the photosynthetic reaction chain.In this way, it generates a new property in the photosynthetic machinery.
Definition 3. The state of the photosynthetic machinery is the emergent property created by the actions of the biochemical regulation system controlling the concentrations of active photosynthetic machinery.
The state of the photosynthetic machinery characterises the complex web of energy capture, biochemical reactions and membrane transport in photosynthesis with one single number.Next, we specify the action of the biochemical regulation system on photosynthetic machinery.
Axiom 1. Synthesis and activation as well as decomposition and deactivation of the photosynthetic machinery are changing the state of the photosynthetic machinery.
Further, we specify the relationship between the environment and the synthesis by the biochemical regulation system.
Axiom 2. The synthesis and activation of the photosynthetic machinery depend linearly on the temperature above freezing point.
We also clarify the behaviour of decomposition and deactivation.
Axiom 3. The decomposition and deactivation of the photosynthetic machinery depends linearly on the state.
Captured light energy may cause damage in chloroplasts in freezing temperatures, when availability of CO 2 is limited for the carbon reactions in photosynthesis.This is why the biochemical regulation system acts strongly to protect against damage.
Axiom 4. The accelerated decomposition and deactivation of the photosynthetic machinery during cold and strong light depends linearly on the product of light and temperature below freezing point.
The concentrations of the photosynthetic machinery affect the performance of photosynthesis.
Definition 4. The efficiency of photosynthetic reactions is the capacity of light and carbon reactions to synthesise sugars.
When we developed the theory of photosynthesis explaining the behaviour in midsummer (Hari et al., 2014), we introduced an axiom stating that the product of the saturating response to the photosynthetically active radiation and CO 2 concentration in the stomatal cavity determines the photosynthesis at a point in space and time.Here, we introduce the annual cycle of photosynthesis into the axioms with the efficiency of photosynthetic carbon and light reactions and the efficiency photosynthetic reactions replacing the parameter b in Eq. (1) in Hari et al. (2014).
Axiom 5.The photosynthesis rate at a point in space and time depends on the product of two terms: (i) the efficiency of photosynthetic light and carbon reactions and (ii) the product of the CO 2 concentration in the stomatal cavity and the saturating response of the light reactions to the photosynthetically active radiation.
The state of the photosynthetic machinery determines the efficiency of photosynthetic light and carbon reactions, which leads to our final axiom.
Axiom 6.The efficiency of photosynthetic light and carbon reactions depends linearly on the state of the photosynthetic machinery.

Mathematical analysis
We introduce mathematical symbols to formulate the axioms in a more exact and compact way.S denotes the state of the photosynthetic machinery, f 1 is the synthesis and activation, f 2 is the decomposition and deactivation, and f 3 is the accel-P.Hari et al.: Scots pine photosynthesis erated decomposition and deactivation of that photosynthetic machinery (i.e.enzymes, membrane pumps and pigments) caused by light at low temperatures.
Axiom 2 states that the relationship between the synthesis and activation and temperature (T ) is linear above the freezing point, which gives where T f is the freezing temperature of needles and a 1 is a parameter.
According to Axiom 3, the relationship between the decomposition and deactivation of the photosynthetic machinery and the state of the photosynthetic machinery, S, is linear: (2) Accelerated decomposition and deactivation takes place to protect the photosynthetic machinery against damage when freezing temperatures hinder the carbon assimilation reactions of photosynthesis (Axiom 4): where I is the intensity of photosynthetically active radiation (PAR).The synthesis, activation, decomposition and deactivation change the state of the photosynthetic machinery as follows: Combining Eqs. ( 1)-( 4), we obtain dS dt = Max 0, a 1 T + T f − a 2 S − a 3 Max T f − T I. (5) Equation ( 5) defines the state of the photosynthetic machinery at any moment t when temperature and solar radiation records are available.The photosynthesis rate, p, is obtained from Axiom 5 as follows: where C S is the CO 2 concentration in the stomatal cavity; f (I ) is the saturating response of the photosynthesis rate to the photosynthetically active radiation (see Hari et al., 2014); and E is the efficiency of photosynthetic carbon and light reactions which, according to Axiom 6, is as follows: When we developed the theory of photosynthesis in midsummer (Hari et al., 2014), we introduced an axiom stating that the product of the saturating response to the photosynthetically active radiation and CO 2 concentration in the stomatal cavity determines the photosynthesis at a point in space and time (A1 in Hari et al., 2014).When we quantified the previous axiom with mathematical notations, we replaced the axiom A1 with the new Axiom 5 that is quite similar to the previous one.The changing efficiency of photosynthetic light and carbon reactions is the novel aspect in Axiom 6.When we quantified the previous axiom with mathematical notations, we introduced a parameter b (Eq. 1 in Hari et al., 2014).Equation ( 6) is very similar to the previous Eq.( 1) in Hari et al. (2014); the only difference is that the efficiency parameter b is replaced with E, which is the state-variable efficiency of photosynthetic carbon and light reactions.We obtain the solution of the optimisation problem in the same way as in the analysis of photosynthesis (p) during midsummer as follows: where g max is stomatal conductance when stomata are open, C a is the CO 2 concentration in the ambient air, r is the rate of respiration and u opt is the so-called seasonal modulated degree of optimal stomatal control given by In Eq. ( 10), λ is a cost of transpiration, i.e. a measure of water-use efficiency.
To summarise, Eqs. ( 5) and ( 7)-( 10) predict the density of the photosynthetic rate when we know the ambient temperature and solar radiation history, density of photosynthetically active solar radiation, and concentrations of water vapour and CO 2 in the air.This prediction is clearly a dynamic version of the formulation by Hari et al. (2014).The changing state of the photosynthetic machinery (i.e.enzymes, membrane pumps and pigments) determines the efficiency of light and carbon reactions, introducing the annual cycle of metabolism into the prediction.Thus, the relationship between light and photosynthesis changes smoothly during the seasons.

Parameter estimation
We tested the new theoretical prediction with field chamber measurements in Scots pine trees in Lapland at the Värriö Subarctic Research Station (SMEAR I; 67 • 46 N, 29 • 35 E; 400 m a.s.l.).We measured the CO 2 exchange of pine shoots with four branch chambers throughout the year in 2011-2014 (Hari et al., 2014).In addition, photosynthetically active radiation (I ) was measured at each chamber, whereas the records for air temperature, air humidity and CO 2 concentration are site specific.Despite the constant supervision, maintenance and malfunction of the measuring system generated some gaps in the data.To obtain maximal data coverage per year, we selected those chambers that measured over the whole year without long maintenance and malfunction periods.
There are four parameters in Eqs. ( 5) and ( 7)-( 10) that describe the annual cycle of photosynthesis (a 1 , a 2 , a 3 and a 4 ).The simultaneous freezing temperatures and sunny weather are quite rare events at our measuring station, occurring only in early spring and very late in autumn.As a result, the parameter a 3 in Eq. ( 3) has a minor role in the predictions and its estimation is based on very scarce data on the CO 2 exchange with the accompanied environmental factors.The residual sum of squares has several local minima, and they hamper the simultaneous estimation of the parameters a 1 , a 2 and a 4 .Therefore, we proceed stepwise; first we fix the value of a parameter.Thereafter we estimate the values of non-fixed parameters with standard numeric methods.We replace the value of the fixed parameter with the one obtained in the estimation.We select another parameter, fix its value with the one obtained in the previous round of estimation and estimate the other two parameters again.We continue the process of fixing, estimating and replacing for several rounds until we get a reasonable fit.In this way, we find the smallest one from a large number of local minima.The estimation resulted into the following values: a 1 = 10, a 2 = 0.065 and a 3 = 2.The values of the parameter a 4 are year and chamber specific.We used a value of −3 • C for T f (T.Hölttä, personal communication, 2017).

Results
We predicted the state of the photosynthetic machinery, i.e. the annual state of enzymes, membrane pumps and pigments with Eq. (5) using the measured temperature and light intensity before the moment in consideration.The predicted annual patterns of the state of the photosynthetic machinery were quite similar between the different years (Fig. 1).There was, however, some weather-driven variation.For example, the very warm August in 2014 generated the large peak in late summer.
The changes in the relationship between light and photosynthesis are characteristic of our theory.Figure 2 depicts the daily patterns of the measured and predicted leaf CO 2 exchange early in the spring (Fig. 2a) and at midsummer (Fig. 2b).The measured and predicted daily patterns generated by the variation in light were very similar to each other, although the level of photosynthesis increased considerably from spring to summer.Our theory predicted the level of this increase during the summer successfully.
Days of intermittent cloudiness dominate our northern climate in the summer (Hari et al., 2014), giving rise to very strong within-day variations in the light levels reaching the canopy.Our theory predicted strong variation in photosynthesis during days of intermittent cloudiness, yet the measured leaf CO 2 exchange seemed to be very similar to the predicted one (Fig. 3a).Heavy clouds tend to cover the sky during rainy days, strongly reducing the light intensity.Our theory predicts strongly reduced photosynthesis during dark rainy days.Again, the measured and predicted leaf CO 2 exchanges were very close to each other when thick clouds covered the sky (Fig. 3b).
Our theory predicts the clear effect of partial closure of stomata on sunny days when the temperature strongly increases during the day.This type of days is, however, a rather rare event at our northern measuring site.Nevertheless, the measurements of leaf CO 2 exchange showed a similar pattern to our prediction on such days (Fig. 3c).
We have continuous measurements for four summers, consisting of more than 130 000 data points during each summer.The predictions of the leaf CO 2 exchange of a shoot were very close to the measured pattern, without exception.Also, the relationships between the measured and predicted leaf CO 2 exchange indicated close correlations between measurements and predictions (Fig. 4).The predictions explained about 95 % of the variance of the measured values.The residuals, i.e. the difference between the measured and predicted leaf CO 2 exchange revealed only slight systematic behaviour (Fig. 5) indicating that the theory was a quite adequate description of the regularities in the photosynthesis of northern Scots pine.

Discussion
Scots pine has a broad distribution range all over Europe, and the local populations have adapted to the regular annual cycle of solar radiation and temperature.The needle metabolism also has a clear annual cycle that alternates between the cold tolerance and very low metabolic activity during winter and the strong metabolism and cold vulnerability in summer.The annual cycle is particularly strong in photosynthesis (Ens-  , 2004b;Kolari et al., 2014;Öquist and Huner, 2003;Pelkonen and Hari, 1980).
We have worked decades with the annual cycle of vegetation from the analysis of daily shoot elongation (Hari and Leikola, 1972;Hari et al., 1977), bud burst of trees (Hari and Häkkinen, 1991) and photosynthesis (Pelkonen and Hari, 1980).Our approach has been dynamic modelling without a clear connection to the physiological background, although we were looking for the metabolic explanations.The strong connection to the light and carbon reactions and their basis on enzymes, membrane pumps and pigments is the novel feature of our theory of the annual cycle of photosynthesis.It provides sound physiological background to our concepts and axioms.We utilised strongly physiological knowledge in the development of our theory.Previously the focus has been in the mathematical formulation of the ideas, whereas the physiological background has been quite unclear.The predictions of our novel theory are close to those obtained previously (Mäkelä et al., 2004) although the fit of the predictions with measurements has improved considerably.
The light and carbon reactions of photosynthesis are downregulated in autumn in order to protect the sensitive machinery against low temperatures and upregulated again in spring.This seasonality has been closely connected to variations in ambient temperatures (Mäkelä et al., 2004;Pelkonen and Hari, 1980) and photoperiod or light intensity changes (Ensminger et al., 2004a;Porcar-Castell et al., 2008).A delayed effect of temperature on photosynthesis recovery in spring is introduced (Mäkelä et al., 2004;Pelkonen and Hari, 1980) and tested with field measurements (Kolari et al., 2009).
The Newtonian approach provided a sound backbone to collect physiological knowledge for the development of our theory of the annual cycle of photosynthesis.The definitions of concepts determine the most important features in the theory and the axioms the critical relationships between the con- cepts.The application of the mathematical analysis and simulations of the behaviour of the system, as defined by the concepts and axioms, proved to be an efficient tool to analyse the consequences of photosynthesis and to derive predictions.We defined new concepts, the biochemical regulation system and the state of the photosynthetic machinery (enzymes, membrane pumps and pigments) that played a very important role in the argumentation and are justified from the basic physiological understanding of the photosynthetic processes.Each step is based on a specific pigment, membrane pump or enzyme.In an efficient metabolic chain, the steps have to be in balance with each other.The biochemical regulation system, with emerged through evolution, generates a balance between the steps in photosynthesis, whereas its actions generate the state of the photosynthetic machinery.The state of the photosynthetic machinery determines the changing efficiency of the light and carbon reactions in photosynthesis.In this way, the action of the biochemical regulation system generates the annual metabolic cycle of photosynthesis and the synchrony with the strong annual cycle of radiation and temperature.
The axioms clarify the action of the biochemical regulation system in the synthesis and decomposition of the photosynthetic machinery.The physiological basis of the actions is clear.Metabolic reactions take place faster at elevated temperatures than at low ones.Thus, synthesis is temperature dependent (Axiom 2).The enzymes, membrane pumps and pigments are non-stable compounds as introduced in Axiom 3.
The increasing temperatures in the spring accelerate the synthesis and activation of the photosynthetic machinery, resulting in increased photosynthesis.The combination of sunny and cold mornings accelerates the decomposition and deactivation and thus decreases photosynthesis.When the spring proceeds, air temperature rises and the synthesis and activation increase the state of the photosynthetic machinery, resulting in enhanced photosynthesis.
The enzymes, membrane pumps and pigments are nonstable compounds, and, consequently, their decomposition and deactivation increases during summer, resulting in a quite stable state of the photosynthetic machinery.When the temperature starts to decrease according to the annual cycle, the synthesis declines, decreasing the pool of these nonstable compounds resulting in a reduction in the light response of photosynthesis.In this way the biochemical regulation system generates the annual metabolic cycle of photosynthesis that is in delayed synchrony with the annual cycle of radiation and temperature.
Our theory predicts a slow recovery in the spring, quite steady maximum in the summer and slow decline in the autumn to be characteristic of the annual cycle of photosynthesis due the synthesis, activation, decomposition and deactivation of the photosynthetic machinery.The observed annual patterns of photosynthesis are in good agreement with the above theoretical prediction.
The diurnal cycle of radiation and temperature is clear in summertime and missing during the polar night at our research site.However, we can omit the polar night in photosynthetic studies due to darkness and low temperatures.Our theory predicts that (i) photosynthesis during a day follows the saturating response to light, since the changes in the concentrations of enzymes, membrane pumps and pigments are so slow that the changes do not affect the behaviour of photosynthesis during a day; and (ii) the action of stomata slows down photosynthesis during most sunny days.Our field measurements are in agreement with this prediction.
Our theory has successfully passed the above qualitative tests.However, quantitative tests are more severe and they can provide stronger corroboration for the theory and show its universal character over a huge number of environmental conditions and several seasons.We tested our theory with field measurements over 4 years including over 130 000 measurements of CO 2 exchange, PAR, temperature, atmospheric CO 2 and water vapour concentration.Our theory predicted the annual and daily patterns of photosynthesis explaining about 95 % of the variance in the measured CO 2 exchange, whereas residuals did not show any clear systematic behaviour.Thus, our theory successfully passed the severe tests also in quantitative terms.As a next step and proof of its universal nature, we attempt to use the model developed for the branch scale to predict ecosystem-scale fluxes in several Scots pine forests in different ecoclimatic regions (see Hari et al., 2017).
The estimation of the parameter values is a challenge since the behaviour of the residual sum of squares is very irregular and there are numerous local minima disturbing the estimation with numeric methods.We therefore developed a method that selected the smallest one from a large number of residual sums of squares resulting in a quite stable solution of the minimisation.Further analyses would benefit from independent data sets from other sites in order to describe the variability in these parameters.
It is evident that the nitrogen availability (fertility) as well as plant species affects the parameter a 4 ; i.e. the higher the nitrogen content in the leaves in general is, the higher the relationship between the state of the photosynthetic machinery and the efficiency of photosynthetic carbon and light reactions is (Eq.7).On the other hand, parameter T f describing the temperature when the areas outside living cells freeze, is species specific and also somewhat site specific depending on the water content (Sperling et al., 2017).These events are rather rare but the sensitivity to such events is reflected in parameter a 3 .
Short field campaigns and statistical analysis of the obtained data dominates photosynthetic research under natural conditions.The often very short and fragmentary measurement series hinder the studies of the annual cycle of photosynthesis.The smoothly changing relationship between light and photosynthesis is missing in most statistical analyses of field measurements.The slow changes in the studied relationship are problematic for the statistical analysis of field data and probably explain why there is not any comparable ecological theory of the annual photosynthesis.
In conclusion, Scots pine has adapted to the regular annual cycles of light and temperature and the effective biochemical regulation system of the photosynthetic machinery has emerged through evolution.The action of the biochemical regulation system generates the delayed annual cycle of photosynthesis by synthesising, activating, decomposing and deactivating enzymes, membrane pumps and pigments.The linear relationship between synthesis and activation on temperature above the freezing point synchronises the metabolic and light cycles with each other.Prevailing light and the annual metabolic cycle determines photosynthesis, although the action by the stomata modifies the photosynthetic response.Our extensive field measurements corroborate the above conclusion.
Data availability.All measurements at SMEAR I including the shoot chamber measurements are available from https://avaa.tdata.fi/web/smart/smear/download.The code is available in Mathematica and can be accessed via the corresponding author (pertti.hari@helsinki.fi).
Competing interests.The authors declare that they have no conflict of interest.Special issue statement.This article is part of the special issue "Pan-Eurasian Experiment (PEEX)".It is not associated with a conference.

Figure 1 .Figure 2 .
Figure 1.The annual pattern of the state of the photosynthetic machinery (S, arbitrary units) during the years 2011-2014 in Finnish Lapland, 68 • N.

Figure 3 .
Figure 3. Measured and predicted leaf CO 2 exchange (a) during a day of intermittent cloudiness (5 August), (b) during a cloudy day (22 July), and (c) during a sunny day when the stomata close partially (7 July) in Finnish Lapland, 68 • N.

Figure 4 .
Figure 4. Relationship between the measured and predicted leaf CO 2 exchange in Finnish Lapland, 68 • N in the year 2013.The dashed line represents the 1 : 1 line.

Figure 5 .
Figure 5.The residuals as a function of temperature and PAR in the year 2013 in Finnish Lapland, 68 • N.