Self-sustained oscillations in plant water regulation: Induction of bifurcations and anomalous rhythmicity


  • Gunnar Prytz,

    1. Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway;
    2. Present address: ABB Corporate Research Center, PO Box 9, N-1375, Billingsted, Norway
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  • Cecilia M. Futsaether,

    1. Department of Agricultural Engineering, Agricultural University of Norway, PO Box 5065, N-1432 Ås, Norway;
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  • Anders Johnsson

    Corresponding author
    1. Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway;
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Author for correspondence: Anders Johnsson Tel: +47 73591854 Fax: +47 73597710 Email:


  • • The complex dynamics of the water regulatory system was explored by studying the induction of period doublings (period-1 to period-2 to period-4) and other complicated oscillatory patterns (period-3 and period-5) in the transpiration of young oat plants.
  • • A gas exchange system was used to record the transpiration of primary leaves of Avena sativa cv. Seger after exposure to various stimuli.
  • • Period-1 oscillations were induced from steady-state transpiration by a light pulse, by increasing the xylem resistance to water flow or by temporarily lowering the root medium temperature. Similarly, period doubling (period-1 to period-2) was induced by lowering the root medium temperature temporarily or by increasing the xylem resistance. The more complicated oscillatory patterns (period-3, period-4, period-5) were induced by including different concentrations of KCl or KNO3 in the root medium of root excised, xylem-compressed plants.
  • • The range of oscillatory curve forms resembles those displayed by mildly chaotic nonlinear systems. The water regulatory system is discussed in relation to these nonlinear systems. Few studies of such complicated dynamics in plant systems exist.


The water regulatory system of plants often oscillates with periods typically in the range of some minutes to some hours (Barrs, 1971; Cowan, 1972; Johnsson, 1973; Hirose et al., 1994; Naidoo & von Willert, 1994; Rose et al., 1994). In addition, daily/circadian patterns are often present, modulating the short-period responses (Brogårdh & Johnsson, 1975). In many cases the periods cluster around 30–60 min, the exact period being dependent on environmental factors.

Oscillations in transpiration can arise seemingly spontaneously or can be induced by sudden changes in the plant's environment, such as perturbations of the light conditions or the osmotic potential of the root medium (Cowan, 1972; Johnsson, 1976; Naidoo & von Willert, 1994). As oscillations occur in plants with anatomically different stomata, the detailed structure of the stomata does not appear to be a critical factor for the occurrence of oscillations. However, an essential feature for the occurrence of oscillations seems to be a restricted water supply to the stomatal apparatus of the leaf. Experiments have shown that if the xylem resistance is increased, the oscillatory tendency is also increased (Brogårdh & Johnsson, 1973). In addition, a complete removal of the root resistance can be compensated by mechanical compression of the xylem, enabling oscillations to be generated even without the presence of a root system (Brogårdh & Johnsson, 1973).

Period doubling in the water regulation of the Avena leaf has been reported by Johnsson & Skaar (1979; denoted ‘alternating pulse response’) and by Johnsson & Prytz (2002). A characteristic feature of period doubling is the transition from period-n oscillatory behaviour to period-2n oscillatory behaviour (Glass & Mackey, 1987; Strogatz, 1994). Period-n oscillatory behaviour is characterized by a pattern that repeats itself every n maximum. The period-2 oscillations will display alternating peak values in successive order (high/low/high, etc.). Frequency analysis of period-2 oscillations will yield the basic frequency f0 as well as the frequency f0/2 (the double period). Examples of period doublings are transitions from period-1 to period-2 oscillations and from period-2 to period-4 oscillations. The period doubling encountered in Avena leaves has been the transition from period-1 to period-2 oscillations (Johnsson & Prytz, 2002). Period doubling has also been found in simulations of a mathematical model of the water regulatory system indicating that, in theory, period doubling can arise in such systems (Gumowski, 1983).

Period doubling bifurcations can be found in many nonlinear mathematical model systems. At a critical value of the system's control or bifurcation parameter, the period-1 oscillations become unstable and period-2 (or ‘subharmonics’) appear (Glass & Mackey, 1987; Strogatz, 1994; Lloyd & Lloyd, 1995). Further changes in the control parameter result in successive period doubling bifurcations yielding period-4, period-8, period-16, and so on. After an infinity of bifurcations the system becomes aperiodic or chaotic. Such chaotic orbits will either have a limiting set called a strange attractor or will move on the strange attractor itself (Frøyland, 1992; Strogatz, 1994). Thus, the period doubling phenomenon is often an indication of a system approaching a chaotic state. It is important to note that in the chaotic region, an infinity of periodic windows such as period-3, period-5, period-6, and so on, is also predicted by theory.

Period doubling sequences have been reported in a number of experimental systems (see Cvitanovic, 1989 and references therein). Observations of period doublings have been made in physical systems (hydrodynamic, electronic, optical and acoustic) and in chemical systems (Belousov–Zhabotinsky reaction) as well as in biological systems (heart muscles) (Guevara et al., 1981; Arecchi et al., 1982; Libchaber et al., 1982; Testa et al., 1982; Coffman et al., 1987). Many of these systems have been explored by systematically varying a specific control parameter and noting the occurrence of period doubling bifurcations (Libchaber et al., 1982; Coffman et al., 1987). In other experiments, a specific control parameter has been pulsed with variable frequency (Guevara et al., 1981; Markus & Hess, 1985; Shabala et al., 1997).

Most of the published recordings of transpiration rhythms in plants indicate a regular and stable rhythmic behaviour, dominated by one basic frequency f0, henceforth referred to as period-1 oscillations (Barrs, 1971). However, few studies have aimed at a systematic variation of the parameters of the water control system to search for period doubling sequences and possibly even more complicated curve forms. One attempt was made by Klockare et al. (1978) who varied the light intensity while recording the rhythmic behaviour of the transpiration. When the light intensity was changed from high to low values, a transition in the period of the oscillations (from approximately 40 min to 100 min) was found at an irradiance of approximately 0.1 mW cm−2 (corresponding to approximately 5 µmol m−2 s−1). At the transition intensities, the transpiration rhythms showed irregular curve shapes and autocorrelation analysis indicated that both components (i.e. periods) were present simultaneously.

In the present report we focus on experimental approaches of inducing transpiration oscillations with features that are principally new and of a complicated nature. The transpiration oscillations experiments reported by Johnsson & Prytz (2002) indicated that perturbations in several experimental parameters can lead to period doubling. As a means of investigating the occurrence of period doubling bifurcations and other complicated oscillatory patterns in Avena plants, we administered different ionic solutions to plants with excised root systems and compressed xylem vessels. The effect of the potassium ion (K+) on the oscillations was of particular interest as the K+ ion is involved in the processes controlling the volume of the guard cells (and presumably also the subsidiary cells). It therefore represents a crucial ion in stomatal dynamics (Assmann, 1993; Willmer & Fricker, 1996). Bifurcations can reveal general and basic properties of the nonlinear system, including features of the basic mechanisms responsible for the oscillations. We demonstrate that the transpiration oscillations in the primary Avena leaf can show complicated curve forms that are sometimes very regular, sometimes highly irregular. These new recordings suggest that conditions may perhaps exist under which the oscillations become chaotic and that Avena may constitute an example of oscillators that have so-called strange attractors. Chemical, physical as well as biological systems having strange attractors have been found (Arecchi et al., 1982; Roux et al., 1983; Markus et al., 1985; Strogatz, 1994). However, studies of such oscillating systems are, to our knowledge, very few in plant physiology (Shabala et al., 1997) and absent when it comes to plant water regulation.

Materials and Methods

Primary leaves of Avena sativa L. cv. Seger (Svalöf Weibull AB, Svalöv, Sweden) were used in the experiments. The plants were cultivated on a metal mesh in contact with tap water containing 1.00 mm CaCl2. The temperature in the cultivation room was 24 ± 2°C and the relative humidity was 25 ± 5%. Two Sylvania GTE Standard F18W/129 warm white tubes (Sylvania, Danvers, MA, USA) positioned above the growth medium containers provided a 13-h photoperiod of 37 ± 10 µmol m−2 s−1 photosynthetically active radiation (PAR) on the leaves.

At the age of 7–10 d the plants were transported to the gas exchange system. They were mounted inside Plexiglas cuvettes (inner dimensions 17.5 × 109 × 10 mm, (width × height × depth) with an air inlet on the bottom and outlet at the top of the cuvette. Care was taken to ensure that as much as possible of the primary leaf (at least over 90%) was mounted in the cuvette. The root system protruded into a container (noncorrosive steel thermocup with double walls and bottom) filled with root medium (temperature 24 ± 2°C). Unless otherwise stated, the same medium was used during cultivation and the experiments. The slight gap between the leaf base and the cuvette was carefully sealed using electrician's sealant to prevent air leakage. The gas exchange system consisted of four cuvettes with independent air supply (relative humidity approximately 5%, 24 ± 2°C) connected to the air inlet and independent humidity sensors connected to the air outlet, thus enabling simultaneous experiments on four individual plants. An Osram Xenophot HLX 64640 FCS 24 V/150 W halogen bulb (Osram GmbH, München, Germany) mounted in a lens system was used as the light source. The standard white light intensity was set to 80 µmol m−2 s−1.

Dried air with a relative humidity of approximately 5% and a temperature of 24 ± 2°C was supplied (Rena Alize 101 air pump; Rena, Annecy Cedex, France) to each cuvette at a flow rate of approximately 100 ml min−1 measured using an Aalborg GFM171 mass flowmeter (Aalborg Instruments & Controls Inc., Orangeburg, NY, USA). The air in the cuvette was not stirred. The relative humidity of the air leaving the cuvettes was recorded with HIH-3605-A capacitive humidity sensors (Honeywell, Morristown, NJ, USA) and provided a measure of the transpiration of the plant in the cuvette. The sampling frequency was 0.2 Hz. The absolute values of the transpiration level can be estimated from the transpiration measurements and varied typically between 0 and 60 mm3 H2O h−1 (3.7 mmol m2 s−1) with leaf areas of approximately 250 mm2.

A special type of Plexiglas cuvette (inner dimensions 20 × 60 × 15 mm) was used in some of the experiments. Using this cuvette type, mechanical pressure could be applied to the xylem or leaf base of the plant to increase the water flow resistance. The mechanical pressure was applied by two small aluminium plates pressed together by a spring. The leaf base was positioned between the plates. Two small parallel bars (with triangular cross-sectional area) 1 mm apart on one of the plates pressed against the xylem or leaf base of the plants with a force that was dependent on the precise position of the spring. The force was of the order of 5 N (Brogårdh & Johnsson, 1973).

The pressure cuvette was used to increase the water flow resistance in either intact plants or in root excised plants. Root excision was done by cutting the coleoptile with a sharp knife below the water level (Falk, 1966). As excision of the root system eliminated the root resistance, the pressure cuvette could be used to obtain a controlled variation of the flow resistance into the leaf (Falk, 1966). Furthermore, the composition of the medium entering the leaf could be varied and was not dependent on the permeabilities of the root barrier to different substances as under normal conditions.

Once mounted in the gas-exchange system, plants were exposed to the following perturbations: (1) high intensity white light pulses from the Osram Xenophot light source for a specific time interval; (2) darkness pulses obtained by turning the light source off; (3) root temperature pulses by exchanging the root medium container with an identical one containing root medium stabilized at a different temperature in a water bath; (4) standard root medium containing 20–80 mm KCl or KNO3 by exchanging the standard root medium (1 mm CaCl2 solution) container with an identical one containing KCl or KNO3 solution. The exchange of root medium container took no more than 5 s.


Induction of period-1 oscillations in the water control system

The following methods were used to induce long-term, period-1 oscillatory transpiration in oat plants displaying steady-state (nonoscillatory) transpiration.

Light treatment Light pulses were routinely used to induce period-1 oscillations. A high-intensity white light pulse (180 µmol m−2s−1) of 12-min duration at the start of the experiment was usually sufficient to initiate self-sustained oscillations. If not, the pulse was repeated after a dark period of at least 12 min. The oscillations became fairly stable in amplitude and period within a few hours.

Temperature treatment The temperature of the root system has been shown to affect the root hydraulic conductivity (Dale et al., 1990; Bolger et al., 1992). Thus, temperature pulse treatment of the root system should affect the water flow resistance and should therefore be capable of inducing oscillations. As shown in Fig. 1, cooling could induce oscillations in plants showing steady-state transpiration.

Figure 1.

Oscillatory transpiration in an intact Avena sativa plant induced by temperature treatment. Cold treatment (the root medium temperature was changed from 24°C to 2°C for 12 min) of the root system occurred during the time interval indicated by the vertical lines. Oscillations were induced, presumably owing to an increase in the water resistance of the root system.

Mechanical pressure application By carefully increasing the pressure on the leaf base, period-1 oscillations could be induced in steady-state transpiring plants (Brogårdh & Johnsson, 1973). Further pressure increases dampened or often eliminated the oscillations.

Induction of period doubling in transpiration oscillations

Period-doubling, in this case the transition from period-1 to period-2 oscillatory transpiration with alternating peak values in successive order (low/high/low, etc.), was first observed by chance. In a study of period-1 oscillatory transpiration, a period-1 to period-2 transition was observed in 41 out of 153 plants (27%). The transitions arose seemingly spontaneously as the experimental conditions were kept constant. The average duration of the period-2 oscillations was 7.7 ± 1.2 h. The periods were approximately 25 ± 5 min and 50 ± 5 min.

Period doubling could also be induced in plants displaying period-1 oscillations. The following methods were used to induce period doubling.

Mechanical pressure application Mechanical pressure applied across the leaf base often induced period doubling in already oscillating plants (Fig. 2a,b). The period-2 oscillations often ceased if the applied pressure was increased further.

Figure 2.

Induction of period doubling in transpiration oscillations of intact plants. Period-2 oscillations were induced after the following perturbations. (a) Increase of the xylem resistance to water flow. Mechanical pressure was applied at the time indicated by the vertical line (the maximum). (b) Increased xylem resistance obtained by the application of mechanical pressure at transpiration minima. (c) Low temperature (2°C) applied to the root system for 12 min, as indicated by the vertical lines. (d) Fourier spectrum found using a fast Fourier transform routine of the time series in (c) from 4.5 h to 15 h. The periods were approximately 40 and 80 min.

Temperature treatment of the root system As shown in Fig. 2c, period doubling could also be induced by short duration temperature changes in the root medium. The frequency content of the time series was analysed using a fast Fourier transform routine. The resulting periods of the series were approximately 40 min and 80 min (see Fig. 2d) corresponding to the two frequency components f0 and f0/2, respectively.

Phase shifting of period-2 rhythms

Perturbations could be used to phase shift the period-2 pattern. Phase shifts of period-1 oscillations, caused by light or darkness pulses, have been studied by Johnsson (1973). Figure 3 shows examples in which the period doubled transpiration rhythms were phase shifted by 180° (i.e. the order of low/high amplitude was reversed after the pulse perturbation). Corresponding period alterations were also found. In Fig. 3a, a dark pulse was administered to the leaf. In Fig. 3b,c, the effects of heat and cold pulses given to the root system are shown, respectively.

Figure 3.

Phase-shifting of period-2 oscillations in intact plants. The highest maxima of the period-2 transpiration oscillations have been marked by black triangles. The applied perturbations reversed the order of the maxima, as is also indicated by the triangles. The triangles indicate that the lowest maxima have become the highest ones after the perturbations. (a) The leaf was placed in darkness for 2 min, as indicated by the vertical lines. (b) High temperature: heat pulse applied to the root system (24°C to 39°C, ΔT = 15°C) for 2 min (c) Low temperature: cold pulse applied to the root system (24°C to 9°C, ΔT = −15°C) for 2 min.

More complicated waveforms observed in intact plants

In long duration experiments, a circadian pattern modulated the oscillations. In addition to this modulation, more complicated waveforms could, in some cases, be observed. We encountered patterns in which every third peak had a smaller amplitude than the others, as shown in Fig. 4. This pattern will be denoted as period-3 oscillations.

Figure 4.

Period-3 oscillations in an intact plant. In intact plants this type of oscillation was rare and only a few cases have been observed during several years of studies.

Period doubling by ion treatment

Figure 5(a) shows an example in which 80 mm KCl was added to the root medium of an intact plant. The immediate osmotic effect consisted of stomatal opening, causing an immediate increase in amplitude, but no period doubling was seen. A slight change of the period did, however, occur: the medium with the lower water potential (80 mm ion solution corresponding to approximately −4.0 bar) produced oscillations with slightly higher frequency.

Figure 5.

Potassium salt effects on the transpiration rhythm. Black triangles (b,c,d) denote the highest transpiration maxima in the pattern. (a) KCl, 80 mm, was added to the root medium (at the vertical line) of an intact plant. Period doubling was not induced. The period-1 oscillations were retained. (b) KCl, 20 mm, was added to the root medium (at the vertical line) of a root excised, xylem-compressed plant. Period doubling was induced, as demonstrated by a transition from period-1 to period-2 oscillations. (c) KCl, 40 mm was added to the root medium (at the vertical line) of a root excised, xylem-compressed plant. Period doubling was induced (transition from period-1 to period-2 oscillations). (d) 40 mm KNO3 was added to the root medium (at the vertical line) of a root excised, xylem-compressed plant. Period doubling was induced (period-1 to period-2 oscillations).

In root excised, xylem-compressed plants, ions can enter the xylem pathway unhindered by the root cell membranes and root resistances. It was therefore possible to study whether different types of ions could induce period doubling in the transpiration oscillations.

Solutions with K+ concentrations in the range 20–80 mm were tested. Period doubling was often observed after application of approximately 20 mm KCl (Fig. 5b), 40 mm KCl (Fig. 5c) and 40 mm KNO3 (Fig. 5d).

The K+ concentration dependence inducing period doubling has not been unequivocally determined. In general, lower (close to zero) concentrations induced period doubling to a lesser extent, whereas higher (60–80 mm) concentrations often resulted in damping of the oscillations.

Period-3, period-4-and period-5 patterns in root excised plants

Various oscillatory patterns were displayed by root excised, xylem-compressed plants. In Fig. 6a, a rhythmic behaviour that appears to repeat every fourth peak is shown. This period-4 pattern followed a period-2 pattern (not shown), thus a period doubling from period-2 to period-4 oscillatory behaviour had occurred. In this case, the medium given to the root excised, xylem-compressed plant contained 40 mm KNO3. A different period-4 pattern is demonstrated in Fig. 6b, where 80 mm KCl was added to the medium. The period-1 oscillation changed into a complicated oscillatory pattern that died out after approximately 8 h.

Figure 6.

Potassium salt effects on the transpiration rhythm of root excised, xylem-compressed plants. Black triangles denote highest transpiration maxima in the pattern. (a) KNO3, 40 mm, was added to the root medium. Period-4 oscillations were observed. This pattern followed a period-2 oscillation, thus a period doubling bifurcation had occurred. (b) KCl, 80 mm, was added to the root medium at t = 2 h. Period-4 oscillations were induced. (c) KCl, 20 mm, was added to the root medium. Period-3 oscillations were observed. (d) KCl, 20 mm, was added to the root medium. Period-5 oscillations were observed. A transition from period-5 to period-3 oscillations was observed as the pattern in (c) immediately followed the pattern in (d). (e) Fourier spectrum of the period-3 time series in (c) found using a fast Fourier transform routine. The periods were approximately 30, 45 and 90 min. (f) Fourier spectrum of the period-5 time series in (d). The periods were approximately 15, 30, 40, 60 and 75 min.

Figure 6c shows part of a long-duration experiment where 20 mm KCl was added to the medium (at t = −15 h relative to the time scale in Fig. 6. As can be seen, the transpiration amplitude showed a period-3 rhythm for approximately 12 h, with periods of approximately 30, 45 and 90 min (see the Fourier spectrum in Fig. 6e).

As shown in Fig. 6d, period-5 oscillations were also occasionally observed in root excised, xylem-compressed plants. The periods were approximately 15, 30, 40, 60 and 75 min (Fig. 6f). Interestingly, the period-5 pattern shown in Fig. 6d was followed by a transition to the period-3 pattern in Fig. 6c. The medium contained 20 mm KCl during the period-5 and period-3 behaviour.

Pulse treatment and phase shifting

As was the case for period-2 oscillations in intact plants, pulse treatment of root excised, xylem-compressed plants could phase shift the oscillations. This is shown in Fig. 7, where 80 mm KCl was added to the root medium for a duration of 1 h.

Figure 7.

Phase shifts of oscillations in root excised, xylem-compressed plants by a potassium pulse. Addition of 80 mm KCl to the standard medium for 1 h (between 5 h and 6 h, as indicated by the vertical lines) phase shifted the period-2 rhythm, thereby reversing the order of the maxima. The highest maxima of the period-2 transpiration oscillations have been marked by black triangles.

Period variations

The period doubling properties of the oscillations were not only observed in the amplitude values of the successive peaks. Successive period estimates, as judged from the time between successive maxima or minima in the recordings also showed regularly changing values. A typical sequence calculated from maximum-to-maximum time intervals, taken from the period-2 recording in Fig. 2c, gives the following values: 36.8, 34.3, 37.5, 35.4, 36.6 and 35.6 min. A second time sequence, also calculated from maximum-to-maximum times, was taken from Fig. 6(c). The values clearly illustrate the period-3 character of this oscillation: 36.4, 28.8, 23.4, 36.1, 29.7, 24.3, 37.1, 28.6 and 25.3 min. The period-2-and period-3 behaviours shown in Figs 2c and 6c, respectively, are also supported by the Fourier spectra shown in Figs 2d and 6e.

The period doubling phenomenon can thus be documented either from amplitude recordings, from frequency analysis or from recordings of maximum-to-maximum (or minimum-to-minimum) interval lengths. In several cases, the interval lengths may provide more reliable estimates of the periodic phenomena than the amplitude values, as the latter may be more easily disturbed than the interval lengths. Frequency analysis may often be complicated by the fact that biological time series are short and nonstationary.


The broad range of oscillatory patterns encountered in the present experiments hints at a possible bifurcation sequence. Period-1-and period-2 oscillations as well as period-4 oscillations were observed. Such period doubling sequences are commonly found in nonlinear systems and are often an indication that the system is approaching a chaotic state (Metropolis et al., 1973; Feigenbaum, 1983; Strogatz, 1994).

An experimental system that displays period doubling sequences is the much-studied Belousov–Zhabotinsky (BZ) reaction (Roux et al., 1983; Coffman et al., 1987; Strogatz, 1994). In the BZ-reaction, oscillations arise in the oxidation of malonic acid by bromate ions in the presence of a suitable catalyst in a reaction sequence with many reacting species. If the liquid is stirred, the system behaves as an overall, lumped oscillatory system without gradients in space.

Coffman et al. (1987) showed that in the BZ reaction, the concentrations passed through certain bifurcation sequences. When the bifurcation parameter, which corresponded to the time the chemicals spent in the reactor vessel, was changed slightly, period-1, period-2, period-4 and period-8 oscillations, as well as a few cycles of period-16 oscillations, were recorded. Given the rapid convergence of the period doubling sequence, it is difficult to observe more than a few transitions experimentally (Cvitanovic, 1989 and references therein). However, other oscillatory patterns such as period-3, period-5 and period-6 oscillations were also observed.

It was found that the BZ reaction could be described mathematically using a so-called unimodal one-dimensional (1D) map (May, 1976; Coffman et al., 1987). The unimodal 1D map is a function that describes the amplitude of a maximum of the oscillation as a function of the amplitude of the preceding maximum (Strogatz, 1994). The mapping function has the characteristics of being smooth, concave down and has a single maximum. For systems obeying an unimodal 1D map, the types of oscillatory behaviour as a function of the bifurcation parameter occur in a particular sequence denoted the universal or U-sequence (Metropolis et al., 1973; Feigenbaum, 1983). As the system's control or bifurcation parameter is changed, the system passes through successive period doubling bifurcations (i.e. from period-1 to period-2 to period-4 to period-8, etc.) (Glass & Mackey, 1987; Strogatz, 1994; Lloyd & Lloyd, 1995). After an infinity of bifurcations the system becomes aperiodic or chaotic. In the chaotic region, an infinity of periodic windows such as period-3, period-5, period-6, and so on, occur in a particular sequence when the bifurcation parameter is progressively changed.

The observed behaviour of the plant water regulatory system appears to be similar to the behaviour of the BZ-reaction. In the plant water regulatory system, period-doubling transitions from period-1 to period-2-and period-2 to period-4 were observed. In addition, period-3 and period-5 oscillations as well as the transition from period-5 to period-3 oscillations were recorded. The transition from period-5 to period-3 oscillations was also found in the BZ reaction (Coffman et al., 1987), where it was induced by a slight increase in the bifurcation parameter. In the present experiments on Avena leaves, the experimental parameters remained constant, once the medium composition was defined. However, it is possible that factors in the plant such as the ionic conditions in and around the guard cells were changing, thus initiating the transition.

Given the similarities between the BZ reaction's transitions and the transpiration oscillatory patterns, it is perhaps possible that the water regulation system dynamics follows a mapping function with similarities to the unimodal map proposed for the BZ-reaction (Coffman et al., 1987). It may therefore be possible to infer some of the mathematical features of the plant water regulatory system from the general properties of systems characterized by period doubling bifurcations and unimodal maps. Such systems are nonlinear and of three or higher dimensions (Cvitanovic, 1989 and references therein; Frøyland, 1992; Strogatz, 1994). A system with an unimodal map will follow Feigenbaum's universality theory, that is, periodic attractors always occur in the same sequence, the U-sequence. The strange attractor of such systems is very flat with a dimension only slightly greater than two. This requires that only two or three degrees of freedom are active in the system. The remaining degrees of freedom follow along. Thus, systems displaying this type of behaviour are mildly chaotic.

Most of the mathematical models of transpiration are three-dimensional, nonlinear models (Cowan, 1972; Delwiche & Cooke, 1977; Gumowski, 1981, 1983; Garcia, 1985). The three dynamic elements correspond to the guard, subsidiary and mesophyll cell volumes. In principle, these models should be capable of simulating the complicated dynamics demonstrated experimentally. Although period-2 behaviour can be simulated in variants of some of the models (Gumowski, 1983; Johnsson & Prytz, 2002), period-3 and higher period oscillations have, to date, not been simulated. Chaotic behaviour has not been observed in model simulations but it has been proposed that, in theory, such behaviour is possible (Gumowski, 1983). Thus, further model studies should be conducted in order to incorporate these new experimental findings and explore under which conditions they can occur. Such studies may further increase our understanding of the basic mechanisms behind stomatal control and transpiration oscillations.

The behaviour and the oscillatory patterns of a nonlinear system are determined by the bifurcation parameter. It is therefore of great interest to identify the bifurcation parameter in such systems. A possible bifurcation parameter in the water regulatory system is the ion concentration, possibly the K+ concentration, in the stomatal region. Oscillatory behaviour characteristic of period doubling and chaotic behaviour was obtained when K was supplied to root excised, xylem-compressed plants. Notably, period-1, period-2, period-4 as well as period-3-and period-5 oscillations were observed. Given that K+ is involved in stomatal control, it is not unreasonable to assume that the K+ concentration in the stomatal region affects the oscillations (Willmer & Fricker, 1996). However, given the effect of ion concentration on osmotic pressure, the osmotic pressure conditions in the stomatal region may also constitute a possible bifurcation parameter. All the methods used to induce period doubling in the transpiration system may, via signalling chains, affect the potassium gradients or osmotic gradients across the stomatal membranes (Raschke et al., 1988). In bifurcation studies of mathematical models of transpiration, the parameter studied corresponded to the osmotic contents of the guard cells (Rand et al., 1981; Garcia, 1985), further supporting the hypothesis of a bifurcation parameter related to the ion concentration in the stomatal region.

It is, however, possible that factors other than the hydraulic interactions also modulate or influence the stomatal oscillations (Johnsson, 1976). Given the importance of calcium signalling in plants cells and the existence of oscillations in guard cell Ca2+ concentration (McAinsh et al., 1995, 1997; McAinsh & Hetherington 1998; Blatt, 2000), the possible role of calcium in the generation of transpiration oscillations should be investigated. The period of Ca2+ oscillations in guard cells is approximately 10–15 min (McAinsh et al., 1995, 1997; Blatt, 2000) whereas the period of stomatal oscillations is in the range 30–40 min (Barrs, 1971). However, period changes can occur if networks of coupled guard cells are considered (Pavlidis, 1973; Bindschadler & Sneyd, 2001). Thus, a network of coupled oscillators displaying short-period oscillations can give rise to oscillations with longer periods. Some preliminary experiments on the effects of Ca2+ in the root medium of root excised, xylem-compressed seedlings did not give rise to period-2, period-3, period-4- and period-5 oscillations as found in the K+ experiments.

Period doubling sequences have been observed experimentally in several physical, chemical and biological systems (see Cvitanovic, 1989 and references therein). However, very few studies of possible period doubling sequences and chaos in plant physiology exist. Two early studies hinted at a possible period doubling in the water regulatory system. Johnsson and Skaar (1979) studied root excised primary Avena leaves in contact with the root medium, as in the present study. The leaves were exposed to repetitive light pulses (so-called alternating light pulses). Successive transpiration responses to the light pulses were different but every second response had the same shape. This response was also similar to the period-2 behaviour discussed in the present study with regard to the period length and the phase shifting by light. The alternating pulse response was affected by increased ion concentration, particularly sodium ions. In the second study, again on Avena leaves, Klockare et al. (1978) found that when the light intensity was varied, a transition in the transpiration oscillation period occurred at a certain irradiance level. At the transition intensities, the transpiration rhythms showed irregular curve shapes and both components (i.e. periods) were present simultaneously. A recent study reports period doubling and tripling in leaf bioelectric and temperature responses to rhythmical light (Shabala et al., 1997). A common factor in these studies is that the responses were induced by alterations in the leaf illumination. These alterations will, in turn, affect the ionic conditions in the stomatal region (Willmer & Fricker, 1996), again hinting at a bifurcation parameter reflecting the ion conditions.

The transpiration from whole, uniformly illuminated primary leaves of oat plants was recorded in the present experiments. Thus, we have implicitly assumed the oscillations to be the result of in-phase mode oscillations (Rand et al., 1982). However, patchy stomatal behaviour has been observed in several plant species (Pospisilova & Santrucek, 1994; Mott & Buckley, 2000). Patchy behaviour can result in misleading interpretations of whole-leaf transpiration measurements. The transpiration rate can appear constant even though different patches are oscillating with large amplitude but slightly out of phase (Siebke & Weis, 1995; Mott & Buckley, 2000). It can therefore be argued that the complicated oscillatory behaviour observed in this report may be a result of out-of-phase regions on the leaf and not of the whole-leaf behaviour. Previous experiments on the coupling between different regions of an Avena leaf indicate that the coupling can be assumed to be fairly strong (cf. experiments by Brogårdh & Johnsson, 1974 on phase locking between different parts of the Avena leaf). Recent experiments using infrared imaging of Avena leaves showed that when a complicated oscillatory pattern consisting of several frequency components was observed in whole-leaf transpiration, then the same frequency components were present in the leaf temperature at all areas of the leaf (Prytz et al., 2003). Further, if the whole-leaf transpiration oscillations were damped out, then the leaf temperature oscillations at all areas of the leaf were damped out simultaneously. Thus, all parts of the leaf displayed approximately the same behaviour. Therefore, there is strong evidence suggesting that a young and small Avena leaf behaves as a single oscillator and that the observed complicated behaviour reflects whole-leaf behaviour.

The present work shows that the dynamics of the transpiration oscillations can result in very complicated oscillatory patterns, not previously reported. These patterns are similar to the period doubling sequences and oscillations displayed by certain chaotic systems. Thus, it is perhaps possible that under certain conditions the water regulatory system behaves as a mildly chaotic system. As has been demonstrated in some physical, chemical and biological systems, the sensitivity of chaotic systems to initial conditions and small parameter changes makes them susceptible to control, thereby enabling finer and faster control of the system (Ditto et al., 1990; Garfinkel et al., 1992; Roy et al., 1992; Petrov et al., 1993; Schiff et al., 1994; Braun et al., 1997). A mildly chaotic water regulatory system may perhaps be advantageous for the plant by allowing for rapid, fine-tuned responses to environmental stimuli. Much experimental work remains, however, before a bifurcation parameter and a mapping function for the water control system can be established. In conclusion, new mathematical features of the transpiration oscillations in plants have been described, further emphasizing the complexity of the plant water regulatory system.


The authors would like to thank Tor Egil Rødahl, M.Sc, for his participation in the experimental work. Financial support from the Research Council of Norway is gratefully acknowledged.