Parameter optimisation of fast chlorophyll fluorescence induction model
Introduction
The intensity of Chl a fluorescence emission in dark-adapted oxygen evolving systems shows a characteristic variation in time, known as fluorescence transient or induction, FI [1]. The fluorescence intensity rises quickly from an initial low value, F0 (the O level), to a higher one, FP (the P level). Two intermediary steps designated as FJ (the “J” level) and FI (the “I” level) normally appear under high light illumination conditions [2], [3], [4]. It is well established that the origin of FI of higher plants and green algae is the antenna system of photosystem II (PS II), and is due to the variation of the quantum yield of fluorescence emission (between 2 and 10%). Chl a fluorescence yield is principally influenced by the redox state of QA, the secondary electron acceptor of PS II, which is assumed to functions as a quencher in the oxidised state. As PS II function is highly sensitive to the environmental conditions (e.g. pollutants, heat and water stress, excess light, increasing carbon-di-oxide concentration, and so on), FI is expected to be very useful in plant biology research and related areas, as ecology, horticulture, agronomy, and biotechnology. In this view, it is essential to develop a special approach to characterise the plant samples, based on our knowledge about the processes leading to fast FI and the effect of external influences on it. Over the last few years a mathematical model of the fast FI has been proposed [5], [6], [7], [8], [9], [10], which is based mainly on the kinetics of the photochemical and redox reactions around the PS II centres. It consists of a reaction scheme that is translated to an ordinary differential equation system (ODEs) for the concentrations of the species using the law of mass action, and of an ‘output’ function describing the fluorescence yield for given concentrations of the redox species. Both, the ODE system and the output function, depend on unknown parameters (e.g. on the rate constants and other specific parameters [10]). After an appropriate choice of parameter values, this model was able to reproduce qualitatively the typical form of FI curves, with the characteristic O–J–I–P phases. But of course, a good model should also be able to reproduce experimental data quantitatively. Here, we present the results of parameter estimations, obtained using the software package PARFIT developed by Bock [11], [12], which consists of a discrete treatment of the DAE model by multiple shooting or collocation as a non-linear constraint of the optimisation problem. The fits of the experimental data are very good, but some of the parameter values deviate from those presented in the literature. We discuss the possible reason for these deviations, and we propose some ideas to be considered in the future development of the model.
Section snippets
The model of fast Chl a fluorescence induction
The model for the fast Chl a fluorescence induction proposed by Stirbet et al. [8] is based on six main assumptions:
(i) The Chl a fluorescence yield reflects the concentration of closed PS II centres with reduced QA.
(ii) The rate constants of QA reduction are modulated by the redox state of the oxygen evolving complex (OEC), the so-called S-states.
(iii) A so-called “two electron gate” process on the acceptor side of PS II unit is working.
(iv) The excitation energy of antenna can be exchanged
Experimental methods
Fast Chl a fluorescence induction measurements were performed at room temperature, on 15 min dark adapted Pisum sativum leaves, with a shutter-less fluorimeter (Plant Efficiency Analyser, built by Hansatech Ltd.).
The samples were illuminated 1 s with continuous light (600 W m−2 power; emission peak at 650 nm) provided by an array of six light-emitting diodes, focussed on a circle of 4 mm diameter of the sample surface. The fluorescence signals were detected using a PIN-photodiode after passing
Parameter optimisation procedure
The dynamic behaviour of the model depends on a set of unknown parameters contained in the right side of the ODEs system and in the output function (i.e. the function that maps the state variables and parameters to the observed quantity, fluorescence intensity F). These parameters have to be determined so that the experimental data are reproduced ‘as good as possible’. Here, we take the usual least squares approach, in which the sum of the squares of the differences between experimental data
Results and discussions
Three identical experiments on pea leaves were carried out. We have observed small, but non-stochastic differences between the three sets of data (results not shown). The numerical computations were carried out with a version of PARFIT based on multiple shooting, applied to the model presented in [8]. After few numerical experiments, we decided to use 10 multiple shooting nodes. During the iterations some of the parameters became negative, making the differential equations unstable. Instead of
Conclusions
The approximations of fast Chl a fluorescence induction curves O–J–I–P measured on pea leaves are quite good (see Fig. 1, Fig. 2). However, even after the introduction of three new insights in the model the estimates of model parameters obtained in this study are not always in agreement with the literature (see Table 2). Of course, in some cases we can blame the literature, as the related values are not always applicable. But there are also several possible explanations, related to the model
Acknowledgements
Supported by Swiss National Foundation Grant No. 3100-057046.99/1, and by SFB 359 of the Deutsche Forschungsgemeinschaft
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