Optimal control of gas exchange

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Optimal control of gas exchange ARTICLE in TREE PHYSIOLOGY · JANUARY 1987 Impact Factor: 3.66 · DOI: 10.1093/treephys/2.1-2-3.169 · Source: PubMed

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4 AUTHORS, INCLUDING: Pertti Hari University of Helsinki 268 PUBLICATIONS 6,802 CITATIONS SEE PROFILE

Maria Holmberg Finnish Environment Institute 28 PUBLICATIONS 387 CITATIONS SEE PROFILE

Available from: Pertti Hari Retrieved on: 03 February 2016

Tree Physiology 0 1986 Heron

Optimal

2, 169-175 Publishing-Victoria,

(1986). Cunudu.

control of gas exchange

PERTTI HARP, HOLMBERG’

ANNIKKI

’ Department of Silviculture, 2 IIASA, A-2361 Loxenburg,

MAKELA2,

Universily Austria

of Helsinki,

EEVA KORPILAHTI’ Unioninkatu

40 B, 00170

and MARIA Helsinki,

Finland

Summary

Introduction The work of Gaastra (1959) began a new line in research on plant gas exchange and the coupling of transpiration and photosynthesis. A new theoretical approach was introduced by Cowan (1977) and Cowan and Farquhar (1977) and further developed by Cowan (1982). According to this treatment, the control of gas exchange is considered optimal when maximal amounts of carbohydrates are produced per unit of water transpired under the prevailing environmental conditions. If A denotes the photosynthetic rate per leaf area, E the transpiration rate per leaf area and t time, the optimal control problem can be expressed as: max

y (A(t) - A E(t)) dt

(1)

t1

where (t,, t2) is the time period considered and A is the amount of carbon consumed per water unit. Using the calculus of variations, Cowan (1977) and Cowan and Farquhar (1977) showed that for the optimal control of gas exchange

-= dA dE

A

The solution obtained by Cowan (1977) and Cowan and Farquhar (1977) seems difficult to relate to empirical observations (cf. Cowan 1982). The aim of the present paper is to reformulate the optimization problem to facilitate comparison of the model with empirical observations. This is achieved by introducing more Dejnition

qf synbols

with units given

at end

of urticle.

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A major difficulty in evaluating the optimization theory of leaf gas exchange under conditions of water deficit has been that of obtaining suitable experimental data. Mathematical solutions to three formulations of optimal stomata1 control are presented which can be tested experimentally. First, it is assumed that the movement of stomata and changes in environmental factors are slow compared to changes in the internal CO, concentration. The optimization problem is solved under this assumption, and the procedures for testing the solution experimentally are described. Second, instantaneous stomata1 response is postulated and the solution suggests that very rapid oscillations provide optimal CO2 uptake. Third, variable stomata1 dynamics are postulated and the mathematical solution shown to be similar to that of the second case. The second and third cases can also be tested empirically.

HARI

170

ET

AL.

detailed assumptions on photosynthesis and transpiration.

Model for gas exchange Photosynthetic carbon assimilation may be treated as a two-step process: first, the diffusion of atmospheric CO* into the leaf intercellular space; and second the biochemical process of carbon fixation. Thornley (1976) and Kaitala et al. (1982) have presented dynamic models describing these processes. Let Ci denote the intercellular CO2 concentration and C, the ambient CO2 concentration. The inflow of CO2 per unit leaf area is g(Ca - Cl)

(2)

f(I>

(3)

Ci

Expressions (2) and (3) can be combined to describe the dynamics of the intercellular carbon dioxide concentration dCi -= dt

g(C, - CJ - f(r) Ci h

(4)

where h denotes the mean thickness of the intercellular space of the leaf, and the concentration Ci is assumed constant throughout the leaf. Equation (4) represents the ideas of Gaastra (1959), Thomley (1976) and Kaitala et al. (1982) in a condensed and simplified form. The flow of water vapor is also a physical event. Let ei and e, denote the intercellular and ambient water vapor concentrations, respectively. The model for transpiration is E = u g(ei - e,)

(5)

where the symbol a is a constant with a value of 1.6. The model for transpiration also follows the approach first developed by Gaastra (1959). Let us assume that the primary regulation of gas exchange involves stomata1 conductance, g. For mathematical simplicity, let us denote: g = ugo

(6)

where g, is the conductance when the stomata are fully open, and u is a control signal, termed here the degree of stomata1 opening, that varies between 0 and 1. Three formulations of stomata1 control are now evaluated.

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where g is the stomata1 conductance (Gaastra 1959). The rate of photosynthetic carbon dioxide assimilation is proportional to leaf intercellular CO? concentration and also depends upon leaf irradiance, I. If f(Z) is the net rate of CO1 assimilation per unit leaf area and per unit of CO2 concentration, then the assimilation of carbon per unit leaf area is

OPTIMALCONTROLOFGAS

EXCHANGE

Slow stomata-first

171

formulation

If the movement of stomata is slow and if changes in the environmental factors affecting photosynthesis are slow, then the intercellular CO2 concentration as given by Equation (4) is relatively stable and can be approximated by the equilibrium value. This allows a simplification of the optimization problem. The equilibrium condition can be mathematically expressed by equating the time derivative dC,/dt to zero. Thus according to Equation (4): U go(Ca - Ci) - f(r) Ci = 0

(7)

The equilibrium value of Ci can be obtained from Equation (7) as a function of the degree of stomata1 opening u and of f(I>: Ci =

U&G wo + m

rn?

y (f(l>Ci - Aag,u(ei - e,)) dt

(9)

t1

The solution of the problem can be found by the calculus of variations. Substituting the equilibrium Ci from Equation (8) to Equation (9) yields the following Lagrangian function L

The optimal degree of stomata1 opening, u*, which maximizes function of Equation (9), can be found by requiring

the objective

(11) This yields: u* =

(J

CC! -1)F Aa(e, - e,)

Since requirement (11) is only a necessary condition for u* to be optimal, the solution must be further analyzed to confirm the optimality. By definition u takes values within the interval O-l, therefore Equation (12) can only be used if 05u*ll. Otherwise, the solution equals either zero or unity, corresponding to fully closed or fully open stomata. As Cowan (1982) points out, the stomata1 optimization hypothesis can only be applied to short intervals because of the diurnal rhythm in light intensity and

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This allows us to express the photosynthetic rate of Equation (3) as a function of the degree of stomata1 opening, U. The optimization problem of Equation (1) can now be expressed as follows

172

HARI

ET AL.

Testing the model Let us summarize the properties of the solution outlined above. 1. There is a threshold temperature above which the stomata start to close. 2. The threshold temperature increases with increasing partial pressure of atmospheric water vapor. 3. The threshold temperature increases with increasing light intensity. 4. The threshold temperature decreases with increasing soil water deficit. 5. The temperature range within which the optimal degree of stomata1 opening decreases from one to zero becomes narrower when the water deficit becomes more severe. Testing these characteristics of the model individually provides an indirect way of testing the model. All these results involve a threshold temperature above which the conductivity of the stomata decreases, and they indicate that the threshold and the slope of the decrease or both depend on light intensity, atmospheric water vapor pressure and soil water content. To test these hypotheses, it is necessary to identify the temperature dependence of stomata1 conductance at a range of values of the environmental variables. This means identifying temperaturedependence curves for different ambient water vapor levels, light intensity levels, and water deficit levels, and assessing whether the families of curves are consistent with the model. This can be done using standard statistical methods. Such considerations are quantitatively well-defined if all the variables involved

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temperature. A suitable time period is 24 hours. Taking this as the basis of further analysis we assume that the amount of water in the soil and water vapor concentration in the air are effectively constant. This allows the assumption that the cost of transpiration, A, measured as the amount of carbon required per unit amount of water transpired, is constant during each optimization period. The optimal degree of stomata1 opening depends on light intensity and temperature. Let us assume that light intensity is not limiting photosynthesis, i.e., f(Z) is maximal. The driving force for transpiration is the water vapor pressure difference between the intercellular space and ambient air. The ambient water vapor concentration changes only slowly in nature. Thus when considering intervals of 24 h, the ambient water vapor concentration can be assumed to be constant. The leaf temperature determines the intercellular water vapor concentration and thus the driving force of transpiration. The optimal degree of stomata1 opening can be examined as a function of temperature using the dependence of saturated water vapor concentration on temperature. The solution has the value 1 near the dew point of air. When temperature rises from the dew point, ei increases and there is a threshold temperature at which U* becomes smaller than unity. Above this temperature U* falls nearly linearly to zero. The threshold temperature decreases rapidly with increasing cost of water, A, and increases with increasing light intensity, I.

OPTIMAL

CONTROL

OF GAS

EXCHANGE

173

Other formulations The assumption of Equation (7) that the intercellular CO2 is at equilibrium cannot always be justified. On the contrary, the canopy is often characterized by rapid variations in the light intensity falling on the leaf. This can be accounted for by utilizing Equation (4) for the flow of carbon dioxide, in its dynamic form. In this formulation the optimization problem of Equation (1) can be solved using the optimal control theory. The Hamiltonian, corresponding to the Lagrangian in Equation (lo), is H = Cif(l> + hUg,(ei - e,) + P (Ugo(Ci - CJ - Cif(Z))

(13)

Because the Hamiltonian is linear with respect to the degree of stomata1 opening, the solution is bang-bang, i.e., the control signal is either zero or one and changes instantaneously (Kirk 1971, Luenberger 1979). Depending on the environmental conditions, this model predicts that optimal control leads to high frequency stomatal oscillations because no guard cell inertia is assumed. A third formulation assumes a maximum speed of stomata1 movement. This gives rise to the following reformulation of the optimization problem. Let g denote the stomata1 conductance. Assume that the plant controls the movement of stomata according to the following:

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can be measured. The degree of stomata1 conductance can be measured indirectly by monitoring photosynthesis and transpiration. Direct measurements can be obtained for temperature, partial pressure of ambient water vapor and light intensity. However, the soil water deficit, which is related to the parameter h in the model, has not been quantified. The water content of the soil can be used as an estimate of the relative changes in A, although the absolute values remain unspecified. When combining the measurements of metabolic processes and environmental factors, we should take special care because the former measurements involve time and space integrals, the latter are instantaneous values (cf. Hari et al. 1983). When designing the experiments, the assumptions of the derivation of the model should not be violated. Since it was assumed that, partial pressure of ambient water vapor, and soil water deficit, are constant over the optimization period, it is necessary that the plant monitored has attained a steady state with respect to those factors. It may take hours or days before the plant can be assumed to have reached a steady state after a change in water deficit. The relatively slow changes in natural partial pressure of water vapor suggest that at least 24 h should elapse between two series of measurements with a major change in water vapor. A threshold temperature for reduction in photosynthesis has been repeatedly found in field data (Hari and Luukkanen 1973, 1974, Hallman et al. 1978, Pelkonen 1980). This is consistent with inference (1) above. Inference (4) above similarly gains support from empirical observation (Hari and Luukkanen 1973).

HARI

174

Ak= dt

ago

ET AL

- RI

where u is a control signal having values between - 1 and + 1. The Hamiltonian of the optimization problem is now H = Cif(O + hg(e,- ea> + PI (g(Ci-

Ca) - Cif(l>) + Pzug(ei - e,> (15)

References Cowan, I.R. 1977. Stomata1 behaviour and environment. Adv. Bot. Res. 4:117-228. Cowan, I.R. and G.D. Farquhar. 1977. Stomata1 function in relation to leaf metabolism and environment. In Integration of Activity in the Higher Plant. Ed. D.H. Jennings. Univ. Press., Cambridge. pp 471-505. Cowan, I.R. 1982. Regulation of water use in relation to carbon gain in higher plants. In Encyclopedia of Plant Physiology, New Series. Vol. 12B. Eds. O.L. Lange, P.S. Nobel and C.B. Osmond. Springer-Verlag, Berlin. pp 589-613. Gaastra, T. 1959. Photosynthesis of crop plants as influenced by light, carbon dioxide, temperature and stomata1 diffusion resistance. Mededelningen van de Landbouwhogeschool de Wageningen, Nederland 13:1-68. Hallman, E., P. Hari, P. Rasanen and H. Smolander. 1978. The effect of planting shock on the transpiration, photosynthesis, and height increment of Scats pine. Acta For. Fennica 161. Hari, P. and 0. Luukkanen. 1973. Effect of water stress, temperature, and light on photosynthesis in alder seedlings. Physiol. Plant. 29:45-53. Hari, P. and 0. Luukkanen. 1974. Field studies of photosynthesis as affected by water stress, temperature and light in birch. Physiol. Plant. 32:97-102. Hari, P., R. Sievanen and R. Salminen. 1983. On measuring in plant ecological studies. Flora 173:63-70. Kaitala, V., P. Hari, E. Vapaavuori and R. Salminen. 1982. A dynamic model for photosynthesis. Ann. Bot. 50:385-396. Kirk, D.E. 1971. Optimal control theory. An introduction. Prentice-Hall, London. 453 p. Luenberger, D.G. 1979. Introduction to dynamic systems theory, models and applications. John Wiley and Sons, New York. 456 p. Pelkonen, P. 1980. The uptake of carbon dioxide in Scats pine during spring. Flora 169:386-397. Thornley, J.H.M. 1976. Mathematical models in plant physiology. Academic Press, London. 318 p.

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The Hamiltonian is again linear in U, the solution of the optimization problem is again bang-bang (Kirk 1971). The difference from the second formulation is that the time required for the movement is explicitly accounted for by Equation (14). These two latter examples illustrate the potential of the optimization theory in analyzing leaf gas exchange. In all three formulations of the optimal control of gas exchange, the predictions can be tested empirically. A more detailed analysis requires a more realistic description of the internal structure of the leaf, and should account for physiological processes other than photosynthesis. Based on the present study the authors believe that further formulations of the optimization hypothesis will yield physiologically plausible models that can be tested empirically.

OPTIMALCONTROLOFGAS

List

of Symbols

A E e, e, h C, C, g go I h u

(gm *s-l) (grnm2s.’ 1 km-7 (g m-7 (gmmZs-’ 1 (g m-7 km-‘1 (m s-l) (m s-‘) (W mmZ) (ml

EXCHANGE

photosynthetic rate per leaf area transpiration rate ambient water vapor concentration intercellular water vapor concentration amount of carbon consumed per water intercellular CO, concentration ambient CO, concentration stomata1 conductance conductance of open stomata light intensity mean thickness of intercellular space degree of stomata1 opening

175

unit

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