Thermodynamic Effects on the Evolution of Performance Curves

vol. 176, no. 2
the american naturalist
august 2010
E-Note
Thermodynamic Effects on the Evolution of
Performance Curves
Dee A. Asbury and Michael J. Angilletta Jr.*
Department of Biology, Indiana State University, Terre Haute, Indiana 47809
Submitted November 30, 2009; Accepted February 21, 2010; Electronically published June 8, 2010
abstract: Models of thermal adaptation assume that warmadapted and cold-adapted organisms can achieve the same fitness,
yet recent comparative studies suggest that warm-adapted organisms
outperform cold-adapted ones. We explored how this thermodynamic effect on performance might influence selective pressures on
thermal physiology. In the absence of a thermodynamic effect, natural
selection favors a thermal optimum for performance that closely
matches the mean (or modal) body temperature. When warmadapted organisms outperform cold-adapted organisms, natural selection can favor a thermal optimum that exceeds the mean body
temperature. The optimal mismatch between the thermal optimum
and the mean temperature increases as does the variation in body
temperature within generations. This result holds regardless of
whether performance affects fitness through fecundity or survivorship. The selective pressures generated by a thermodynamic effect
might explain the substantial mismatch between thermoregulatory
behavior and thermal physiology that has been observed in some
species.
Keywords: heterogeneity, performance, temperature, thermodynamics.
Introduction
Since the seminal work of Levins (1968), biologists have
used the concept of a performance curve to model the
effects of environmental heterogeneity on fitness. Most
models focus on two aspects of these curves: the environment that permits maximal performance (environmental
optimum of performance) and the range of conditions
over which performance can occur (environmental
breadth of performance). Both spatial and temporal variations in environmental conditions influence the evolution of a performance curve (Lynch and Gabriel 1987;
Gabriel 1988; Gilchrist 1995, 2000). Predictable environmental cues select for genotypes that can developmentally
alter their performance curves in the face of environmental
heterogeneity (Gabriel and Lynch 1992; Gabriel 1999,
* Corresponding author. Present address: School of Life Sciences, Arizona
State University, Tempe, Arizona 85287; e-mail: michael.angilletta@asu.edu.
Am. Nat. 2010. Vol. 176, pp. E40–E49. 䉷 2010 by The University of Chicago.
0003-0147/2010/17602-51761$15.00. All rights reserved.
DOI: 10.1086/653659
2005). Prompted by the ecological impacts of human population growth, some biologists have extended the basic
models to evaluate the evolution of performance curves
in environments that change directionally over time (Huey
and Kingsolver 1993; Lynch and Lande 1993; Burger and
Lynch 1995). Most recently, other biologists have investigated the coevolution of performance curves among interacting species (de Mazancourt et al. 2008; Johansson
2008). These models make a common prediction about
the outcome of natural selection: the environmental mode
of performance should closely match the most frequent
state of the environment. This prediction stems from the
assumption that adaptation enables organisms to perform
equally well at any environmental optimum.
In contrast to current models, adaptation does not always confer the same capacity for performance in all selective environments. In particular, thermodynamics has
been hypothesized to constrain adaptation to thermal variation (Angilletta et al. 2010b). Because temperature affects
the mean kinetic energy of molecules, biochemical reactions proceed more quickly at higher temperatures (Mortimer 2000; Hochachka and Somero 2002). On the basis
of this thermodynamic effect, some researchers have argued that adaptation to warm environments should confer
greater performance than adaptation to cold environments, leading to the notion that “hotter is better” (Bennett 1987; Huey and Kingsolver 1989; Kingsolver and
Huey 2008). In support of this hypothesis, maximal rates
of population growth for warm-adapted genotypes generally exceed those for cold-adapted genotypes (Eppley
1972; Frazier et al. 2006; Knies et al. 2009). Both comparative and experimental studies of other performances
also suggest that hotter is usually better (reviewed in Angilletta 2009; Angilletta et al. 2010b). Although the magnitude of this thermodynamic effect varies considerably
among traits and taxa, the metabolic theory of ecology
postulates a precise relationship between body temperature
and maximal performance (Savage et al. 2004).
To explore the influence of thermodynamics on adaptation, we introduced a thermodynamic effect into a model
of optimal performance curves. We modeled this ther-
The Evolution of Performance Curves E41
modynamic effect according to the Boltzmann factor,
which describes how reaction kinetics vary with temperature (Gillooly et al. 2001). As with previous models, we
imposed a trade-off between specialization and generalization (Angilletta et al. 2003). Because many performance
curves appear asymmetric (Huey and Stevenson 1979; Angilletta et al. 2002; Angilletta 2009), we permitted the shape
of the performance curve to evolve along with the thermal
optimum and thermal breadth. Our model suggests that
thermodynamic constraints can favor the evolution of a
thermal optimum that exceeds the mean body temperature. This mechanism could partially explain the frequent
mismatch between thermoregulatory behavior and thermal physiology in ectotherms (see Martin and Huey 2008).
The Model
We developed a model to examine the optimal performance curve in an environment that varies seasonally and
stochastically. The thermal sensitivity of performance (Z)
was described by a nonlinear function:
Z(Tb ) p
c#
[(Tb ⫺ a)/b](g/b)⫺1[1 ⫺ (Tb ⫺ a)/b][(1⫺g)/b]⫺1G(1/b)
G(g/b)G[(1 ⫺ g)/b]
(1)
# e⫺E/kTb,
where Tb equals the body temperature in Kelvin, k equals
the Boltzmann constant, and E equals the energy of activation (0.6 eV; Gillooly et al. 2001). The variables a, b,
and g determine the mode, breadth, and skewness of the
performance curve, respectively (fig. 1A–1C). The parameter b sets the maximal breadth, and the parameter c scales
the value of performance. The second term, based on the
metabolic theory of ecology (Brown et al. 2004), represents
a hypothetical thermodynamic effect on mean performance (Savage et al. 2004; Frazier et al. 2006). In the
absence of this thermodynamic effect, the integral of the
performance curve would equal bc despite changes in a,
b, and g; this constraint would force a trade-off between
specialization and generalization (Angilletta et al. 2003).
However, the thermodynamic effect enables greater mean
performance when the maximal performance occurs at
higher temperatures. Because empirical performance
curves usually appear asymmetrical, our function provides
a more realistic representation of these curves than does
the Gaussian function used by other researchers (see Lynch
and Gabriel 1987).
We modeled both seasonal and stochastic variations in
body temperature. Seasonal variation was modeled according to a sinusoidal function, such that daily mean
temperatures (T) were calculated as
[ ( ) ]
T p d sin d #
p 1
⫺ ⫹ 293.15,
s
2
(2)
where s equals the length of the active season in days, d
equals the day of the season, and d equals the seasonal
range of daily mean temperatures (fig. 1D–1F). Stochastic
variation within days was modeled using a Gaussian function (F):
Fp
{
[
(Tb ⫺ T)
1
exp ⫺
2j2
j冑2p
]
2
0
if F(Tb ) 1 1 # 10⫺3,
(3)
otherwise,
where j determines the magnitude of daily variation (fig.
1G–1I). This discontinuous function maintained the
Gaussian distribution used in previous work while constraining the range of body temperatures; this constraint
became critical when we modeled the effect of performance on survivorship (see below). Following Gilchrist
(1995), we assumed an organism with a short life span (5
days) relative to the length of the active season (180 days).
Because the mean temperature varies among days, both
seasonal and stochastic changes contribute to thermal variation within generations, whereas only seasonal change
contributes to thermal variation among generations. We
considered nine patterns of thermal heterogeneity, based
on combinations of j and d (j p {0.5, 3.5, 6.5} and d p
{0, 10, 20}). These assumptions about generation time and
body temperature captured a range of thermal variations
within and among generations (fig. 1). In principle, the
model applies to ectotherms or endotherms, as long as
one chooses appropriate values of j and d (see Angilletta
et al. 2010a).
We modeled the fitness of phenotypes (w) as the product
of survivorship and fecundity:
写写
L
wpR#
Tmax
jp1 TbpTmin
S(T(d(g, j)), Tb ),
(4)
where R equals the fecundity at maturation, S equals survivorship, g equals the generation, j equals the age of the
organism in the current generation, L equals the life span,
and Tmin and Tmax equal the minimal and maximal body
temperatures experienced during the current day. This
function provides a valid estimate of fitness for a semelparous organism in a stable population (Kozłowski et al.
2004). Using our fitness function, we modeled fitness landscapes for two scenarios. In the first scenario, we assumed
that performance contributed directly to fecundity without
affecting survivorship (S p 1 for all values of Tb), such
that fitness simply equaled the fecundity at maturation.
E42 The American Naturalist
Figure 1: A, Parameter a determines the position of the performance curve along the thermal axis. All else being equal, increasing a by 1 will
increase the thermal optimum by 1⬚C. Because of the thermodynamic effect (eq. [1]), increasing the value of a also increases the maximal performance.
B, Parameter b determines the breadth of the performance curve. An increase in b causes a proportional increase in the performance breadth and
a proportional decrease in the maximal performance. Because of the thermodynamic effect, increasing the value of b also increases the thermal
optimum. C, Parameter g determines the skew of the performance curve. Values of g ! 0.5 cause a positive skew, and values 10.5 cause a negative
skew. Because of the thermodynamic effect, increasing the value of g also increases the maximal performance. D–F, Parameter d determines the
variation in body temperature experienced among days. These plots show seasonal variations in body temperature for the values of d used to generate
fitness landscapes. G–I, Parameter j determines the variation in body temperature experienced within days. These plots show distributions of body
temperatures for the values of j used to generate fitness landscapes.
As per Gilchrist (1995), we assumed that fecundity scaled
proportionally to the sum of performance throughout life:
冘冘
L
wpRpm#
because probabilities of survival generally remain high and
constant over a range of temperatures and drop sharply
outside this range (van der Have 2002; Angilletta 2009).
Thus, fitness for this scenario was calculated as
Tmax
F(T(d(g, j)), Tb ) # Z(Tb ),
(5)
jp1 TbpTmin
where F equals the frequency of a given Tb and m coverts
units of performance into the number of offspring. In
biological terms, one can envision performance as a measure of energy assimilation, egg maturation, or some other
process that limits fecundity.
In the second scenario, we assumed that fecundity and
survivorship depend on body temperature. We modeled
the thermal sensitivity of survivorship as a step function,
[
Tmax
jp1 TbpTmin
[写 写
L
#
冘冘
L
wp m#
Tmax
jp1 TbpTmin
where
]
]
F(T(d(g, j)), Tb ) # Z(Tb )
F(T(d(g, j)), Tb ) # S(Z(Tb )) ,
(6)
The Evolution of Performance Curves E43
Sp
Z(T ) 1 # 10
{10 ifotherwise.
1
b
⫺12
,
(7)
Equation (7) ensures that the evolution of a narrow
breadth of performance also reduces the range of temperatures tolerated by the organism. In biological terms,
this relationship could stem from a positive genetic correlation between the thermal sensitivities of fecundity and
survivorship.
For either scenario, the long-term mean fitness of a
phenotype was calculated as
1/N
[写 ]
N
w̄ p
w(g)
,
(8)
gp1
where N equals the number of generations in the active
season. Using equation (8), we constructed a fitness landscape for the performance curve in each hypothetical environment. To ease interpretation, we plotted optimal performance curves on a scale of degrees Celsius instead of
Kelvin.
Because distributions of body temperatures for real organisms usually differ from Gaussian distributions, we also
constructed fitness landscapes for several bimodal distributions. To create these bimodal distributions, we followed
Gilchrist (1995), who summed the frequencies generated
by two Gaussian functions with different modes:
Fp
[
(
)] [
(
)]
1
(T ⫺ T )2
1
(T ⫺ T )2
exp ⫺ b 2 L ⫹
exp ⫺ b 2 U
2j
2j
j冑2p
j冑2p
(9)
if F(Tb ) 1 1 # 10⫺3, and F p 0 otherwise, where TL and
TU define the lower and upper modes of the distribution,
respectively, for a given magnitude of thermal variation
(j). We chose values of TL and TU such that the modes
of the distribution were 20⬚ and 30⬚C for each value of j.
Results and Discussion
A thermodynamic effect qualitatively alters the optimal
performance curve in a heterogeneous environment. In
the absence of a thermodynamic effect, natural selection
favors a thermal optimum that equals or slightly exceeds
the mean body temperature (Gilchrist 1995). When body
temperature varies little within generations, the performance breadth should scale proportionally to the variation
in body temperature among generations. Given a thermodynamic effect, variation in body temperature within
generations favors a greater mismatch between the mean
temperature and the thermal optimum (fig. 2). This phenotype imposes some loss of performance at the mean
body temperature but confers a substantial increase in
performance at higher body temperatures. Importantly, an
organism realizes this benefit only when its body temperature varies throughout its life. Thus, the optimal mismatch between the thermal optimum and the mean temperature increases with increasing thermal heterogeneity.
A thermodynamic effect has no impact on the optimal
performance curve when body temperature remains
constant.
A thermodynamic effect imposes a similar selective pressure when performance affects survivorship as well as fecundity (fig. 3). When performance determines fecundity
only, fitness depends on the sum of performance over time.
In this scenario, a genotype can fail to perform for extended periods, as long as favorable temperatures occur
at some point in each generation. When performance determines survivorship (as in eq. [7]), an individual must
perform at all times to achieve any fitness. Therefore,
greater variation in body temperature within generations
favors a wider performance breadth (fig. 3), as described
by Lynch and Gabriel (1987). Close inspection of the fitness landscapes in figures 2 and 3 also reveals that stronger
stabilizing selection of the thermal optimum occurs when
performance determines survivorship (e.g., the steepness
of the fitness landscape along the Y-axis of fig. 3C exceeds
that of fig. 2C). Nevertheless, variation within generations
still favors a thermal optimum that exceeds the mean body
temperature (fig. 4A, 4C). Thus, the qualitative impact of
a thermodynamic effect does not depend on the mechanism by which performance affects fitness.
Changing the distribution of body temperatures from
unimodal to bimodal did not alter the basic predictions
of our model (fig. 5). In the absence of a thermodynamic
effect, natural selection would drive the thermal optimum
of performance toward either the lower or the upper mode
of body temperature (Gilchrist 1995). But when hotter is
better, selection favors a thermal optimum that equals or
exceeds the upper mode (fig. 4B, 4D). When body temperature varies little within generations, the thermal optimum should lie close to the upper mode. Increasing the
thermal heterogeneity within generations leads to selection
for a thermal optimum that exceeds the upper mode. This
result does not depend on whether performance contributes to fitness via fecundity or survivorship; however, the
optimal performance breadth depends strongly on the relationship between performance and fitness (cf. fig. 4B,
4D).
Several factors could quantitatively alter our predictions
about the optimal mismatch between the mean temperature and the thermal optimum. First, the actual ther-
E44 The American Naturalist
Figure 2: Fitness landscapes for nine patterns of thermal heterogeneity when the performance of an organism determines its fecundity. To reduce
the landscape to three dimensions, we plotted results only for g p 0.5 . Values of parameters were as follows: b p 80 , c p 1 , L p 5 , N p 36. Contour
lines depict the relative fitness, ranging from 0 to 1 (interval p 0.1). When body temperature varies within generations, a thermodynamic effect
on performance favors a thermal optimum that exceeds the mean body temperature (see fig. 4A). As in Gilchrist’s model, thermal specialists are
favored whenever the thermal variation within generations equals or exceeds the thermal variation among generations.
modynamic effect on performance likely differs from the
one modeled here. We used the Boltzmann factor to model
a thermodynamic effect, but estimated thermodynamic effects vary considerably among traits and taxa (Angilletta
et al. 2010b); for example, the thermodynamic effect on
the population growth rate of insects exceeded the effect
described by the Boltzmann factor (Frazier et al. 2006). A
change in either the form or the magnitude of the thermodynamic effect would alter the fitness landscapes described by our model. Second, reversible acclimation of
the performance curve would effectively reduce the impact
of thermal heterogeneity on performance (see Gabriel
2005). Although we did not model this phenomenon, our
intuition suggests that the mismatch between the mean
The Evolution of Performance Curves E45
Figure 3: Fitness landscapes for nine patterns of thermal heterogeneity when the performance of an organism determines its fecundity and survivorship.
To reduce the landscape to three dimensions, we plotted results only for g p 0.5 . Values of parameters were as follows: b p 80 , c p 1, L p 5,
N p 36. Contour lines depict the relative fitness, ranging from 0 to 1 (interval p 0.1). In this case, generalists are favored whenever temperature
varies greatly within generations. Nevertheless, variation within generations still favors a thermal optimum that exceeds the mean body temperature
(see fig. 4C).
temperature and the thermal optimum should depend on
the time lag for acclimation. For an organism with no lag,
the thermal optimum should instantaneously track the
body temperature. As the time lag increases, the optimal
mismatch should approach that of an organism without
the ability to acclimate. Finally, a closer match between
the mean temperature and the thermal optimum could
result from complex relationships between performance
and fitness. In our model, we considered a relatively simple
fitness function, in which the timing of performance had
no consequence for fitness. However, real organisms probably need to perform at certain levels during specific periods. Such a requirement would translate to a more complex fitness function than those considered to date. By
E46 The American Naturalist
Figure 4: When a thermodynamic constraint exists, natural selection can favor a thermal optimum that substantially exceeds the mean temperature
of the organism. This conclusion holds when performance contributes to either fecundity alone (A, B) or both fecundity and survivorship (C, D).
The conclusion also holds for both Gaussian (left) and bimodal (right) distributions of body temperatures. Each plot shows optimal performance
curves for three levels of thermal heterogeneity within generations (j p {0.5, 3.5, 6.5}). In all cases, we assumed that distributions of body temperatures
remained constant among generations (i.e., d p 0). Dotted vertical lines denote mean or modal body temperatures.
refining our assumptions about the magnitude of the thermodynamic effect, the capacity for thermal acclimation,
and the relationship between performance and fitness, we
can better resolve the generality of the selective pressures
described by our model.
By varying the shape of the performance curve, we compared the fitnesses of symmetric and asymmetric curves.
When performance affects fecundity only, a strongly
skewed curve confers the greatest fitness in all environments (see fig. 4A, 4B). Yet thermal heterogeneity selects
for weaker skews (and wider breadths) when performance
affects survivorship (see fig. 4C, 4D). These results have
important implications for the interpretation of previous
theoretical and empirical work. Most evolutionary models
assume a symmetric curve, because the functions that describe such curves possess convenient mathematical properties (Lynch and Gabriel 1987; Gabriel 1988, 2005; Gabriel
and Lynch 1992; Kopp and Gavrilets 2006). For example,
the Gaussian function has two variables that independently
determine the mode and the breadth of the curve. By
contrast, we used a function with multiple parameters that
interact to determine the mode and breadth of the curve
(eq. [1]). Although both approaches yield qualitatively
similar predictions about certain aspects of thermal physiology, our approach led to novel predictions about the
selective pressures on the shape of a performance curve.
These predictions accord well with empirical observations.
Asymmetric functions nicely describe the thermal sensitivities of assimilation, growth, development, and locomotion in many ectotherms (Huey and Stevenson 1979;
Angilletta et al. 2002; Angilletta 2009). Asymmetric functions even describe the thermal sensitivities of common
estimates of fitness, such as r and R0 (Huey and Berrigan
2001). Nevertheless, symmetric functions better describe
thermal sensitivities of survivorship (e.g., see Angilletta et
al. 2008). In light of our model, the symmetry of performance curves for survivorship and the asymmetry of other
performance curves might reflect evolutionary adaptations
rather than physical constraints.
Our model potentially explains some known discrepancies between the thermoregulatory behaviors and the
thermal physiologies of ectotherms. Martin and Huey
The Evolution of Performance Curves E47
Figure 5: A–C, Bimodal distributions of body temperatures approach the shape of a Gaussian distribution as the magnitude of thermal variation increases
within generations. Values of j were 0.5, 3.5, and 6.5 for low, medium, and high variation, respectively. For all three distributions, the intermodal distance
equals 10⬚C. D–F, Fitness landscapes corresponding to the distributions of body temperatures depicted in A–C, respectively, when the performance of
an organism determines its fecundity. To reduce the landscape to three dimensions, we plotted results only for g p 0.5 . Values of parameters were as
follows: b p 80, c p 1, L p 5, N p 36, d p 0. Contour lines depict the relative fitness, ranging from 0 to 1 (interval p 0.1 ). When body temperature
varies little within generations, natural selection favors a thermal optimum that matches the higher modal body temperature (see fig. 4B). When body
temperature varies greatly within generations, a thermodynamic effect on performance favors a thermal optimum that exceeds the higher modal body
temperature (see fig. 4B). G–I, Fitness landscapes corresponding to the distributions of body temperatures depicted in A–C, respectively, when the
performance of an organism determines its fecundity and survivorship. Values of parameters were the same as those used for D–F. When body temperature
varies little within generations, natural selection favors a thermal optimum near the higher modal body temperature (see fig. 4D). As the variation in
body temperature increases within generations, natural selection favors an increase in the thermal optimum and the performance breadth (see fig. 4D);
these changes in the optimal performance curve result from an increase in the optimal value of b as variation in body temperature increases within
generations (fig. 1B illustrates this effect of b on the performance curve).
(2008) showed that the optimal temperature for sprinting
usually exceeds the preferred body temperature. They attributed this mismatch to a selective pressure imposed by
imperfect thermoregulation. Their argument goes as fol-
lows. Most performance curves possess an asymmetric
shape, such that relatively high temperatures impair performance more than relatively low temperatures do. If an
animal regulates its temperature imprecisely, periods of
E48 The American Naturalist
high temperature impose a severe cost. This cost would
be minimized if genotypes expressed a thermal optimum
that exceeded the mean body temperature. This particular
thermal physiology provides a safety margin in the event
of overheating; in fact, unusually high temperatures would
enhance rather than impair performance when the thermal
optimum exceeds the mean body temperature. According
to their model, a greater degree of asymmetry in the performance curve favors a greater shift in the thermal optimum. In support of their model, Martin and Huey
showed that the mismatch between thermoregulatory behavior and thermal physiology was significantly correlated
with the degree of asymmetry among species of reptiles.
Nevertheless, the preferred temperature and the thermal
optimum differed by as much as 8⬚C, while their model
predicts a discrepancy of only 1⬚–2⬚C. Large mismatches
between thermoregulatory behavior and thermal physiology might be explained partly by the selective pressures
imposed by thermodynamics.
Interestingly, Martin and Huey (2008) concluded that
a thermodynamic effect has no influence on the optimal
mismatch between thermoregulatory behavior and thermal physiology. This counterintuitive result occurred because these researchers optimized the distribution of body
temperatures for a fixed performance curve, whereas we
optimized the performance curve for a fixed distribution
of body temperatures. In Martin and Huey’s model, selection favors a relatively low mean body temperature,
which increases performance at high body temperatures
while decreasing performance at the mean body temperature. In our model, selection favors an increase in the
thermal optimum to enhance performance at high body
temperatures without necessarily impairing performance
at the mean body temperature. In fact, the two models
describe independent selective pressures associated with
variation in body temperature: selection imposed by the
negative skew of a performance curve and selection imposed by a thermodynamic effect on the mean
performance.
The selective pressures described by our model might
account for a very important macrophysiological pattern.
By analyzing global patterns of thermal physiology for
several groups of animals, Deutsch et al. (2008) discovered
an interesting contrast between temperate and tropical ectotherms. Temperate ectotherms perform best at body
temperatures that greatly exceed the mean air temperatures
of their environments. By contrast, tropical ectotherms
perform best at body temperatures that lie close to their
mean environmental temperatures. Deutsch et al. (2008)
defined two metrics to quantify the relative impacts of
warming on these species. The first metric, termed the
safety margin, equals the difference between the thermal
optimum for performance (or fitness) and the mean en-
vironmental temperature. The second metric, termed the
warming tolerance, equals the difference between the critical thermal maximum and the mean environmental temperature. Both the safety margin and the warming tolerance tend to increase with increasing latitude (Deutsch et
al. 2008). Therefore, climate change could pose a greater
threat to tropical ectotherms, which possess narrow safety
margins and warming tolerances (Tewksbury et al. 2008).
Importantly, this macrophysiological pattern accords with
a prediction of our model: the safety margin and warming
tolerance should correspond to the variation in body temperature within generations (see fig. 4). Given that temperate ectotherms likely experience more variation in body
temperature than do tropical ectotherms (Ghalambor et
al. 2006), we should expect temperate species to enjoy
wider safety margins and warming tolerances. This connection between our evolutionary model and a macrophysiological pattern underscores the value of theoretical
development by evolutionary physiologists. Ultimately, a
sound theory of physiological evolution would help biologists to address pressing ecological issues, such as the
potential loss of biodiversity during climate change
(Chown and Gaston 2008; Gaston et al. 2009; Chown et
al. 2010).
Acknowledgments
We thank J. Boyles, S. Lima, W. Mitchell, and M. Sears
for valuable discussions related to the models presented
here. This research was supported by a grant from the
National Science Foundation (IOS 0616344).
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Associate Editor: Ary A. Hoffmann
Editor: Mark A. McPeek