domain: sdv sde sde.mcg

text:
Shijuan Han

Stephen C Maberly

Brigitte Gontero

Zhenfei Xing

Wei Li

Hongsheng Jiang

Wenmin Huang
email: huangwm@wbgcas.cn

Structural basis for C 4 photosynthesis without Kranz anatomy in leaves of the submerged freshwater plant Ottelia alismoides

Keywords: aerenchyma, bicarbonate use, CAM, CO2 acclimation, CO2concentrating mechanism (CCM), chloroplast ultrastructure, freshwater macrophyte, Hydrocharitaceae

Background and Aims Ottelia alismoides (Hydrocharitaceae) is a freshwater macrophyte that, unusually, possesses three kinds of carbon dioxide-concentrating mechanisms. Here we describe its leaf anatomy and chloroplast ultrastructure, how they are altered by CO2 concentration and may underlie C4 photosynthesis.

• Methods Light and transmission electron microscopy were used to study the anatomy of mature leaves of O. alismoides grown at high and low CO2 concentrations.

Diel acid change and the activity of PEP carboxylase were measured to confirm that CAM activity and C4 photosynthesis were present.

• Key Results When O. alismoides was grown at low CO2 the leaves performed both C4 and CAM photosynthesis whereas with high CO2 leaves used C4 photosynthesis.

The leaf comprised an upper and lower layer of epidermal cells separated by a large air space occupying about 22% of the leaf transverse-section area, and by mesophyll cells connecting the two epidermal layers. Kranz anatomy was absent. At low CO2, chloroplasts in the mesophyll cells were filled with starch even at the start of the photoperiod, while epidermal chloroplasts had small starch grains. The number of chloroplasts in the epidermis was greater than in the mesophyll cells. At high CO2, the structure was unchanged but the thickness of the two epidermal layers, the air space, mesophyll and the transverse-section area of cells and air space were greater.

• Conclusions Leaves of O. alismoides have epidermal and mesophyll cells that contain chloroplasts and large air spaces but lack Kranz anatomy. The high starch content of mesophyll cells suggests they may benefit from an internal source of CO2, for example via C4 metabolism, and are also sites of starch storage. The air spaces may help in the recycling of decarboxylated or respired CO2. The structural similarity of leaves from low and high CO2 is consistent with the constitutive nature of bicarbonate and C4 photosynthesis. There is sufficient structural diversity within the leaf of O. alismoides to support dual-cell C4 photosynthesis even though Kranz anatomy is absent.

INTRODUCTION

In their evolution from terrestrial ancestors, freshwater plants have traded-off problems of water shortage for problems of carbon-shortage [START_REF] Maberly | Trade-offs and synergies in the structural and functional characteristics of leaves photosynthesizing in aquatic environments[END_REF]. Carbon-shortage is mainly caused by low rates of CO2 diffusion across boundary layers surrounding aquatic leaves [START_REF] Black | Resistances to carbon dioxide fixation in four submerged freshwater macrophytes[END_REF]. Additionally, especially in productive lakes, generation of CO2 concentrations below airequilibrium and, oxygen concentrations above air-equilibrium, can together stimulate photorespiration [START_REF] Maberly | Diel, episodic and seasonal changes in pH and concentrations of inorganic carbon in a productive lake[END_REF][START_REF] Sand-Jensen | Shallow plant-dominated lakes-extreme environmental variability, carbon cycling and ecological species challenges[END_REF]. In response to the absence of water shortage and the presence of carbon limitation, the leaves of submerged freshwater plants have thin cuticles, lack stomata and sub-stomatal spaces and have chloroplasts in epidermal cells [START_REF] Sculthorpe | The biology of aquatic vascular plants[END_REF]. Laminar leaves are generally thin with a high specific leaf area [START_REF] Enríquez | Broad-scale comparison of photosynthetic rates across phototrophic organisms[END_REF][START_REF] Poorter | Causes and consequences of variation in leaf mass per area (LMA): a meta-analysis[END_REF] and the lamina often comprises only two or three cell layers [START_REF] Maberly | Trade-offs and synergies in the structural and functional characteristics of leaves photosynthesizing in aquatic environments[END_REF].

Aerenchyma is a common feature of aquatic plants [START_REF] Sculthorpe | The biology of aquatic vascular plants[END_REF] and is present in the roots, leaves and stems of most aquatic species [START_REF] Silveira | Anatomical development of roots of native and non-native submerged aquatic macrophytes in different sediment types[END_REF].

In addition to these structural changes, freshwater macrophytes employ a number of avoidance, exploitation and amelioration strategies to overcome carbon limitation [START_REF] Klavsen | Crassulacean acid metabolism in the context of other carbon-concentrating mechanisms in freshwater plants: a review[END_REF]. Amelioration strategies involve active CO2concentrating mechanisms (CCMs) that increase CO2 around the primary carboxylase, ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco). There are three known CCMs in freshwater macrophytes: of the species investigated, about half have the ability to use HCO3 -, 8% have CAM and 4% have C4 [START_REF] Maberly | Ecological imperatives for aquatic CO 2 -concentrating mechanisms[END_REF][START_REF] Iversen | Catchment properties and the photosynthetic trait composition of freshwater plant communities[END_REF]. Bicarbonate is the dominant form of inorganic carbon when pH is between about 6.3 and 10.1 and even when CO2 becomes depleted as pH increases, bicarbonate can be present at appreciable concentrations. C4 photosynthesis and CAM depend on temporary fixation of bicarbonate by phosphoenolpyruvate carboxylase (PEPC) to form a C4 compound that is subsequently decarboxylated, raising the CO2 concentration around the active site of Rubisco [START_REF] Keeley | Isoetes howellii: a submerged aquatic CAM plant?[END_REF][START_REF] Bowes | C 4 mechanisms in aquatic angiosperms: comparisons with terrestrial C 4 systems[END_REF].

The carboxylation and decarboxylation processes are separated spatially in C4 photosynthesis and temporally in CAM. C4 photosynthesis reduces photorespiration during the day in aquatic and terrestrial plants. In addition, CAM can reduce the loss of carbon from dark respiration and extend the duration of carbon uptake to the night [START_REF] Maberly | Freshwater angiosperm carbon concentrating mechanisms: processes and patterns[END_REF][START_REF] Pedersen | Crassulacean acid metabolism enhances underwater photosynthesis and diminishes photorespiration in the aquatic plant Isoetes australis[END_REF].

Function and structure are often closely associated [START_REF] Smith | Leaf structural responses to preindustrial, current and elevated atmospheric [CO 2 ] and temperature affect leaf function in Eucalyptus sideroxylon[END_REF]. In freshwater macrophytes, use of bicarbonate can be associated with polar leaves with high pH produced at the adaxial/upper surface and low pH at the abaxial/lower surface [START_REF] Nielsen | Photosynthesis of aquatic plants with special reference to the carbon-sources[END_REF][START_REF] Prins | Bicarbonate utilization: function and mechanism[END_REF]. Terrestrial CAM plants have large vacuoles to store the C4 compound, often in the form of malate, accumulated during the night [START_REF] Nelson | Functional constraints of CAM leaf anatomy: tight cell packing is associated with increased CAM function across a gradient of CAM expression[END_REF][START_REF] Silvera | Evolution along the crassulacean acid metabolism continuum[END_REF]. Terrestrial C4 plants, typically have a specialised 'Kranz anatomy', that comprises mesophyll cells with C4 photosynthesis surrounding bundle sheath cells where CO2 is concentrated and assimilated by the Calvin-Benson-Bassham cycle [START_REF] Sage | C 4 plant biology[END_REF]. In addition to different photosynthetic enzymes, these two types of cell also differ in starch content and chloroplast ultrastructure [START_REF] Edwards | Single-cell C 4 photosynthesis versus the dual-cell (Kranz) paradigm[END_REF]) so that the initial carboxylation and the subsequent decarboxylation can be spatially separated in these two distinctive types of photosynthetic cell [START_REF] Raghavendra | Introduction[END_REF][START_REF] Sage | A portrait of the C 4 photosynthetic family on the 50th anniversary of its discovery: species number, evolutionary lineages, and hall of fame[END_REF]. However, a few terrestrial plants within the dicot Amaranthaceae family, e.g.

Bienertia cycloptera, Bienertia sinuspersici and Suaeda aralocaspica (formerly

Borszczowia aralocaspica), operate C4 photosynthesis through the spatial separation of dimorphic chloroplasts within a single cell [START_REF] Voznesenskaya | Kranz anatomy is not essential for terrestrial C 4 plant photosynthesis[END_REF][START_REF] Voznesenskaya | Proof of C 4 photosynthesis without Kranz anatomy in Bienertia cycloptera (Chenopodiaceae)[END_REF][START_REF] Voznesenskaya | Development of biochemical specialization and organelle partitioning in the singlecell C 4 system in leaves of Borszczowia aralocaspica (Chenopodiaceae)[END_REF][START_REF] Edwards | Single-cell C 4 photosynthesis versus the dual-cell (Kranz) paradigm[END_REF][START_REF] Akhani | Bienertia sinuspersici (Chenopodiaceae): a new species from Southwest Asia and discovery of a third terrestrial C 4 plant without Kranz anatomy[END_REF].

C4 photosynthesis is also present in freshwater plants from the monocot Hydrocharitaceae family: Egeria densa, Ottelia alismoides and Ottelia acuminata [START_REF] Casati | Induction of a C 4 -like mechanism of CO 2 fixation in Egeria densa, a submersed aquatic species[END_REF][START_REF] Lara | CO 2 -concentrating mechanisms in Egeria densa, a submersed aquatic plant[END_REF][START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF], as well as in the wellstudied species Hydrilla verticillata, where it takes place within a single cell [START_REF] Bowes | Plasticity in the photosynthetic carbon metabolism of submersed aquatic macrophytes[END_REF][START_REF] Bowes | Single-cell C 4 photosynthesis in aquatic plants[END_REF]. In E. densa and H. verticillata, C4 photosynthesis is induced by carbon limitation whereas in O. alismoides it is present in mature leaves regardless of the CO2 concentration [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF][START_REF] Shao | Responses of Ottelia alismoides, an aquatic plant with three CCMs, to variable CO 2 and light[END_REF][START_REF] Huang | Different CO 2 acclimation strategies in juvenile and mature leaves of Ottelia alismoides[END_REF]. In contrast, the CAM activity of O. alismoides is facultative and is induced at low CO2 but absent at high CO2 [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF]. In addition, O. alismoides has a constitutive ability to use bicarbonate and is the only plant known to have three different CCMs [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF][START_REF] Shao | Responses of Ottelia alismoides, an aquatic plant with three CCMs, to variable CO 2 and light[END_REF][START_REF] Huang | Different CO 2 acclimation strategies in juvenile and mature leaves of Ottelia alismoides[END_REF].

In terrestrial plants, elevated CO2 concentrations can alter anatomical structure [START_REF] Pritchard | Elevated CO 2 and plant structure: a review[END_REF][START_REF] Uprety | Effect of elevated CO 2 concentration on leaf structure of Brassica Juncea under water stress[END_REF]. Leaves of freshwater plants grown in air and water can have very different morphologies and structure [START_REF] Maberly | Trade-offs and synergies in the structural and functional characteristics of leaves photosynthesizing in aquatic environments[END_REF]. However, the anatomical response of leaves of freshwater plants to different CO2 concentrations has not been studied. Since O. alismoides is the only known species with three kinds of CCMs, we hypothesized that its leaf anatomy and chloroplast ultrastructure might be peculiar and reflect the integration of these three different processes and that they might be affected by CO2 concentration.

MATERIALS AND METHODS

Plant material

On 22 March 2018, seeds of O. alismoides were sown in plastic pots (11 cm in diameter and 7 cm deep) containing sterile soil from nearby Donghu Lake and covered with 2 cm of sterile tap water. The chemical composition of the tap water was analyzed. The concentrations of total nitrogen (TN) and total phosphorus (TP) were determined spectrophotometrically after digestion with K2S2O8 [START_REF] Huang | Survey, observation and analysis of lake ecology[END_REF].

The concentrations of Na + , K + , Ca 2+ and Mg 2+ were determined using optical emission spectrometry (ICP-OES) Optima 8000DV (Perkin Elmer, MA, USA) after addition of 200 μL HNO3 to 10 mL tap water. The concentrations of Cl -, SO4 2-and NO3 -were measured using Dionex ICS-5000 + HPIC system (Thermo Fisher Scientific, MA, USA). The composition is shown in Supplementary Data Table S1.

The pots were placed in a plant growth chamber at 28 °C, 12/12 hours photoperiod (150 μmol photon m -2 s -1 , photosynthetically active radiation). The water level was increased as the seedlings grew to keep them submerged. When the seedlings were about 4 cm tall, the pots were placed in 1 L glass beakers in the growth chamber. After 40 days, three to five seedlings (~8 cm tall) were transplanted into another plastic pot (15 cm diameter, 12 cm deep) containing the sterile soil. These containers were placed in a 400 L tank (64 cm deep) located in a glasshouse receiving natural daylight. The tap water in the tank was changed weekly and snails were removed daily.

Response of mature O. alismoides leaves to different CO 2 concentrations

In August 2018, when the plants had produced many broad, oval-shaped mature leaves, one pot was placed in each of eight white plastic buckets (25 × 25 × 35 cm).

These were filled with tap water and placed in the rooftop tanks, surrounded by water to keep the water temperature consistent among all the buckets, but the solution in each bucket was independent from the others. High and low CO2 concentrations were produced using the method described in [START_REF] Shao | Responses of Ottelia alismoides, an aquatic plant with three CCMs, to variable CO 2 and light[END_REF] with four replicates of each treatment. The water in the buckets was changed twice during the 40-days acclimation.

Low CO2 (LC) was produced by allowing plant photosynthesis to deplete inorganic carbon and increase pH while high CO2 (HC) was produced by adding a CO2 solution to produce a set pH twice each day. In the morning (between 08:00 and 09:00) and afternoon (between 17:30 to 19:00), the water was gently stirred to thoroughly mix it and pH was measured with a pH electrode and temperature was measured with a thermometer. Alkalinity was measured every two days by Gran titration [START_REF] Shao | Responses of Ottelia alismoides, an aquatic plant with three CCMs, to variable CO 2 and light[END_REF]. On each sampling occasion, CO2-saturated tap water was added to the HC treatment, to bring the pH to 6.8. Concentrations of CO2 were calculated from pH, alkalinity and temperature [START_REF] Maberly | Diel, episodic and seasonal changes in pH and concentrations of inorganic carbon in a productive lake[END_REF]. In the LC treatment, the pH increased from 8.0 to over 9.8 and the CO2 concentration varied between 0.1 and 13 μmol L -1 with a mean of 2.4 μmol L -1 . In the HC treatment, the pH varied between 6.6 and 6.9, producing CO2 concentrations between 481 μmol L -1 and 1110 μmol L -1 with a mean of 720 μmol L -1 . Water temperature ranged between 25 and 35 ºC with a mean of 30 ºC. As a reference, the CO2 concentration in equilibrium with 400 ppm atmospheric CO2 will be about 14 μmol L -1 at 30 °C. Changes in CO2 concentration in the two treatments are shown in Supplementary Data Fig. S1. After 40-days acclimation to LC and HC treatments, fully expanded mature leaves that had been produced during the experiment and appeared to be of similar age were sampled for structure and physiology studies.

Light Microscopy and Transmission Electron Microscopy

Leaf segments (3 mm × 3 mm) sampled at 0500 and 1800 were fixed overnight in 2.5% glutaraldehyde in 0.1 M phosphate buffer (pH 7.4) at 4°C and post-fixed in 1% OsO4 at 4°C for 2.5 h [START_REF] Farnese | The involvement of nitric oxide in integration of plant physiological and ultrastructural adjustments in response to arsenic[END_REF]. They were dehydrated using a stepwise ethanol series (30, 50, 70, 90 and 100%) and processed with a mixed solution of ethanol and acetone, then infiltrated in a mixture of acetone and epoxy resin (1: 1 for 1 h, and then 1: 2 for 8 h) and finally embedded using SPI-PON 812 at 60°C for 48 h.

Semi-thin sections (1.5 μm) and ultrathin sections (72 nm) were obtained on a Leica EM UC7 ultramicrotome. Semi-thin sections were stained with methylene blue and observed using a Motic BA310 digital light microscope. Quantitative characteristics of leaf structures were measured using the Motic Images Plus 2.0 ML software. For the ultrathin sections, after staining with uranyl acetate and lead citrate, they were observed and photographed with a HT7700 transmission electron microscope (Hitachi High-Tech, Japan). Leaf chloroplasts and starch grains in electron micrograph were measured using Image J software. The distribution of chloroplasts in cells was assessed by measuring the number of chloroplasts per unit length of cell wall at different locations from the semi-thin sections under the light microscope, including the upper epidermis next to water, the upper epidermis next to air space, the upper mesophyll cells, the lower mesophyll cells, the lower epidermis next to air space and the lower epidermis next to water.

Enzyme activity measurement

Mature leaves were collected at 0500 and 1800 using the extraction and assay protocols for Rubisco and PEPC activities as described previously [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF][START_REF] Shao | Responses of Ottelia alismoides, an aquatic plant with three CCMs, to variable CO 2 and light[END_REF][START_REF] Huang | Different CO 2 acclimation strategies in juvenile and mature leaves of Ottelia alismoides[END_REF]. Enzyme activities were calculated from the rate of disappearance of NADH at 340 nm and 25°C measured with a microplate reader (Tecan M200 PRO, Austria).

CAM activity measurement

CAM activity was assessed by calculating the daily change in titratable acidity, in leaves harvested at 0500 and 1800 as previously described [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF].

Briefly, to measure leaf acid content, 10 mL CO2-free deionized water were added to the leaf samples of known fresh weight (0.2 ~ 0.5 g) and then boiled for 30 min. The acidity was titrated to pH 8.3 using 0.01 N NaOH.

Statistical analysis

SPSS 20 (SPSS Inc., Chicago, IL, USA) was used to analyze the data. An independent sample t-test was used for two groups of data with normal distribution, the Mann-Whitney U-test was used for two groups of data where the distribution was not normal and the Kruskal-Wallis test was used for three or more groups of data where the distribution was not normal. The significance level of the statistics was accepted at P < 0.05.

RESULTS

C 4 photosynthesis and CAM activity

The enzyme activities of PEPC and Rubisco in mature leaves under HC and LC treatments were measured to confirm that C4 photosynthesis was present. There was no significant difference in PEPC activities between leaves collected at dusk and dawn regardless of the treatments, but the PEPC activities were significantly higher in leaves grown at a low versus a high concentration of CO2 (Fig. 1A). Rubisco activity was significantly lower in leaves collected at dawn versus dusk, but was not affected by CO2 concentration (Fig. 1B). The ratio of PEPC to Rubisco was between 1.6 and 4.2 (Fig. 1C) within a range typical of terrestrial C4 plants [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF]. This ratio was significantly higher at low than at high CO2 both at dawn and dusk. At dawn, the ratio was about 2.6-fold higher in low compared to high CO2 leaves suggesting that leaves from low CO2 had more active C4 photosynthesis than those from high CO2. Leaves at low CO2 had a marked diel change in acidity that was significantly higher (58 μequiv g -1 FW) than at high CO2 concentrations (19 μequiv g -1 FW, Fig. 2). These results confirm that these mature O. alismoides leaves perform both C4 photosynthesis and CAM when acclimated to low CO2 and C4 photosynthesis at high CO2.

Effects of CO 2 concentration on leaf anatomy

Transverse sections showed that there were no major differences in basic structure and types of cell in leaves grown at high and low concentrations of CO2 (Fig. 3).

Leaves from both treatments comprised an upper and lower layer of epidermal cells separated by a large air space and between one and three but most often two stacked mesophyll cells connecting the two epidermal layers. In transverse section, the epidermal cells varied in shape between rectangular and elliptical while the mesophyll cells were less elongated (Fig. 3B,E). Parenchymal cells containing some chloroplasts occurred around the vascular bundle, the bundle sheath, but this was not surrounded by mesophyll cells and thus Kranz anatomy was absent (Fig. 3C,F).

Leaf width was significantly greater at high (8.56 (mean) cm ± 0.50 (SD)) compared to low CO2 concentration (5.44 cm ± 0.18), but leaf length was unaltered at ~12 cm (Table 1). Consequently, the length to width ratio was greater at low CO2.

The leaves of O. alismoides were thicker at high than at low CO2 concentration (Fig. 3, Table 1). The ratio of air space area to leaf area in transverse section was 0.22 to 0.23 but was not affected significantly by CO2 concentration (Table 1). The air spaces were bounded top and bottom by the epidermal cells and surrounded at the sides by a network of mesophyll cells and so were not connected to one another (see also Supplementary Data Fig. S2).

The epidermis and mesophyll of plants grown at high CO2 concentration were significantly thicker and of greater cell area than in plants grown at low CO2 concentration (Fig. 4). Furthermore, the upper epidermis was thicker than the lower epidermis, both at low and high CO2 (P < 0.001), but the area was not different. The thickness and area of an air space were correspondingly greater at high CO2 concentration.

Effects of CO 2 concentration on chloroplasts

Chloroplasts were present in epidermal and mesophyll cells (Fig. 3). Since chloroplasts can differ in cells carrying out different processes, such as in C4 photosynthesis, we characterized chloroplast size, shape and ultrastructure in leaves grown at high and low CO2. In leaves grown at high CO2, the chloroplasts in epidermis and mesophyll cells were all nearly spherical with large starch grains (Fig. 5A,C,D,E), which occupied ~65 and ~80% of the chloroplast area both at dusk and dawn (Fig. 6E &F). In contrast, in leaves grown at low CO2, epidermal chloroplasts were spindle-shaped (Fig. 5F), while the mesophyll chloroplasts were spherical (Fig. 5I) and consequently, their minor axis length was shorter (Fig. 6B). In the leaves collected at dusk, starch occupied 40% and 64% of the chloroplast area in the epidermal and the mesophyll cells, respectively (Fig. 6E). The same trend was also observed in the leaves sampled at dawn (Fig. 6F), small and large starch grains were distributed in the chloroplasts of the epidermal and mesophyll cells, respectively (Fig. 5H &J). Grana with many thylakoids were clearly visible in epidermal chloroplasts (Fig. 5B &G) whereas in mesophyll chloroplasts, the grana thylakoids were indistinct because of the high starch content.

The average area of a chloroplast was greater in leaves grown at high versus low CO2 in both types of cell (Fig. 6C). The area of a chloroplast was greater in mesophyll than in epidermal cells for both CO2 treatments. At high CO2, the number of mitochondria within 1 µm around a chloroplast in mesophyll and epidermal cells was not significantly different (Fig. 6D). However, at low CO2, the number of mitochondria around chloroplasts in mesophyll cells was significantly higher than in the epidermal cells.

The distribution of chloroplasts in epidermal and mesophyll cells was measured under different CO2 concentrations (Fig. 7). The frequency of chloroplasts in the upper and lower epidermis was significantly greater than in the mesophyll cells.

The chloroplast frequency at the cell walls next to air spaces in the lower epidermis was less than the equivalent location in the upper epidermis, at high and low CO2. The frequency of chloroplasts at the upper epidermis next to an air space was significantly lower in leaves acclimated to low compared to high CO2.

DISCUSSION

Adaptation to photosynthesis underwater

The presence of chloroplasts in epidermal cells is a common feature in submerged angiosperms, but rare in their terrestrial ancestors [START_REF] Sculthorpe | The biology of aquatic vascular plants[END_REF][START_REF] Rascio | The underwater life of secondarily aquatic plants: some problems and solutions[END_REF][START_REF] Maberly | Trade-offs and synergies in the structural and functional characteristics of leaves photosynthesizing in aquatic environments[END_REF]. Mature leaves of O. alismoides have two types of photosynthetic cell, epidermal and mesophyll. In contrast, leaves from three other species of Hydrocharitaceae, H. verticillata, E. densa and Elodea callitrichoides, only have two layers of epidermal cells and no mesophyll cells [START_REF] Falk | Zellfeinbau bei Plasmolyse[END_REF][START_REF] Pendland | Ultrastructural characteristics of Hydrilla leaf tissue[END_REF][START_REF] Hara | Organ-level analysis of idioblast patterning in Egeria densa Planch. leaves[END_REF]. As a consequence, the leaf thickness of O. alismoides, especially in the leaves grown at high CO2, at 196 µm, is much greater than the median (95 µm) for submerged leaves from a range of freshwater macrophytes [START_REF] Maberly | Trade-offs and synergies in the structural and functional characteristics of leaves photosynthesizing in aquatic environments[END_REF] and greater than E. callitrichoides and E. densa estimated from published images to be about 65 µm. While [START_REF] Black | Resistances to carbon dioxide fixation in four submerged freshwater macrophytes[END_REF] estimated that the internal resistance to diffusion of CO2 within the relatively thin leaves of four Potamogeton species (48 to 63 µm) was only 3 to 4% of the total, the internal resistance in the thicker leaves of O. alismoides could be greater but could be offset by the high porosity (ratio of air space to transverse sectional leaf area, 0.22 to 0.23). In O. alismoides, the sizes of the upper and lower epidermal cells are similar while in H. verticillata, the upper epidermal cells are noticeably larger than the lower cells [START_REF] Pendland | Ultrastructural characteristics of Hydrilla leaf tissue[END_REF]. In E. callitrichoides the transverse sectional area of the upper epidermal cells is about four-times that of the lower epidermis [START_REF] Falk | Zellfeinbau bei Plasmolyse[END_REF], assuming the orientation in Fig. 1 in [START_REF] Falk | Zellfeinbau bei Plasmolyse[END_REF] is correct since the adaxial and abaxial layers are not specifically labelled. Similarly, in E. densa, the transverse sectional area of the upper epidermal cells is about five-times larger than that of the lower epidermal cells [START_REF] Hara | Organ-level analysis of idioblast patterning in Egeria densa Planch. leaves[END_REF]. The functional significance of these differences is currently unknown but they could be linked to the use of bicarbonate involving different processes in upper and lower cell layers [START_REF] Nielsen | Photosynthesis of aquatic plants with special reference to the carbon-sources[END_REF][START_REF] Prins | Bicarbonate utilization: function and mechanism[END_REF].

Well-developed aerenchyma is a common feature of aquatic plants [START_REF] Jung | Anatomical patterns of aerenchyma in aquatic and wetland plants[END_REF]. O. alismoides contains large air spaces between the epidermal layers while in H. verticillata and E. callitrichoides, there are numerous small intercellular spaces between the upper and lower epidermal cells that comprise a small proportion of the leaf volume [START_REF] Falk | Zellfeinbau bei Plasmolyse[END_REF][START_REF] Pendland | Ultrastructural characteristics of Hydrilla leaf tissue[END_REF]. In many isoetids, the aerenchyma is a large proportion of the leaf volume and is continuous from the roots to the leaves, allowing sedimentary CO2 to be taken up and fixed [START_REF] Madsen | Carbon acquisition and carbon dynamics by aquatic isoetids[END_REF] and oxygen to be supplied to the roots [START_REF] Sand-Jensen | Oxygen exchange with the lacunae and across leaves and roots of the submerged vascular macrophyte, Lobelia dortmanna L[END_REF]. Since in O. alismoides the air spaces are discrete (Supplementary Data Fig. S2), and so not connected to the roots, these two processes are unlikely to occur. Air spaces comprise about 22 to 23% of the transverse section leaf area in O. alismoides, which is within the broad range recorded for terrestrial leaves (3 -73%, [START_REF] Slaton | Mesophyll architecture and cell exposure to intercellular air space in alpine, desert, and forest species[END_REF][START_REF] Earles | Beyond porosity: 3D leaf intercellular airspace traits that impact mesophyll conductance[END_REF].

In terrestrial leaves, the intercellular air spaces are connected to the atmosphere via stomata and help to maximize the mesophyll surface area in contact with atmospheric CO2. Aquatic plants generally lack functional stomata, so the air spaces within a leaf are not connected to the exterior. The air spaces provide buoyancy, allowing the leaves to float towards the surface where light is higher, but could also act as a reservoir of respiratory CO2 [START_REF] Wetzel | Photosynthesis of submersed macrophytes in acidified lakes. I. Carbon fluxes and recycling of CO 2 in Juncus bulbosus L[END_REF] or photorespiratory CO2 [START_REF] Søndergaard | Photorespiration and internal recycling of CO 2 in the submersed angiosperm Scirpus subterminalis[END_REF] as can also occur in terrestrial C3 plants [START_REF] Busch | C 3 plants enhance rates of photosynthesis by reassimilating photorespired and respired CO 2[END_REF]. However, a calculation for low CO2 leaves based on a one-sided specific leaf area of 100.5 cm 2 g -1 FW and net rates of photosynthesis at around air-equilibrium CO2 of 2 µmol g -1 FW h -1 [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF], air space area as a proportion of leaf area of 0.23 (this study) and maximal CO2 partial pressure in the air space of 10,000 ppm [START_REF] Madsen | Interactions between internal and external CO 2 pools in the photosynthesis of the aquatic CAM plants Littorella uniflora (L.) Aschers and Isoetes lacustris L[END_REF] suggests that air-space CO2 could only support net photosynthesis for about 5 minutes. Nevertheless, if C4 decarboxylation occurs in the mesophyll cells, as seems possible given the relatively high chloroplast density in these cells in relation to their distance from external inorganic carbon sources, then the air spaces may serve to trap and recycle CO2 produced by C4 decarboxylation. This could be efficient as loss of CO2 from the air spaces to the outside will be limited by the chloroplasts in the epidermis and the low rate of CO2 exchange between the leaf and the bulk water as a result of transport limitation across the boundary layer. At 25 °C the air-equilibrium molar ratio of oxygen to CO2 in water is about 28-fold less than in air, so the air spaces may also provide a means to reduce the concentration of photosynthetically produced oxygen within mesophyll and epidermal cells.

C 4 photosynthesis and absence of Kranz anatomy

In the Hydrocharitaceae, four species are known to have constitutive or facultative C4 photosynthesis, H. verticillata, E. densa, O. acuminata and O. alismoides [START_REF] Bowes | C 4 mechanisms in aquatic angiosperms: comparisons with terrestrial C 4 systems[END_REF][START_REF] Lara | CO 2 -concentrating mechanisms in Egeria densa, a submersed aquatic plant[END_REF][START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF][START_REF] Yin | Photosynthetic inorganic carbon acquisition in 30 freshwater macrophytes[END_REF] and lack Kranz anatomy. There is evidence that H. verticillata has single cell C4 photosynthesis [START_REF] Bowes | C 4 mechanisms in aquatic angiosperms: comparisons with terrestrial C 4 systems[END_REF]. In this respect, it is similar to single-cell terrestrial C4 plants [START_REF] Voznesenskaya | Kranz anatomy is not essential for terrestrial C 4 plant photosynthesis[END_REF][START_REF] Voznesenskaya | Proof of C 4 photosynthesis without Kranz anatomy in Bienertia cycloptera (Chenopodiaceae)[END_REF] such as B. cycloptera and S. aralocaspica where C4 production and decarboxylation occur in different parts of a cell [START_REF] Edwards | Single-cell C 4 photosynthesis versus the dual-cell (Kranz) paradigm[END_REF][START_REF] Edwards | C 4 Photosynthesis: Kranz forms and single-cell C 4 in terrestrial plants[END_REF][START_REF] Sharpe | One decade after the discovery of single-cell C 4 species in terrestrial plants: what did we learn about the minimal requirements of C 4 photosynthesis?[END_REF][START_REF] Von Caemmerer | Single cell C 4 photosynthesis in aquatic and terrestrial plants: a gas exchange perspective[END_REF]. In these terrestrial plants, C4 decarboxylation occurs in mitochondria using NAD malic enzyme (NAD-ME). In O. alismoides, [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF] suggested, on the basis of enzyme activity measurements, that NAD-ME is also the decarboxylating enzyme, unlike in H. verticillata where NADP-ME is believed to be involved [START_REF] Bowes | Single-cell C 4 photosynthesis in aquatic plants[END_REF]. The location of primary CO2 fixation and decarboxylation in O. alismoides, and whether they take place in a single cell, is unknown. Here we show that the mesophyll cells of O. alismoides have a high starch content which is consistent with these cells either being the site of high rates of photosynthesis resulting from decarboxylation, producing CO2 locally or being sites of starch storage. However, the high frequency of mitochondria around mesophyll chloroplasts in leaves acclimated to low CO2 supports the possibility that these cells are the site of decarboxylation by the mitochondrial NAD-ME, as does the presence of chloroplasts in mesophyll cells despite their distance from external sources of inorganic carbon. In this dual-cell model, the O. alismoides epidermal cells would then perform the function of the terrestrial C4 mesophyll cells by producing a C4 product and the O. alismoides mesophyll cells would perform the function of the terrestrial C4 bundle sheath cells in decarboxylating the C4 product. However, it is also possible that both processes could be occurring within the mesophyll cells. Some C4 terrestrial plants have dimorphic chloroplasts in a single cell [START_REF] Voznesenskaya | Kranz anatomy is not essential for terrestrial C 4 plant photosynthesis[END_REF][START_REF] Voznesenskaya | Proof of C 4 photosynthesis without Kranz anatomy in Bienertia cycloptera (Chenopodiaceae)[END_REF][START_REF] Voznesenskaya | Development of biochemical specialization and organelle partitioning in the singlecell C 4 system in leaves of Borszczowia aralocaspica (Chenopodiaceae)[END_REF][START_REF] Akhani | Bienertia sinuspersici (Chenopodiaceae): a new species from Southwest Asia and discovery of a third terrestrial C 4 plant without Kranz anatomy[END_REF][START_REF] Sharpe | One decade after the discovery of single-cell C 4 species in terrestrial plants: what did we learn about the minimal requirements of C 4 photosynthesis?[END_REF]. The different forms of chloroplasts in the Kranz anatomy cells are linked to different energy requirements and fixation of carbon [START_REF] Edwards | Single-cell C 4 photosynthesis versus the dual-cell (Kranz) paradigm[END_REF]. Dimorphic chloroplasts differing in size and ultrastructure also occur in leaves of freshwater plants. In Cabomba caroliniana (Cabombaceae), the chloroplasts in mesophyll cells have larger starch grains, more thylakoids per granum, and are larger than epidermal chloroplasts [START_REF] Galati | Chloroplast dimorphism in leaves of Cabomba caroliniana (Cabombaceae)[END_REF] Table 2). Species within the aquatic angiosperm family Podostemaceae also have two types of chloroplasts. Small chloroplasts with a normal grana ultrastructure and very small starch grains occur at the upper tangential walls of epidermal cells, while large chloroplasts with more thylakoids per granum and many well-developed starch grains occur at the lower tangential walls of these cells and also in mesophyll cells [START_REF] Fujinami | Dimorphic chloroplasts in the epidermis of Podostemoideae, a subfamily of the unique aquatic angiosperm family Podostemaceae[END_REF]; Table 2). However, there is no evidence for CCMs in C. caroliniana [START_REF] Yin | Photosynthetic inorganic carbon acquisition in 30 freshwater macrophytes[END_REF]. In the Podostemaceae, although carbon isotope values range widely between -12.8 and -38.6‰ [START_REF] Ziegler | Carbon isotope fractionation in species of the torrenticolous families Podostemaceae and Hydrostachyaceae[END_REF] at different locations suggesting that there could be differences in discrimination resulting from C4 photosynthesis or bicarbonate use, differences in carbon isotope signature caused by the source inorganic carbon cannot be excluded so the presence or absence of CCMs in these plants is currently unknown.

In terrestrial C4 plants, cells or parts of cells where C4 fixation occurs have smaller chloroplasts with less starch than chloroplasts in cells or parts of cells where decarboxylation occurs (Table 2). Chloroplasts from the epidermal cells of O. alismoides are similar to those involved in C4 fixation while chloroplasts from mesophyll cells are similar to those involved in C4 decarboxylation, suggesting that C4 photosynthesis in this species may involve both cell types. However, although the biochemical evidence suggests that O. alismoides belongs to the NAD-ME sub-type of C4 [START_REF] Zhang | Biochemical and biophysical CO 2 concentrating mechanisms in two species of freshwater macrophyte within the genus Ottelia (Hydrocharitaceae)[END_REF], its pattern of thylakoids per granum in epidermal compared to mesophyll cells is closer to that of terrestrial NADP-ME C4 plants, suggesting that there is not a direct 'read-across' between C4 photosynthesis in aquatic and terrestrial plants.

Responses to CO 2

To maximise plant productivity there is an intricate relationship between leaf structure and function [START_REF] Oguchi | Leaf anatomy and function[END_REF]. In terrestrial plants, where water can be limiting, there is an evolutionary pressure to maximise the ratio of carbon gain to water loss.

Cell size and leaf thickness are dependent on environmental conditions [START_REF] Zeiger | The biology of stomatal guard cells[END_REF][START_REF] Jones | Adaptive significance of leaf development and structural responses to environment[END_REF][START_REF] Radoglou | Effects of CO 2 enrichment on four poplar clones. I. Growth and leaf anatomy[END_REF]. In Liquidambar styraciflua, Pinus taeda, and Brassica juncea, high CO2 also has a structural effect, increasing the thickness of the upper and lower epidermis and mesophyll cell of leaves [START_REF] Rogers | Response of agronomic and forest species to elevated atmospheric carbon dioxide[END_REF][START_REF] Uprety | Effect of elevated CO 2 concentration on leaf structure of Brassica Juncea under water stress[END_REF]. Elevated CO2 generally increases the size of terrestrial plants [START_REF] Pritchard | Elevated CO 2 and plant structure: a review[END_REF]. We found the same here for the freshwater plant O. alismoides, suggesting that the increase of this resource has a universal effect on plant size.

Conclusions

O. alismoides has three CCMs that requires structural and functional coordination to operate efficiently. Unlike terrestrial plants, the anatomy of O. alismoides is relatively simple, and spongy and palisade tissues are absent, as they are in submerged leaves of all aquatic plants [START_REF] Maberly | Trade-offs and synergies in the structural and functional characteristics of leaves photosynthesizing in aquatic environments[END_REF]. The leaf comprises two types of photosynthetic cell, epidermal and mesophyll. The conceptual overview summarizing the structure of O. alismoides leaves acclimated to high CO2 and low CO2 concentrations is shown in Supplementary Data Fig. S3. Epidermal cells, containing chloroplasts, maximise uptake of external CO2, aided by bicarbonate use, while the mesophyll cells may be sites where CO2 is concentrated by decarboxylation.

Abundant discrete air spaces provide buoyancy but may also trap (photo)-respiratory CO2, or CO2 produced by decarboxylation, permitting its refixation. Overall, there is sufficient structural diversity within the leaf of O. alismoides to support dual-cell C4 photosynthesis even though Kranz anatomy is absent. However, further studies are needed to conclude definitively if O. alismoides has dual-cell C4 with the mesophyll cells representing the site of decarboxylation. Work is underway to test this by locating key photosynthesis enzymes in the epidermal and mesophyll cells.

SUPPLEMENTARY DATA

Supplementary data consist of the following.

  Figure S1: Fluctuations of CO2 concentration in high CO2 and low CO2 treatments during the 40-days acclimation.

Figure S2 :

 S2 Figure S2: Photographs of the surface of a mature O. alismoides leaf from the low CO2 treatment using a laser scanning confocal microscope.Figure S3: Conceptual

  FIG. 1. Influence of CO2 concentration on activities of enzymes from O. alismoides leaves collected at dusk (1800) and dawn (0500). (A) PEPC activity. (B) Rubisco activity. (C) Ratio of PEPC to Rubisco activity. The mean values (n = 3-4), with their SD are shown. The statistical differences between means were tested using independent sample t-tests in panel A and B and independent sample t-tests and the Mann-Whitney U test in panel C. The statistic above the horizontal line compares leaves exposed to high CO2 (black bars) and low CO2 (white bars) collected at the same time (NS not significant, * P < 0.05, *** P < 0.001); data with different small and large letters are significantly different between dusk and dawn, under high and low CO2 treatments respectively (P < 0.05).

  Calculated according to axis length in reference b Plants treated with low CO 2 concentrations c Chloroplasts have structural characteristics like those in mesophyll d Chloroplasts have structural characteristics like those in bundle sheath cell -Lack grana/starch; + Rudimentary grana/starch; + + Contain grana/starch; + + + Welldeveloped grana/starch; BS = Bundle sheath cell; C = Chlorenchyma cell; E = Epidermal cell; M = Mesophyll cell; P = Palisade cell; nd = not determined.

Table 1 .

 1 Influence of CO2 concentration on characteristics of O. alismoides leaves.The mean values are given with SD in parenthesis. Significant differences between leaves treated with different CO2 concentration are shown based on independent sample t-test (NS not significant, * P < 0.05, *** P < 0.001).

	Characteristics	High CO 2	Low CO 2	Significance n
	Leaf length (cm)	12.08 (0.36)	11.58 (0.68)	NS	5
	Leaf width (cm)	8.56 (0.50)	5.44 (0.18)	***	5
	Length-width ratio	1.41 (0.06)	2.13 (0.12)	***	5
	Leaf thickness (μm)	196 (17)	161 (18)	*	4
	The ratio of air space to leaf area in transverse section	0.22 (0.03)	0.23 (0.02)	NS	4

Table 2 .

 2 Characteristics of dimorphic chloroplasts in aquatic and terrestrial plants.

	Environment /type/ species	Location Area (μm 2 )	Thylakoids per granum	Starch	References
	Aquatic C 3
	Hydrobryum khaoyaiense	E	3.7 ± 1.8	3~4	+	Fujinami et al. (2011)
	Cabomba caroliniana	E E M	23.7 ± 11 3 . 8 a ~33.2 a	4~5 ~9 ~29	+ + + + + +	Galati et al. (2015)
	Aquatic NAD-ME C 4
	Ottelia alismoides b	E	9.0 ± 2.3	+ +	+ +	This study
		M	13.8 ± 3.3	+	+ + +
	Terrestrial NADP-ME C 4
	Zea mays	M	Small	+ + +	+ +	Hodge et al. (1955); Laetsch (1968)
		BS	Large	+	+ + +
	Saccharum officinarum	M	Small	+ + +	+	Laetsch and Price (1969)
		BS	Large	-	+ + +
	Sorghum bicolor	M	nd	+ + +	nd	Woo et al. (1970)
	Salsola australis	BS M BS	nd 6.67~8.2 a 14.6~16.1 a	-6~12 2~5	nd + + + +	P'yankov et al. (1997)
	Terrestrial NAD-ME C 4
	Amaranthus edulis	P	Small	+ +	+	Laetsch (1968)
		BS	Large	+ +	+ + +
	Atriplex lentiformis	P	nd	2~3	+	Laetsch (1968)
		BS	nd	+ +	+ + +
	Atriplex spongiosa	M	nd	+	nd	Woo et al. (1970)
		BS	nd	+ + +	nd
	Terrestrial Single-cell
	NAD-ME C 4 Bienertia cycloptera	C c C d	nd nd	~ 2 3~5	-+ +	Voznesenskaya et al. (2002)
	Suaeda aralocaspica	C c C d	3.48~4.0 a 7.18~8.0 a	+ + + +	+ + + +	Voznesenskaya et al. (2003)
	a

ACKNOWLEDGEMENTS

We thank Yuan Xiao for providing the TEM service and Jun Men for assistance in the chemical analysis of water (Analysis and Testing Center, Institute of Hydrobiology, Chinese Academy of Sciences). This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB31000000), Chinese Academy of Sciences President's International Fellowship Initiative to SCM and BG (2015VBA023, 2016VBA006), and the National Scientific Foundation of China (31860101, 31970368).

1 2

Supplementary Data Table S1. The chemical composition of the tap water used in the growth experiments.


R Bachelard

C Chandre

M Vittot

Hamiltonian description of a self-consistent interaction between charged particles and electromagnetic waves

Keywords: numbers: 45.20.Jj, 52.65.Ff, 41.60.Cr, 52.35.Qz

The Hamiltonian description of the self-consistent interaction between an electromagnetic planewave and a co-propagating beam of charged particles is considered. We show how the motion can be reduced to a one-dimensional Hamiltonian model (in a canonical setting) from the Vlasov-Maxwell Poisson brackets. The reduction to this paradigmatic Hamiltonian model is performed using a Lie algebraic formalism which allows us to remain Hamiltonian at each step of the derivation.

I. INTRODUCTION

The interaction between electromagnetic fields and a beam of charged particles exhibit a dynamics which is nowadays not fully understood, mainly due to the complexity inherent to the infinite dimensional phase space. Although some attention has been devoted to controlling those systems, a preliminary step is to shed light on the dynamics by analysing for instance phase space structures and transport properties. In this ambitious framework, reduced models have proved to be very valuable for this task. The reduction is of course guided by the physics of a particular setting. For instance, a beam of ultra-relativistic electrons interacts with plane waves (whether these are external ones or produced by the beam moving into an array of magnets, called undulator) has shown to be a way for producing a coherent light source. In what follows, we consider such a beam of electrons moving inside an undulator which produces a static (but non-uniform) magnetic field. The acceleration produced by the external magnetic field makes them emit a synchrotron radiation, which is self-consistently interacting with the particles. Under some resonance condition, the intensity of this electromagnetic wave grows exponentially and then saturates. In order to capture this effect, a one-dimensional Hamiltonian model has been proposed. The reduced Hamiltonian describes the evolution of the position θ j and relative momentum p j (around a resonant value) of the j-th particle. The N particles interact through a wave described by its intensity I and phase φ. It reads :

H = N j=1 p 2 j 2 + 2 √ I sin(θ j -φ) , (1)
where (θ j , p j ) and (φ, I) are canonical pairs of conjugate variables.

In the Free Electron Laser (FEL) configuration, this model has been derived [START_REF] Bonifacio | [END_REF]2] from the equations of motion of charged particles and Maxwell's equations. Some approximations were involved during the course of the derivation, and were guided by the physics of the device, and among them, a specific form for the radiated fields (as a plane wave), a onedimensional reduction (obtained by reducing the dynamics in the transverse plane), an expansion around a resonance fixed by the characteristics of the undulator. In addition, this Hamiltonian model was also proposed to describe the wave-particle interaction in other contexts, such as the beam-plasma instability [3,[START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF], or the Collective Atomic Recoil Laser [START_REF] Bonifacio | [END_REF].

In this article, we propose a derivation of the one-dimensional Hamiltonian (1) for the selfconsistent interaction from the Vlasov-Maxwell equations in a Hamiltonian setting. This allows us to show that at each step of the derivation, the Hamiltonian structure of the problem is conserved. Another advantage is that the conserved quantity (namely the total momentum of the system) is also easily deduced from the conserved quantity of the Vlasov-Maxwell equations. Our approach follows from this algebraic framework: Instead of working with the equations of motion for the derivation, we consider the Hamiltonian and its associated Poisson bracket (for an introduction, see Refs. [6,7]). By using a canonical version, the Poisson bracket remains canonical (or generalized canonical in a broader sense).

Therefore the main approximations and computations have to be done on a scalar function, the Hamiltonian which aims at simplifying the derivation.

In Sec. II we recall some basics of the Hamiltonian formulation of the Vlasov-Maxwell equations for a continuous description of the particle distribution. In Sec. III, we apply to this Hamiltonian system the approximations and reduction necessary for the derivation

II. HAMILTONIAN FORMULATION OF VLASOV-MAXWELL EQUATIONS

The dynamics of Hamiltonian (1) follows from Hamilton's equations for each pair of canonically conjugate variables. More generally the dynamics of an observable F (function of phase space coordinates ({θ i , p i }, I, φ)) is given by :

dF dt = {H, F }, (2)
where the Poisson bracket between two observables is given by

{F, G} = N i=1 ∂F ∂p i ∂G ∂θ i - ∂F ∂θ i ∂G ∂p i + ∂F ∂I ∂G ∂φ - ∂F ∂φ ∂G ∂I .
In a continuous setting, this Hamiltonian model can be extended in a straightforward way.

The beam is now described by a distribution function f (θ, p) which constitutes a dynamical field, i.e., for each location in phase space (θ, p), the density of particles f (θ, p), labelled by the phase space coordinates of the particles, evolves dynamically. The one-dimensional Hamiltonian model is generalized from Hamiltonian (1)

H[f, I, φ] = dθdpf (θ, p) p 2 2 + 2 √ I sin (θ -φ) , (3)
where the dynamical variables are now I and ϕ, and a field of variables f (θ, p). The dynamics of f , I and θ are obtained using the canonical Poisson bracket:

{F, G} = dθdpf (θ, p) ∂ ∂p δF δf ∂ ∂θ δG δf - ∂ ∂θ δF δf ∂ ∂p δG δf + ∂F ∂I ∂G ∂φ - ∂F ∂φ ∂G ∂I , (4)
i.e. it leads to a Vlasov equation for f

∂f ∂t + p ∂f ∂θ -2 √ I cos(θ -φ) ∂f ∂p = 0,
where we notice that df /dt = ∂f /∂t since we use an Eulerian description for the observables.

This equation has been used to estimate quantitatively some features of the dynamics, like the derivation of a reduced dimensional model [8] or the characteristics of the bunching in the saturated regime [9,10].

If f is a Klimontovitch distribution, that is the distribution function is a sum of Dirac representing some point particles:

f (θ, p) = j δ(θ -θ j (t))δ(p -p j (t)),
we recover the equations for Hamiltonian [START_REF] Bonifacio | [END_REF].

The continuous formalism (3)-( 4) is particularly well-suited for an algebraic treatment of the dynamics (see e.g. [11]). In what follows we use Vlasov-Maxwell equations to derive Hamiltonian (3). In order to do this we use a Hamiltonian formulation of these equations.

First let us recall that the interaction between electromagnetic fields and charged particles (of normalized mass m = 1 and charge e = 1) is given as the sum of the kinetic energy of the particles plus the energy of the field [12,13,14]:

H = d 3 qd 3 pf (q, p) 1 + p 2 + d 3 q |E(q)| 2 + |B(q)| 2 2 , (5)
where f (q, p) describes the distribution of particles in phase-space. Even though the kinetic energy of the particles and the electromagnetic energy appear to be decoupled in the Hamiltonian, the interaction between the matter and the fields comes from the bracket which gives the dynamics :

{F, G} = d 3 qd 3 p f ∂ ∂p δF δf • ∂ ∂q δG δf - ∂ ∂q δF δf • ∂ ∂p δG δf - d 3 qd 3 p f B • ∂ ∂p δF δf × ∂ ∂p δG δf + d 3 qd 3 p δF δf ∂f ∂p • δG δE - δG δf ∂f ∂p • δF δE + d 3 q ∇ × δF δB • δG δE - δF δE • ∇ × δG δB . (6)
This bracket satisfies the antisymmetry property, the Leibnitz product rule and the Jacobi identity. Here, the Lie algebra on which this bracket operates is the set of smooth functionals

F [f (q, p), E(q), B(q)].
Using Hamiltonian ( 5) and the brackets (6), Eq. ( 2) allows one to retrieve Maxwell's equations for E and B, as well as Vlasov equation for f :

∂f ∂t ≡ ḟ = {H, f } = -v • ∇f -(E + v × B) • ∂f ∂p , ∂E ∂t ≡ Ė = {H, E} = ∇ × B -d 3 pvf, ∂B ∂t ≡ Ḃ = {H, B} = -∇ × E,
where v is the velocity

v = p 1 + p 2 . ( 7
)
We notice that the first line of the right hand side of Eq. ( 6) refers only to the particles (and it is canonical), the second and third lines are the field-particle interaction terms (noncanonical terms) and the last line is a field-only term (which is also canonical).

III. INTERACTION BETWEEN A PLANE WAVE AND A CO-PROPAGATING BEAM OF PARTICLES A. Expression of the Hamiltonian system in a canonical way

The Vlasov-Maxwell equations ( 5)-( 6) can also be described using the potentials instead of the fields [13]. The Lie algebra is now a set of functionals F [f mom (q, p), A(q), Y(q)]. The Hamiltonian and the bracket become :

H = d 3 qd 3 pf mom 1 + (p -A) 2 + d 3 q |Y| 2 + |∇ × A| 2 2 , (8)
{F, G} =

d 3 qd 3 p f mom ∂ ∂p δF δf mom • ∂ ∂q δG δf mom - ∂ ∂q δF δf mom • ∂ ∂p δG δf mom + d 3 q δF δY • δG δA - δF δA • δG δY . (9)
This can be obtained from Eqs. ( 5)-( 6) using the change of coordinates

f (q, p) = f mom (q, p + A), E = -Y, B = ∇ × A.
We notice that this time, the Poisson bracket is canonical and there is no term in this bracket which couples the particles and the field. However, the coupling term is present in Hamiltonian ( 8).

We translate the potential vector by a static A w (q), which is imposed externally (as in an undulator). We notice that a translation of A by a quantity A w is a canonical transformation, which implies that the bracket ( 9) is not changed. The new Hamiltonian reads :

H = d 3 qd 3 pf mom 1 + (p -A w -A) 2 + d 3 q |Y| 2 + 2∇ × A w • ∇ × A + |∇ × A| 2 2 ,
where we have dropped the constant quantity

d 3 q|∇ × A w | 2 /2.
In particular, we notice that the dynamics of the radiated field is

Ȧ = {H, A} = δH δY = Y,
which is equivalent to the equation for the radiated electric field E r :

E r = - ∂A ∂t .
For a wave co-propagating with the electrons in the z-direction, one can define the k-mode of the wave as follows:

A k (q ⊥ ) = 1 L dze -ikz A(q), Y k (q ⊥ ) = 1 L dze -ikz Y(q),
where q ⊥ = (x, y) and L is the length of the cavity where the interaction takes place.

This gives the Fourier expansion in the propagation direction Y(q) = k Y k (q ⊥ )e ikz and A(q) = k A k (q ⊥ )e ikz . Furthermore, since

δA k (q ⊥ ) δA(q ′ ) = 1 L e -ikz δ(q ⊥ -q ′ ⊥ ),
which is obtained from the definition and linearity of the functional derivative, it follows from the bracket (9) that

{Y k (q ⊥ ), A -k ′ (q ′ ⊥ )} = 1 L δ kk ′ δ(q ⊥ -q ′ ⊥ ).
Since we also have:

{Y k (q ⊥ ), Y k ′ (q ′ ⊥ )} = {A k (q ⊥ ), A k ′ (q ′ ⊥ )} = 0,
the field part of the bracket turns into

d 3 q δF δY • δG δA - δF δA • δG δY = 1 L k d 2 q ⊥ δF δY k • δG δA -k - δF δA k • δG δY -k .
We now consider the paraxial approximation both for the radiated and external fields, i.e.

we neglect their spatial variations in the x and y directions, so that they are homogeneous in the transverse section S of interaction, and null outside it. We notice that this paraxial approximation is the strongest approximation involved in the derivation process. The dimensional reduction crucially depends on it. In addition to this approximation, we restrict the derivation to a monochromatic wave, i.e. we only take into account one Fourier mode k in the propagation direction, and consider the case of a circularly polarized radiated wave.

Other modes can be included in the derivation in a very similar way, but we have only kept one mode for the sake of clarity of the derivation. These two approximations allow us to define the complex amplitude of the wave a such that

A = - i √ 2 ae ikz ê -a * e -ikz ê * , Y = - k √ 2 ae ikz ê + a * e -ikz ê * , (10)
with ê = (x + iŷ)/ √ 2. Conversely, a can be defined as

a = 1 kV d 3 qe -ikz (-Y + ikA) • ê * = 1 kS d 2 q ⊥ (-Y k + ikA k ) • ê * ,
where V is the volume of the interaction domain, i.e. V = LS with the above notations.

Then, since

δa δY k (q ⊥ ) = - 1 kS ê * , δa δA k (q ⊥ ) = i S ê * , we get {a, a * } = i kV ,
so that a and a * are the new conjugate variables describing the radiated field. Hence, the bracket turns into:

{F, G} = d 3 qd 3 pf mom ∂ ∂p δF δf mom • ∂ ∂q δG δf mom - ∂ ∂q δF δf mom • ∂ ∂p δG δf mom + i kV ∂F ∂a ∂G ∂a * - ∂F ∂a * ∂G ∂a .
As for the Hamiltonian, since we have ∇ × A = kA and |Y| 2 = k 2 |A| 2 = k 2 aa * for the vector potential A, the energy of the radiated wave now reads:

d 3 q |Y| 2 + |∇ × A| 2 2 = k 2 V aa * ,
where we have used the relations ê • ê = 0 and ê • ê * = 1. So that the Hamiltonian becomes:

H = d 3 qd 3 pf mom 1 + p 2 + aa * -i √ 2(ae ikz ê -a * e -ikz ê * ) • A w +|A w | 2 -2p ⊥ • (A w + A) 1/2 +k 2 V a * a - ikS √ 2 dz(ae ikz ê -a * e -ikz ê * ) • (∇ × A w ).

B. Reduction to a one-dimensional model

Here we assume that the external field A w created by the undulator only depends on the longitudinal variable z. If the beam of electrons has been injected in a proper way, we show below that the motion is exactly described by a one-dimensional Hamiltonian, allowing for a reduced -but exact -description of the dynamics.

This reduction follows from the properties of the Liouville operator H = {H, •}. Recalling that:

δH δf mom = 1 + p 2 +aa * -i √ 2(ae ikz ê -a * e -ikz ê * ) • A w +|A w | 2 -2p ⊥ • (A w + A) 1/2
, the Liouville operator reads:

H = d 3 qd 3 p f mom 1 + p 2 + (A + A w ) 2 (p ⊥ -A w -A) • ∂ ∂q ⊥ δ δf mom +p z ∂ ∂z δ δf mom -(A + A w ) • ∂ ∂z (A + A w ) ∂ ∂p z δ δf mom + i kV ∂H ∂a ∂ ∂a * - ∂H ∂a * ∂ ∂a .
The ∂ ∂p ⊥ δ δfmom term has disappeared since its factor ∂(A + A w )/∂q ⊥ vanishes, as the fields are assumed not to depend on the transverse direction q ⊥ . As a consequence, there is no evolution for the particles distribution f mom (q, p) along the direction p ⊥ . This can be seen by considering a distribution function of the following form f mom (q, p) = f(q, p z )δ(p ⊥ ).

Under the Liouville operator, we see that Hf mom is also proportional to δ(p ⊥ ), using an integration by parts and the fact that δf mom (q, p) δf mom (q ′ , p ′ ) = δ(qq ′ )δ(pp ′ ).

We recall that p ⊥ has been translated by A w in Sec. III A, so in other words, if the beam is initially injected with the transverse velocity A w / 1 + |A w | 2 [from Eq. ( 7)], it remains with this specific transverse velocity. Then, once restricted to the δ(p ⊥ ) distribution, it comes that the transverse profile of f does not act on the longitudinal dynamics any more. In other words, the set F of observables F which do not depend on the transverse component of the distribution, i.e. F such that

∂ ∂q ⊥ δF δ f = 0,
is stable by H (i.e. HF ∈ F if F ∈ F ), since A and A w do not depend on q ⊥ . This allows one to focus on the longitudinal dynamics by defining a reduced dynamics on F , with f (z, p z ) the longitudinal distribution as new variable associated with the particles. The bracket reduces to :

{F, G} = dzdp z f ∂ ∂p z δF δ f ∂ ∂z δG δ f - ∂ ∂z δF δ f ∂ ∂p z δG δ f + i kV ∂F ∂a ∂G ∂a * - ∂F ∂a * ∂G ∂a ,
and the Hamiltonian becomes

H = dzdp z f 1 + p 2 z + aa * -i √ 2(ae ikz ê -a * e -ikz ê * ) • A w + |A w | 2 (11) +k 2 V aa * - ikS √ 2 dz(ae ikz ê -a * e -ikz ê * ) • (∇ × A w ).
Since the motion is now one-dimensional, we drop the label z of the momentum in what follows.

C. Particles-field phase frame

While Hamiltonian ( 11) is reduced to distribution functions with one dimension (one in space and one in momentum), it still contains some terms which are not specific to the interaction like the term aa * . The emphasis can be put on the interaction between the particles and the wave by considering the dynamics into the particles-field phase frame. We consider a specific medium for the interaction between the particles and the radiated field.

For example, we can consider a linear undulator, such as in a Free Electron Laser. The transverse field produced by such an undulator reads :

A w = a w √ 2 e -ikwz ê + e ikwz ê * . ( 12
)
First we neglect the effects of finite size : The last term in Eq. ( 11) vanishes since it is a sum of terms proportional to dze ±i(k+kw)z terms. The Hamiltonian becomes

H = dzdp f 1 + p 2 + aa * -ia w (ae i(k+kw)z -a * e -i((k+kw)z) ) + a 2 w + k 2 V aa * .
In this Hamiltonian, the last term simply yields the propagation of the electromagnetic wave. Indeed, since {k 2 V aa * , a} = -ika, it generates an e -ikt factor for a(t), and so it is a pure propagation term. This remark calls for a time-dependent change of coordinates. This procedure is fairly standard, e.g., in quantum mechanics, and corresponds to the interaction representation [15]. Since time is not a variable for this model, we first need to extend phase space to add a new pair of canonically conjugate variables with one being similar to time. More precisely, we define the pair of conjugate variables (τ, E), such that the new Hamiltonian and bracket read

H ext [f, a, a * , E, τ ] = H[f, a, a * ] + E, {F, G} = dzdp f ∂ ∂p δF δ f ∂ ∂z δG δ f - ∂ ∂z δF δ f ∂ ∂p δG δ f + i kV ∂F ∂a ∂G ∂a * - ∂F ∂a * ∂G ∂a + ∂F ∂E ∂G ∂τ - ∂F ∂τ ∂G ∂E ,
so that τ = {H ext , τ } = 1, which means that, practically, τ is identical to the evolution variable t. We now consider the canonical change of variables ( f , a, a * , τ, E)

→ ( f, â, â * , τ , Ê) such that â = ae ikτ , Ê = E + k 2 V aa * f (z, p) = f (z, p), τ = τ.
It can be checked that the symplectic form ikV da * ∧ da + dE ∧ dτ is conserved by this transformation. This results in a conservation of the Poisson bracket, while the Hamiltonian now reads:

Ĥ = dzdp f 1 + p 2 + ââ * -ia w (âe i((k+kw)z-kτ ) -âe -i((k+kw)z-kτ ) ) + a 2 w + Ê.
Finally, the dynamics can be studied in the particles-field phase frame, the latter phase being defined as θ = (k + k w )z -kτ , by considering the change of variables

f (θ, p) = f (z, p)/(k + k w ), ā = â, Ē = Ê + k k + k w dθdp fp, τ = τ,
where f has been divided by k + k w for normalization purposes. We notice that the term -k/(k + k w ) dθdp f p has been added to Ê in order to ensure the canonicity of the change of coordinates which translates here into the condition { Ē, f} = 0. Here we have used the following properties of the functional derivative: If we perform a change of variables of f ,

denoted f = f • g, then the functional derivative of F [ f ] = F [f ] satisfies δ F δ f = δF δf g ′ • g -1 ,
which is obtained in a straightforward way from the definition of the functional derivative.

Now that we have performed the time-dependent change of coordinates and that the Hamiltonian is time-independent, the (τ , Ē) variables are somehow artificial and decoupled from the other ones. It is more convenient to work in the reduced space and drop this additional pair of variables. For notation purposes, we drop the bars over the other variables.

The Hamiltonian now reads:

H = dθdp f 1 + p 2 + a 2 w -ia w (āe iθ -ā * e -iθ ) + āā * - k k + k w p . (13)
The

D. Resonance condition and high-gain amplification

The next step is to expand Hamiltonian (13) around a resonant value. Following Eq. ( 2), Hamiltonian (13) and bracket (14) yield the following equations of motion:

df dt = -(k + k w ) p 1 + p 2 + a 2 w -ia w (ae iθ -a * e -iθ ) + aa * - k k + k w ∂f ∂θ + a w (k + k w )(ae iθ + a * e -iθ ) 1 + p 2 + a 2 w -ia w (ae iθ -a * e -iθ ) + aa * ∂f ∂p da dt = - k V dθdpf ia -a w e -iθ
1 + p 2 + a 2 w -ia w (ae iθ -a * e -iθ ) + aa * From these equations it can be seen that the system is at equilibrium for a = 0 and f (θ, p) = δ(p -p R )F (θ), where F (θ) is a distribution which satisfies dθe -iθ F (θ) = 0, and p R is given by

p R 1 + a 2 w + p 2 R - k k + k w = 0.
This resonant momentum p R can be linked to a resonant energy γ R for the particles, defined by

γ R = 1 + a 2 w + p 2 R = 1 + a 2 w k + k w k w (2k + k w ) . (15)
In the limit k ≫ k w , the definition (15) of the resonant energy yields the usual definition γ R = k(1 + a 2 w )/(2k w ) (see for example [2]). However, this equilibrium is unstable, and exposed to small perturbations, the wave starts growing and destabilizes the particles at p = p R . This instability is responsible for the highgain growth of the wave, which is taken advantage of in devices such as FEL. The dynamics can be linearized around this equilibrium point: Assuming the momenta of the particles remain close from the resonant one p R , we shift p by p R , by defining f (θ, p) = f (θ, p) with p = p -p R . We also consider that the amplitude of the radiated field is weak compared to the resonant energy |a| ≪ γ R .

Then, Hamiltonian ( 13) expands, at the first order in a and second order in p (we have dropped the hat over p and f ), as follows:

H lin = dθdpf 1 + a 2 w γ 3 R p 2 2 - ia w 2γ R ae iθ -a * e -iθ ,
associated with the bracket (14).

The equations of motion can be normalized through the following change of variables

f ′ (θ ′ , p ′ ) = 1 β f (θ = θ ′ , p = p ′ /β), a ′ = ǫa.
Moreover, we include a rescaling of time t ′ = αt, i. Using

α = 1 γ R a 2 w k w (k + k w /2) 2kV 1/3 , β = 2 k + k w 2kV k 2 w (k + k w /2) 2 a 2 w 1/3 , ǫ = 4k 2 V 2 k w (k + k w /2) a w 1/3 , ν = 2 2kV k 2 w (k + k w /2) a 2 w 1/3
, the Hamiltonian and the bracket become :

H = dθdpf p 2 2 -i dθdpf ae iθ -a * e -iθ , (16)
{F, G} = dθdpf ∂ ∂p δF δf ∂ ∂θ δG δf - ∂ ∂θ δF δf ∂ ∂p δG δf + i ∂F ∂a ∂G ∂a * - ∂F ∂a * ∂G ∂a .
Finally, the canonical change of variables (a, a * ) → (φ, I) such that a = √ Ie -iφ (so that {φ, I} = 1) allows one to retrieve Hamiltonian (3) associated with the bracket (4).

E. Conserved quantities

Apart from H as given by Eq. ( 5), the Hamiltonian system of charged particles interacting self-consistently with electromagnetic fields has its total momentum as conserved quantity, as it was reported in Ref. [14]:

P[f, E, B] = d 3 qd 3 pf p + d 3 qE × B.
We perform the same approximations and reductions as the ones done on Hamiltonian [START_REF] Bonifacio | [END_REF].

In this way, we recover the conserved quantity of the reduced model. For example, in the canonical formulation ( 8) and ( 9), the total momentum turns into:

P[f mom , A, Y] = d 3 qd 3 pf mom (p -A) -d 3 qY × (∇ × A) .
Furthermore, when considering a monochromatic, circularly polarised plane-wave such as the one given by Eq. ( 10), the total momentum is decomposed into a transverse and a longitudinal component:

P ⊥ [f, A] = d 3 qd 3 pf (p ⊥ -A -A w ), P z [f, A, Y] = d 3 qd 3 pf p z + k 2 V aa * + k d 3 q (A w × Y) • êz .
When focusing on the longitudinal dynamics (see Sec. III B), the conserved quantities P ⊥ can be dropped since they provide information on the transverse dynamics. Then, when considering the specific external field (12), if we neglect the finite-size effect, the last term of the longitudinal momentum can be dropped, so that we get, for Hamiltonian (13), the following conserved quantity P z [f, a, a * ] = d 3 qd 3 pf p + k 2 V aa * .

Using the same normalization as in Sec. III D, P z becomes a conserved quantity for Hamiltonian ( 16):

P = dθdpf p + aa * .
Finally, using the canonical change of variables (a, a * ) → (φ, I), it becomes:

P = dθdpf p + I,
i.e. the average momentum plus the intensity is conserved by Hamiltonian (3).

Conclusion

In this paper, we derived a reduced Hamiltonian for the interaction between a wave and a beam of charged particles driven by an external field, on the sole assumptions of transverse fields and on-axis injection for the particles. A resonance condition -around an unstable equilibrium point -has been identified, allowing for a linearization of the dynamics. Finally, under the extra hypotheses of high-energy particles and weak radiated field, a paradigmatic Hamiltonian has been retrieved within a fully Hamiltonian treatment. The main advantage of the present derivation is a fully algebraic framework which is well suited to include additional effects, like for instance, higher order terms in the expansions, or strategies using other harmonics of the radiated field [16]. We have shown here that this treatment allows one to recover some general features of the Vlasov-Maxwell equations in a very natural way, like a conserved quantity of the flow.

of the reduced model ( 1 )

 1 by expressing first the Vlasov-Maxwell system into a canonical setting (Sec. III A), then performing the one-dimensional reduction (Sec. III B), changing the reference frame (Sec. III C), and expanding the resulting Hamiltonian around a resonance condition (Sec. III D). Finally, we deduce (Sec. III E) a conserved quantity of the reduced model (1) from a conserved quantity of the Vlasov-Maxwell equations by following the same procedure as for the reduction of the Hamiltonian.

  resulting Poisson bracket writes {F, G} = (k + k w )

  e. H ′ = H/α, and we consider a new Hamiltonian νH ′ with a new Poisson bracket ν -1 {., .} (which does not change the dynamics).

Acknowledgements

We acknowledge useful discussions with G. De Ninno, E. Allaria, D. Fanelli, Y. Elskens, R. Paškauskas, and the Nonlinear Dynamics group at CPT. This work is supported by