Articular surfaces of joints can provide important clues about an animal's locomotor behavior, because the size and shape of an articular surface should be correlated with habitual postures and movements (Jungers, 1991). More directly, the size and shape of joint surfaces should reflect constraints and requirements for strength, mobility, and stability of the joint—all requisite for joint function and normal locomotion (Hamrick, 1996). Mobility is the potential range of motion of the joint, and stability is the capacity of the joint to prevent unwanted motion, including motion outside normal kinematics and joint dislocations. The strength, the magnitude, and the frequency of loading a joint can withstand without suffering damage to the avascular articular cartilage or immediate subchondral bone (the shape of which we examine herein), may be a special important parameter in patterning joint morphology, because irreparable damage can impede joint function and hinder proper movement. More precisely, the strength of an articular surface is the amount of stress (force divided by area) it can withstand before yielding (irreversible deformation). Articular surfaces must be strong enough to accommodate specific levels of stress, which are a function of body size, form of locomotion, and the size of the articular surfaces available to transmit forces. Thus a primary determinant of joint strength is the contact area between opposing surfaces, and ultimately the maximum force that can be accommodated safely is determined by the smaller of the two joint surfaces.
As with many skeletal parameters relevant to locomotion, the pervasive influence of body size must be considered when examining articular surfaces. Extraordinarily, well documented in organismal biology are the proportional decreases in linear dimensions and areas with increasing body size (volume) (Alexander et al., 1979; Calder, 1984; Schmidt-Nielsen, 1984; Biewener, 1989). If scaled isometrically, joint surface areas scale is proportional to two-third power of body mass, and thus are proportionally smaller in larger animals (Biewener, 1982). Despite the proportional decrease in articular surface area, joint surface linear dimensions have been found to scale very close to isometry in hominoids (Jungers, 1991), whereas overall skeletal isometry has been noted for mammals in general (Alexander et al., 1979). One potential way to accommodate increasing body size (and thus decrease the magnitude of localized peak stresses) is through a decrease in joint radius of curvature (i.e., flattening of the articular surface; Latimer et al., 1987; Latimer and Lovejoy, 1989). This solution, however, is predicated on the equivalency of joint postures and limb excursion angles within or among the taxa being compared (Biewener, 1983; McMahon, 1984).
Flattening of articular surfaces with increased size results in greater joint contact area and more uniform pressure distributions across the entire joint surface (i.e., more of the articular surface lies normal to joint forces), which effectively reduces stress concentrations in localized areas of the articular cartilage (Latimer et al., 1987). Regulating the stress across articular surfaces is especially important, as articular cartilage is avascular and cannot actively repair itself after damage (Mow et al., 1989). Thus, an allometric shape change of this sort would afford larger-bodied animals the ability to attenuate proportionally higher transarticular loads and regulate contact pressure within and upon the articular cartilage-effect, and this morphology would reduce the risk of irreparable cartilage damage in larger-bodied animals. This hypothesis has been evaluated both experimentally and theoretically, and has received mixed results, further underscoring the complexity of joint surface (and articular cartilage) ontogeny (Frost, 1979, 1994, 1999; Carter, 1987; Carter and Wong, 1988, 1990; Hamrick, 1999; Plochocki et al., 2006, 2009; Organ and Ward, 2006; Hammond et al., 2010).
A complementary hypothesis to the joint flattening hypothesis is Hamrick's (1999) theory of chondral modeling (based on earlier work by Frost [ 1979, 1994, 1999]), which predicts that high levels of hydrostatic pressure within a frequently-loaded region of articular cartilage inhibits chondrocyte mitosis in that region, preventing articular cartilage growth in the heavily loaded region. Adjacent regions of articular cartilage experiencing lower (moderate) levels of hydrostatic pressure, however, will not be inhibited, and will therefore respond to the loading through cartilage deposition (chondral modeling). This model explicitly predicts that the central-most regions of an articular surface should bear levels of hydrostatic pressure that are too high to stimulate chondrocytic cell division and should therefore grow at a much slower rate than adjacent regions experiencing lower peak pressures. Thus, according to this model, adult morphologies should reflect the ontogenetic magnitude and direction of joint loading. Because articular cartilage develops from the ossifying subchondral growth front of the long bone, changes in the contours of the articular cartilage (induced by differential levels of hydrostatic pressure within the articular cartilage) should be reflected in the contours of the underlying subchondral bone, and thus detectable from our analysis of articular joint surfaces. (Frost, 1999). Differential growth rates in different regions of articular cartilage should account for observable differences in subchondral joint contours in extant hominoids, where central regions of a joint surface should be flatter than peripheral regions, with even more flattening (i.e., even higher hydrostatic pressure in the central regions of the joint surface) in larger-bodied animals. Cartilage thickness itself varies only slightly across joint surfaces (Eckstein et al., 1995; Cohen et al., 1999). Among species with different patterns and histories of joint loadings, such as those examined here (see later), there should also be predictable differences in articular topographies in cases with relatively little soft tissue variation. Flattening of the regions of the joint surface is hypothesized to result, however, only if the distribution of loading is not uniformly distributed across the joint surface. If loadings are experienced uniformly across the joint surface, then all regions will be equally inhibited (or promoted), and flattening will not occur.
In large part, the question of articular surface size isometry has been resolved, but only across taxa; it has not been studied extensively within taxa. Biewener (1989) demonstrated that larger animals use more extended limb postures and lower angular excursions (Biewener, 1983) to proportionally decrease the muscular forces required to support and propel body mass and, hence, the forces exerted on the skeletal system. Thus, larger animals preferentially load articular surfaces at one particular point in the joint's full range of motion, leading to flattened articular surfaces in that region. Changes in posture associated with increased body masses have not been demonstrated within a species. Hence, it remains possible that within a species joints get flatter with increasing body size. Flattening joint surfaces could also be important in the context of hominid evolution. If hominids used modern extended limb postures soon after the evolution of bipedalism, then they could not have taken advantage of changes in limb posture to attenuate changes in musculoskeletal forces that would have accompanied increasing body size. Hence, even though joint flattening does not appear to be an issue across quadrupedal species, it remains possible that it does occur within quadrupedal species, and/or that it also occurs within hominids.
A recent study by Organ and Ward (2006) explored the relationship between curvature and size in the lateral condyle of tibiae of large-bodied hominoids finding modest positive allometry. The impetus for that study centered on alleged variation in anteroposterior joint curvature of the lateral tibial condyle of Australopithecus, which had been argued to reflect differences in locomotor repertoire. Although the study by Organ and Ward (2006) did not find support for the chondral modeling hypothesis as an explanation for these alleged differences in joint shapes, it did show no systemic differences in joint shape among Australopithecus taxa: large-bodied australopiths were no more human- or ape-like in their lateral tibial condyle morphologies than were small-bodied australopiths, including the “chimpanzee-like tibia from Sterkfontein,” Australopithecus africanus (Berger and Tobias, 1996).
The purpose of this study is to test the proposition that joint surfaces become flatter with increasing body size in three large-bodied African hominoids: Gorilla, Pan, and Homo. Gorilla and Pan typically load their knees across varying degrees of flexion and extension during quadrupedal, knuckle-walking locomotion, whereas Homo typically loads its knees in a fully extended posture during bipedal locomotion. Differential function of the knee joint across these taxa makes this group amenable to testing hypotheses of joint flattening in response to increased body size only within each taxon.
MATERIALS AND METHODS
The sample consists of 120 African great ape and modern human specimens, divided equally among gorillas (Gorilla gorilla, n = 40), common chimpanzees (Pan troglodytes, n = 40), and modern humans (Homo sapiens, n = 40) curated as part of the Hamman-Todd skeletal collection housed at the laboratory of physical anthropology, Cleveland museum of natural history. Each sample is comprised of an equal number of males and females, and sexes were combined for analysis because joint sizes do not appear to be sexually dimorphic (Jungers, 1991). All individuals are free from pathology and skeletally adult. Nonhuman specimens are all wild-shot. In the case of the modern human sample, the sample is distributed equally between American blacks and whites as defined in the Hamman-Todd osteological collection in Cleveland.
Differential Function of the Knee Joint in African Apes and Humans
The morphological differences between bipedal hominoids (Homo) and the great apes merit a brief description to better understand the measurements that are assessed herein. These morphologic differences develop ontogenetically as a result of habitual knee postures during weight-bearing activities (as described previously). In essence, human knee joints have undergone dramatic reorganization to accommodate high peak transarticular loads incurred at heel strike in an extended limb.
The first significant difference is the presence of a bicondylar angle in the human distal femur, which places the femur in a valgus position relative to the tibia. The human bicondylar angle normally falls between 8-degree angle and 12-degree angle (Lovejoy, 1984). The valgus position allows humans to place their center of mass more directly over their base support (the stance leg and foot), preventing the necessity of shifting their trunk over the hip during bipedal gait. On the contrary, when chimpanzees and gorillas engage in brief episodes of (facultative) bipedal locomotion, they need to shift their body weight over their hip to place their center of mass over their base support.
Related to the presence of a valgus knee in humans is a morphologic change in the patellar groove on the distal femur, which acts as a guide for the patella during flexion and extension. The bicondylar angle of the femur creates a tendency for the patella to laterally subluxate from its contact area on the femur (Dougherty et al., 1976). Specifically, during extension, the tendency is for the patella to deviate laterally because the net contraction of the quadriceps femoris mm. is nearly parallel to the axis of the femur (Lovejoy, 1984). To counteract this effect, a lateral lip on the patellar surface of the femur is required to retain femoropatellar contact.
Finally, grooves on the femoral condyles of humans, caused by femoral articulation with the menisci, show significantly greater tibial contact in humans than is seen in gorillas and chimpanzees using more flexed-knee locomotion. The meniscal grooves define the anterior-most extent on each femoral condyle where contact with the tibia occurs and separate the femoral condyles into contact areas reserved for tibial contact and patellar contact. The meniscal grooves are asymmetrically located on the human femur, where the medial condyle has a much more distinct and more anterolaterally curved meniscal groove. This suggests that significant lateral rotation of the tibiofemoral joint (i.e., medial rotation of the femur upon a fixed tibia) occurs during full or nearly full extension in humans (Lovejoy, 1984).
All three of these human morphologies are consistent with locomotor behavior that is highly dependent on quadriceps femoris mm. contraction. They also describe how human knees have been structured to best attenuate high compressive loads on the tibia (tibial dominance), unlike African great apes who have knees structured to best comply with high compressive force at the patellofemoral joint (patellar dominance). When the quadriceps femoris mm. contracts a large component of the joint reaction force must be absorbed at the patellofemoral joint, which requires a considerable amount of patellofemoral congruity (Lovejoy, 1984). However, as the knee approaches full extension, the compression of the patellofemoral joint is reduced and compression increases at the tibiofemoral joint (Lovejoy, 1984). Especially, this is an important fact because animals that locomote with a more bent-knee gait, like gorillas and chimpanzees, require less tibiofemoral congruity and more patellofemoral congruity. The femoral and tibial condyles of African apes and humans are considerably different, which reflects this mechanical adaptation.
The femoral condyles in humans are elliptical in their anteroposterior profiles, maximizing the area of contact with the tibia when the knee is extended and minimizing the load on the knee (Heiple and Lovejoy, 1971). The femoral condyles are not identical to each other, as the lateral condyle is much flatter in profile than the medial (Aiello and Dean, 1990). However, both of the femoral condyles in humans are symmetrical about a parasagittal plane, reflecting that they are equally loaded with similar weight transfer during bipedal locomotion, and comparable tibial contact between the two condyles. The articular surfaces of the femoral condyles in humans, despite the comparable size of the condyles themselves, are very distinct. The lateral articular surface of the femur is significantly smaller than the medial articular surface.
Tibial condyles in humans also exhibit a high degree of symmetry in both size and shape (Spencer, 1989), which reflects comparable weight transfer during bipedal locomotion (Aiello and Dean, 1990). As the joint extends, the lateral femoral condyle reaches the edge of its articular surface before the medial condyle does, resulting in conjunct rotation of the knee because of its valgus position. At this point, the anterior cruciate ligament reaches its maximum tension and acts as a pivot for the tibia to rotate laterally on the femur (Kapandji, 1987).
In gorillas and chimpanzees, knee joint function is significantly different because the femoral condyles are more rounded in profile and do not show symmetry about a parasagittal plane (Aiello and Dean, 1990). They are also more comparable in size to one another, disabling them from contributing to a locking knee and allowing a significantly lesser degree of medial rotation of the tibia.
Polyvinylsiloxane molds (President Jet Regular, Coltene-Whaledent) of the articular surface of the medial condyle of each specimen were prepared. Casts were created using dental plaster and scanned using a NextEngine laser scanner. The resulting three-dimensional virtual surfaces were saved in virtual reality modeling language format (*.vrml) and exported to the software package Amira 4.1.1. In Amira, surfaces were converted to triangulated-mesh models of between ∼25,000 and 60,000 vertices (depending on the size of the condyle). All joint surfaces were anatomically aligned and nonarticular surface was removed using the Surface Editor module. Files were saved in open inventor format (*.iv) and all subsequent measurements were automated using custom programs written for Matlab.
Measurements included three-dimensional surface area and two-dimensional projected area. Projected two-dimensional area was calculated by projecting the articular surface onto a transverse plane after anatomical orientation. Anteroposterior arcs and chords were taken at 50% of total width because the central portion of the joint surface is thought to be most affected by high hydrostatic pressure (Hamrick, 1999; Organ and Ward, 2006). This was accomplished by extracting vertices that describe the contour of an anteroposterior transect across the joint surface along an anteroposterior line located at 50% of total joint width. The extracted points are the intersection between a joint surface (more precisely the lines connecting model vertices) and a vertical plane. The same technique was used to extract mediolateral arcs and chords at 50% of total anteroposterior length. The mediolateral measurements are taken from 25% to 75% of total width to exclude the intercondylar eminence and then to make the curve symmetric with respect to the joint surface.
Curvature was evaluated using three metrics. First, the ratio of the three-dimensional surface area and the two-dimensional projected area. Second, the radius of curvature was calculated for each contour by fitting a circle with the lowest geometric error to the points that describe a particular transect contour across the joint surface. Only the middle 50% of the articular surface was used to calculate the radius of curvature. The circle fit to the points minimizes the squared distances between the points of the extracted contours and the perimeter of the circle. Frequently, curvature is expressed as 1/radius of curvature to reflect the fact that larger radii are less curved. This convention was not used because isometry would be represented by a slope of –1, and positive allometry by more negative values, thus contrary to normal convention. Instead all analyses were performed directly on the radius of curvature.
In addition, the ratio of arc and chord lengths was calculated following the rationale in Organ and Ward (2006) that as the surface departs from flat (increased curvature) the ratio will depart from one. The 50% arcs and chords were measured from anterior peak (most superiorly projecting point along anterior half of curve) to posterior peak (homologous point on posterior half of curve). If the arc and chord for an individual were modeled as a circle, the ratio would be the equivalent of the included angle. Measurements are illustrated in Fig. 1. It is important to note that neither metric will distinguish between a convex and concave surface. All surfaces were inspected prior to analysis, and none were convex.
Exploring relationships with body size without a direct measure of body size requires a surrogate that is highly correlated with body size. Several researchers have suggested that the most biomechanically relevant estimator of body size for locomotor studies is body mass (Jungers, 1985). Unfortunately recorded body masses were not available for the skeletal sample used in this study. In place of body mass superoinferior femoral head diameter (caliper measurement) was used, because it has been shown to be highly correlated with body mass in modern humans (Ruff et al., 1991, 1997; McHenry, 1992; Grine et al., 1995). Modern humans are known to have relatively large femoral heads compared to other hominoids because of their habitual bipedal gait and thus this body size surrogate cannot be used as a body size estimate for interspecific comparisons. Our objective was to explore scaling relationships within a species, and thus we rely on previous research positively correlating femoral head diameter and body mass in each of the taxa included (Jungers, 1991; Ruff, 2003; Holliday and Franciscus, 2009, for justifications within each taxon examined here).
Radius of curvature was regressed against femoral head diameter using reduced major axis regression after taking the natural log of all data. Isometry is indicated by a slope of one. Positive allometry (proportionally larger radius of curvature, and thus less curved) will be indicated by a slope greater than one, and negative allometry (proportionally smaller radius of curvature, and thus more curved) will be indicated by slopes less than one. Ratio metrics were also regressed against femoral head diameter. Because these are shape variables, isometry is indicated by a slope of zero, and positive and negative allometry by positive and negative slopes. Nonsignificant correlations would indicate no linear correlation between size and shape. As RMA regression slope is calculated as the ratio of the standard deviations of dependent and independent variables, it is possible to get a nonzero slope value, but still have a zero correlation. The 95% confidence interval for each slope estimate was calculated to determine if it included isometry and the test for isometry (slope = 1), advocated by Zar (1999), was performed.
Plots and regression estimates for the analyses of radius of curvature are provided in Table 1 and Figs. 2, 3. Correlation coefficients for all analyses are very low, ranging from −0.053 to 0.300. Regression lines were not included on the bivariate scatter plots because all the correlations are low and do not indicate any significant linear relationships between femoral head diameter and radius of curvature.
Table 1. Radius of curvature regressed against femoral head diameter
Plots and regression estimates for the analyses of the arc-to-chord ratios and ratios of three-dimensional to two-dimensional surfaces are provided in Table 2 and Figs. 4–6. Again, correlation coefficients for all analyses are very low, ranging from −0.162 to 0.352. As above, regression lines were not included in the bivariate scatter plots, because all the correlations are so low and do not indicate any significant linear relationships between femoral head diameter and the curvature ratios.
Table 2. Ratios regressed against femoral head diameter
Pan AP50 Ratio
Pan ML50 Ratio
Gorilla AP50 Ratio
Gorilla ML50 Ratio
Human AP50 Ratio
Human ML50 Ratio
The overwhelming result of these analyses is that within a species, medial tibial condyle curvature appears to be largely independent of body size as estimated by femoral head diameter. In the curvature metrics examined, the correlations between the curvature and femoral head size were simply, too low to propose any meaningful biological relationship between the two variables. As a result, the chondral modeling hypothesis as it relates to articular surface morphology is not supported by these findings (Hammond et al., 2010, for evidence of chondral modeling in the epiphyseal growth plate of the developing long bone). The mechanism of joint flattening in larger animals as proposed by Latimer and colleagues (Latimer et al., 1987; Latimer and Lovejoy, 1989) also remains unsupported by these findings.
Latimer and Lovejoy (1989, p 379) specifically suggest that flatter surfaces would decrease shear forces that could significantly damage articular cartilage. Examination of how shear forces would transfer across joints provides a potential explanation for why we were unable to find flatter surfaces in larger individuals. Shear forces are forces that exist parallel to the surface of a material. The scale of interest is an important point when considering shear within an articulation. One level of scale might be considered the “whole joint,” in this case the tibia and femur. Shear forces would be forces that are perpendicular to the axial loading between the two bones, and would act to destabilize the joint, leading to subluxation or dislocation. Because of the extremely low friction coefficient between articular surfaces and synovial fluid, shear forces of this type are resisted by joint capsules, ligaments, muscles, and joint topography and not by the articular cartilage itself (Charnley, 1960). Highly curved (not flatter) joint surfaces are more suited to resisting forces from multiple directions (Hamrick, 1996).
Another level of scale about which one might be concerned is shear within an infinitesimally small piece of articular cartilage. In this case, forces transferred across the joint may be decomposed into forces that act normal and parallel to the joint surface at that specific point. On highly curved surfaces (e.g., hemispherical), axial compressive forces (forces normal to the surface) would be higher near the center of the joint because the joint surface at this point is normal to the applied load, and the compressive forces would approach zero near the edge of the joint (Brinckmann et al., 2002). Shear forces (forces parallel to the surface) would follow an opposite pattern, lower in the center and higher towards the edges. For shear forces to transfer across the joint surface, however, the frictional force between the two joint surfaces must be sufficiently high. Frictional forces are the product of frictional coefficients and the force normal to the joint surface at that point. They will be uniformly low in articular joints because the coefficient of friction is low, but they must also follow the pattern of the compressive forces, which in a highly curved joint is higher in the center of the joint and approaches zero toward the edges. Thus, in areas of the joint where shear forces would be higher, the frictional forces would be too low (approaching zero) for them to propagate across the joint.
Another possible explanation for the lack of correlation between the curvature of the medial tibial condyle and the femoral head diameter is the presence, and biomechanical effect, of the medial meniscus. The medial meniscus is a semicircular band of fibrocartilage that contributes to the concavity of the tibial portion of the knee joint. The anterior portion of the medial meniscus attaches to the anterior tibial intercondylar eminence whereas the posterior portion attaches to the posterior portion of the tibial intercondylar eminence. The menisci (of which there is a medial and a lateral one) are known to affect the mechanics of the knee. Thambyah et al., (2006) recently demonstrated that articular cartilage under the meniscus has different mechanical properties and thickness than articular cartilage not covered by meniscus. They also found that the subchondral bone under the meniscus is thinner and of lower apparent density. Loading in the knee may be mediated to such an extent by the medial meniscus that the articular surface does not respond in ways predicted by the chondral modeling hypothesis.
It also remains possible that the medial tibial condyle is responding to changes in body size in ways predicted by the chondral modeling hypothesis and by the allometric relationship proposed by Latimer and Lovejoy (1989), although we are unable to detect it using the methods employed in this study. Flattening in joints might increase the area of the articulation that is perpendicular to loading, and as a result increase the effective area (thus lowering pressure and strain). Operating under this assumption, the chondral modeling hypothesis would predict a flatter femoral condyle, but a more curved tibial condyle that conforms more tightly to the distal femur. These changes in joint curvature are hypothesized to occur because sufficiently high levels of hydrostatic pressure in regions of the articular cartilage (caused by frequent loading) will inhibit the activity of chondrocytes, thus reducing cartilage growth in the regions. In adjacent regions that experience lower levels of hydrostatic pressure, chondrocyte activity is enhanced and stimulates chondral modeling (Hamrick, 1999). Thus femora loaded within a specific range of motion will transmit forces across a particular portion of the articular cartilage more frequently, inhibiting growth in this region and stimulating in adjacent regions, resulting in joint surface flattening. In the tibia, the preferential loading of one area will lead to inhibited growth in this region, whereas stimulating growth in neighboring regions, thus leading to a more curved surface (Hamrick, 1999). These two effects will lead to greater joint congruity thereby distributing the pressure more evenly across the joint surface and reducing the maximum pressure. Thus, as body size increases within a species, the tibial condyle may become more curved, simply to maintain sufficient joint congruity through the entire range of flexion and extension with a proportionally flatter femoral condyle. These two effects may cancel each other out, and lead to the isometry reported here. This effect cannot be established without examination of the associated distal femora.
On the basis of the chondral modeling hypothesis as articulated by Hamrick (1999), joint flattening would only occur if growth-inhibiting loading is not uniformly distributed across the range of motion. If the appropriate magnitude of loading is uniformly distributed across the joint surface then growth will be equally inhibited and promoted across the surface. During the stance phase of chimpanzee walking, the knee moves through a large range of motion, ranging from 65-degree angle to 140-degree angle of knee flexion (D'Aout et al., 2004). Climbing activities in chimpanzees also require loading in many different degrees of knee flexion. If the magnitude of compressive force that inhibits growth is equally distributed across the joint surface, it will not result in joint flattening because all portions are equally promoted and/or inhibited. The range of normal angles during the stance phase of human walking is much smaller, 10° to 45° (Kozanek et al., 2009). Thus forces transmitted across the joint represent a much smaller portion of the total range of motion. In the human femur, this would lead to flattening of the femoral condyles because the central portion of the articulation is loaded with greater frequency. Loading in the tibial condyle is primarily on the center (deepest portion) of the surface (Lovejoy, 2007), and should result in a more curved surface.
Larger-bodied animals should experience greater joint reaction forces, and thus one advantage of flatter joints (i.e., decreased curvature) would be an increase in contact area between opposing joint surfaces. Biewener (1989) demonstrates, however, that larger animals use more extended limb postures than smaller bodied animals, and as a result, stresses on the musculoskeletal system become proportionally smaller in larger animals (across taxa). This allows musculoskeletal elements to scale very near isometry. This should also reduce joint reaction forces, compressive, shear or otherwise during normal kinematics, obviating joint flattening. This mechanism, however, may not have been possible within the course of hominid evolution. If early hominids already used modern extended limb postures as has been suggested (Latimer, 1991), then the mechanism demonstrated by Biewener (1989) may not have been an available evolutionary solution to increased body size. So, when joint surfaces can scale isometrically in quadrupeds that can change limb postures with increasing size, joint surfaces may have scaled with positive allometry in the course of hominid evolution.
The authors wish to thank Bruce Latimer and Lyman Jellema and the rest of the staff of the Cleveland Museum of Natural History for providing access to the human and ape collections. The authors also thank Patricia Kramer and Heather Garvin for valuable conversations during the course of the project. This project would have not been possible without the support of Carol Ward. The authors are grateful to Valerie DeLeon, Timothy Smith, Qian Wang, and two anonymous reviewers, whose comments greatly improved this manuscript.