text:
Lampreys, one of the only two surviving groups of agnathan (jawless)vertebrates, contain several anadromous species that, during their life cycle,thus migrate from fresh to seawater and back to freshwater. Lampreys have independently evolved the same overall osmoregulatory mechanisms as the gnathostomatous (jawed) and distantly related teleost fishes. Lamprey gills thus likewise play a central role in taking up and secreting monovalent ions. However, the ultrastructural characteristics and distribution of their epithelial cell types [ammocoete mitochondria-rich (MR) cell, intercalated MR cell, chloride cell and pavement cell] differ in several respects from those of teleosts. The ultrastructural characteristics of these cells are distinctive and closely resemble those of certain ion-transporting epithelia in other vertebrates, for which the function has been determined. The data on each cell type, together with the stage in the life cycle at which it is found, i.e. whether in fresh or seawater, enable the following proposals to be made regarding the ways in which lampreys use their gill epithelial cells for osmoregulating in hypo- and hypertonic environments. In freshwater, the intercalated MR cell takes up Cl and secretes H+,thereby facilitating the uptake of Na+ through pavement cells. In seawater, the chloride cell uses a secondarily active transcellular transport of Cl to provide the driving force for the passive movement of Na+ through leaky paracellular pathways between these cells.

The agnathan (jawless) stage in vertebrate evolution is represented in the contemporary fauna only by the hagfishes (Myxiniformes) and lampreys(Petromyzontiformes) (Hardisty,1982). Although analyses of morphological and physiological data strongly indicate that lampreys are more closely related to the gnathostomatous (jawed) vertebrates than to the hagfishes(Hardisty, 1982; Forey and Janvier, 1993), the results of certain molecular studies support the view that the hagfishes and lampreys constitute a monophyletic group(Stock and Whitt, 1992; Mallatt and Sullivan, 1998; Kuraku et al., 1999; Delarbre et al., 2002). Yet,there can be no doubt that these two agnathan groups have been separated for a very long period and probably from as long ago as the lower Cambrian(Janvier, 1999).

Hagfishes, which are found only in marine habitats, are osmoconformers(McFarland and Munz, 1965; Cholette et al., 1970). The concentrations of Na+ and Cl in their sera approximate those in full-strength seawater, a situation unique amongst vertebrates and which results in their internal milieu being essentially iso-osmotic with that of their marine environment(Robertson, 1974). These characteristics imply that hagfishes have always lived in marine habitats(Robertson, 1957, 1974; Lutz, 1975).

In contrast to hagfishes, the anadromous species of lampreys spend a substantial part of their life cycle in freshwater(Hardisty and Potter, 1971; Hardisty et al., 1989). The efficient osmoregulatory mechanisms evolved by lampreys enable the concentrations of Na+ and Cl in their internal milieu to be maintained at levels well above that of freshwater, when the animal is living in rivers, and well below that of full-strength seawater,when the animal is residing in marine environments(Morris, 1972; Beamish et al., 1978). Since the osmolality of their serum is far lower than that of seawater, it has been concluded that lampreys have an ancient freshwater history and that the marine parasitic phase of their life cycle was developed relatively late in their evolution (Hardisty et al.,1989). However, the recent discovery of a lamprey-like fossil in lower Cambrian marine deposits (Shu et al., 1999) strongly suggests that the initial evolution of lampreys occurred in a marine environment at a very early date, i.e. over 545 million years ago, and thus prior to the time when this group invaded freshwater.

The marked differences between both the ionic composition and osmolality of the body fluids of hagfishes and lampreys are consistent with the long period that these two agnathan groups are believed to have been separated. Indeed, as long ago as 1932, Homer Smith stated that “these two groups lead back to a parting of the ways in the evolution of body fluids”.

The teleost fishes are found in a wide range of fresh, brackish and marine habitats and thus, as a group, are faced with the same variety of osmotic problems as those experienced by the anadromous species of lampreys during the course of their life cycle. The diversity and abundance of teleosts account for this group having been the subject of the majority of the studies aimed at elucidating the mechanisms by which fish regulate the ionic composition and osmolality of their serum (Smith, 1930, 1932; Krogh, 1939; Karnaky, 1980, 1986; Zadunaisky, 1984; Perry, 1997; Wilson et al., 2000a,b). Although lampreys are not closely related to teleosts, their gills and kidneys likewise constitute the main organs responsible for osmoregulation, and their overall mechanisms for regulating the concentrations of Na+ and Cl in their body fluids are similar(Morris, 1972; Beamish, 1980; Hardisty et al., 1989). However, ultrastructural studies have demonstrated that, particularly when lampreys are in freshwater, the cellular composition of their gill epithelium differs from that of teleosts (Bartels et al., 1996, 1998). These studies have also revealed that certain epithelial cells in the lamprey gill possess highly distinctive ultrastructural features, which they share with particular cell types in specific ion-transporting epithelia in other vertebrates and for which a function has been determined.

In the present review, the ultrastructural characteristics of each of the cell types present in the gill epithelium at the different stages in the life cycle of lampreys are described. The characteristics of each cell type are then considered in the context of both the type of environment in which the animal containing that cell is found, i.e. freshwater or seawater, and the roles played by analogous cells in other vertebrates. This then enables each of these various cell types to be assigned a presumptive role in lamprey osmoregulation.

When in freshwater, fish are subjected to an osmotic influx of water and efflux of ions across their skin and gills. This problem is overcome by an active uptake of monovalent ions across the gills and the excretion of a copious and dilute urine by the kidney(Fig. 1A; Smith, 1932; Krogh, 1939; Parry, 1966; Perry, 1997). The uptake of Na+ and Cl by various vertebrates in freshwater,i.e. under in vivo conditions, is stoichiometrically balanced and occurs independently by exchanging H+ for Na+ and HCO3 for Cl(Krogh, 1939; Kirschner, 1983). The models developed for these ion exchange mechanisms were derived mostly from the results of studies carried out in Ussing-like chambers. Since such studies require the use of flat sheets of tissues, they have been performed mainly on amphibian epidermis, which, in contrast to the fish gill, has a flat surface(Garcia-Romeu and Ehrenfeld,1975; Kirschner,1983; Larsen,1988). The results of these studies have led to the general acceptance that the exchange of Na+ for H+ involves an active secretion of H+ by an electrogenic H+-ATPase and the uptake of Na+ through an epithelial Na+ channel(ENac), rather than occurring simply through an Na+/H+-antiport. This mechanism is dependent on the activities of both cytosolic carbonic anhydrase in the H+ secreting cell and Na+/K+-ATPase in the basolateral membrane of the cell whose apical membrane contains the ENac(Harvey and Ehrenfeld, 1986; Harvey et al., 1988; Nagel and Dörge, 1996; Ehrenfeld and Klein, 1997). Furthermore, the toad and turtle urinary bladders, in which the epithelium has a cellular composition comparable with that of amphibian skin and consists also of granular cells and mitochondria-rich (MR) cells(Wade et al., 1975; Wade, 1976; Rick et al., 1978; Durham and Nagel, 1986; Brown and Breton, 1996),employ essentially the same mechanisms for reabsorbing Na+ from the urine and for secreting H+ into that urine(Stetson and Steinmetz, 1985; Durham and Nagel, 1986; Lang, 1988). In addition, the immunocytochemical demonstration that H+-ATPase and ENac are present in the gill epithelium of teleosts(Sullivan et al., 1995; Wilson et al., 2000a; Marshall, 2002) is consistent with the hypothesis that this mechanism for taking up Na+ from a hypotonic environment is universal amongst vertebrates and invertebrates such as crustaceans, annelids and molluscs(Kirschner, 1983).

In contrast to the indirect coupling of Na+ uptake to H+ secretion, as described above, the uptake of Cl from an hypotonic environment by all epithelia studied thus far occurs directly by means of an HCO3/Cl antiport(Garcia-Romeu and Ehrenfeld,1975; Larsen,1991; Marshall et al.,1997). The cells engaged in Cl uptake are further characterised by the presence of cytosolic carbonic anhydrase and a Cl channel in their basolateral membrane(Larsen, 1991).

When in seawater, teleosts and lampreys are confronted with the reverse situation to that in freshwater, i.e. they experience an osmotic loss of water to the environment. The further loss of water through the kidneys is minimised by reducing excretion from this source to just a small amount of concentrated urine. The osmotic loss of water is overcome by swallowing seawater, resorbing monovalent ions and water from the gut and retaining the free' water that remains following the secretion of an hypertonic solution of Na+and Cl across the gills(Fig. 1B; Smith, 1930; Karnaky, 1980, 1986; Loretz, 1995). As long ago as 1932, Keys and Willmer considered that Cl is secreted by the gill epithelium through a large eosinophilic cell that resembles the parietal cell in the gastric mucosa of mammals and which was consequently termed thechloride secreting cell' (Keys and Willmer, 1932). The cytoplasm of this cell contains numerous mitochondria and a system of membranous tubules(Philpott and Copeland, 1963; Karnaky et al., 1976a; Pisam and Rambourg, 1991). This membranous tubular system, which represents a vast intracellular amplification of the basolateral cell membrane, is the site of Na+/K+-ATPase activity and an Na+/K+/2Cl cotransport system(Karnaky et al., 1976b;Karnaky, 1980, 1986; Eriksson et al., 1985). The mechanism by which Na+ and Cl are transported by the gill epithelium of marine teleosts has been resolved through work on the opercular epithelium, which, in some teleost species, contains numerous chloride cells and, in contrast to the gill surface, is flat and thus suitable for studies in an Ussing-like chamber(Karnaky and Kinter,1977).

Concomitant morphological and electrophysiological studies using the opercular epithelium have confirmed that, when teleosts are in seawater,Cl is secreted by this cell type through a secondary active transport that provides the driving force for the passive transport of Na+ through leaky pathways between adjacent chloride cells(Karnaky et al., 1977; Ernst et al., 1980; Foskett and Scheffey, 1982; Foskett and Machen, 1985). The channel through which Cl is secreted by the teleost chloride cell has now been identified as a homologue of the mammalian cystic fibrosis transmembrane conductance regulator (CFTR; Singer et al., 1998; Wilson et al., 2000b; Marshall et al., 2002). Identical mechanisms for hypertonic salt secretion are also present in the secretory cells of both the rectal gland of elasmobranch fishes and the nasal salt gland of some species of marine birds (Kirschner, 1977, 1980; Riddle and Ernst, 1979; Ernst et al., 1981). It is thus assumed that this mechanism is universally employed by vertebrates for osmoregulation in hypertonic environments.

The life cycle of all species of lampreys contains a microphagous larval phase, which is spent in freshwater(Potter, 1980). After a number of years, the larvae (ammocoetes) undergo a radical metamorphosis into young adults that, in the case of anadromous species, migrate downstream to the sea where they feed on predominantly teleost fishes(Potter and Hilliard, 1987). Following the completion of their marine trophic phase, adult lampreys re-enter freshwater and migrate to upstream areas, where spawning and then death occur (Hardisty and Potter,1971).

Ammocoetes of the sea lamprey Petromyzon marinus are unable to osmoregulate in environments in which the osmolality exceeds that of their own serum, i.e. ∼225 mosmol kg–1(Beamish et al., 1978; Morris, 1980). Although∼50% of larval P. marinus died within 24 h of being placed in water of 350 mosmol kg–1, i.e. about one-third of full-strength seawater, the recently metamorphosed young adults of anadromous species can readily be acclimated to full-strength seawater and are then able to maintain their serum osmolality at ∼260 mosmol kg–1(Beamish et al., 1978). This ability to change from hyper-osmotic regulation in freshwater to hypo-osmotic regulation in seawater is so effective that more than 80% of young adult P. marinus can even survive direct transfer from fresh to full-strength seawater (Potter and Beamish, 1977). Similar results have been obtained with young adults of the anadromous lampreys Lampetra fluviatilis and Geotria australis (Potter and Huggins, 1973; Potter et al.,1980). The ability to osmoregulate in hypertonic environments is lost by adult lampreys soon after they have completed their marine trophic phase and embarked on their upstream spawning migration (Morris, 1956, 1958; Pickering and Morris,1970).

The gill epithelium undergoes marked changes during the life cycle of anadromous lampreys and, in particular, as the animal migrates from fresh to seawater and vice versa (Table 1). These changes include alterations to the composition and spatial relationships of the cells and to the structure of their tight junctions. The outer cellular layer of the gill filaments and lamellae of ammocoetes comprises two types of MR cell (i.e. the ammocoete MR cell and the intercalated MR cell) and pavement cells(Bartels et al., 1998). During metamorphosis, the ammocoete MR cells disappear and the chloride cells develop(Peek and Youson, 1979b). Thus, following the completion of metamorphosis, the gill epithelium of the downstream migrating young adult possesses chloride cells, pavement cells and intercalated MR cells. The last of these cells disappears after the young adult has entered seawater and does not reappear until the adult has finished growing and has re-entered freshwater on its upstream spawning migration(Bartels et al., 1998). By contrast, the chloride cells gradually disappear during the spawning run(Morris, 1957; Morris and Pickering,1976).

The presence or absence of the different cell types at the surface of the gills at different phases in the life cycle of anadromous lampreys can now be used to propose which cells are involved in osmoregulation in hypo- and hyper-osmotic environments. For example, since the intercalated MR cell is the only cell type present throughout all freshwater phases, and is absent during the marine phase, it presumably plays a crucial role in osmoregulation when lampreys are in hypo-osmotic media. Likewise, since the chloride cell develops just prior to the marine phase and disappears soon after the completion of this phase, it can only be involved in osmoregulation when lampreys are in an hyper-osmotic medium. Although the pavement cell is the only cell type that is present on the gill surface throughout the entire life cycle of lampreys and is thus a potential candidate as an osmoregulatory cell in both fresh and seawater, there is currently no evidence that this cell type is required for osmoregulation in seawater (see below). The fact that the ammocoete MR cell is never found after the completion of the larval phase demonstrates that any role that it plays in osmoregulation in ammocoetes must be undertaken by other cell type(s) during the freshwater phases of post-larval life.

### The freshwater phases in the life cycle

#### The intercalated MR cell

The cell in the lamprey gill epithelium that we have termed the intercalated MR cell (Bartels et al.,1998) corresponds to that designated by Youson and Freeman(1976) and Mallatt and Ridgway (1984) as a chloride cell in the ammocoete gill. Our choice of the term intercalated MR cell is based on the fact that, while these cells differ from the chloride cells in the gills of adult lampreys in that they lack a membranous tubulous system(see above), they do have the same ultrastructural characteristics as the MR cells that, for example, are intercalated in the epithelium of both the skin and urinary bladder of amphibians, of the urinary bladder of reptiles and of the collecting duct of the amphibian and mammalian kidney (Figs 2A, 3; Brown and Breton, 1996). Thus,the branchial intercalated MR cell of lampreys is characterized by the presence of numerous membranous vesicles and tubules between the mitochondria and also often immediately beneath the apical membrane(Fig. 2A). Moreover, the apical surface of the intercalated MR cell is enlarged by slender, branching microfolds that, from scanning electron microscopy, can be seen to produce a honeycomb appearance (Figs 2A, 3C). As primarily shown in the epithelium of the turtle urinary bladder, the extent of such enlargements,which varies markedly amongst intercalated MR cells, is inversely related to the number of membranous tubules and vesicles in the apical cytoplasm, which are incorporated into and removed from the apical membrane by exo- and endocytosis, respectively (Stetson and Steinmetz, 1983; Brown,1989; Brown and Breton,2000). A coat of studs projects about 12 nm outwards from the cytoplasmic side of the apical membrane(Fig. 2B). In freeze-fracture replicas, the cell membrane and the membranes of the cytoplasmic vesicles and tubules of the intercalated MR cell are characterized by the presence of rod-shaped particles on the protoplasmic (P) face and by complementary pits on the exoplasmic (E) face (Fig. 3A,B). These particles, which are either 16–18 nm or 24–27 nm in length and 8–9 nm in width, consist of two or three globular subunits. They are located in the apical membrane of almost all of the intercalated MR cells and in the basolateral membrane of a few of these cells (Bartels and Welsch,1986; Bartels et al.,1998).

In the lamprey gill epithelium, these cuboidal intercalated MR cells are generally confined to the filaments, where they typically occur singly at their base and between the lamellae(Bartels et al., 1998). They are intercalated between ammocoete MR cells in larval lampreys(Fig. 3C) and either between chloride and pavement cells or between pavement cells in downstream migrating lampreys and between pavement cells in upstream migrating lampreys.

The intercalated MR cells in urinary epithelia and the amphibian epidermis are rich in cytosolic carbonic anhydrase (CA II) and contain, within their plasma membrane and the membranes of cytoplasmic vesicles, a vesicular type of proton pump (Brown and Breton, 1996, 2000). The peripheral cytoplasmic subunit V1 of this pump has been immunolocalized to the studs on the cytoplasmic side of the MR cell membrane(Brown et al., 1987). Furthermore, a combination of physiological and morphological studies in various urinary epithelia indicates that the rod-shaped particles are either the transmembrane portion or an intimate associate of this pump(Stetson and Steinmetz, 1986; Brown et al., 1987; Kohn et al., 1987). This view is supported by the presence of H+ V-ATPase activity in all of those membranes that have been shown by freeze-fracture replicas to contain rod-shaped particles (Brown and Breton,1996).

On the basis of differences in the locations of the H+ V-ATPase,rod-shaped particles and a bicarbonate exchanger, two subtypes of intercalated MR cells (A and B) were initially distinguished in the collecting duct of the mammalian kidney and turtle urinary bladder(Stetson and Steinmetz, 1985; Brown et al., 1988; Brown and Breton, 1996). The subtype A contains the H+ V-ATPase and rod-shaped particles in its apical and cytoplasmic vesicular membranes and possesses, in its basolateral membrane, the anion exchanger, identified as an alternatively spliced kidney form of the band 3 protein AE-1 (Fig. 4A). This subtype is responsible for electrogenic H+secretion. The subtype B of the intercalated MR cell is characterized by an apical membrane that possesses a bicarbonate exchanger, which, although functionally detectable, does not immunoreact with antisera against AE-1 or any other anion exchanger (Fig. 4B). This subtype is responsible for HCO3 secretion. Although the AE-1-negative subtype B was originally distinguished from subtype A by the presence of H+ V-ATPase and rod-shaped particles in its basolateral membrane,immunocytochemical studies have now shown that the H+ V-ATPase can occur in various locations in these AE-1-negative MR cells(Brown and Breton, 1996). This has led to the identification of a third subtype (C) of the intercalated MR cell in the amphibian epidermis (Fig. 4C), which is characterised by the presence of an anion exchanger and the H+ V-ATPase in its apical membrane and thus provides a mechanism for the uptake of Cl from a dilute solution(Larsen et al., 1992).

The ultrastructural and functional characteristics of intercalated MR cells have been conserved in certain ion-transporting epithelia of vertebrates as diverse as amphibians, reptiles and mammals(Brown and Breton, 1996). Since the intercalated MR cells in the lamprey gill epithelium have the same unique ultrastructural characteristics as the other members of this group of cells, they presumably perform the same basic functions as those cells. The vast majority of the intercalated MR cells in the lamprey gill epithelium contains rod-shaped particles in their apical membrane and thus belong to the subtypes A or C of these cells. It is assumed that both subtypes, which cannot be distinguished on the basis of the location of the rod-shaped particles (and H+ V-ATPase) in their apical membranes alone, are present in the lamprey gill epithelium. Subtype A is envisaged as actively secreting H+, whereas subtype C is responsible for taking up Cl. The presence of an anion exchanger in parallel with the H+ V-ATPase in the apical membrane, as is the case in subtype C cells, would enable the actively secreted hydrogen ions to bind the HCO3 as it leaves the cell and thereby establish an HCO3 gradient across this membrane, which would drive the uptake of Cl. The low external pH generated by nearby subtype A intercalated MR cells would further enhance this effect and help overcome the unfavourable Cl gradient, with the intracellular Cl concentration being approximately 10–20-fold greater than the extracellular concentration. Thus, the characteristics of subtype C make it much more efficient for taking up Cl from the environment than would those of subtype B, whose apical membrane contains the anion exchanger but not the H+V-ATPase. It is assumed that the primary role of the subtype B of the intercalated MR cell is HCO3 secretion during alkalosis, e.g. in urinary epithelia where these cells frequently occur(Alper et al., 1989), rather than the uptake of Cl from a dilute solution. This conclusion would explain why the subtype B is very rare in the lamprey gill epithelium.

Finally, there is the question of whether epithelial Na+channels are present in the apical membrane of intercalated MR cells and provide a route for the uptake of Na+ through these cells. Ehrenfeld et al. (1989)proposed that, in the frog skin under natural' conditions, i.e. low external Na+ concentrations and open circuit, a significant amount of Na+ is taken up through the MR cells and that only under Ussing-like conditions, i.e. high mucosal Na+ concentration and short circuit, is Na+ taken up through the granular cells. By contrast, Nagel and Dörge(1996) concluded that, even under natural conditions, the uptake of Na+via MR cells is negligible and occurs almost exclusively through the granular cells.

#### The pavement cell

The cells that form the surface of the lamellae in adult lampreys were called pavement cells by Bartels(1989) to be consistent with the terminology used for comparable cell types in the gills of teleosts and other fishes (Laurent, 1984; Wilson and Laurent, 2002). These cells correspond to those which, in ammocoetes, were designated as mitochondria-poor cells by Youson and Freeman(1976), mucous-platelet cells by Mallatt and Ridgway (1984)and mucous pavement cells by Mallatt et al.(1995). In ammocoetes, the pavement cells are restricted to the apex of the lamellae and thus, as will be described later, beyond the region occupied by the ammocoete MR cells(Youson and Freeman, 1976),whereas in adults they cover the entire lamellar surface and, at least in those in freshwater, also part of the interlamellar region of the filament. They are squamous on the lamellae and columnar in the interlamellar region,particularly in upstream migrants.

The pavement cell is ultrastructurally very similar to the granular cell in the amphibian urinary bladder (Wade et al., 1975; Bartels,1989). It is thus characterized by the presence of numerous ovoid mucous granules, which are located in the apical cytoplasm, a rough endoplasmic reticulum with few cisternae, a well-developed Golgi apparatus and less mitochondria than are present in the two types of MR cells(Fig. 5A). The apical surface bears microplicae or short microvilli. Freeze-fracture replicas have shown that the majority of the intramembranous particles of the apical membrane are located on its E face and that few particles are present on its P face(Fig. 5B,C). This pattern of distribution of intramembrane particles differs from that found in the plasma membrane of most other vertebrate cells, including the basolateral membrane of both the lamprey pavement cell and the granular cell of the amphibian urinary bladder. Most of the particles on the E face of the apical membrane are relatively large (diameter 10–13 nm) and of greater size than those on the corresponding P face (diameter 6–8 nm).

The granular cell in the toad urinary bladder epithelium facilitates ionic and osmotic regulation by acting as the effector cell through which water and Na+ are taken up independently. These two functions are hormonally controlled by the antidiuretic hormone (ADH) and the mineralocorticosteroid aldosterone. The permeability of the apical membrane of the granular cell to water is very low, unless this cell has been stimulated by ADH(DiBona et al., 1969; Harris et al., 1991). This low permeability has been related to the presence of only a few particles on the P face of the apical membrane (Bourguet et al., 1976), which in turn may reflect an unusual composition or arrangement of lipids in the exoplasmic half of this membrane(Harris et al., 1991). Since this unusual and highly specialised membrane structure is shared by the pavement cell in the lamprey gill epithelium, the apical membrane of the latter cell is likewise assumed to be relatively impermeable to water. This conclusion is supported by the observation by Bentley(1962) that the permeability of the body surface of adult Lampetra fluviatilis to water is far lower than that of either the isolated toad urinary bladder or even the frog skin. It thus appears relevant that, particularly in adult lampreys, the contribution made by pavement cells to the area of the body surface that is exposed to the environment is far greater than that of any other cell type. Thus, when adult lampreys are in freshwater, the possession by these cells of a relatively impermeable apical membrane helps protect these animals against an osmotic influx of water across the gills.

During periods of water shortage in toads, ADH increases the permeability of the apical membrane of the granular cells to water by recruiting water channels (aquaporin 2) into this membrane from a cytoplasmic pool(Wade et al., 1981; Brown et al., 1989). Thus, since lampreys are not threatened by dehydration when in freshwater and thus do not require an ADH-mediated mechanism for either conserving or taking up water, it is hardly surprising that this hormone does not elicit an hydro-osmotic response in these animals(Bentley and Follett,1962).

The uptake of Na+ from the urine by the granular cell in the toad bladder, which is regulated by aldosterone and ADH, occurs through ENac in the apical membrane and is driven by Na+/K+-ATPase activity in the basolateral membrane(Macknight et al., 1980; Garty, 1986; Garty and Palmer, 1997). Although lampreys do not produce aldosterone(Bentley and Follett, 1962),the administration of this hormone (but not of neurohypophyseal hormones) to these agnathans likewise increases the extrarenal uptake of Na+(Bentley and Follett, 1962, 1963). This finding raises the possibility that, as with the granular cells of Na+-resorbing epithelia such as the toad urinary bladder and epidermis, ENac are present in the lamprey pavement cell. The pavement cell thus becomes the main candidate for the uptake of Na+ and the target of unidentified mineralocorticoid hormone(s) in lampreys.

#### The ammocoete MR cell

The cell that we term the ammocoete MR cell(Bartels et al., 1998)corresponds to the mitochondria-rich cell of Morris and Pickering(1975), the mitochondria-rich platelet cell of Youson and Freeman(1976) and the ion-uptake cell of Mallatt and Ridgway(1984). The latter three groups of authors considered that this cell is responsible for taking up ions from the environment.

The ammocoete MR cells represent up to 60% of the cells at the surface of the gill lamellae of larval lampreys. They are arranged in large groups at the base of the lamellae and in the region between the lamellae. Their mitochondria, which occupy about one-third of the cell volume(Mallatt et al., 1995), have an unusually electron-dense matrix (Fig. 2C). Elongated mucous granules, which are smaller than those of pavement cells, lie directly beneath the apical membrane of some of these cells. In freeze-fracture replicas, the apical membrane of the ammocoete MR cell exhibits the typical characteristics of vertebrate cell membranes, i.e. the P face of the cleaved membrane contains most of the globular particles(diameter 8–9 nm) whereas the E face contains few particles but numerous pits (Bartels et al.,1998).

The presence of numerous mitochondria and a positive histochemical reaction for carbonic anhydrase by the ammocoete MR cell(Conley and Mallatt, 1988) are consistent with the view that, in larval lampreys, this cell is responsible for exchanging Na+ for H+ and/or Clfor HC03(Morris and Pickering, 1975; Youson and Freeman, 1976; Mallatt and Ridgway, 1984). However, our preliminary studies show that, when ammocoetes of G. australis are held in distilled water, in which the uptake of Na+ and Cl is maximally stimulated, the characteristics of the ammocoete MR cells (and pavement cells) are unaffected whereas the density of the intercalated MR cells increases significantly beyond that found with these cells in ammocoetes held in 10% seawater (H. Bartels, J. Rosenbruch and I. C. Potter, unpublished observations). Furthermore, adult lampreys in freshwater still possess the osmoregulatory capacity to overcome the same osmotic challenges as ammocoetes even though they do not possess ammocoete MR cells. The ammocoete MR cell may thus have a function additional to or other than osmoregulation. Since lampreys feed in freshwater only during their larval phase, it is possible that the ammocoete MR cell, which apparently has no morphological counterpart in other vertebrate epithelia, may be involved in excreting ions and/or waste products that have been derived from the digestion of their algal and detrital food(Bartels et al., 1998). The ability to take up ions via the branchial epithelium would be particularly valuable during the upstream spawning run since, during that migration, the lamprey ceases feeding and its gut degenerates(Youson, 1981) and thus no longer has the potential to acquire ions from food.

#### Models for Na+ uptake by the branchial epithelium of lampreys in freshwater

Since the intercalated MR cell and pavement cell are the only two cell types that are present on the gill surface of both larval and adult lampreys in freshwater, it has been assumed that Na+ and Cl must be taken up through either one or both of these cell types (Bartels, 1989; Bartels et al., 1998). Since the putative role of the subtype C of the intercalated MR cell in the uptake of Cl has been discussed above, we will now focus on the uptake of Na+.

The arrangement and ultrastructure of the intercalated MR cell and pavement cell in the gill epithelium of adult lampreys in freshwater bear a striking resemblance to those of the intercalated MR cells and granular cells in the amphibian skin and urinary bladder, respectively. This led to the proposal that, in the lamprey gill, the intercalated MR cells (subtypes A and C)likewise facilitate the uptake of Na+ by active H+secretion and that the pavement cell is the primary candidate for the uptake of Na+ (Fig. 6; Bartels, 1989; Bartels et al., 1998). Except in very tight' epithelia, such an indirect coupling of H+secretion and Na+ uptake requires a close spatial relationship between the cell types involved. This is the case in both of the freshwater stages of adult lampreys, i.e. during their downstream and upstream migration,when the MR cells are intercalated between (or directly neighbouring) the granular cells. However, in larval lampreys, the pavement cells are always separated from the intercalated MR cells by large groups of ammocoete MR cells. The distance between these two cell types would thus presumably be too large to produce an effect on the electrical potential generated by the active H+ secretion through intercalated MR cells on the pavement cells. Although the electrical resistance of the gill epithelium of larval lampreys has not yet been determined, the tight junctions, as revealed in freeze-fracture replicas, do not possess the structural characteristics considered responsible for epithelial tightness, e.g. a large number of superimposed junctional strands, exhibiting a high degree of branching and anastomosing and consisting of solid fibrils rather than particles (see fig. 3A in Bartels et al., 1998; Claude and Goodenough, 1973; Claude, 1978; Cerejido et al., 1989).

Provided that the gill epithelium of larval lampreys is not very tight and the ammocoete MR cells do not contribute to the uptake of Na+,there are two possible alternative mechanisms for this transport in ammocoetes. The first hypothesis is based on the assumption that, in ammocoetes, Na+ uptake is coupled to H+ secretion. Under this condition, only the intercalated MR cell can be responsible for the uptake of Na+. This model, which is consistent with that proposed for the MR cells in the frog skin under natural conditions (see above; Ehrenfeld et al., 1989; Ehrenfeld and Klein, 1997),has the advantage of being equally applicable to larval and adult lampreys. It requires the presence both of the H+-ATPase and the ENac in the apical membrane and of significant amounts of Na+/K+-ATPase in the basolateral membrane of the intercalated MR cell. However, there is as yet no direct evidence from patch–clamp or immunocytochemical studies that the ENac is present in intercalated MR cells of the amphibian epidermis or in any other location in which these cells occur. Furthermore, studies on the cellular distributions of Na+/K+-ATPase in the toad and turtle urinary bladders and frog skin have shown that the vast majority of Na+ pumps are localised in the granular cells and that the intercalated MR cell possesses only a few of these pumps. This distribution pattern is consistent with the model that the granular cell compartment is responsible for Na+uptake (Rick et al., 1978; Durham and Nagel, 1986; Nagel and Dörge, 1996)and suggests that the few Na+ pumps present in the intercalated MR cells are sufficient to maintain the balance of the Na+ and K+ gradients across the basolateral membrane but do not significantly contribute to Na+ uptake.

The alternative model assumes that Na+ is taken up by pavement cells and is energized only by the activity of the Na+ pump,located in the basolateral membrane of this cell, and is not facilitated by H+ secretion by intercalated MR cells. The question then arises as to whether the rate of this branchial Na+ uptake, possibly in concert with intestinally resorbed dietary Na+, is sufficient to substitute the passive loss of Na+. Under such a condition,H+ secretion might provide the energy for Cluptake through the subtype C of the intercalated MR cells (see above). By contrast, since the requirements for a coupling of Na+ uptake and H+ secretion are met in adults, H+ secretion through the subtypes A and C could also facilitate Na+ uptake in the adult stages of the life cycle in addition to driving the uptake of Cl. This model implies that the mechanisms for regulating the Na+ concentration are not the same in larval and adult lampreys. It also takes into account the fact that the composition and arrangement of the branchial epithelial cells in larval and adult lampreys differ and that the larval lamprey feeds whereas the adult lamprey in freshwater does not.

### The marine phase in the life cycle

#### The chloride cell

Lamprey chloride cells are disc-like and form long continuous rows that extend throughout the interlamellar region of the filament and into the basal region of the filament where lamellae are absent (Figs 7A, 8; Bartels et al., 1996). The cytoplasm of these cells contains numerous membranous tubules, between which large mitochondria are intercalated (Fig. 7A), thereby paralleling the situation with the chloride cells of teleost fishes. Since these tubular membranes represent a vast intracellular amplification of the basolateral cell membrane, the lumen of the tubules constitutes part of the extracellular space(Philpott and Copeland, 1963; Philpott, 1966; Nakao, 1974; Karnaky et al., 1976a; Peek and Youson, 1979a). The particles in the membranes of these tubules have been shown by freeze fractures to be located mainly on their P face and to be arranged in linear and helicoidally twisted arrays (Fig. 7B; Hatae and Benedetti,1982; Bartels and Welsch,1986).

When young adult lampreys are still in freshwater, the surface of all but a small circular central region of their chloride cells is covered by the flanges of adjacent pavement cells (Fig. 8A). After young adults have entered the marine environment, their pavement cell flanges retract, with the result that the surface of each chloride cell that is exposed to the environment then occupies a rectangle,the length of which corresponds to the entire width of the interlamellar region (Fig. 8B). In addition,the apical surface loses its short microvilli and thus becomes relatively smooth (Fig. 8B;Bartels et al.,1996). Freeze-fracture replicas demonstrate that, when these lampreys are still in freshwater, the particles in the apical membrane of the chloride cells are randomly distributed on the P face. However, after the lamprey has entered seawater, most of the particles present on both fracture faces are hexagonally arranged into clusters with a periodic spacing of approximately 19 nm (Fig. 7C,D; Bartels et al., 1993).

As a consequence of the changes at the gill epithelium surface, the length of the paracellular pathway between adjacent chloride cells is increased by four to five times. At the same time, the tight junctions between chloride cells become greatly reduced in depth and their strands decline in number from four to either one or two (Fig. 7C), whereas those between chloride and pavement cells remain deep and the number of their strands declines only from four to three(Bartels and Potter, 1991). Finally, the membranous tubules in the cytoplasm, which are convoluted when the lamprey is still in freshwater, become aligned and organised into tight bundles when the animal enters seawater. After the fully grown lamprey has left the sea and embarked on its spawning run, the chloride cells become covered by the flanges of adjacent pavement cells and undergo apoptosis.

In the gills of marine teleosts, the site of Na+/K+-ATPase activity lies predominantly in the membranous tubules in the cytoplasm of their chloride cells(Karnaky et al., 1976b), which contains intramembranous particles that form a tight and regular(cobblestone') arrangement. Since these particles are considered to correspond to the Na+ pumps(Sardet et al., 1979; Sardet, 1980), it has been proposed that the particles of the helicoidally twisted linear arrays in the corresponding membrane of lamprey chloride cells are also the sites of Na+/K+-ATPase activity(Hatae and Benedetti, 1982; Bartels and Welsch, 1986). Furthermore, in teleosts, this membrane contains a furosemide-sensitive Na+/K+/2Cl cotransporter(Eriksson et al., 1985),through which Cl enters the cell on its basolateral side, a process driven by the steep Na+ gradient maintained across this membrane by the activity of the Na+ pump(Karnaky, 1986).

The type of clusters of intramembranous particles, which are present in the apical membrane of the chloride cells of lampreys in seawater, has not been observed in any of the Cl secretory cell types engaged in osmoregulation in marine environments. However, crystalline-like plaques',whose particles are of a similar size and spacing as those in the lamprey chloride cell membrane, are present in the apical membrane of the principal cells in the renal collecting tubule of the salamander Amphiuma means(Biemesderfer et al., 1989),i.e. in a cell engaged in taking up rather than secreting Na+ and Cl (Hunter et al.,1987). Since the changes in the apical membrane of the lamprey chloride cell described earlier take place at the time when the animal becomes faced with the need to excrete monovalent ions in a hypertonic environment(and when an extended leaky paracellular pathway is developed between the chloride cells), the clusters of particles in the apical membrane of lamprey chloride cells are likely to be associated with the transport of Cl across this membrane. The occurrence of these clusters of particles in cell membranes, across which Cl is transported in opposite directions, implies that the direction of Clmovement depends on the electrochemical gradient.

The above description of the lamprey chloride cell demonstrates that, when lampreys are in seawater, this cell possesses the morphological characteristics that, in ion-secretory cells in epithelia such as those of the teleost gill and operculum, the elasmobranch rectal gland and the avian nasal gland, have been shown to be involved in secreting excess monovalent ions when the animal is in hypertonic environments(Kirschner, 1980). In this regard, the three most salient features are: (1) the secretory cells are organised into multicellular units, (2) there is an extended leaky paracellular pathway (shunt) between the secretory cells and (3) the(baso)lateral membrane is greatly amplified, which thereby provides space for a large number of Na+ pumps. It is thus concluded that, as in the gills of marine teleosts, the chloride cell of lampreys in hypertonic environments is responsible for secreting the excess component of the Na+ and Cl that has been absorbed through the alimentary canal (Fig. 9). The mechanisms used for such secretion by lamprey chloride cells are assumed to be essentially the same as those employed by the chloride cells of teleosts in seawater. They thus involve a secondary active transport of Cl through the chloride cells, which drives the passive movement of Na+ through the leaky pathways between these cells(Bartels and Potter, 1991; Bartels et al., 1996).

#### The pavement cell

The only morphological change observed in the pavement cells of young adult lampreys as these animals migrate from fresh to seawater was a reduction from five to four in the number of strands in their tight junctions(Bartels and Potter, 1993). Since the number of strands of the tight junctions between pavement and chloride cells also declined, in this case from four to three(Bartels and Potter, 1991), the paracellular pathway in the lamprey gill epithelium becomes more leaky during the transition from fresh to seawater. Although such a conclusion is consistent with the results of studies on teleosts, which have shown that the permeability of their gills to Na+ is greater in sea than freshwater (Potts, 1984), any such increase in the permeability to Na+, when young adult lampreys enter seawater, would be due mainly to the pronounced changes that occur to the structure and length of the occluding junctions between the chloride cells. Since, as in marine teleosts, the chloride cell of lampreys apparently possesses all of the structural characteristics required to fulfil the osmoregulatory function of gills in seawater, there are at present no indications that the osmoregulation of lampreys in seawater requires an involvement of the pavement cell.

Although the gills of lampreys and teleosts are assumed to perform similar,if not identical, osmoregulatory functions, the cellular composition of their epithelia differs considerably in some respects(Laurent, 1984; Wilson and Laurent, 2002). Thus, in contrast to the situation in lampreys, the gill epithelia of teleosts always only contain pavement and chloride cells, irrespective of whether the fish is living in fresh or seawater. Furthermore, teleosts possess two types of chloride cell. There is thus a freshwater chloride cell, which is typically singly intercalated between the pavement cells, as is the case with the intercalated MR cells in lampreys, and a seawater chloride cell, which is similar to the chloride cells of adult lampreys in seawater (see below).

The freshwater chloride cell of teleosts differs from the intercalated MR cell of lampreys through its possession of a tubular system, which is less elaborate than that of the chloride cell in seawater, and by the absence of rod-shaped particles in its plasma and cytoplasmic membranes. Although the term `MR cell' has been proposed for the freshwater chloride cell of teleosts(Pisam and Rambourg, 1991),the gills of teleosts in freshwater do not contain cells with the cytological characteristics of either the intercalated MR cells of lampreys and higher vertebrates or the ammocoete MR cell. Experimental studies indicate that, in teleosts, the freshwater chloride cell exchanges Cl for HCO3 while the pavement cell exchanges Na+ for H+ (Goss et al., 1992; Perry,1997). The latter conclusion is supported by the results of immunocytochemical studies, which showed that the pavement cell is the only cell in the teleost gill epithelium that contains both H+-ATPase and ENac (Sullivan et al.,1995; Wilson et al.,2000a). The freshwater chloride cell of teleosts thus performs the role that we consider is carried out in lampreys by the subtype C of the intercalated MR cell. The function performed by the pavement cell of teleosts in freshwater, in turn, is ascribed in lampreys to the activities of the subtypes A and C of the intercalated MR cell and the pavement cell, coupled in adults and possibly uncoupled in ammocoetes.

The differences between the gill epithelium of teleosts and lampreys in marine environments are less pronounced than those described above for freshwater and are mainly associated with chloride cells, which in both groups are responsible for excreting excess Cl and Na+. They thereby constitute the main effector cells for osmoregulation in seawater. Although the chloride cells of teleosts and lampreys are both located close to the afferent filament artery and form multicellular complexes, the arrangement and size of these complexes in the two groups differ. Thus, the chloride cells in the gills of teleosts form small groups of 2–4 cells whereas those of lampreys form long rows. Furthermore, while accessory cells, which lack an extensive tubular system and associated Na+/K+-ATPase, also frequently contribute to the groups of chloride cells in teleosts (Hootman and Philpott, 1980), such cells are not found in lampreys(Bartels and Potter, 1991). Moreover, in teleosts, each group of chloride cells and of chloride and accessory cells share an apical crypt(Karnaky, 1986), a structure not found in association with these cells in lampreys(Nakao, 1974; Peek and Youson, 1979a; Bartels and Potter, 1991;Bartels et al., 1993, 1996). The paracellular pathways between the chloride cells in both lampreys and teleosts and between chloride and accessory cells in teleosts contain leaky occluding junctions through which Na+ enters the environment passively(Sardet et al., 1979; Ernst et al., 1980; Bartels and Potter, 1991). However, the increase that occurs in the length of this shunt, following the migration of individuals from fresh to seawater, is achieved in teleosts through the development of interdigitations amongst the cells that form multicellular complexes (Sardet et al.,1979; King et al.,1989) and in lampreys by the retraction of pavement cells(Bartels et al., 1996). Finally, the apical membrane of the chloride cells of teleosts in seawater does not contain the prominent clusters of particles that are found in this membrane in lampreys (Sardet et al.,1979; Ernst et al.,1980; Bartels et al.,1993).

As mentioned earlier, the fact that the osmolality of the sera of all stages in the life cycle of lampreys is far less than that of seawater argues that this agnathan group spent a considerable period in freshwater(Hardisty et al., 1989). However, the discovery of a lamprey-like fossil in Cambrian marine deposits from 545 million years ago strongly indicates that the Petromyzontiformes evolved in marine environments (Janvier,1999; Shu et al.,1999). If this is the case, the specialised parasitic phase in the life cycle of contemporary anadromous lampreys represents a secondary return to a marine environment. By contrast, the teleosts are believed to have evolved in freshwater and then undergone extensive adaptive radiation in the sea during the Mesozoic, with some groups subsequently reinvading freshwater(Lutz, 1975). The possession of the same basic mechanisms for osmoregulation by lampreys and teleosts thus presumably represents the result of convergent evolution. Such independent evolution of the same mechanisms would account for differences between the characteristics of the cell types used for osmoregulation by these two divergent groups.

Gratitude is expressed to Prof. Wolfram Nagel for helpful discussions and to Ursula Fazekas, Pia Unterberger and Horst Ruß for expert technical assistance. Financial support was provided by the Australian Research Grants Committee and Deutsche Forschungsgemeinschaft.

Alper, S. L., Natale, J., Gluck, S., Lodish, H. J. and Brown,D. (
1989
). Subtypes of intercalated cells in rat kidney collecting duct defined by antibodies against erythroid band 3 and renal vacuolar H+-ATPase.
86
,
5429
-5433.
Bartels, H. (
1989
). Freeze-fracture study of the pavement cell in the lamprey gill epithelium. Analogy of membrane structure with the granular cell in the amphibian urinary bladder.
Biol. Cell
66
,
165
-171.
Bartels, H. and Potter, I. C. (
1991
). Structural changes in the zonulae occludentes of the chloride cells of young adult lampreys following acclimation to seawater.
Cell Tissue Res.
265
,
447
-457.
Bartels, H. and Potter, I. C. (
1993
). Intercellular junctions in the water–blood barrier of the gill lamella in the adult lamprey (Geotria australis, Lampetra fluviatilis).
Cell Tissue Res.
274
,
521
-532.
Bartels, H. and Welsch, U. (
1986
). Mitochondria-rich cells in the gill epithelium of cyclostomes. A thin section and freeze fracture study. In
Proceedings of the Second International Conference on Indo-Pacific Fishes
(ed. T. Uyeno, R. Arai, T. Taniuchi and K. Matsuura), pp.
58
-72. Tokyo:Ichthyological Society of Japan.
Bartels, H., Schewe, H. and Potter, I. C.(
1993
). Structural changes in the apical membrane of lamprey chloride cells after acclimation to seawater.
Am. J. Physiol.
265
,
C733
-C739.
Bartels, H., Moldenhauer, A. and Potter, I. C.(
1996
). Changes in the apical surface of chloride cells following acclimation of lampreys to seawater.
Am. J. Physiol.
270
,
R125
-R133.
Bartels, H., Potter, I. C., Pirlich, K. and Mallatt, J.(
1998
). Categorization of the mitochondria-rich cells in the gill epithelium of the freshwater phases in the life cycle of lampreys.
Cell Tissue Res.
291
,
337
-349.
Beamish, F. W. H. (
1980
). Osmoregulation in juvenile and adult lampreys.
Can. J. Fish. Aquat. Sci.
37
,
1739
-1750.
Beamish, F. W. H., Strachan, P. D. and Thomas, E.(
1978
). Osmotic and ionic performance of the anadromous sea lamprey, Petromyzon marinus.
Comp. Biochem. Physiol. A
60
,
435
-443.
Bentley, P. J. (
1962
). Permeability of the skin of the cyclostome Lampetra fluviatilis to water and electrolytes.
Comp. Biochem. Physiol.
6
,
95
-97.
Bentley, P. J. and Follett, B. K. (
1962
). The action of neurohypophysial and adrenocortical hormones on sodium balance in the cyclostome, Lampetra fluviatis.
Gen. Comp. Endocrinol.
2
,
329
-335.
Bentley, P. J. and Follett, B. K. (
1963
). Kidney function in a primitive vertebrate, the cyclostome Lampetra fluviatilis.
J. Physiol. (Lond.)
169
,
902
-918.
Biemesderfer, D., Stanton, B., Wade, J. B., Kashgarian, M. and Giebisch, G. (
1989
). Ultrastructure of Amphiuma distal nephron: evidence for cellular heterogeneity.
Am. J. Physiol.
256
,
C849
-C857.
Bourguet, J., Chevalier, J. and Hugon, J. S.(
1976
). Alterations in membrane-associated particle distribution during antidiuretic challenge in frog urinary bladder.
Biophys. J.
16
,
627
-639.
Brown, D. (
1989
). Membrane recycling and epithelial cell function.
Am. J. Physiol.
256
,
F1
-F12.
Brown, D. and Breton, S. (
1996
). Mitochondria-rich, proton-secreting epithelial cells.
J. Exp. Biol.
199
,
2345
-2358.
Brown, D. and Breton, S. (
2000
). H+V-ATPase-dependent luminal acidification in the kidney collecting duct and the epididymis/vas deferens: vesicle recycling and transcytotic pathways.
J. Exp. Biol.
203
,
137
-145.
Brown, D., Gluck, S. and Hartwig, J. (
1987
). Structure of the novel membrane-coating material in proton-secreting epithelial cells and identification as an H+-ATPase.
J. Cell Biol.
105
,
1637
-1648.
Brown, D., Hirsch, S. and Gluck, S. (
1988
). An H+-ATPase is present in opposite plasma membrane domains in subpopulations of kidney epithelial cells.
Nature
331
,
622
-624.
Brown, D., Katsura, T., Kawashima, M., Verkman, A. S. and Sabolic, I. (
1995
). Cellular distribution of the aquaporins:a family of water channel proteins.
Histochem. Cell Biol.
104
,
1
-9.
Cerejido, M., Gonzalez-Mariscal, L. and Contreras, G.(
1989
). Tight junction: barrier between higher organisms and environment.
NIPS
4
,
72
-75.
Cholette, C., Gagnon, A. and Germain, P.(
1970
). Isosmotic adaption in Myxine glutinosa L. I. Variations of some parameters and role of the amino acid pool of the muscle cells.
Comp. Biochem. Physiol.
33
,
333
-346.
Claude, P. (
1978
). Morphological factors influencing transepithelial permeability: a model for the resistance of the zonula occludens.
J. Membr. Biol.
39
,
219
-232.
Claude, P. and Goodenough, D. A. (
1973
). Fracture faces of zonulae occludentes from “tight” and“leaky” epithelia.
J. Cell Biol.
58
,
390
-400.
Conley, D. M. and Mallatt, J. (
1988
). Histochemical localization of Na+-K+-ATPase and carbonic anhydrase activity in gills of 17 fish species.
Can. J. Zool.
66
,
2398
-2405.
Delarbre, C., Gallut, C., Barriel, V., Janvier, P. and Gachelin,G. (
2002
). Complete mitochondrial DNA of the hagfish, Eptatretus burgeri: The comparative analysis of mitochondrial DNA sequences strongly supports the cyclostome monophyly.
Mol. Phylog. Evol.
22
,
184
-192.
DiBona, D. R., Civan, M. M. and Leaf, A.(
1969
). The cellular specificity of the effect of vasopressin on toad urinary bladder.
J. Membr. Biol.
1
,
79
-91.
Durham, J. H. and Nagel, W. (
1986
). Evidence for separate cellular origins of sodium and acid-base transport in the turtle bladder.
Am. J. Physiol.
250
,
C609
-C616.
Ehrenfeld, J. and Klein, U. (
1997
). The key role of the H+ V-ATPase in acid–base balance and Na+ transport processes in frog skin.
J. Exp. Biol.
200
,
247
-256.
Ehrenfeld, J., Lacoste, I. and Harvey, B. J.(
1989
). The key role of mitochondria-rich cells in Na and H transport across the frog skin epithelium.
Pflügers Arch.
414
,
59
-67.
Eriksson, Ö., Mayer-Gostan, N. and Wistrand, P. J.(
1985
). The use of isolated fish opercular epithelium as a model tissue for studying intrinsic activities of loop diuretics.
Acta Physiol. Scand.
125
,
55
-66.
Ernst, S. A., Dodson, W. B. and Karnaky, K. J., Jr(
1980
). Structural diversity of occluding junctions in the low-resistance chloride-secreting opercular epithelium of seawater-adapted killifish (Fundulus heteroclitus).
J. Cell Biol.
87
,
488
-497.
Ernst, S. A., Hootman, S. R., Schreiber, J. H. and Riddle, C. V. (
1981
). Freeze-fracture and morphometric analysis of occluding junctions in rectal glands of elasmobranch fish.
J. Membr. Biol.
58
,
101
-114.
Forey, P. L. and Janvier, P. (
1993
). Agnathans and the origin of jawed vertebrates.
Nature
361
,
129
-134.
Foskett, J. K. and Machen, T. E. (
1985
). Vibrating probe analysis of teleost opercular epithelium: correlation between active transport and leak pathways of individual chloride cells.
J. Membr. Biol.
85
,
25
-35.
Foskett, J. K. and Scheffey, C. (
1982
). The chloride cell: definitive identification as the salt-secretory cell in teleosts.
Science
215
,
164
-166.
Garcia-Romeu, F. and Ehrenfeld, J. (
1975
). In vivo Na+ and Cl independent transport across the skin of Rana esculenta.
Am. J. Physiol.
228
,
839
-844.
Garty, H. (
1986
). Mechanisma of aldosterone action in tight epithelia.
J. Membr. Res.
90
,
193
-205.
Garty, H. and Palmer, L. G. (
1997
). Epithelial sodium channels: function, structure and regulation.
Physiol. Rev.
77
,
359
-396.
Goss, G. G., Laurent, P. and Perry, S. F.(
1992
). Evidence for a morphological component in the regulation of acid–base balance in hypercapnic catfish (Ictalurus nebulosus).
Cell Tissue Res.
268
,
539
-552.
Hardisty, M. W. (
1982
). Lampreys and hagfishes:an analysis of cyclostome relationships. In
The Biology of Lampreys
, vol.
4B
(ed. M. W. Hardisty and I. C. Potter), pp.
165
Hardisty, M. W. and Potter, I. C. (
1971
). The general biology of adult lampreys. In
The Biology of Lampreys
, vol.
1
(ed. M. W. Hardisty and I. C. Potter), pp.
127
Hardisty, M. W., Potter, I. C. and Hilliard, R. W.(
1989
). Physiological adaptations of the living agnathans.
Trans. Roy. Soc. Edinburgh Earth Sciences
80
,
241
-254.
Harris, H. W., Jr, Strange, K. and Zeidel, M. L.(
1991
). Current understanding of the cellular biolgy and molecular structure of the antidiuretic hormone-stimulated water transport pathway.
J. Clin. Invest.
88
,
1
-8.
Harvey, B. J. and Ehrenfeld J. (
1986
). Regulation of intracellular sodium and pH by the electrogenic H+pump in frog skin.
Pflügers Arch.
406
,
362
-366.
Harvey, B. J., Thomas, R. and Ehrenfeld, J.(
1988
). Intracellular pH controls cell membrane Na+and K+ conductances and transport in frog skin epithelium.
J. Gen. Physiol.
92
,
767
-791.
Hatae, T. and Benedetti, E. L. (
1982
). Mosaic structure in the plasma membrane: spiral arrays of subunits in the cytoplasmic tubules of lamprey chloride cells.
J. Cell Sci.
56
,
441
-452.
Hootman, S. R. and Philpott, C. W. (
1980
). Accessory cells in teleost branchial epithelium.
Am. J. Physiol.
238
,
R199
-R206.
Hunter, M., Horisberger, J.-D., Stanton, B. and Giebisch, G.(
1987
). The collecting tubule of Amphiuma. I. Electrophysiological characterisation.
Am. J. Physiol.
253
,
F1263
-F1272.
Janvier, P. (
1999
). Catching the first fish.
Nature
402
,
21
-22.
Karnaky, K. J., Jr (
1980
). Ion-secreting epithelia: chloride cells in the head region of Fundulus heteroclitus.
Am. J. Physiol.
238
,
R185
-R198.
Karnaky, K. J., Jr (
1986
). Structure and function of the chloride cell of Fundulus heteroclitus and other teleosts.
Am. Zool.
26
,
209
-224.
Karnaky, K. J., Jr and Kinter, W. B. (
1977
). Killifish opercular skin: a flat epithelium with a high density of chloride cells.
J. Exp. Zool.
199
,
355
-364.
Karnaky, K. J., Jr, Ernst, S. A. and Philpott, C. W.(
1976a
). Teleost chloride cell. I. Response of pupfish Cyprinodon variegatus gill Na, K-ATPase and chloride cell fine structure to various high salinity environments.
J. Cell Biol.
70
,
144
-156.
Karnaky, K. J., Jr, Kinter, L. B., Kinter, W. B. and Stirling,C. E. (
1976b
). Teleost chloride cell. II. Autoradiographic localization of gill Na, K-ATPase in killifish Fundulus heteroclitusadapted to low and high salinity environments.
J. Cell Biol.
70
,
157
-177.
Karnaky, K. J., Jr, Degnan, K. J. and Zadunaisky, J. A.(
1977
). Chloride transport across isolated opercular epithelium of killifish: a membrane rich in chloride cells.
Science
195
,
203
-205.
Keys, A. B. and Willmer, E. N. (
1932
). Chloride secreting cells in the gills of fishes with special reference to the common eel.
J. Physiol. (Lond)
76
,
368
-378.
King, J. A. C., Abel, D. C. and DiBona, D. R.(
1989
). Effects of salinity on chloride cells in the euryhaline cyprinodontid fish Rivulus marmoratus.
Cell Tissue Res.
257
,
367
-377.
Kirschner, L. B. (
1977
). The sodium chloride excreting cells in marine vertebrates. In
Transport of Water and Ions in Animals
(ed. B. L. Gupta, R. B. Moreton, J. L. Oschman and B. J. Wall), pp.
427
-452. London, New York, San Francisco: Academic Press.
Kirschner, L. B. (
1980
). Comparison of vertebrate salt-excreting organs.
Am. J. Physiol.
238
,
R219
-R223.
Kirschner, L. B. (
1983
). Sodium chloride absorption across the body surface: frog skins and other epithelia.
Am. J. Physiol.
244
,
R429
-R443.
Kohn, O. F., Hand, A. R., Mitchell, P. P. and Steinmetz, P. R. (
1987
). Intra- and submembrane particle densities during CO2 stimulation of H+ secretion in turtle bladder.
Am. J. Physiol.
272
,
F491
-F497.
Krogh, A. (
1939
).
Osmotic Regulation in Aquatic Animals
. London: Cambridge University Press.
Kuraku, S., Hoshiyama, D., Katoh, K., Suga, H. and Miyata,T. (
1999
). Monophyly of lampreys and hagfishes supported by nuclear DNA-coded genes.
J. Mol. Evol.
49
,
729
-735.
Lang, F. (
1988
). NaCl transport in the kidney. In
Advances in Comparative and Environmental Physiology
, vol.
1
(ed. R. Greger), pp.
153
-188. Berlin, Heidelberg:Springer-Verlag.
Larsen, E. H. (
1988
). NaCl transport in amphibian skin. In
Advances in Comparative and Enviromental Physiology
, vol
1
(ed. R. Greger), pp.
189
-248. Berlin, Heidelberg:Springer-Verlag.
Larsen, E. H. (
1991
). Chloride transport by high-resistance heterocellular epithelia.
Physiol. Rev.
77
,
235
-283.
Larsen, E. H., Willumsen, N. J. and Christoffersen, B. C.(
1992
). Role of proton pump of mitochondria-rich cells for active transport of chloride ions in toad skin epithelium.
J. Physiol.
450
,
203
-216.
Laurent, P. (
1984
). Gill internal morphology. In
Fish Physiology
, vol.
10
,part A (ed. W. S. Hoar and D. J. Randall), pp.
73
Loretz, C. A. (
1995
). Electrophysiology of ion transport in teleost intestinal cells. In
Cellular and Molecular Approaches to Fish Ionic Regulation
(ed. C. M. Wood and T. J. Shuttleworth), pp.
25
Lutz, P. (
1975
). Adaptive and evolutionary aspects of the ionic content of fishes.
Copeia
1975
,
369
-373.
Macknight, A. D. C., DiBona, D. R. and Leaf, A.(
1980
Physiol. Rev.
60
,
615
-715.
Mallatt, J. and Ridgway, R. L. (
1984
). Ultrastructure of a complex epithelial system: the pharyngeal lining of the larval lamprey Petromyzon marinus.
J. Morphol.
180
,
271
-296.
Mallatt, J. and Sullivan, J. (
1998
). 28S and 18S rDNA sequences support the monophyly of lampreys and hagfishes.
Mol. Biol. Evol.
15
,
1706
-1718.
Mallatt, J., Bailey, J. F., Lampa, S. J., Evans, M. A. and Tate,W. (
1995
). Quantitative ultrastructure of gill epithelial cells in the larval lamprey Petromyzon marinus.
Can. J. Fish Aquat. Sci.
52
,
1150
-1164.
Marshall, W. S. (
2002
). Na+,Cl, Ca2+ and Zn2+ transport by fish gills: retrospective review and prospective synthesis.
J. Exp. Zool.
293
,
264
-283.
Marshall, W. S., Bryson, S. E., Darling, P., Whitten, C.,Patrick, M., Wilkie, M., Wood, C. M. and Buckland-Nicks, J.(
1997
). NaCl transport and ultrastructure of opercular epithelium from a freshwater-adapted euryhaline teleost, Fundulus heteroclitus.
J. Exp. Zool.
277
,
213
-237.
Marshall, W. S., Lynch, E. M. and Cozzi, R. R. F.(
2002
). Redistribution of immunofluorescence of CFTR anion channel and NKCC cotransporter in chloride cells during adaptation of the killifish Fundulus heteroclitus to sea water.
J. Exp. Biol.
205
,
1265
-1273.
McFarland, W. and Munz, F. W. (
1965
). Regulation of body weight and serum composition by hagfish in various media.
Comp. Biochem. Physiol.
14
,
383
-398.
Morris, R. (
1956
). The osmoregulation ability of the lampern (Lampetra fluviatilis L.) in sea water during the course of its spawning migration.
J. Exp. Biol.
23
,
235
-248.
Morris, R. (
1957
). Some aspects of the structure and cytology of the gills of Lampetra fluviatilis.
Q. J. Microsc. Sci.
98
,
473
-485.
Morris, R. (
1958
). The mechanism of marine osmoregulation in the lampern (Lampetra fluviatilis L.) and the causes of its breakdown during the spawning migration.
J. Exp. Biol.
35
,
649
-665.
Morris, R. (
1972
). Osmoregulation. In
The Biology of Lampreys
, vol.
2
(ed. W. M. Hardisty and I. C. Potter), pp.
192
Morris, R. (
1980
). Blood composition and osmoregulation in ammocoete larva.
Can. J. Fish Aquat. Sci.
37
,
1665
-1679.
Morris, R. and Pickering, A. D. (
1975
). Ultrastructure of presumed ion-transporting cells in the gills of ammocoete lampreys, Lampetra fluviatilis (L.) and Lampetra planeri(Bloch).
Cell Tissue Res.
163
,
327
-341.
Morris, R. and Pickering, A. D. (
1976
). Changes in the ultrastructure of the gills of the river lamprey, Lampetra fluviatilis (L.), during the anadromous spawning migration.
Cell Tissue Res.
173
,
271
-277.
Nagel, W. and Dörge, A. (
1996
). The role of mitochondria-rich cells in sodium transport across amphibian skin.
Pflügers Arch. Eur. J. Physiol.
433
,
146
-152.
Nakao, T. (
1974
). Fine structure of the agranular cytoplasmic tubules in the lamprey chloride cells.
Anat. Rec.
178
,
49
-62.
Parry, G. (
1966
Biol. Rev.
41
,
392
-444.
Peek, W. D. and Youson, J. H. (
1979a
). Ultrastructure of chloride cells in young adults of the anadromous sea lamprey, Petromyzon marinus L., in fresh water and during adaptation to sea water.
J. Morph.
160
,
143
-163.
Peek, W. D. and Youson, J. H. (
1979b
). Transformation of the interlamellar epithelium of the gills of the anadromous sea lamprey, Petromyzon marinus L., during metamorphosis.
Can. J. Zool.
57
,
1318
-1332.
Perry, S. F. (
1997
). The chloride cell:structure and function in the gills of freshwater fishes.
Annu. Rev. Physiol.
59
,
325
-347.
Philpott, C. W. and Copeland, D. E. (
1963
). Fine structure of chloride cells from three species of Fundulus.
J. Cell Biol.
18
,
389
-404.
Pickering, A. D. and Morris, R. (
1970
). Osmoregulation of Lampetra fluviatilis L. and Petromyzon marinus (Cyclostomata) in hyperosmotic solutions.
J. Exp. Biol.
53
,
231
-243.
Pisam, M. and Rambourg, A. (
1991
). Mitochondria-rich cells in the gill epithelium of teleost fishes: an ultrastructural approach.
Int. Rev. Cytol.
130
,
191
-232.
Philpott, C. W. (
1966
). The use of horseradish peroxidase to demonstrate functional continuity between the plasmalemma and the unique tubular system of the chloride cell.
J. Cell Biol.
31
,
86
.
Potter, I. C. (
1980
). Ecology of larval and metamorphosing lampreys.
Can. J. Fish. Aquat. Sci.
37
,
1641
-1657.
Potter, I. C. and Beamish, F. W. H. (
1977
J. Zool. (Lond.)
181
,
113
-130.
Potter, I. C. and Hilliard, R. W. (
1987
). A proposal for the functional and phylogenetic significance of differences in the dentition of lampreys (Agnatha: Petromyzontiformes).
J. Zool.(Lond.)
212
,
713
-737.
Potter, I. C. and Huggins, R. J. (
1973
). Observations on the morphology, behaviour and salinity tolerance of downstream migrating river lamprey (Lampetra fluviatilis).
J. Zool.(Lond.)
169
,
365
-379.
Potter, I. C., Hilliard, R. W. and Bird, D. J.(
1980
). Metamorphosis in the Southern Hemisphere lamprey, Geotria australis.
J. Zool. (Lond.)
190
,
405
-430.
Potts, W. T. W. (
1984
). Transepithelial potential in fish gills. In
Fish Physiology
, vol.
10
, part B (ed. W. S. Hoar and D. J. Randall), pp.
105
Rick, R., Dörge, A., v. Arnim, E. and Thurau, K.(
1978
). Electron microprobe analysis of frog skin epithelium:evidence for a syncytial sodium transport compartment.
J. Membr. Biol.
39
,
313
-331.
Riddle, C. V. and Ernst, S. A. (
1979
). Structural simplicity of the zonula occludens in the electrolyte secreting epithelium of the avian salt gland.
J. Membr. Biol.
45
,
21
-35.
Robertson, J. D. (
1957
). The habitat of the early vertebrates.
Biol. Rev.
32
,
156
-187.
Robertson, J. D. (
1974
). Osmotic and ionic regulation in cyclostomes. In
Chemical Zoology
, vol.
8
(ed. M. Florkin and B. T. Scheer), pp.
149
Sardet, C. (
1980
). Freeze-fracture of the gill epithelium of euryhaline teleost fishes.
Am. J. Physiol.
238
,
R207
-R212.
Sardet, C., Pisam, M. and Maetz, J. (
1979
). The surface epithelium of teleostean fish gills. Cellular and junctional adaptations of the chloride cell in relation to salt adaptation.
J. Cell Biol.
80
,
96
-117.
Shu, D.-G., Luo, H.-L., Conway Morris, S., Zhang, X.-L., Hu,S.-X., Chen, L., Zhu, M., Li, Y. and Chen, L.-Z.(
1999
). Lower Cambrian vertebrates from south China.
Nature
402
,
42
-46.
Singer, T. D., Tucker, S. J., Marshall, W. S. and Higgins, C. F. (
1998
). A divergent CFTR homologue: highly regulated salt transport in the euryhaline teleost F. heteroclitus.
Am. J. Physiol.
274
,
C715
-C723.
Smith, H. W. (
1930
). The absorption and excretion of water and salts by marine teleosts.
Am. J. Physiol.
93
,
485
-505.
Smith, H. W. (
1932
). Water regulation and its evolution in the fishes.
Q. Rev. Biol.
7
,
1
-26.
Stetson, D. L. and Steinmetz, P. R. (
1983
). Role of membrane fusion in CO2 stimulation of proton secretion by turtle bladder.
Am. J. Physiol.
245
,
C113
-C120.
Stetson, D. L. and Steinmetz, P. R. (
1985
).α and β types of carbonic anhydrase-rich cells in turtle bladder.
Am. J. Physiol.
249
,
F553
-F565.
Stetson, D. L. and Steinmetz, P. R. (
1986
). Correlation between apical intramembrane particles and H+ secretion rates during CO2 stimulation in turtle bladder.
Pflügers Arch.
407
,
S80
-S84.
Stock, D. W. and Whitt, G. S. (
1992
). Evidence from 18S ribosomal RNA sequences that lampreys and hagfishes form a natural group.
Science
257
,
787
-789.
Sullivan, G. V., Fryer, J. N. and Perry, S. F.(
1995
). Immunolocalization of proton pumps (H+-ATPase)in pavement cells of rainbow trout gill.
J. Exp. Biol.
198
,
2619
-2629.
1976
). Membrane structural specializations of the toad urinary bladder revealed by the freeze-fracture technique. II. The mitochondria-rich cell.
J. Membr. Biol.
29
,
111
-126.
Wade, J. B., Di Scala, V. A. and Karnovsky, M. J.(
1975
). Membrane structural specialization of the toad urinary bladder revealed by the freeze fracture technique. I. The granular cell.
J. Membr. Biol.
22
,
385
-402.
Wade, J. B., Stetson, D. L. and Lewis, S. A.(
1981
). ADH action: evidence for a membrane shuttle mechanism.
372
,
106
-117.
Wilson, J. M. and Laurent, P. (
2002
). Fish gill morphology: inside out.
J. Exp. Zool.
293
,
192
-213.
Wilson, J. M., Laurent, P., Tufts, B. L., Benos, D. J.,Donowitz, M., Vogl, A. W. and Randall, D. J. (
2000a
). NaCl uptake by the branchial epithelium in freshwater teleost fish: an immunological approach to ion-transport protein localization.
J. Exp. Biol.
203
,
2279
-2296.
Wilson, J. M., Randall, D. J., Donowitz, M., Vogl, A. W. and Ip,Y. K. (
2000b
). Immunolocalisation of ion-transport proteins to branchial epithelium mitochondria-rich cells in the mudskipper(Periophthalmodon schlosseri).
J. Exp. Biol.
203
,
2297
-2310.
Youson, J. H. (
1981
). The alimentary canal. In
The biology of Lampreys
, vol.
3
(ed. M. W. Hardisty and I. C. Potter), pp.
95
Youson, J. H. and Freeman, P. A. (
1976
). Morphology of the gills of larval and parasitic adult sea lamprey, Petromyzon marinus L.
J. Morphol.
149
,
73
-104.


text:
# What does black hole evaporation correspond to in the accelerating universe / black hole analogy?

In the same way, as a black hole emits Hawking radiation corresponding to its temperature

$$T = \frac{\hbar}{r_{BH}}$$

the accelerating universe can be described as all of the stuff falling towards the cosmic horizon, such that it is analogue to a black hole where the cosmic horizon takes the role of the event horizon with the black hole "interior" being outside.

The corresponding temperature of the accelerating universe is given by

$$T = \hbar H = \frac{\hbar}{R_{H}}$$

where H is the Hubble constant. Therefore, the cosmic horizon emits something similar to Hawking radiation too.

The fate of a black hole is to finally evaporate. My question is: What does black hole evaporation correspond to in the accelerating universe when making use of this analogy?

-
Dear downvoter, what is wrong with my question, can you please be a bit more specific and explicit thant just pushing the button ? –  Dilaton Apr 21 '14 at 11:14

## 1 Answer

An analogy in the settings of the question would be a deceleration down to an expansion rate, such that eventually light from anywhere in the universe can reach us, meaning the cosmic horizon can vanish. A constant expansion rate of the universe above the speed of light would be the analogy of a black hole of fixed mass.

In an infinite universe this would mean, that its expansion needs to slow down to an asymptotic speed for infinite distance below the speed of light. For a finite 3-spherical universe for a fixed cosmic time a slow-down to below the speed of light would be sufficient. This would correspond to a massive follow-up object of the black hole without event horizon.

For an analogy to a completely evaporating black hole the expansion rate needs to slow down to zero. The slow down between speed of light and zero corresponds to the very last phase of black hole evaporation, when its mass drops below the Planck mass, then without event horizon.

The analogy to Hawking radiation would be radiation or matter emerging from behind the former cosmic horizon.

Thinking in stereographic projections may help for a better understanding: Project infinity of a plane expanding universe to the singularity of a black hole at the north pole of an expanding sphere, as an initial point of further reasoning.

-

format: plain


text:
# Fixed point iteration-finding $g(x)$

I have an equation $f(x)= x^4 +2x^3 -5x^2 -7x-5=0$ and I want to solve it (numerically) at [1.5,3] using the fixed point iteration-method, so I want to find a function g(x) such that $f(x)=0$ is equivalent to $g(x)=x$. However I've tried many different $g(x)$'s but can't seem to find one that converges to the solution (which is about 2.19). First of all, anyone has any idea which $g(x)$ I could use? Second is there a trick to finding it?

To get the solution to converge, besides $g(x)=x$ you need $|g'(x)| \lt 1$ at the root. The distance from the root is multiplied by about $|g'(x)|$ every iteration, so if that is less than $1$ it will converge. Powers change quickly and roots slowly, so a natural try is to write $$x^4 +2x^3 -5x^2 -7x-5=0\\x^4=-2x^3+5x^2+7x+5\\ x=\sqrt[4]{-2x^3+5x^2+7x+5}$$ It converges nicely starting at $0, 1$ and $3.5$ but fails starting from $4$ because the piece under the root goes negative. Another try, which I have not tested, would be to write $$x^4 +2x^3 -5x^2 -7x-5=0\\x^4=-2x^3+5x^2+7x+5\\ x=(-2x^3+5x^2+7x+5)/x^3$$ It is a bit of an art. If you don't know the root, you can't evaluate the derivative of your proposed $g(x)$ at it.

The scholars of old had a philosophy of avoiding differences if manually computing roots of equations. That way a treatment of quadratic or cubic equations could fill multiple pages considering the various cases, but avoided the discussion of roots of negative numbers.

In your equation one would separate by this philosophy $$x^4+2x^3=5x^2+7x+5$$ and solve the higher degree side, for instance as $$x=g(x)=\sqrt[\Large3\,\!]{\frac{5x^2+7x+5}{x+2}}=\sqrt[\Large3\,\!]{5x-3+\frac{11}{x+2}}$$ This as fixed-point iteration maps the interval $$[2,3]$$ into itself an has thus a fixed point. The derivative of $$g$$ is smaller than $$\frac{5}{12}<\frac12$$ providing sufficient speed of convergence. A numerical series confirms this result.

 k         x[k]        x[k]-x[k-1]     1.4*0.3^k

0      2.0
1      2.1363293408   0.1363293408     0.138
2      2.1786508930   0.04232155218    0.0414
3      2.1915435027   0.0128926097     0.01242
4      2.1954485170   0.003905014239   0.003726
5      2.1966292396   0.001180722584   0.0011178
6      2.1969860556   0.0003568160362  0.00033534
7      2.1970938687   0.0001078131484  0.000100602
8      2.1971264433   3.25745335e-05   3.01806e-05
9      2.1971362852   9.841886902e-06  9.05418e-06
10      2.1971392587   2.973559468e-06  2.716254e-06
11      2.1971401571   8.984094482e-07  8.148762e-07
12      2.1971404286   2.714387324e-07  2.4446286e-07


text:
Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł książki

Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 325 wydano: 1993
Zawartość
Warianty tytułu
Abstrakty
EN
Some generalizations of the approximation theorem of Wong-Zakai type for stochastic differential equations are examined. One of them deals with functional stochastic differential equations defined on some spaces of continuous functions. The second one concerns the situations when the state space and the Wiener process have values in some Hilbert spaces. The comparison of these results as well as some examples are also included. The correction terms computed here are then applied to the derivation of the relation between the Itô and Stratonovich integrals. Other important applications of the above theorems are indicated.
EN

CONTENTS
1. Introduction...........................................................................................................................................5
1.1. The Wong-Zakai theorem and its generalizations.............................................................................5
1.2. Approximation methods for stochastic differential equations.............................................................7
1.3. Extensions of the Wong-Zakai theorem and their applications..........................................................9
2. Approximation theorem of Wong-Zakai type for functional stochastic differential equations................10
2.1. Introductory remarks........................................................................................................................10
2.2. Definitions and notation...................................................................................................................10
2.3. Description of the model..................................................................................................................11
2.4. Approximation theorem....................................................................................................................15
2.5. Examples.........................................................................................................................................24
3. An extension of the Wong-Zakai theorem to stochastic evolution equations in Hilbert spaces............26
3.1. Introductory remarks.......................................................................................................................26
3.2. Definitions and notation..................................................................................................................26
3.3. Description of the model.................................................................................................................27
3.4. The main theorem...........................................................................................................................31
3.5. Examples.........................................................................................................................................41
3.5.1. Equations satisfying the assumptions of Theorem 3.4.1.............................................................41
3.5.2. Stochastic delay equations.........................................................................................................43
3.5.3. Stochastic wave equations..........................................................................................................45
4. Comparison of the results...................................................................................................................46
4.1. Finite-dimensional case..................................................................................................................46
4.2. Stochastic delay equations.............................................................................................................47
5. On the relation between the Itô and Stratonovich integrals in Hilbert spaces.....................................47
6. Conclusions........................................................................................................................................49
References.............................................................................................................................................50
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 325
Liczba stron
54
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, CCCXXV
Daty
wydano
1993
otrzymano
1992-03-27
poprawiono
1992-07-13
Twórcy
autor
• Institute of Mathematics, Warsaw Technical University, Pl. Politechniki 1, 00-661 Warszawa, Poland
Bibliografia
• [1] P. Aquistapace and B. Terreni, An approach to Itô linear equations in Hilbert spaces by approximation of white noise with coloured noise, Stochastic Anal. Appl. 2 (1984), 131-186.
• [2] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York 1974.
• [3] S. S. Artem''ev, Numerical solution of stochastic differential equations, in: Proc. Third Conf. Differential Equations and Applications, Ruse 1985, VTU A. K''nchev'', 1985, 15-18 (in Russian).
• [4] V. Bally, Approximation for the solutions of stochastic differential equations, III. Jointly weak convergence, Stochastics Stochastics Rep. 30 (1990), 171-191.
• [5] H. T. Banks and F. Kappel, Spline approximation for functional differential equations, J. Differential Equations 34 (1979), 496-522.
• [6] D. R. Bell and S. E. A. Mohammed, On the solution of stochastic ordinary differential equations via small delays, Stochastics Stochastics Rep. 28 (1989), 293-299.
• [7] Z. Brze/xniak, M. Capi/nski and F. Flandoli, A convergence result for stochastic partial differential equations, Stochastics 24 (1988), 423-445.
• [8] H.-F. Chen and A. J. Gao, Robustness analysis for stochastic approximation algorithms, Stochastics Stochastics Rep. 26 (1989), 3-20.
• [9] A. Chojnowska-Michalik, Representation theorem for general stochastic delay equations, Bull. Acad. Polon. Sci. 26 (7) (1978), 635-642.
• [10] A. Chojnowska-Michalik, Stochastic differential equations in Hilbert spaces and their applications, thesis, 1976.
• [11] J. M. C. Clark, An efficient approximation for a class of stochastic differential equations, in: Advances in Filtering and Optimal Stochastic Control, W. Fleming and L. G. Gorostiza (eds.), Proceedings of the IFIP Working Conference, Cocoyoc, Mexico, 1982, Lecture Notes in Control and Inform. Sci. 42, Springer, Berlin 1982, 69-78.
• [12] J. M. C. Clark and R. J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, in: Stochastic Differential Systems-Filtering and Control, B. Grigelionis (ed.), Proceedings of the IFIP Working Conference, Vilnius, Lithuania, USSR, 1978, Lecture Notes in Control and Inform. Sci. 25, Springer, Berlin 1980, 162-171.
• [13] R. F. Curtain and P. F. Falb, Itô's lemma in infinite dimensions, J. Math. Anal. Appl. 31 (1970), 431-448.
• [14] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer, Berlin 1978.
• [15] G. Da Prato, Stochastic differential equations with noncontinuous coefficients in Hilbert space, Rend. Sem. Mat. Univ. Politec. Torino, Numero Speciale 1982, 73-85.
• [16] G. Da Prato, S. Kwapień and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics 23 (1987), 1-23.
• [17] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1991.
• [18] A. L. Dawidowicz and K. Twardowska, On the Stratonovich and Itô integrals with integrands of delay argument, submitted to Stochastica (1991).
• [19] H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré 13 (2) (1977), 99-125.
• [20] G. Ferreyra, A Wong-Zakai type theorem for certain discontinuous semimartingales, J. Theoret. Probab. 2 (3) (1989), 313-323.
• [21] A. Greiner and W. Strittmatter, Numerical integration of stochastic differential equations, J. Statist. Phys. 51 (1-2) (1988), 95-108.
• [22] I. Gyöngy, On the approximation of stochastic differential equations, Stochastics 23 (1988), 331-352.
• [23] I. Gyöngy, On the approximation of stochastic partial differential equations, Part I, ibid. 25 (1988), 59-85, Part II, ibid. 26 (1989), 129-164.
• [24] I. Gyöngy, The stability of stochastic partial differential equations and applications. Theorems on supports, in: Lecture Notes in Math. 1390, Springer, Berlin 1989, 91-118.
• [25] J. Hale, Theory of Functional Differential Equations, Springer, Berlin 1977.
• [26] N. Ikeda, S. Nakao and Y. Yamato, A class of approximations of Brownian motion, Publ. RIMS Kyoto Univ. 13 (1977), 285-300.
• [27] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam 1981.
• [28] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ. 4 (1) (1964), 1-75.
• [29] R. Janssen, Difference methods for stochastic differential equations with discontinuous coefficient, Stochastics 13 (1984), 199-212.
• [30] R. Janssen, Discretization of the Wiener process in difference methods for stochastic differential equations, Stochastic Process. Appl. 18 (1984), 361-369.
• [31] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin 1966.
• [32] S. Kawabata, On the successive approximation of solutions of stochastic differential equations, Stochastics Stochastics Rep. 30 (1990), 69-84.
• [33] P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin 1992.
• [34] F. Koneczny, On the Wong-Zakai approximation of stochastic differential equations, J. Multivariate Anal. 13 (1983), 605-611.
• [35] P. Kotelenez, A submartingale type inequality with applications to stochastic evolution equations, Stochastics 8 (1982), 139-151.
• [36] H. H. Kuo, Gaussian Measures in Banach Spaces, Springer, Berlin 1975.
• [37] H. Kushner, Jump-diffusion approximations for ordinary differential equations with wide-band random right hand sides, SIAM J. Control Optim. 17 (1979), 729-744.
• [38] H. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York 1977.
• [39] H. J. Kushner and G. Yin, Stochastic approximation algorithms for parallel and distributed processing, Stochastics 22 (1987), 219-250.
• [40] R. Liptser and A. Shiryayev, Statistics of Random Processes, Springer, Berlin 1977.
• [41] W. Mackevičius, SP-stability of symmetric stochastic differential equations, Litovsk. Mat. Sb. 25 (4) (1985), 72-84.
• [42] X. Mao, Approximate solutions for a class of delay stochastic differential equations, Stochastics Stochastics Rep. 35 (1991), 111-123.
• [43] S. Marcus, Modeling and approximation of stochastic differential equations driven by semimartingales, Stochastics 4 (1981), 223-245.
• [44] E. J. McShane, Stochastic differential equations and models of random processes, Proc. 6th Berkeley Sympos. Math. Statist. Probab., Vol. 3, University of California Press, Berkeley 1972, 263-294.
• [45] J. Memin and L. Słonimski, Condition UT et stabilité en loi des solutions d'équations différentielles stochastiques, to appear in Séminaire de Probabilités 1991.
• [46] M. Métivier, Semimartingales, Walter de Gruyter, Berlin 1982.
• [47] M. Métivier and J. Pellaumail, Stochastic Integration, Academic Press, London 1980.
• [48] G. N. Milshteĭn, Approximate integration of stochastic differential equations, Theor. Veroyatnost. i Primenen. 19 (1974), 583-588 (in Russian).
• [49] G. N. Milshteĭn, Weak approximation of solutions of systems of stochastic differential equations, ibid. 30 (1985), 706-721 (in Russian).
• [50] S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Marshfield 1984.
• [51] S. E. A. Mohammed, Retarded Functional Differential Equations, Pitman, London 1978.
• [52] S. Nakao and Y. Yamato, Approximation theorem on stochastic differential equations, in: Proc. Internat. Sympos. SDE Kyoto 1976, Tokyo 1978, 283-296.
• [53] N. J. Newton, An asymptotically efficient difference formula for solving stochastic differential equations, Stochastics 19 (1986), 175-206.
• [54] N. J. Newton, An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times, ibid. 29 (1990), 227-258.
• [55] E. Pardoux and D. Talay, Discretization and simulation of stochastic differential equations, Acta Appl. Math. 3 (1985), 23-47.
• [56] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin 1983.
• [57] J. Picard, Approximation of stochastic differential equations and application of the stochastic calculus of variations to the rate of convergence, in: Lecture Notes in Math. 1316, Springer, 1988, 267-287.
• [58] E. Platen, An approximation method for a class of Itô process, Lietuvos Matematikos Rinkinys 21 (1) (1981), 121-133.
• [59] E. Platen, Approximation of the first exit times of diffusions and approximate solutions of parabolic equations, Math. Nachr. 111 (1983), 127-146.
• [60] E. Platen, A Taylor formula for semimartingales solving a stochastic equation, in: Lecture Notes in Control and Inform. Sci. 36, Springer, 1981, 157-174.
• [61] P. Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab. 13 (3) (1985), 716-743.
• [62] A. Shimizu, Approximate solutions of stochastic differential equations, Bull. Nagoya Inst. Tech. 36 (1984), 105-108.
• [63] A. V. Skorokhod, K-martingales and Stochastic Equations, in: Proc. School-Seminar of the Theory of Random Processes (Druskininkai 1974), Part II, Inst. Fiz. Mat. AN Litovsk. SSR, Vilnius 1975, 195-234 (in Russian).
• [64] K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer, Dordrecht 1991.
• [65] J. L. Solé and F. Utzet, Stratonovich integrals and trace, Stochastics 29 (1980), 203-220.
• [66] R. L. Stratonovich, A new representation for stochastic integrals and equations, SIAM J. Control Optim. 4 (2) (1966), 362-371.
• [67] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in: Proc. 6th Berkeley Sympos. Math. Statist. Probab., Vol. 3, University of California Press, Berkeley 1972, 333-359.
• [68] H. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab. 6 (1) (1978), 19-41.
• [69] D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic differential systems, submitted to INRIA, Report 965 (1989).
• [70] D. Talay, Efficient numerical schemes for the approximation of expectations of functionals of S.D.E., in: Filtering and Control of Random Process, H. Korezlioglu, G. Mazziotto and J. Szpirglas (eds.), Proceedings of the ENST-CNET Colloquium, Paris 1983, Lecture Notes in Control and Inform. Sci. 61, Springer, Berlin 1984, 294-313.
• [71] D. Talay, Résolution trajectorielle et analyse numérique des équations différentielles stochastiques, Stochastics 9 (1983), 275-306.
• [72] D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law, Stochastics Stochastics Rep. 29 (1990), 13-36.
• [73] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887-919.
• [74] C. Tudor, On stochastic evolution equations driven by continuous semimartingales, Stochastics 24 (1988), 179-195.
• [75] C. Tudor, On the successive approximations of solutions of delay stochastic evolution equations, An. Univ. Bucureşti Mat. 34 (1985), 70-86.
• [76] C. Tudor, On weak solutions of Volterra equations, Boll. Un. Mat. Ital. 7 1-B (1987), 1033-1054.
• [77] C. Tudor, Some properties of mild solutions of delay stochastic evolution equations, Stochastics 17 (1986), 1-18.
• [78] K. Twardowska, An extension of Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces, Stochastic Anal. Appl. 10 (4) (1992), 471-500.
• [79] K. Twardowska, On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probab. Math. Statist. 12 (2) (1991), 319-334.
• [80] K. Twardowska, On the relation between the Itô and Stratonovich integrals in Hilbert spaces, submitted to Statist. Probab. Lett. (1991).
• [81] R. B. Vinter, A Representation of Solutions to Stochastic Delay Equations, Imperial College Report, 1975.
• [82] R. B. Vinter, On the evolution of the state of linear differential delay equations in M²: properties of the generator, J. Inst. Math. Appl. 21 (1) (1978), 13-23.
• [83] N. N. Vakhaniya, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions in Banach Space, Nauka, Moscow 1985 (in Russian).
• [84] E. Wagner, Unbiased Monte-Carlo estimators for functionals of weak solutions of stochastic differential equations, Stochastics Stochastics Rep. 28 (1989), 1-20.
• [85] E. Wong, Stochastic Processes in Information and Dynamical Systems, McGraw-Hill, 1971.
• [86] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965), 1560-1564.
• [87] E. Wong and M. Zakai, Riemann-Stieltjes approximations of stochastic integrals, Z. Warsch. Verw. Gebiete 12 (1969), 87-97.
• [88] J. Zabczyk, On decomposition of generators, SIAM J. Control Optim. 16 (4) (1978), 523-534.
Języki publikacji
 EN
Uwagi
1991 Mathematics Subject Classification: 34G20, 34K50, 35R15, 60H05, 60H10, 60H15, 60H30.


text:
## Elementary Geometry for College Students (7th Edition)

$\frac {25 gallons}{1 mile}$
$\frac {200 gallons}{8 miles}$ reduces by 8 in both the numerator and the denominator to $\frac {25 gallons}{1 mile}$.