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1 The SRIV algorithm is particularly appropriate for this kind of analysis, as it is based on a special ‘instrumental variable’ technique, which includes adaptive prefiltering of the IO data. This ensures that the identification and estimation analysis is very robust to noise on the experimental data, and works particularly well if the input stimuli are step (or impulsive) in form, as in the present study.
Single leaves of 3-month-old Cedrella odorata seedlings were exposed to a step reduction in the ambient dew point. The resultant time series of dynamic variations in leaf surface water vapour concentration, leaf surface water vapour concentration gradient, transpiration rate and stomatal conductance to water vapour, are analysed using the data-based mechanistic (DBM) modelling methodology of Young (e.g. Young & Lees 1992; Minchin et al. 1996 ). It is shown that the identified second-order, dynamic model between transpiration rate (as the input) and stomatal conductance (as the output) provides an appropriate, physiologically meaningful, description of the system. In particular, the dynamic relationship between these two variables is remarkably linear and can be resolved in terms of two parallel, first-order, subsystems; a model which complements the results of Cowan (1977) for cotton. The model is also compared with the recently published simulation model of Haefner, Buckley & Mott (1997).
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Because ambient humidity exerts a significant influence on stomatal aperture and, hence, the conductance of leaves to either water vapour (g(t)) or CO2, much effort has been focused on gas exchange measurements of g(t) in response to variations in humidity. Most of these measurements have been made in the steady state, but it is often observed that, under certain conditions, an excitation in humidity causes oscillations in g(t) [see Barrs (1971)]. This dynamic behaviour alludes to the functioning of feedback(s) in the response of stomata to changes in humidity and, therefore, provides a seductive insight into the mechanisms governing the system. Here and subsequently, the use of the term ‘system’ relates to the overall description of the response of stomata and the rate of water loss, E(t), to changes in humidity.
Because a wholly static analysis cannot address the fundamental dynamic nature of stomatal homeostasis, it seems more appropriate to utilize a dynamic systems approach to investigate the nature of the dynamic mechanisms that underlie the stomatal regulatory process. Surprisingly, however, there are relatively few examples of research into this topic that have exploited dynamic systems methodology. Farquhar (1973) described some experimental system identification of stomatal responses to both humidity and CO2 using sinusoidal perturbations in association with a frequency domain analysis. Also, Farquhar & Cowan (1974) manipulated the ‘environmental gain’ of the system in order to control the stability of the system response. Several simulation models of stomatal movement dynamics have been published which generate response characteristics that are qualitatively similar to the observed system dynamics (e.g. Cowan 1972; Delwiche & Cooke 1977; Haefner, Buckley & Mott 1997).
Cowan (1977), developing Farquhar’s (1973) earlier work, recognized the potential of using a more objective systems identification methods based on the estimation of transfer functions (TFs) from experimental data. Cowan (1977) assumed that g(t) responds to E(t) and, therefore, he estimated the dynamic relationship between the resultant perturbations of E(t) and g(t) [which we will denote by ΔE(t) and Δg(t)] following a perturbational step increase in the leaf surface water vapour concentration gradient, ΔD(t). He proposed that, for cotton, this relationship was well described by a second-order TF composed of two, first-order, subsystems operating in parallel to one another: namely, a rapid stomatal opening and a persistent stomatal closure (with estimated time constants of 1·6 and 16 min, respectively). Cowan’s (1977) choice of E(t) as the input is upheld by the findings of Mott & Parkhurst (1991) and the analysis of Monteith (1995), but it can be criticized at higher values of D(t) [see Farquhar (1978)]. Also, Cowan (1977) concluded that, unless the modeller is very careful when identifying dynamic relationships between variables, it is easy to derive TFs that fit the data well yet may bear little relation to the processes being described.
Young and coauthors (e.g. Young & Lees 1992; Young 1993, Young & Beven 1994; Minchin et al. 1996 ; Young, Parkinson & Lees 1996) have developed a data-based mechanistic (DBM) method for modelling stochastic, dynamic systems which appears to satisfy Cowan’s (1977) requirement for a ‘proper statistical approach’ that avoids the kind of TF identification problems he describes. This DBM approach attempts to infer the nature of the dynamic system from the experimental data without undue reliance on a priori assumptions about the nature and order of the system. In this manner, it seeks to avoid over-reliance on current paradigms and minimizes the possibility that prejudicial judgements will be made in the formulation of the model. In particular, because the order of the TF model is identified statistically, by direct reference to the data, the DBM approach tends to yield a minimally parameterized or ‘parsimonious’ model that is well identified from the data and so avoids the possibility of over-parameterization and non-uniqueness.
The present paper employs DBM modelling methods to analyse data obtained from gas exchange experiments on leaves of Spanish cedar (Cedrella odorata). The aim of the analysis is to define the appropriate system inputs and outputs and to identify what mechanistic information is observable within a particular set of gas exchange time series. In this manner, we show that the dynamics of the stomatal response to a decrease in humidity is remarkably linear, with ΔE(t), rather than ΔD(t), being the appropriate input for the applied conditions ( Monteith 1995). Also, a similar parallel pathway model to that identified for cotton by Cowan (1977) is observed for Cedrella.
MATERIALS AND METHODS
Seedlings of C. odorata were sown in a 2:2:1 mix of John Innes, peat and fine grit with a granular slow release fertilizer (Osmocote 16:8:12; Scotts, Nottingham, UK). The plants were raised for 2 months in a cabinet with conditions of 12 h day, 450 μmol m–2 s–1 PPFD at the top of the plant supplied by metal halide lamps, 40–50% relative humidity, 350–400 p.p.m. CO2 and 22–28 °C. At 2 months single plants were placed in a controlled environment where the conditions around the shoot were 600 μmol m–2 s–1 PPFD at the top of the plants (supplied by metal halide lamps), 24 ± 0·3 °C day/night temperature, 400 ± 10 p.p.m. [CO2] and 50 ± 2% relative humidity.
Single, fully expanded leaflets were sealed into a 0·25 dm3 leaf cuvette which formed part of an open gas exchange system. Input and output air dew points were measured using Edgetech 911 dew point mirrors (Edgetech, Milford, USA), leaf temperatures were measured using an Omega 40 gauge thermocouple (Omega, Stanford, USA) and a Comark 6600 thermocouple meter (Comark, Rustington, UK). The cuvette temperature was 22·5 °C. The outlet dew point was controlled by means of a small pump diverting a proportion of the humidified inlet air through a desiccant column. The pump rate was controlled by the PC used to data log all instrument outputs. Outlet CO2 concentrations were monitored by a WMA1 single channel IRGA (PP Systems, Hitchin, UK) and controlled at 400 p.p.m. by varying the input of 1% CO2 from a FC260 mass flow controller (Tylan, Swindon, UK) again controlled by the PC. The dry air flow rate was set at 1·4 dm3 min–1. The cuvette boundary layer conductance was estimated as 1·1 mol m–2 s–1 using wet filter paper leaf replicas. Calculations of E(t) and g(t) were made using the equations of von Caemmerer & Farquhar (1981). In addition, because the measurements are non-steady state, the rate of change of water vapour within the gas exchange system during the initial reduction in cuvette dew point has been accounted for. This was achieved by estimating the apparent mixing volume [see Young & Lees (1993)] of the gas exchange system from the plant-free response to the reduction in cuvette dew point. The mixing volume is ‘apparent’ as it accommodates not only the leaf chamber volume but also desorption of water from the walls of the system and the lag of the dew point mirrors.
When equilibrium was reached in g(t) 3 h after installing the leaf in the leaf chamber, the dew point of the air within the cuvette was decreased from 16·5 to 12·5 °C which, on average, corresponded to a steady-state increase in D from 0·011 to 0·014 mol mol–1. The discrete-time transient responses of E(t) and g(t), at a regular sampling interval of Δt = 2 min [denoted here by E(k) and g(k)], were then logged for approximately 90 min until a new equilibrium was approached.
The quantitative analysis utilized here assumes that the observed inputs and outputs of a system are a manifestation of the dominant dynamic processes occurring within the system under study. On the basis of a fairly general class of TF models (linear or non-linear, depending upon the nature of the system), it is then possible to statistically identify and estimate an objective and parsimonious mathematical description of these processes, as excited by the input stimulus. Following the DBM modelling philosophy, the resultant input–output (IO) model is then open to detailed examination, with the object of inferring the nature of the physical processes it represents. In the present study, the IO models are formulated as discrete-time TFs relating present and past regularly sampled values of the input and output variables [although the DBM analysis is not restricted to such models and could directly utilize alternative continuous-time, differential equation models: see, e.g. Young (1996)]. Because the data to be analysed here are discretely sampled at intervals of Δt = 2 min, the TFs are estimated and expressed in finite difference form where the output Y(k) is related to past values of itself and the input U(k),
where d is any pure time delay that may exist between the input and its first effect on the output. Introducing the backward shift operator, z–1, where z–1Y(k) = Y(k–1), this model can be represented in the following TF form,
where Y(k) and U(k) are, respectively, the system outputs and inputs at the kth sampling instant and the numerator and denominator polynomials in z–1 define the TF between the variables. The steady-state or equilibrium condition for Eqn 2 is defined when Y(k) = Y(k – 1) = z–1Y(k), so that the steady-state gain (SSG) of the TF is obtained by setting z–1 = 1·0 in the TF, i.e.
Also, if the denominator polynomial in the TF has positive real roots λi, i = 1, 2, . . . n, the time constants of the n dynamic modes are given by [see, e.g. Young (1984)],
An appropriate or ‘well-identified’ TF is defined as one which provides an adequate explanation of the data and yet uses the minimum number of a and b parameters to do so. This goal is achieved by using the simplified refined instrumental variable (SRIV) estimation algorithm [see, e.g. Young (1984, 1985)]1 to estimate a broad range of TFs, as defined by the integers n, m and d; and then ranking the TFs with respect to an index which ensures a well-identified, parsimonious model. Here, such TF order identification is made principally, but not solely, on the basis of the logarithmically defined Young information criterion [YIC: see, e.g. Young (1991)], the most negative value of which normally provides a good balance between the model’s ability to explain the experimental data (as indicated by a coefficient of determination r2 near to unity) and the need for well-defined parameter estimates (i.e. low parameter estimation error variance). To simplify the estimation procedure, the experimental input stimulation is applied to the system once quasi-steady-state conditions have been reached; and the initial value of the inputs and outputs are subtracted from the input and output time series prior to SRIV estimation. In this manner, all the initial conditions are zero and the analysis is based on perturbational variables. The analysis is of the mean response of seven excitations performed on single leaves of seven different plants.
Figure 1 shows the time series of the leaf surface water vapour concentration (W(k)), together with D(k), E(k) and g(k), following the decrease in cuvette dew point from 16·5 to 12·5 °C. Table 1a shows the three TFs estimated with ΔE(k) as the input and Δg(k) as the output (ΔE(k) = E(k)–E(0) and Δg(k) = g(k)–g(0)) that gave the lowest YIC values out of the 108 TF structures estimated over the range n = 1:6; m = 1:6; d = 1:3. Table 1b shows the three TFs estimated with ΔD(k) as the input and Δg(k) as the output (ΔD(k) = D(k)–D(0)) that gave the lowest YIC values out of the 108 TF structures estimated over the same range of n, m and d.
Table 1. . (a) Identification results using the first 35 data points in Fig. 1c and d, with ΔE(k) as the input and Δg(k) as the output where ΔE(k) and Δg(k) are the perturbations in E(k) and g(k) following the reduction in cuvette dew point from 16·5 to 12·5 °C. The three transfer functions (TFs) shown gave the lowest values for the Young information criterion (YIC) index out of a total 108 estimated. (b) Identification results using the first 35 data points in Fig. 1b and d, with ΔD(k) as the input and Δg(k) as the output where ΔD(k) and Δg(k) are the perturbations in D(k) and g(k) following the reduction in cuvette dew point from 16·5 to 12·5 °C. Again, the TFs shown gave the lowest YIC index out of a total 108 estimated. Parameter estimates were obtained using an en bloc implementation of the simplified refined instrumental variable (SRIV) algorithm and are accompanied by associated standard deviations in parentheses. The steady-state gain (SSG) of the identified TFs is also given. The mean response of seven excitations taken from single leaves of different plants was used for the analysis. Δt = 2 min. Information on YIC and SRIV is given in Young (1984, 1991)
From Table 1a we see that the estimated YIC index indicates that the dynamic relationship between ΔE(k) and Δg(k) in Fig. 1 is well described by a n = 2, m = 2, d = 0 TF (denoted by the triad [2,2,0]). There is a marginal improvement in r2 with the inclusion of an additional b3 parameter; however, this is more than offset by an increase in uncertainty on parameter estimates, as indicated by the increased YIC value for the [2,3,0] TF (see Table 1a). Similarly, from Table 1b, we see that the estimated YIC index indicates that the dynamic relationship between ΔD(k) and Δg(k) in Fig. 1 is well described by a [2,2,0] TF. Again, there is a marginal improvement in r2 with the inclusion of an additional b3 parameter, which is more than offset by an increase in parameter estimate uncertainty associated with the [2,3,0] TF (see Table 1b).
From Tables 1a and b, we can conclude that, for the decrease in cuvette dew point from 16·5 to 12·5 °C, the dynamic relationship between ΔE(k) and Δg(k) may be provisionally described by,
whilst the dynamic relationship between ΔD(k) and Δg(k) may be provisionally described by,
Figure 2 shows the respective fits of both identified TFs. The denominator of Eqn 5 has two real roots of 0·8364 and 0·6907, giving time constants of 11·20 and 5·41 min, respectively (see Eqn 4); while the denominator of Eqn 6 has two complex roots of 0·7838 ± 0·5159i. Consequently, decomposition of Eqn 5 into two first-order systems yields two possible real solutions: one a parallel configuration; and one a feedback configuration, as shown diagrammatically in Figs 3a and b, respectively. On the other hand, Eqn 6 can only be decomposed into one system involving first-order processes with real roots: namely, a feedback configuration as shown in Fig. 3c.
The model identification, estimation and decomposition into parallel and feedback structures provides the basis for the last stage of DBM modelling: namely, the mechanistic interpretation of the identified system in terms of plant physiology. In this section, we consider various aspects of this interpretation and compare the results with previous published research on this topic.
What is being observed?
Having identified and estimated TFs between potential input and output states, it is important to understand what Eqns 5 and 6 may represent in terms of stomatal physiology. g(k) is a measure of the mean dynamic behaviour of some 1 million stomata (stomatal density ≈ 366 mm–2, leaf area ≈ 30 cm–2). The first question we must ask, therefore, is: are the dynamic variations we have observed in g(k) a result of a true shift in the population mean behaviour with approximately constant variance ( Laisk, Oja & Kull 1980), i.e. stomatal movements are synchronized across the observation leaf; or, are we observing the de-synchronization over time of an initially synchronized stomatal response? In the absence of any measure and because the experimental conditions (moderate D(k), well-watered plants) were such that the probability of patchy stomatal behaviour is reduced, it will be assumed that stomatal movements are synchronized across the observation leaf. In other words the TFs identified in Table 1 are assumed to represent a leaf average estimate of the physiological effect of D(k), or E(k), on the mean stomatal aperture, with approximately constant variance ( Laisk et al. 1980 ). This last statement highlights another important assumption: namely, that g(k) is an appropriate measure of stomatal aperture and, hence, accurately reflects the stomatal mechanisms which we wish to identify. Parlange & Waggoner (1970) demonstrated analytically that, in the absence of an antechamber and when the interstomatal spacing is greater than three times the stomatal pore length, then a degree of linearity between stomatal aperture and g(k) is to be expected. With Cedrella, scanning electron microscopy showed no antechamber and an interstomatal spacing greater than three times the stomatal pore length.
Defining the appropriate system input
We need to decide which of the functional relationships in Eqns 5 and 6 is the most appropriate description of the physiology of the stomatal response to humidity. E(k) and D(k) are measured variables, whilst g(k) is inferred from these measurements under the assumption that evaporation from the leaf surface obeys Fick’s law, i.e.
In considering how Eqn 7 may relate to the physiological response of stomata to humidity, two scenarios can be envisaged: one ( Fig. 4a) demonstrating feedback through the effects of the physiological output g(k) on the input E(k); the other ( Fig. 4b) suggesting feedforward of the physiological effects of D(k) on g(k) and E(k), or some combination of the two ( Farquhar 1978). Because Eqn 6 can only be resolved in terms of feedback, it is likely that it is a linear approximation of the scenario in Fig. 4a, incorporating both the physiology of the stomatal response and the effects of stomata on the rate of water loss, as described by Eqn 7. If this is so, then Eqn 5 would represent the open-loop physiological behaviour of the system.
A test of the validity of Eqns 5 and 7 and Fig. 4a as complete system descriptions is to estimate the closed-loop response of the system using only the measured variables, i.e. a measure of humidity and E(k). For this analysis, we can assign the perturbations in the leaf surface water vapour concentration, ΔW(k) = W(k)–W(0), as the input and ΔE(k) as the output, as this results in a convenient cancellation in the initial conditions W(0) and E(0), as shown in Appendix A. Here, the water vapour mole fraction inside the leaf, Wl, is treated as a constant (0·0305 ± 0·0007 mol mol–1), as the measured variations in leaf temperature were small (< 0·4 °C). If Eqns 5 and 7 are valid descriptions of the system, then one would expect to identify a [2,3,0] TF between ΔW(k) and ΔE(k); and, from the parameter estimates, comparable estimates for the parameters in Eqn 5, as well as an estimate for g(0), should be obtained (see Appendix A). Table 2 shows that, indeed, a [2,3,0] TF does yield the lowest YIC out of the range of 108 estimated TFs. From Eqn A8, a1 and a2 are not true, fixed parameters but are, instead, dependent on D(k). However, the variance associated with these estimates is small (see Table 2), implying negligible variation because, in absolute terms, the changes in D(k) are also small (0·011–0·014 mol mol–1).
Table 2. . Identification results using the data in Fig. 1a and c, with ΔW(k) as the input and ΔE(k) as the output where ΔW(k) and ΔE(k) are the perturbations W(k) and E(k) following the reduction in cuvette dew point from 16·5 to 12·5 °C. The three transfer functions (TFs) shown gave the lowest for the Young information criterion (YIC) index out of a total 108 estimated. Parameter estimates were obtained using an en bloc implementation of the simplified refined instrumental variable (SRIV) algorithm and are accompanied by associated standard deviations in parentheses. The steady-state gain (SSG) of the identified TFs is also given. The mean response of seven excitations taken from single leaves of different plants was used for the analysis. Δt = 2 min. Information on YIC and SRIV is given in Young (1984, 1991)
The estimated TF between ΔW(k) and ΔE(k) takes the form,
Equating the parameters in Eqn 8 with those in Eqns A8a–A8e yields = 0·125 mol m–2 s–1 (note g(0) = 0·126 mol m–2 s–1 in Fig. 2) and,
which is in reasonable agreement with Eqn 5 as can be seen by the respective step responses in Fig. 5, indicating the efficacy of Eqns 5 and 7 as representations of the closed-loop system response. The slight differences in the parameter estimates between Eqns 5 and 9 falls within the variance on the parameter estimates. Therefore, for the given set of conditions considered in our experiments, we propose that Eqn 5 can be considered as the appropriate description of the open-loop physiological aspects of the system response; whilst Eqn 6 is a linearized description of the overall closed-loop system response, i.e. Eqns 5 and 7 combined.
From Fig. 2 and Table 1, we see that fixed parameter, linear TFs are able to provide an excellent, parametrically efficient explanation of the observed dynamic behaviour. Although there is considerable potential for non-linearities to be present within the system response (e.g. the effects of additional inputs, such as variations in the rate of photosynthesis elicited by variations in g(k); Jarvis & Davies 1998), these have apparently not been greatly excited by the applied level of disturbance in the cuvette dew point. Subsequently, any such processes can be represented as fixed parameters within the identified TFs. Following a re- analysis of many published data sets, Monteith (1995) noted that, for many species, linear relationships between increases in leaf transpiration rate E(k) and reductions in stomatal conductance to water vapour g(k) are commonly observed in steady-state experiments when D(k) is varied. This re-enforces the findings of Mott & Parkhurst (1991) that g(k) can be shown to respond to a disturbance in E(k) even when D(k) is constant. It has been observed that, in the steady state, the linearity between E(k) and g(k) can break down at high values of D(k) (e.g. Schulze et al. 1972 ), a condition represented by Figs 4a and b combined. The analysis presented here suggests that the imposed increase in D(k) used in our experiments is not sufficient to make such a response observable within the data in Fig. 1. What is observed, however, is that the linearity observed in the steady state ( Monteith 1995) is also observed in the transient response. The observed linearity could result from looking at the aggregated behaviour of a large population of stomata. However, observations of single stomata have revealed behaviour similar to that shown in Fig. 1d following a reduction in humidity (e.g. Mott, Denne & Powell 1997). A possible explanation for the observed dynamic linearity would be that, as with man-made systems, robustness attained from a degree of linearization of the control of transpiration by stomata could be viewed as advantageous.
Inspection of the fit of Eqn 5 in Fig. 2 shows a small but statistically significant over-estimate of g(n) with time at the end of the response. Because Eqn 5 was estimated from the first 35 data points, the parameter estimates are associated with these data and the model error we see in Fig. 2 could suggest a slow change in the TF parameter values after 35 samples, or approximately 1 h, implying the physiological gain between E(k) and g(k) becomes more negative. This effect is accentuated by the feedback within the system implicit in Eqn 7. Equation 6 is less prone to drift in the physiological gain as g(k) has much less of an effect on D(k) than E(k). Change in the physiological gain of the system could explain hysteresis when humidity is decreased then increased ( Franks, Cowan & Farquhar 1997). Because the conditions around the shoot as a whole are near constant, we are led to conclude that the disturbance in the transpiration rate of the observation leaf results in a slow change in the internal state of that leaf and, hence, the system behaviour. A possible candidate for this trend might be increased synthesis and release of antitranspirants, such as ABA, within the leaf in response to an increase in E(k) and, hence, water potential-induced stress. The likelihood that the imposed increase in E(k) affected the shoot as a whole is small, as the observation leaf constitutes less than 2% of the total transpiring surface area of the plant.
Parallel pathway model
Equation 5 has two distinct time constants of 11·20 and 5·41 min, with associated gains of –586 and 386 when decomposed into two parallel first-order subsystems by partial fractions (see Fig. 3a). Using an alternative approach to model identification and estimation based on frequency domain methods, Cowan (1977) identified a similar second-order continuous-time TF for cotton in response to a step in D(k) from 0·014 to 0·018 mol mol–1, with time constants of 16 and 1·6 min and associated SSGs of – 43 and 36, resulting in an overall SSG of –7. As noted earlier, the partial fraction decomposition of Eqn 5 is not a unique solution in terms of first-order subsystems, as one could equally decompose this TF model into a feedback structure ( Fig. 3b). However, the feedback decomposition is conceptually harder to resolve in terms of plant physiology than the parallel structure, as it suggests that there is an unstable effect of g(k) on the input, independent of Eqn 7. Therefore, we will provisionally accept and develop Cowan’s (1977) chosen decomposition, whilst bearing in mind the possibility that an alternative solution does exist. Finally, Cowan (1977) also estimated a 0·7 min pure time delay associated with the 16 min time constant response for cotton. Here the sampling interval (Δt) has been restricted to 2 min to avoid the uncertainties associated with the initial gas exchange measurements immediately following the applied excitation in cuvette dew point. Consequently, any delays within the plant response less than 2 min will not be identified by our analysis.
Environmental gain and system dynamics
Equations 5 and 7 describe a simple feedback system for controlling water loss from leaves. In the absence of the boundary layer effects, D(k) represents the gain (termed the ‘environmental gain’; Farquhar 1973; Farquhar & Cowan 1974; Cowan 1977) of stomatal aperture (and hence g(k)) in eliciting a change in E(k). As pointed out by Cowan (1977), the dynamic behaviour of the system’s closed-loop response is determined by the denominator of the TF in Eqn A7 which is a function of D(k) as shown in Eqn A8. Figure 6 shows the effect of increasing D(k) on the natural frequency (ωn) and damping ratio (ζ) of the TF in Eqn A8 where ωn is given by,
and ζ is given by,
Assuming that the system is linear across a range of D(k), sustained oscillations in the closed-loop response (i.e. ζ = 0) will be reached when D(k) = 0·0178 mol mol–1 (see Fig. 6), which is close to 0·019 mol mol–1 estimated experimentally by Farquhar & Cowan (1974) for cotton. D(k) will vary in time and space, depending on the prevailing environmental conditions. This may present a considerable problem to the plant, as the efficiency of the stomatal aperture as an actuator for controlling water loss must also vary accordingly. Farquhar (1973) speculated that the oscillatory behaviour of the system arising from the non-minimum phase or ‘wrong way’ behaviour of stomata in association with environmental gain might act to search for the appropriate system position, given the uncertainties in variations in the plant’s environment.
Choice of system excitation
From Eqn 7 we see that E(k)∝g(k) if D(k) is constant. Subsequently information on the dynamic relationship between E(k) and g(k) will be restricted to the initial, most uncertain data if a highly controlled step in D(k) is used as a means of exciting and identifying the dynamic relationship between E(k) and g(k). Such an excitation is, therefore, a rather poor choice; whilst varying D(k) by, say, conducting a step in E(k) would be more appropriate. Indeed, the identification of Eqn 5 here has only been facilitated through the inadequacies in the control of cuvette dew point resulting in variations in D(k).
Recently, Haefner et al. (1997) presented a simple, linearized dynamic model of stomatal function in response to changes in D(k), a one-dimensional expression of which can be compared with Eqn 5 and Fig. 3a. Although the Haefner et al. (1997) model is speculative and, unlike Eqn 5, is not obtained by rigorous identification and estimation from experimental data, it does provide a framework within which we might offer a provisional but testable mechanistic interpretation to Eqn 5. First, in order to directly compare the two models, we convert Eqn 5 into the approximate, continuous-time equivalent form assuming the input remains constant over each sampling interval, i.e. the continuous-time system is in series with a zero-order hold, which is equivalent to the analogue to digital converter used for data logging by the PC [see, e.g. Nise (1995, pp. 718–719)]. With time units in minutes this yields,
where s is the Laplace operator, equivalent to d/dt if there are zero initial conditions, and s = (– ln z–1)/Δt. The partial fraction expansion of the continuous-time TF yields,
Considering perturbations about the steady state, the model presented by Haefner et al. (1997) can be expressed (see Appendix B) as,
where q scales the stomatal aperture to conductance; φ converts volumes to water potentials; Cg and Ce are the mechanical influence coefficients of the guard cells and epidermal cells, respectively; L is the hydraulic conductivity of the water flux pathway to the epidermis from some arbitrary reference; α–1 is the time constant for osmotic equilibrium of the guard cells; and β is the sensitivity of guard cell solute resting levels to epidermal turgor. In words, Haefner et al. (1997) follow Cowan (1972) and attribute the non-minimum phase rapid opening response to the mechanical advantage of the epidermis over the guard cells, following transpiration from the epidermis; whilst the slower closing response arises from the loss of solute from the guard cell to the epidermis, in response to epidermal turgor ( Haefner et al. 1997 ). The structural configuration of Eqns 13 and 14 is summarized in Fig. 7.
From Eqns 13 and 14 and Fig. 7, we see that there is a structural similarity between the mechanistic and data-based models, as both can be expressed as the sum of two subsystems (although it will be noted that none of the mechanistic parameters in Eqn 14 is uniquely identifiable). In the Haefner et al. (1997) model, because the slow time constant closing of stomata occurs in response to the faster time constant changes in epidermal water relations, one of the subsystems is serially configured and second order; so that the overall physiological response is third order. Although a third-order system has not been identified for the data in Fig. 1, it is possible that the identified slow time constant of Eqn 5 could be viewed as an approximation of a serially configured second-order subsystem. However, a simpler assumption that is consistent with the data would be a dependence of the desired set point guard cell osmotic pressure, πgd, on E(k), independent of epidermal turgor, i.e.
where πgm is the maximum guard cell osmotic pressure and β is the sensitivity of πgd to E. A mechanistic justification for this assumption might lie with a direct effect of E(k) on the mass balance of antitranspirants in the proximity of the guard cells ( Jarvis & Davies 1997). Equation 14 then becomes,
Now α is an identifiable parameter (α = 0·0893) from our model Eqn 13. If, as suggested by Eqn B5, the rate-limiting step for stomatal closure is determined by the rate of solute leakage from the guard cell, then α would relate closely to the conductivity of the guard cell plasmalemma to this solute flux; a process modulated by ion channel activity. A perturbation in epidermal water relations at constant E should provide an experimental basis for testing not only whether Eqns 14 or 16 hold, but also whether the observed non-minimum phase behaviour is attributable to hydraulic processes.
If the provisional mechanistic interpretation offered here for Eqn 5 is correct, then the observation that stomatal closure, following a decrease in humidity, is dominated by a slow time constant process appears to support the hypothesis that the humidity response of stomata is predominantly a non-hydraulic event ( Bunce 1996; Jarvis & Davies 1997; Haefner et al. 1997 ); whilst the hydraulic aspects of the response introduce non-minimum phase behaviour resulting in the characteristic oscillatory nature of the system response.
The acceptance of Eqn 5 over Eqn 6 as a description of the physiological response of stomata to evaporative demand provides objective support for the argument that stomata respond to evaporation from leaves via some intermediary internal state within the plant. Although some of this controlling water loss may not be under stomatal control, for the conditions applied in this experiment the greater proportion must be.
The analysis presented in this paper demonstrates the potential of the DBM modelling approach in the objective identification of dynamic physiological processes from leaf gas exchange time series. The applicability of linear TFs to perturbations about a given condition indicates a degree of linearization of the dynamic response of stomata to humidity. Robustness stemming the partial linearization of gas exchange control has been offered as one possible explanation for this observation.
The estimated time constants for both the opening and closing phases of stomatal movements for Cedrella following the reduction in cuvette humidity are reasonably similar to those estimated by Cowan (1977) for cotton, but the gains of these processes are significantly different, as might be expected.
Controlled steps in the water vapour content of air over leaves has been a preferred excitation of g(k) in this and previous studies of this kind with the notable exception of the sinusoidal input used by Farquhar (1973). However, it appears that a step in D(k) is a poor choice of excitation for the stomatal response to water loss, a step in E(k) being likely to be a more appropriate choice.
For Cedrella, the SSG of the relationship between E(k) and g(k) appears to be dominated by slow closure. Subsequently, it seems reasonable that functional relationships for forecasting g(k) should be expressed in terms of processes giving rise to this response. The simulation model recently presented by Haefner et al. (1997) bears some similarities to the data-based model identified here. However, it becomes simpler and more comparable with the data-based model if it was assumed that guard cell osmotic potential responds more directly to E(k).
We would like to thank Professor Graham Farquhar and two anonymous referees for helpful comments made during the preparation and revision of this paper. Andrew Jarvis was supported by a Natural Environment Research Council grant under the Terrestrial Initiative in Global Environmental Research programme, grant number GST/02/986. Part of this work was supported under a Biotechnology and Biological Sciences Research Council grant number E06813.
APPENDIX A: DERIVATION OF THE TF BETWEEN W(K) AND E(K)
If the initial rates of change of all states are zero Eqn 5 can be expressed,
where g(0) is the initial resting value of g at time k = 0 and G(z–1) is the identified second-order, [2,2,0] TF in the backward shift operator z–1, i.e.
Combining Eqns 7 and A1 and noting E(k) = ΔE(k) + E(0) gives,
Gathering the terms in ΔE(k) gives,
Because D(k) = Wl(k)–W(k), where Wl and W are the water vapour mole fractions inside and at the surface of the leaf, respectively, and W(k) = ΔW(k) + W(0), we can write,
If Wl(k) is constant because of little change in leaf temperature then,
and Eqn A5 simplifies to,
Equation A7 is a form of the closed-loop response of the system ( Cowan 1977). If, as indicated by the data in Fig. 1 and the analysis of Cowan (1977), G(z–1) is a second-order TF, then substituting in Eqn A2 yields,
APPENDIX B: DERIVATION OF PERTURBATIONAL TF FORM OF HAEFNER ET AL. (1997) STOMATAL MODEL
Using similar notation to Haefner et al. (1997) , one-dimensional stomatal conductance, g(t), is defined as a linear function of the difference between guard cell (Pg(t)) and epidermal (Pe(t)) turgor.
where q is an empirical scalar which converts aperture to conductance; and Cg and Ce are the guard cell and epidermal mechanical influence coefficients, respectively. Turgor, P(t), is given by the sum of the water potential ψ(t) and osmotic pressure (π(t)), i.e.
assuming the reflection coefficient across the guard cell plasmalemma is near unity. The water potential of the epidermis, ψe(t), and the guard cells, ψg(t), are assumed to be equal so that g(t) is determined by differences in osmotic potentials and mechanical influence coefficients between the guard cells and the epidermis. There are two dynamic components to the model. First, there is the mass balance of water in the epidermis, giving rise to epidermal water potential, ψe(t).
Here φ converts volumes to water potentials; while F(t) is the flux of water into the epidermis, which is determined by the gradient in potential between the epidermis and some reference point, at constant potential ψr, and the hydraulic conductivity of the water flux pathway, L, i.e.
Second, there is the mass balance of solute in the guard cells, which is approximated by the difference between the current osmotic pressure (πg(t)) and some desired set point (πgd), i.e.
where α is inversely proportional to the time constant for this equilibrium. Lastly, Haefner et al. (1997) define πgd as a linear function of Pe(t), i.e.
where πgm is the maximum guard cell osmotic pressure and β is the sensitivity of πgd to epidermal turgor pressure. To enable comparison with Eqn 13, we need to combine Eqns B1–B6 in the form of a TF relating Δg(s) to ΔE(s) where s is the Laplace operator and s = d/dt with zero initial conditions. Combining Eqns B3 and B4 and assuming zero initial conditions, we can derive the TF between Δψe(s) and ΔE(s) in the form,
Similarly, combining Eqns B2, B5 and B6 and again assuming zero initial conditions, we can derive the TF between Δπg(s) and Δψe(s) as,
Now, from Eqns B1 and B2 we can write,
assuming Δπe(t) = 0. So that finally, substituting Eqns B7 and B8 into Eqn B9 we obtain the overall continuous-time TF relating Δg(s) to ΔE(s), which takes the following form: