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0 Section of Integrative Biology, BIO LABS 311, The University of Texas, Austin, Texas 78712 USA
Received for publication June 8, 1999. Accepted for publication November 2, 1999.
ABSTRACT
Surface-to-volume (S/V) ratios of drought-adapted plants affect transpiration, photosynthesis, and water-storage capacity. The S/V ratio of cladodes and flat leaves is S/V = 2/T, where T is thickness: even slight thickening greatly reduces S/V. During rain/drought cycles succulent stems swell and shrink without tearing by having flexible ribs, but ribs increase S/V above that of a smooth cylindrical stem with equal volume: the increased surface area is Sribbed/Scylindrical = N
/
), where N is number of ribs and x is rib height relative to the radius of the inner stem. Numerous low ribs provide moderate expandability (storage volume) with little increase in S/V and are adaptive where droughts are short. Tall ribs provide greater expandability but greatly increase S/V and probably are adaptive only in mesic habitats. Having ~8–15 ribs, each about as tall as the inner stem radius, provides large storage capacity and intermediate increase in S/V. By increasing absolute size, S/V is reduced so greatly that even large ribs can have an S/V smaller than that of a narrow cylindrical or spherical stem with less volume.
Key Words: adaptation • Cactaceae • cactus • cladode • desert • evolution • succulent • surface-to-volume ratio • xeric
Many plant species have adapted to xeric conditions by becoming succulent, and during their evolution, several problems had to be solved. First, the transpirational surface area could be reduced either temporarily by leaf abscission or permanently by evolutionary reduction of leaves. Second, sufficient water storage capacity had to be available to allow persistent organs such as buds, roots, and the stem axis to survive droughts. Third, seasonal rain/drought cycles caused the plants' volume to increase and decrease cyclically. These three factors affect a plant's surface-to-volume (S/V) ratio.
Simply reducing a plant's S/V ratio to a minimum may not be an optimal survival strategy. Because the transpirational surface is typically also a photosynthetic surface, reducing surface area reduces photosynthesis (Fig. 1). If the habitat has high atmospheric humidity and only short periods without rain, then maintaining extra surface area (a non-minimum S/V ratio) may be advantageous (Figs. 2–6). Conversely, if a plant will experience droughts that last a year or longer, large amounts of succulent tissue and a low S/V ratio may be necessary (Figs. 7, 8). Several species of cactus survive up to three years without water (Szarek and Ting, 1975
; Smith and Madhaven, 1982
).
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; Barcikowski and Nobel, 1984
; Mauseth, 1995
). The epidermis and hypodermis must accommodate this, but whereas young, growing dermal tissues are extremely extensible, mature ones are not: the total surface area of a region of mature stem tends to be constant. Many succulent stems have contiguous ribs that can widen or shrink at the base whenever the stem swells or contracts (Fig. 9; Porembski, Martens-Aly, and Barthlott, 1991
; Felger and Henrickson, 1997
). When dry, the stem has lost volume and the ribs are narrow; when hydrated, the stem is swollen and its ribs are broad. Thus, volume cycles, while surface area remains constant. Ribbed stems occur in Asclepiadaceae, Cactaceae, Euphorbiaceae, and Vitaceae as well as other families.
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MATERIALS AND METHODS
The values listed in Table 1 and the plants illustrated in Figs. 1–8 are part of a large data set of almost 200 species that were studied in the field, in botanical gardens or obtained from nurseries (Abbey Garden Nursery, P. O. Box 2249, La Habra, California 90632-2249 USA, phone: 562-905-3520; Mesa Garden Nursery, P. O. Box 72, Belen, New Mexico 87002 USA, phone: 505-864-3131; Miles to Go Nursery, P. O. Box 6, Cortaro, Arizona 85652 USA, phone: 520-682-7272; also see www.cactus-mall.com). All measurements were taken on mature, living plants before fixation and dehydration.
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Leaves
For ordinary flat, non-succulent leaves, the surface area S of the upper surface is approximately equal to that of the lower surface, and the total leaf surface = 2S. Leaf volume V can be computed by multiplying leaf thickness T by either the upper or lower surface: Vleaf = ST. Therefore:
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r (r is the radius), whereas the cross-sectional area Ac-s = r2. For a given length L of stem, the surface area S = PL and volume V = Ac-sL, so the ratio of P/Ac-s is identical to the ratio of S/V: S/V = PL/Ac-sL = P/Ac-s. The ratio of perimeter to cross-sectional area (and thus S/V) is
 |
). This equation is basically identical to S/Vleaf = 2/T: a cylindrical leafless, ribless stem has the same S/V as a flat thin leaf if r = T. Of course, most leaves are thinner (~0.5 mm, S/V = 4.0 mm2/mm3) than most stems (radius of at least 2 mm, S/V of at most 1.0 mm2/mm3), and consequently leaves have higher surface-to-volume ratios. Most plants could reduce their S/V ratio by merely abscising their leaves or losing them evolutionarily. But if a plant has succulent, thick leaves on slender stems (e.g., T = 1.0 mm and r = 1.0 mm, as occurs in many Mesembryanthemaceae and Sansevieria species; Koller and Rost, 1986
; Sajeva and Costanzo, 1994
), then S/Vleaf = S/Vstem and the plant could reduce its total surface area but not its S/V ratio by becoming leafless. Furthermore, such a plant could reduce its S/V ratio by one-half by either doubling leaf thickness or stem radius—doubling leaf thickness would double leaf volume and have only a small effect on surface, whereas doubling stem radius would double the stem surface and quadruple its volume. Decreasing the S/V ratio by increasing stem radius actually increases total surface area, unless the stem simultaneously becomes shorter.
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Stems with ribs
In a transverse section of a stem with ribs, each rib appears as a triangle with its base Brib located on the surface of a cylindrical stem (which will be called the inner stem) and its height H projecting radially away from the inner stem (Fig. 12). The triangular transverse section can be considered as two right triangles back to back with their hypotenuses constituting the rib perimeter. Ribs are usually symmetrical, so the dimensions of one half of the rib are identical to those of the other. As discussed above, rib perimeter in cross section multiplied by length results in surface area of the rib, and cross-sectional area multiplied by length results in rib volume, therefore P/Ac-s = S/V for ribbed stems just as for ribless ones. Typically each rib contacts two others along its base (Figs. 2–7), and the surface of the stem consists of just rib surface. In some species with only two or three thin ribs, there is a strip of ordinary stem cortex and epidermis separating each rib from its neighboring ribs.
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r/N is a bit longer than Brib, which is the cord of the arc. With this value for the base, the cross-sectional area of ribs can be calculated as 
In a stem with ribs, total stem cross-sectional area (CSA) and thus volume occur in both inner stem and in ribs:
The photosynthetic cortex of succulent stems occurs as a layer of cells just below the rib epidermis or hypodermis, so water stored within ribs themselves is closer to the cells that must be kept hydrated. From a water distribution standpoint, it may be more advantageous to have a large fraction of the water storage capacity located within the ribs rather than the inner stem. This fraction can be calculated as
Rib perimeter and surface are calculated as the hypotenuses of two right triangles constituting the rib (Fig. 12), with the height H of each triangle equaling Hrib and base of each triangle Btriangle equaling half of Brib = (1/2)(2r/N) = r/N. Each hypotenuse is 
, and this must be doubled to get the two hypotenuses of one rib:
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rH/N (Eq. 4), and the P/Ac-s ratio of one rib is obtained by dividing this into the perimeter (Eq. 9): 
rH; Eq. 5):
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Equations 9–12 demonstrate several important points. Rib number affects P/Ac-s and S/V ratios: if two stems are identical in r and H but one has more ribs (and thus Brib = 2r/N of each rib is less), total CSA of all the ribs is the same in the two stems (Eq. 5), but total perimeter is greater in the stem with more ribs (Eq. 10), so its S/V ratio is greater. Consequently, when considering plants with similar inner stem radius and similar rib height, plants with more ribs have more transpirational and photosynthetic surface area relative to volume and might be expected in more mesic habitats.
Rib shape affects P/Ac-s and S/V ratios. Being composed of two right triangles, the hypotenuse is at a minimum relative to CSA when Htriangle = Btriangle, that is, when each half of the rib is a right isosceles triangle and the whole rib itself has Hrib = 1/2Brib (Fig. 13). For any rib that is taller relative to its base (H >> B) or shorter (H << B), the P/Ac-s and S/V ratios are higher: plants with tall thin ribs or low flat ribs have higher S/V ratios than those in which rib height is half that of width, if their cross-sectional areas are equal. Ribs provide flexibility to the stem surface but automatically increase the S/V ratio above that of a ribless cylindrical stem (P/Ac-s = 2/r; Eq. 2), but ribs with H 
1/2Brib cause minimum increase in stem S/V ratio. It might be expected that species adapted to arid conditions would have ribs with H 1/2Brib, whereas those in more mesic environments could have tall, thin ribs or short, broad ones, both of which would increase the S/V ratio, increasing photosynthetic surface relative to water storage volume.
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1/2Brib if their absolute size is sufficiently larger. Similarly, if two stems have the same number and height of ribs but one stem is broader than the other (and thus has broader ribs), the broader stem has a lower S/V ratio due to absolute size. If a stem must have ribs, minimum S/V ratio is achieved by having the largest possible ribs with H 1/2Brib.
Cyclical changes due to water absorption and loss
For any particular mature stem, N is constant but r, H, Brib and the S/V ratio vary as the stem absorbs water after a rain and loses it during drought (Fig. 9). Rib shape also changes: although it is theoretically possible for H and Brib to expand or shrink symmetrically such that rib shape remains constant, ribs on real plants are significantly narrower when desiccated than when hydrated, but not significantly shorter. As tissues of the inner stem dehydrate and shrink, stem radius r decreases as do inner stem circumference (2r) and thus rib bases (2r/N). Rib height could decrease as well, but that would require that the perimeter (the two hypotenuses) also decrease, but the epidermis and hypodermis tend to have thick walls in ribbed succulents, and probably can neither stretch nor shrink significantly (Mauseth, 1996
; Mauseth, Terrazas, and Loza-Cornejo, 1998
).
As a stem absorbs water and swells, ribs expand and their bases become broader. As the inner stem expands, its original radius r1 increases to r2 and its original circumference C1 increases to C2; the base of each rib 2r1/N increases to 2r2/N. The total perimeter of all N ribs is constant, as given by Eq. 10, which shows that the perimeter is the same whether the stem is dehydrated and the ribs are thin and sitting on a shrunken inner stem or whether the stem is fully hydrated and the ribs are expanded and broad. Under ideal conditions, ribs would be capable of expanding so much that the stem would become circular in transverse section. But for this to happen, the ribs would have to be so flexible that they could swell until they become flat and the base of each rib is as long as the two hypotenuses (Fig. 14); rib height would have to either shrink or be very much smaller than the radius of the inner cortex (x of H = xr would have to be small even when the stem is dehydrated). If the stem could swell to a new circular perimeter, then its new radius r2 would be the radius of a circle whose circumference equals that of the perimeter of all the original ribs. It is possible to express the new radius r2 as a factor of r1 (for example, r2 = yr1), and the circumference of yr1 equals the perimeter of all N ribs (given by Eq. 10):
r21(1 + x), which is obtained by letting H = xr1 in Eq. 6: r1(r1 + H)] would be 
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r(r + H), and if H is expressed as xr, we get 
r2(1 + x); Eq. 18] as the ribbed stem: 
When comparing two stems with the same CSA, it may seem logical to compare the S/V ratios of the two, but because the two have the same CSA and volume, the denominators in any comparison cancel, and by simply considering only the difference in surface we automatically consider the difference in S/V ratio.
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Low ribs provide flexibility without greatly increasing the amount of surface or the S/V ratio. All values in the x = 0.1 column of Table 2 are <2.0, indicating that even if a stem had 50 very low ribs, its surface and S/V ratio would not even be double that of a ribless cylindrical stem with the same CSA and volume. Thus, having these numerous low ribs would not greatly increase the risk of excessive transpiration in xeric habitats. Conversely, low ribs, even when there are many of them, do not permit much expansion of the stem, so if an abundance of water would be available after a heavy rain, the stem would not have much expandability to store it. Both tendencies are more extreme if there are just a few low ribs: the S/V ratio is almost the same as that of a cylindrical stem but expandability is almost zero. Plants with this type of ribbing might be expected to occur in habitats that are extremely dry but which receive small amounts of rain periodically throughout the year: low S/V ratio and considerable water storage capacity are advantageous, but the plant does not need to absorb a year's worth of water in a brief rainy season (Figs. 2, 5, 6).
Ribs of moderate height (x = 0.6) have little effect on the surface area and the S/V ratio unless they are quite numerous (N > 25 or so; Table 2). A stem with ten ribs of height 60% of the radius of the inner stem (x = 0.6) has a S/V ratio that is only 1.7 times larger than a cylindrical stem with the same CSA and volume, and the increase does not double until a plant has 12 or 13 ribs like this. These plants have low S/V ratios, moderate expandability, and some extra photosynthetic surface area. This might be adaptive in habitats that are not so xeric that the S/V ratio is an overriding factor and in which rainfall is strongly episodic yet reasonably reliable—rains may occur twice a year, and completely dry years are rare. In such a habitat, plants would need to store sufficient water for four or five very dry months but would rarely draw down their stored water during a full year or two without rain. Felger and Lowe (1967)
found that in the columnar cactus Lophocereus schottii, populations in drier habitats had fewer ribs and thus lower S/V ratios than populations in more mesic regions.
If ribs of moderate height (x = 0.6) are numerous (N > 25), then the S/V ratio becomes high, as much as 4–7.5 times higher than that of a cylindrical stem with the same volume. Such high S/V ratios might be advantageous in habitats such as dry forests of Mexico and Brazil, habitats with enough moisture to support the growth of tall trees but which have a very dry, hot season in which leaves are abscised. During the rainy season, the surrounding trees are leafy and shade the succulent plants but during the dry, leafless season, the succulents receive full sunlight and are healthy and metabolically active, relying on their stored water. In such habitats the increased surface area of the ribs may be advantageous in facilitating greater photosynthesis in the brief sunny period, and the increased transpiration is tolerable because the dry season is not prolonged. Succulents in these regions may actually not conserve water very well, absorbing and losing large amounts each year, and thus needing the increased expandability that the numerous moderate-sized ribs provide. Coryphantha vivipara undergoes cycles in which it survives losing as much as 91% of its water (Nobel, 1981
).
Tall ribs greatly increase the S/V ratio, even if there are few ribs. Ribs with x = 1.0 double the S/V ratio of a stem even if there are as few as eight or nine ribs, and as few as five ribs with x = 2.0 doubles the S/V ratio over that of a cylindrical stem with the same CSA. Fifty tall ribs would increase the S/V ratio by as much as 11–18 times. Plants with a few tall ribs would be expected to be restricted to extremely mesic habitats that experience either brief or no drought. Very tall ribs (x > 2.0) essentially would be acting like thick succulent leaves (ribs are never as thin as ordinary leaves: minimum Brib = 1.0 mm in Table 1), with the significant difference that ribs cannot be abscised during drought, whereas leaves can be. Certainly the three or four very tall ribs of Deamia (Fig. 4) could never expand fully to make the plant cylindrical, and such a huge water storage capacity would be completely unnecessary in their mesic habitat in Central America (Backeberg, 1977
). Instead, the large surface area provided by these ribs undoubtedly increases the photosynthetic surface area. Plants with many tall ribs (N > 25, x = 2.0) might not be expected to occur at all: with their extremely high S/V ratio they would be restricted to very humid habitats, and whereas a few tall ribs might be effective at photosynthesis, large numbers of tall ribs—crowded on a relatively narrow stem—would shade each other (Nobel, 1980
). It is difficult to conceive of a habitat in which these features would be selectively advantageous. Very tall ribs do occur (Table 1), with a maximum of x = 10.0 in Rhipsalis (N = 2, H = 10 mm) and H = 55 mm in Dendrocereus (N = 3 or 4; x = 7.86). As might be expected, these are both plants of very humid habitats where "droughts" would last at most a week or two—Rhipsalis is a rainforest epiphyte and Dendrocereus occurs in rainy areas of Hispaniola and Cuba. However, it is not yet known if these two species are exceptional, and more actual species must be examined.
The effect of absolute size may be the most significant factor in maximizing water storage and minimizing the S/V ratio. Increasing the radius r of the inner stem dramatically decreases the S/V ratio (Fig. 15), and similarly, increasing the absolute size of a rib while holding its shape constant causes its S/V ratio to decrease (Fig. 7). In both cases, as S/V is decreasing, the actual surface area is increasing, so photosynthetic capacity is increasing. Consequently, maximum water storage capacity and minimum S/V ratio can be achieved primarily by having stems become very large. Some cacti (Soehrensia) have enormous stems (CSA = 116 802 cm2; Table 1). If a large number of ribs of moderate height is added to a large body, expandability is achieved without increasing the S/V ratio too severely, again because of the large absolute size of the ribs. In contrast, stems with the same relative dimensions (the same N and x) would have a much higher S/V ratio if they are narrow in absolute size (r is small). The larger stems could survive in very xeric habitats, whereas the narrow stems—with the same shape—would be restricted to mesic ones (Figs. 2, 6).
The presence of ribs also affects the stem's interaction with wind and its strength. Air moves smoothly across a large flat surface, creating a thick boundary layer, but ribs create turbulent air flow and a thinner boundary layer (Nobel, 1988
). It may be that all ribbed stems have so much more turbulence than smooth cylindrical stems that number and size of ribs are inconsequential. Turbulent air flow carries away excess heat more rapidly than laminar air flow, but it also removes transpired water vapor more rapidly. Presence of ribs increases the strength and rigidity of stems as well, in some cases providing as much support as the wood (Cornejo and Simpson, 1997
; Niklas, Molina-Freaner, and Tinoco-Ojanguren, 1999
). The mechanical aspects of ribbed stems should be an important phenomenon for further studies, especially in species with very tall ribs.
Unlike ribbed stems, flattened cladodes can swell and shrink without tearing or wrinkling. By being broad and thick, their S/V ratio = 2/T and as Fig. 15 shows, even moderately thick cladodes have low S/V ratios, although not nearly as low as cylindrical stems with the same CSA. Benson (1982)
mentions several platyopuntias (O. chlorotica, O. ficus-indica, O. lindheimeri, O. littoralis) that have cladodes 20–25 mm thick and that would have an S/V ratio of 0.08–0.1 mm2/mm3. In cladodes of platyopuntias (Figs. 8, 17), not only is the stem flattened, but so is the ring of vascular bundles and pith. Water is stored in both cortex and pith, and volume change requires flexion of the wood; this is possible while the cladodes are young, but as they age and develop more wood, they must become rigid. Ribbed cylindrical stems provide flexibility despite the woodiness of the stem.
During the course of their evolutionary history, succulent plants of various families have become adapted to an extremely wide range of habitats that differ in their rainfall patterns. It is to be expected that a variety of rib shapes and numbers and of plant sizes will have evolved, with certain combinations of characters being adaptive in certain environments and other combinations adaptive elsewhere. A sample of almost 200 species of cacti is being studied to determine whether actual species correspond to the principles outlined here.
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1 This research was funded in part by grants from the Mellon Foundation through the Institute of Latin American Studies at the University of Texas and from the Research Committee of the Cactus and Succulent Society of America. 
2 E-mail: j.mauseth{at}mail.utexas.edu
LITERATURE CITED
Backeberg, C. 1977 Cactus lexicon. Blandford Press, Dorset, UK.
Barcikowski, W., and P. S. Nobel. 1984 Water relations of cacti during desiccation: distribution of water in tissues. Botanical Gazette 145: 110–115.[CrossRef]
Benson, L. 1982 The cacti of the United States and Canada. Stanford University Press, Stanford, California, USA.
Cornejo, D. O., and B. B. Simpson. 1997 Analysis of form and function in North American columnar cacti (Tribe Pachycereeae). American Journal of Botany 84: 1482–1501.[Abstract]
Felger, R., and J. Henrickson. 1997 Convergent adaptive morphology of a Sonoran desert cactus (Peniocereus striatus) and an African spurge (Euphorbia cryptospinosa). Haseltonia 5: 77–85.
———, and C. H. Lowe. 1967 Clinal variation in the surface-volume relationships of the columnar cactus Lophocereus schottii in northwestern Mexico. Ecology 48: 530–536.[CrossRef][ISI]
Koller, A. L., and T. L. Rost. 1986 The microscopic anatomy of Sansevieria leaves. Cactus and Succulent Journal (U. S.) 58: 30–33.
Mauseth, J. D. 1995 Collapsible water-storage cells in cacti. Bulletin of the Torrey Botanical Club 122: 145–151.[CrossRef][ISI]
———. 1996 Comparative anatomy of Tribes Cereeae and Browningieae (Cactaceae). Bradleya 14: 66–81.
———, T. Terrazas, and S. Loza-Cornejo. 1998 Anatomy of relictual members of Subfamily Cactoideae, IOS Group 1a (Cactaceae). Bradleya 16: 31–43.
Niklas, K. J., F. Molina-Freaner, and C. Tinoco-Ojanguren. 1999 Biomechanics of the columnar cactus Pachycereus pringlei. American Journal of Botany 86: 767–775.
Nobel, P. S. 1980 Interception of photosynthetically active radiation by cacti of different morphology. Oecologia 45: 160–166.[CrossRef][ISI]
———. 1981 Influence of freezing temperatures on a cactus, Coryphantha vivipara. Oecologia 48: 194–198.
———. 1988 Environmental biology of agaves and cacti. Cambridge University Press, Cambridge, UK.
Porembski, S., B. Martens-Aly, and W. Barthlott. 1991 Surface/volume-ratios of plants with special consideration of succulents. Beiträge zur Biologie der Pflanzen 66: 189–209.
Sajeva, M., and M. Costanzo. 1994 Succulents, the illustrated dictionary. Cassell plc, Villiers House, London, UK.
Smith, B. N., and S. Madhaven. 1982 Carbon isotope ratios in obligate and facultative CAM plants. In I. P. Ting and M. Gibbs [eds.], Crassulacean acid metabolism, 231–243. American Society of Plant Physiologists, Rockville, Maryland, USA.
Szarek, S. R., and I. P. Ting. 1975 Photosynthetic efficiency of CAM pants in relation to C3 and C4 plants. In R. Marcelle [ed.], Environmental and biological control of photosynthesis, 289–297. Dr. W. Junk, The Hague, The Netherlands.
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[x2 + (
/{
. The row for 14 ribs and the column for x = 0.6 are boldface because these are the mean values for rib number and rib height (expressed as x) in Table 1