Insect eggs at a transition between diffusion and reaction limitation: Temperature, oxygen, and water

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ARTICLE IN PRESS
Journal of Theoretical Biology 243 (2006) 483–492
www.elsevier.com/locate/yjtbi
Insect eggs at a transition between diffusion and reaction limitation:
Temperature, oxygen, and water
H. Arthur Woodsa,Ã, Roger T. Bonnecazeb
aSection of Integrative Biology, The University of Texas at Austin, Austin, TX 78712, USA
bDepartment of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, USA
Received 22 November 2005; received in revised form 30 June 2006; accepted 11 July 2006
Available online 15 July 2006
Abstract
In diverse animal taxa, eggs and embryos are incapable of transporting oxygen by convection. In such cases, internal oxygen
distributions are determined jointly by rates of oxygen consumption and diffusive transport. Here we develop a mathematical model of
oxygen consumption and transport in insect eggs, with the goal of understanding—for eggs in variable-temperature environments—the
interactive effects of the two processes on development. We fit the model to previously published data on development time of eggs of a
sphingid moth, Manduca sexta. The fitted coefficients suggest that eggs develop at a transition point between reaction- and diffusion-
limitation. We test then this conclusion with independent data on development times of eggs distributed across a set of temperatures
generated by a thermal gradient bar. Finally, we develop an extension of the model that considers tradeoffs between oxygen transfer to
eggs versus water loss from them. The model results provide both a rationale for why development is often mass-transfer limited and a set
of new predictions about oxygen–water tradeoffs.
r 2006 Published by Elsevier Ltd.
Keywords: Temperature; Eggshell; Diffusion; Manduca sexta; Water loss
1. Introduction
new theory about the scaling of metabolism with body size
(West et al., 1997).
In aerobic organisms, metabolism is fueled by oxygen
We focus here on biological systems in which diffusion
consumption. This self-evident assertion belies a set of
and reaction, rather than convection, are known to
interesting and general questions about internal oxygen
dominate oxygen distribution. Essentially all animals go
economy—namely, how do organisms balance supply and
through one or more stages in which internal convection is
demand, and how often and under what circumstances
unimportant—early in ontogeny, before organs or circula-
does supply versus demand limit metabolism? Such
tory systems have developed (Kranenbarg et al., 2000).
questions motivate work on diverse phenomena, including
Moreover, oxygen diffusion and consumption are differ-
invertebrate (Po¨rtner, 2001, 2002) and human physiology
entially sensitive to temperature (Snyder, 1908, 1911;
(Richardson, 2000; Roca et al., 1989), the composition of
Raven and Geider, 1988): diffusion is relatively insensitive
freshwater invertebrate communities (Chapman et al.,
whereas reactions, including overall metabolism, often are
2004), and the design of reactors for biotechnology
quite sensitive. For organisms exposed to rapid or high-
applications (
ÃCorresponding
Author's personal copy
Fassnacht and Po¨rtner, 1999). Moreover,
magnitude temperature variation relative levels of oxygen
supply–demand considerations have stimulated important
supply and demand may vary profoundly (Po¨rtner, 2001,
2002). This problem is faced by any metabolizing structure
author. Present address: Division of Biological
dependent on diffusive oxygen supply (Woods, 1999).
Sciences, University of Montana, Missoula, MT, USA.
We have recently begun to investigate oxygen supply to
Tel.: +512 471 6201; fax: +512 471 3878.
eggs of an insect, Manduca sexta (Lepidoptera: Sphingidae)
E-mail addresses: art.woods@mso.umt.edu,
(Woods and Hill, 2004; Woods et al., 2005), a hawkmoth
art.woods@mail.utexas.edu (H. Arthur Woods),
bonnecaze@che.utexas.edu (R.T. Bonnecaze).
occurring widely in North and South America (Ferguson
0022-5193/$ - see front matter r 2006 Published by Elsevier Ltd.
doi:10.1016/j.jtbi.2006.07.008

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H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
et al., 1999; Rothschild and Jordan, 1903). Oxygen flux
strated that oxygen gradients were steep across the
into Manduca eggs occurs by diffusion, and the main
eggshell, but much flatter within the egg interior.
resistance to flux appears to stem from just one or two
Reaction and diffusion will be assumed to occur at
subchoral layers in the eggshell (Woods et al., 2005). Once
steady state (or quasi-steady state) throughout develop-
across the shell, oxygen is consumed by embryonic tissues
ment. This assumption too is reasonable. Development of
and, to a lesser extent, by cellularized yolk (Lamer and
the embryo requires 3–8 days depending on temperature,
Dorn, 2001). Despite their small size ($1.5 mg), eggs at
whereas reaction and diffusion change on much shorter
realistically warm temperatures (32–37 1C) exhibited clear
time-scales (minutes to hours). Oxygen transients could
symptoms of oxygen limitation arising from inadequate
arise even over these short time-scales if metabolic rates
diffusive transport—extra oxygen supplied externally
changed much more rapidly than diffusive transport could
stimulated metabolism (Woods and Hill, 2004). At lower
establish new, equilibrium oxygen gradients. Such a
temperatures (22 and 27 1C), extra oxygen had no effect,
situation, however, is unlikely, as metabolic rates increase
implying that metabolism was limited instead by internal
only gradually over the course of several days (Woods
rates of reaction or by the supply or removal of some other
et
al.,
2005).
Moreover,
unpublished
measurements
metabolite.
At
all
temperatures,
moderate
hypoxia
(B. Zrubek and H. A. Woods) indicate that experimental
(9–15 kPa O2) extended total developmental time. These
changes in ambient oxygen availability are detectable
results together suggest that eggs are oxygen-transport
within seconds by oxygen microelectrodes inside an egg—
limited under some conditions but not others.
implying that diffusion would rapidly drive oxygen levels
Here, we develop a mathematical model of insect egg
to new steady-state equilibria.
development, with the goal of understanding the relative
The overall rate of mass-transfer of oxygen Rm from the
importance of oxygen consumption and transport, and
atmosphere across the shell and into the egg is proportional
their interaction, to development. The model is derived
to the concentration difference and is given by the
from simple expressions describing mass-transfer by diffu-
expression
sion and consumption by saturating metabolic reactions
R
À HC
(Michaelis–Menten kinetics). A non-dimensional form of
m ¼ kmAðPO2
I Þ,
(1)
the model is derived that contains four coefficients, which
where km is the mass-transfer coefficient, PO is the ambient
2
we fit to previously published data on development time of
partial pressure of oxygen in the atmosphere, CI is the
embryos of M. sexta as a function of ambient oxygen
concentration of oxygen in the interior of the egg, and H is
availability (Woods and Hill, 2004). The fitted coefficients
the Henry’s law constant describing the effective equili-
suggest that eggs develop at a transition point between
brium partitioning of oxygen between the bulk atmosphere
reaction- and diffusion-limitation. To test this conclusion
and the liquid interior of the egg. Expressing the mass-
with independent data, we measure development times of
transfer in terms of ambient partial pressure of oxygen is
eggs distributed across a finely-incremented set of tem-
most convenient since it is a directly controlled experi-
peratures generated by a thermal gradient bar and analyse
mental parameter. Further, because of the linear relation-
the data by means of an Arrhenius plot. Finally, an
ship between concentration and partial pressure assumed
extension of the model that considers tradeoffs—between
by Henry’s law, concentration can be expressed in terms of
oxygen transfer to eggs versus water loss from them—
an equivalent partial pressure, i.e., Pegg ¼ HC
O
I . Finally, the
2
provides both a rationale for why development is often
Henry’s law constant is not dimensionless. It is most
mass-transfer limited and a set of testable predictions
commonly used in the form expressed above, relating
about oxygen–water tradeoffs.
partial pressure of a species in the vapor phase to its
concentration in the liquid phase (see Smith et al., 2005).
2. Model
The rate of oxygen consumption by the growing embryo
is assumed to follow Michaelis–Menten kinetics for some
Consider the egg of M. sexta as a spherical container of
rate-limiting step, given by the expression:
radius a, surface area A ¼ 4pa2and volume V ¼ 4pa3=3.
k
The eggshell is composed of the chorion, trabecular and
R
rCI V
r ¼
,
(2)
K
wax layers, and several other layers (see Woods et al., 2005,
M þ CI
for details). The
Author's personal copy
embryo, yolk and water reside in the
where kr and KM are rate constants. At steady-state the rate
interior. We consider the egg as a reaction–diffusion
of mass-transfer of oxygen must be balanced by its
system, where all mass-transfer resistance is localized in
consumption by reaction, or
the shell, and reaction occurs in the interior, where the
k
oxygen concentration is assumed to be constant. These
k
rCI V
mAðPO À HC
.
(3)
2
I Þ ¼ K
assumptions are reasonable in light of previously published
M þ CI
data (Woods et al., 2005) showing that most of the
This equation is quadratic in CI. It is convenient to
resistance to oxygen flux occurs across the eggshell itself.
express this equation non-dimensionally by introducing the
Moreover, direct measurement of radial oxygen profiles
dimensionless
concentration
j ¼ HCI =PO , where j
2
within eggs of M. sexta (Woods and Hill, 2004) demon-
ranges from zero (total oxygen starvation of the embryo)

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H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
485
to unity (complete equilibrium or infinitely fast mass-
K2 ¼ KM H ¼ aPO ,
(11)
2
transfer with the ambient atmosphere). Substituting this
and
variable and collecting terms gives the dimensionless
rb
equation
K3 ¼
.
(12)
krV
j2 þ ða þ b À 1Þj À a ¼ 0,
(4)
Finally, for fitting to experimental data correlating
where the dimensionless constants a and b are defined by
development time and oxygen pressure, it is convenient
to rewrite Eq. (7) in a form that explicitly reveals the
K
a
M H
¼
(5)
dependence of j on the partial pressure of oxygen in the
PO2
ambient atmosphere. For this case, it becomes.
À
Á qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
À
Á2
À ðK2=PO Þ þ ðK
Þ À 1 þ
ðK
Þ þ ðK
Þ À 1
þ 4ðK
Þ
2
4=PO2
2=PO2
4=PO2
2=PO2
j ¼
,
(13)
2
and
where
k
krV
b
rV
K
.
(14)
¼
.
(6)
4 ¼
k
kmA
mAPO2
Thus, Eqs. (9)–(14) parameterize development time and
The parameter a represents the degree of saturation of
partial pressure of oxygen in the ambient air. It is clear that
the reaction with respect to oxygen: a51 corresponds to a
four constants, K
completely saturated reaction rate, effectively zeroth-order
1; . . . ; K 4 are necessary to fit the data. In
addition, it is convenient to define a fifth constant,
with respect to oxygen concentration, and ab1 corre-
K
sponds to an unsaturated, effectively first-order reaction
5 ¼ ðK 1K 4ÞÀ1 ¼ kmA=k, which can be thought of as a
scaled eggshell conductance.
with respect to oxygen concentration. The parameter b is a
A non linear least-squares fit to previously published
measure of the relative rates of reaction to mass-transfer:
development time data (Woods and Hill, 2004) was
forb51, the development of the embryo is limited by the
performed to determine the constants. The results of the
rate of the reaction and for bb1, development is limited by
fit are summarized in Table 1, and theoretical predictions
the rate of mass-transfer. The dimensionless concentration
compared to the experimental observations in Fig. 1. The
of oxygen in the interior of the egg is given by the solution
fit is clearly very good (all R240.99). A few details
of Eq. (4) with the quadratic equation, namely
regarding the values of constants are worth pointing out.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
First, the base oxygen consumption as represented non-
Àða þ b À 1Þ þ
ða þ b À 1Þ2 þ 4a
j ¼
,
(7)
dimensionally by K3 is very small, and practically
2
negligible. Thus, although all eggs appear to die at PO o
2
where only the non-negative root is selected.
9 kPa (Woods and Hill, 2004), the model here predicts that
Now, the development time t
the development time of the egg will increase to infinity as
D of the egg is assumed to
be inversely proportional to the rate of oxygen consump-
the oxygen partial pressure vanishes.
tion R
Second, the value of b ¼ K4=PO ranges from 0.36 to 1.2,
r less base oxygen consumption rb for maintenance of
2
the egg. That is,
with the larger values at lower partial pressures of oxygen.
This indicates that mass-transfer of oxygen to the egg
k
t
contributes to development time, especially at low partial
D ¼
,
(8)
Rr À rb
pressures. Indeed, the concentration of oxygen in the egg is
where k is some unknown dimensionless stoichiometric
low (PO o2 kPa as measured with a microelectrode,
2
constant that expresses how many units of development
Woods and Hill, 2004), and the value of its dimensionless
occur per unit of
concentration j can be approximated by
Author's personal copy
oxygen consumed. Expressing Rr with
Eq. (2) and non-dimensionalizing the right-hand-side of
a
j %
.
(15)
Eq. (8), the development time is given by the expression
b À 1
At these low concentrations, the development time is
K
Þ þ jÞ
t
1ððK 2=PO2
D ¼
,
(9)
then approximated by
ð1 À K3Þj À K3
k
1
where
tD % K1b ¼
,
(16)
kmA PO2
k
K
which shows clearly that mass-transfer entirely determines
1 ¼
,
(10)
krV
the development time.

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H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
Table 1
Constants K1; . . . ; K4 determined by a non linear, least-squares fit of Eq. (9) to the experimental data in Fig. 1
Temperature (1C)
K1 (h)
K2 (kPa O2)
K3
K4 (kPa O2)
K5 (mol O2/kPa O2 – hr)
R2 of model fit
22
136
0.244
5.52 Â 10À5
10.82
6.8 Â 10À4a
0.997
27
85
0.444
9.00 Â 10À5
13.42
8.8 Â 10À4
0.999
32
67
0.644
8.94 Â 10À5
11.98
12.4 Â 10À4
0.992
The constant K5 ¼ ðK1K4ÞÀ1 ¼ kmA=k.
200
given 30% honey water and potted tobacco plants for
oviposition.
160
3.2. Aluminum bar experiment
22°C
Eggs were exposed to different, constant temperatures
120
from day 2 to hatching. For comparison, total develop-
ment times (laying to hatching) measured in a previously
27°C
published experiment (Woods and Hill, 2004) ranged from
80
70 h at 32 1C to 140 h at 22 1C.
Temperatures were imposed by placing eggs into an
32°C
Development time (hours)
aluminum thermal gradient bar (Fig. 2) (custom built by
the Department of Chemistry’s Machine Shop at The
40
University of Texas). To facilitate video imaging (see
below), the surface of the bar was spray-painted dull black.
The ends of the bar were fixed at 34 and 22 1C, respectively,
0
by circulating water from constant temperatures baths
0
10
20
30
through chambers milled in the end. Each of 12 evenly
PO2 (kPa)
spaced rows 2.54 cm (1 in) apart of the bar contained eight
Fig. 1. Development time versus ambient oxygen partial pressure for eggs
0.48 cm (3/16 in) diameter chambers drilled through the
of Manduca sexta. Solid symbols are experimental observations from
bar. Eggs were supported $1.3 cm (1/2 in) below the bar’s
Woods and Hill (2004, Fig. 1B) and lines represent fits to the model
surface by small pieces of cotton pushed into the wells. We
(Eq. (9), constants listed in Table 1).
used all 96 wells for eggs (1 egg wellÀ1), and into four of the
The rate constants KÀ1
wells (top wells of rows 1, 4, 9, and 12) we also placed
1 ,
K2 and K5 at the three
temperatures fit well to Arrhenius expressions, which are
T-type thermocouples. Temperatures were noted twice
listed in Table 1. The activation energy for the reaction rate
per day.
(KÀ1
Eggs in the aluminum bar were exposed to ambient room
1 ) is about 59.8 kJ/mol, and the activation energy for
mass-transfer (K
air. This design had two consequences. First, eggs were
5) is 44.8 kJ/mol. As usual the activation
energy for reaction is greater than the activation energy for
always exposed to normoxia (21 kPa oxygen). Although we
mass-transfer, but not hugely so.
did not directly monitor oxygen levels adjacent to eggs, a
The fit shows a transition from reaction-limited to mass-
brief calculation shows that oxygen levels at egg level
transfer-limited development with decreasing P
would be indistinguishable from ambient. Imagine a well as
O . The data
2
used to parameterize the model, however, used only three
a single giant aeropyle. The oxygen concentration at the
temperatures. To test the conclusion that eggs are designed
bottom of the tube (in this case at the egg’s surface) is given
on a temperature-dependent transition point between
by a rearrangement of Fick’s equation, Cegg ¼ CairÀ
reaction and diffusion
Jl=AD, where C
Author's personal copy
limitation, we used a larger range
air
is the concentration of oxygen in
of temperatures with finer increments.
ambient air (8.43 mol mÀ3 at 30 1C), J is the metabolic rate
of a single egg (3.3–10 mol eggÀ1 sÀ1 between 22 and 32 1C;
Woods and Hill, 2004), l is the length of the tube (.013 m),
3. Material and methods
A is the cross-sectional area of the tube (1.8 Â 10À5 m2),
and D is the diffusion coefficient of oxygen in air
3.1. Animals
(20.3 Â 10À6 and 21.5 Â 10À6 m2 sÀ1 at 20 and 30 1C, res-
pectively). Substituting these values into the equation
Eggs were derived from a laboratory colony of M. sexta,
above shows that at no biological temperature could the
exposed to a 14L:10D photoperiod. Eggs, larvae, and
oxygen concentration at the egg’s surface have differed
pupae were kept at 25 1C and adults at 24 1C. Adults were
from ambient by more than 0.05%.

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H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
487
Water
Water
outlet
outlet
Egg wells
8.6cm (3.8 in)
Water
Water
inlet
30.5 cm (12 in)
inlet
Fig. 2. Schematic of aluminum thermal gradient bar. The bar was 2.5 cm thick (1 in). To minimize heat loss from the bar’s faces, it was placed into molded
insulation such that only the upper surface was exposed to air.
A second issue was relative humidity, which was
35
uncontrolled during the experiment but generally ranged
y = -1.09x + 35.1
between 30% and 40% (at room temperature, $23 1C)
during the experiment. This range of relative humidities
°
C)
R2 = 0.998
30
corresponds to a vapor density of about 7 g mÀ3. Because
warmer air has a higher vapor capacity, eggs at the warm
end of the aluminum bar were exposed to lower relative
25
humidities (but the same vapor density). However, the
Temperature (
change was slight—a vapor density of 7 g mÀ3 corresponds
to $20% relative humidity at 32 1C—and was therefore
20
ignored.
0
2
4
6
8
10
12
Oviposition times were determined to within 4 h by
Position along bar
introducing potted tobacco plants into the adult flight cage
and removing them 4 h later. Eggs were stripped from the
Fig. 3. Temperature gradient generated by thermal gradient bar. Each
leaves and kept for 24 h at room temperature ($21 1C).
point represents the mean7S.D. over the six days of the experiment. The
line was fitted to the data by least-squares regression.
Subsequently, eggs were weighed individually on a micro-
balance (Sartorius MC5, Goettingen, Germany, 71 mg)
and assigned randomly to positions in the bar.
Two digital video cameras (Hitachi KP-D50, Tokyo,
4. Results
Japan, 1/2’’ CCD, 768 Â 494 pixels) with magnifying lenses
(Samsung 604CN, South Korea, 6–12 mm vari-focal zoom
The aluminum thermal gradient bar produced a stable,
lens) were mounted above the aluminum bar and adjusted
linear
temperature
gradient,
from
34.070.29 1C
so that eggs in all wells were visible. The experiment was
(mean7S.D.) at the hot end to 22.070.23 1C at the cold
carried out under constant illumination so that larvae
end (Fig. 3), or $1.09 1C per row of wells. Additional
would be visible immediately when they hatched. Images
measurements (not shown) indicated that temperature
from each camera were captured at 6-min (first 3 days) or
within a row variedo0.1 1C from well to well.
12-min (days 4–6) intervals using a frame grabber (Model
No eggs hatched in the two hottest rows (34.0 and
3153, Data Translation, Marlboro, MA, USA) controlled
32.9 1C), though subsequent observations suggested that
by image processing software (Global Lab Image/2 V3.0,
embryos in these rows completed most of development
Data Translation). Images were written to a hard drive,
before dying (pharate first-instar larva visible through the
assembled into movies, and analysed for hatching time.
clear eggshell). In each of the remaining rows, 4–8 eggs
Statistical analysis was done using S-Plus v. 6.1
hatched. Most of those that did not hatch (14/80 ¼ 17.5%)
(Insightful, Seattle, WA, USA). Data on bar temperatures
were sterile, showing no signs of development (remained
were pooled across Author's personal copy
days and analysed by least-squares
dark green). Only 2 of 80 eggs (2.5%) developed but did
linear regression. Developmental times were analysed two
not hatch. Non-hatching eggs were discarded from further
ways. First, total development time (from laying to
analysis.
hatching) was analysed as a function of temperature, with
The 64 remaining eggs had an average fresh mass
a summary description provided by fitting (least squares) a
(7S.D.) of 1.4370.08 mg. At 31.8 1C, the warmest
third-order polynomial to the data. Second, they were
temperature at which hatching occurred, eggs developed
visualized as an Arrhenius plot, with rates calculated as the
in about half the time (3.2 d70.15) that they required at
reciprocal of total development time versus inverse
22.0 1C (6.4 d70.15) (Fig. 4). The data were described well
absolute temperature. Throughout, error bars refer to
(R2 ¼ 0:997) by a third-order polynomial (y ¼ À0:0027x3þ
standard deviations.
0:256x228:27x þ 92:7).

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H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
7
Table 2
Arrhenius fit to model constants K1, K2 and K5
6
KÀ1ðhÞ ¼ 2 Â 107 eÀ6381=TðKÞ
1
K2ðkPaO2Þ ¼ 2 Â 1012 eÀ8764=TðKÞ
5
K5ðmol O2=kPa O2 À hÞ ¼ 58420 eÀ5383=TðKÞ
The units for temperature are Kelvin.
4
3
Development time (days)
ably good (R2 ¼ 0:70). The theoretical predictions do not
show as much of a variation in the slope as seen in the
2
experimental data. This is not surprising considering that
20
25
30
35
the fitted activation energies for mass-transfer and reaction
Temperature (°C)
rates differ by only about 50%. The experimental results
Fig. 4. Development time of eggs (of Manduca sexta) in the thermal
seem to show a much more dramatic variation, with an
gradient bar. All eggs were kept at room temperature (21 1C) for the first
approximately four-fold difference between the two activa-
24 h, then placed into the gradient bar until hatching. There are ten data
tion energies.
points rather than 12 (number of rows in the bar) because no eggs hatched
in the two warmest rows (34 and 32.9 1C).
5. Discussion
Temperature (°C)
For aerobic biological structures relying on diffusive
32
30
28
26
24
22
oxygen supply, metabolic rate will, in the extreme, be
-0.4
limited by either of two processes: the rate of oxygen
supply or the rate at which supplied oxygen is consumed.
-0.5
The former is transfer limitation and the latter reaction
limitation. Metabolism in most biological structures likely
-0.6
is limited by both processes—i.e., with omnipotent experi-
mental powers, one could stimulate metabolic rate by
-0.7
increasing either diffusive oxygen supply or the quantity or
-0.8
concentration of enzymes and other reactants. For such
biological structures, the interesting questions are: What
log (1/development time)
-0.9
are the relative strengths of transfer and reaction limita-
3.26
3.28
3.3
3.32
3.34
3.36
3.38
3.4
tion? And how do the relative strengths change across
1000/K
environments?
The model above analyses these questions in terms of
Fig. 5. Arrhenius plot of developmental data shown in Fig. 4. The curve
specific expressions for oxygen transfer and consump-
(solid line) represents the relationship computed from the fitted model
(coefficients in Table 2).
tion. Mass transfer is modeled as Fickian diffusion and
consumption as a saturating reaction with Michaelis–
Menten kinetics. After non-dimensionalization, the steady-
An Arrhenius plot (Fig. 5) revealed systematic but
state equation contained just four parameters, which we
modest changes in slope across temperatures—shallow at
fitted to previously published data (from Woods and Hill,
warm temperatures and steep at cool temperatures. In
2004) on development time of eggs of an insect, M. sexta,
Arrhenius plots, the slope of the relationship between
subjected to different combinations of temperature and
log(rate process) and inverse temperature estimates the
ambient oxygen (factorial design with 3 temperatures  8
activation energy (Ea) of the rate process (specifically,
oxygen levels). The parameter b, which provides a measure
the slope is ÀEa=R, where Ea is in kJ molÀ1 and R is the
of the relative rates of reaction to mass-transfer, took on
universal gas constant, 0.008314 kJ molÀ1 KÀ1). A linear
values ranging from 0.36 to 1.2, indicating that eggs of M.
least-squares fit to the Author's personal copy
first four (warm temperatures) and
sexta develop at the transition between reaction and
last four points (cool temperatures) indicated that Ea was
transfer limitation.
8.19 and 37.9 kJ molÀ1, respectively.
To test this result with a new data set, we measured total
The inverse development time versus inverse temperature
development time of eggs subjected to finely-incremented
can also be computed based on the model (fitted
temperatures between 22 and 34 1C. The rationale for the
parameters in Table 2), and these results are also shown
experiment was that increasing temperature usually has a
in Fig. 5. Note that in the model here, it was assumed that
more pronounced effect on rates of biochemical reaction
the eggs developed for one day at 21 1C before developing
than on rates of mass transfer by diffusion (Raven and
at elevated temperatures. The agreement between theore-
Geider, 1988; Snyder, 1908, 1911; Woods, 1999). Raven
tical predictions and experimental observations is reason-
thus, eggs at higher temperatures should show a greater

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H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
489
degree of limitation by transfer. This effect will appear in
Fitted K5 at: 22°C 27°C 32°C
an Arrhenius plot as an increasingly shallow slope at high
0.5
22°C
temperatures, corresponding to increasing domination by
the lower activation energy of diffusion compared to
27°C
0.4
15
reaction. The experimental development times showed just
32°C
this relationship (Fig. 5). The model predicted similar
behavior, though not as dramatic as the experimental data.
0.3
22°C
10
5.1. Why are structures as small as eggs of M. sexta ever
TWL
0.2
mass-transfer limited?
27
en inside egg (kPa)
°C
g
32°C
5
The answer likely involves physical constraints on
Oxy
0.1
producing biological structures that prevent water efflux
while allowing adequate oxygen influx, a difficulty that
stems from the molecules’ relative sizes. Barriers that
0
0
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
permit passage of oxygen (molecular diameter $3 A˚) will
K
also allow passage of water vapor (molecular diameter
5
$2.8 A˚). Consequently, eggs are subject to a tradeoff. High
Fig. 6. Predicted concentrations of oxygen in the interior of the egg (filled
eggshell conductance provides sufficient oxygen under
symbols) and total water loss (unfilled symbols) as functions of scaled
diverse environmental conditions, but at the risk of
eggshell conductance, K5. Vertical lines indicate fitted K5’s at three
desiccation. Low eggshell conductance protects water
different temperatures.
reserves, but at the cost of obtaining oxygen slowly.
This tradeoff has been studied in the context of avian
Empirical evidence of two kinds indicates that an
eggs by Rahn, Ar, Paganelli and colleagues. They found
oxygen–water tradeoff is real and pressing for Manduca
that, across a 2000-fold range of bird egg masses, eggshell
eggs. First, Woods et al. (2005) measured developmental
conductance to water vapor scaled to egg mass with an
changes in metabolism and water loss in batches of
exponent of 0.81 (Ar and Rahn, 1978), a value close to the
M. sexta eggs, using a flow-through respirometry system.
exponent (0.71–0.77) describing the scaling of metabolic
From laying to hatching, metabolic rate (as CO2 emission)
rate to egg mass (Hoyt and Rahn, 1980; Rahn et al., 1974).
and water loss rate increased in parallel, both rising $ 3-
This close correspondence was hypothesized to stem from
fold. [We note that Fig. 6A in Woods et al. (2005) contains
the following considerations: (i) eggshells must have certain
an error; the left axis should read ‘Water vapour flux  105
minimum conductance to guarantee sufficient oxygen
(mol sÀ1 mÀ2)’]. These correlative data suggest that as
supply late in development; (ii) diffusive transport of both
embryonic oxygen demand rises, eggshell conductance
water vapor and oxygen occurs through air-filled pores
increases to accommodate larger oxygen fluxes—and the
crossing the shell; (iii) the pores provide the main resistance
associated cost is greater water efflux. Second, Zrubek and
to transport; and (iv) to conserve water, selection
Woods (2006) recently tested the tradeoff hypothesis
minimizes shell conductance as much as possible without
experimentally by subjecting 3-d old eggs of M. sexta to
causing severe oxygen stress. Subsequent direct measures of
15%, 21% (normoxia), and 35% oxygen for 24 h while
the conductance of chicken eggshells to water vapor and
measuring rates of metabolism (as CO2 emission) and
oxygen, and independent measures of PO2 inside the egg,
water loss. Hypoxia depressed egg metabolic rates but led
supported these ideas (Paganelli et al., 1978).
to pronounced, rapid increases in water loss. By contrast,
Together these results imply a physical tradeoff—
hyperoxia had no significant effect on metabolism or water
mediated by the air-filled pores crossing the shell—between
loss. These data demonstrate that insect eggs are active
ability to acquire oxygen and propensity to lose water. We
participants in balancing oxygen gain and water loss, and
propose, by analogy, that selection for water conservation
they constitute direct empirical support for the tradeoff
has driven the conductance of insect eggshells low enough
hypothesis. Specifically, it appears that to allow sufficient
that they lie at the edge of oxygen limitation. A significant
oxygen influx under hypoxia eggs must increase eggshell
functional differen Author's personal copy
ce between insect and bird eggs is the
conductance, which allows greater water efflux.
location and nature of the primary flux-resisting layer. In
eggs of M. sexta, it occurs not in the air-filled pores
5.2. Modeling oxygen–water tradeoffs
crossing the outermost layer (as it does in bird eggshells)
but in solid and semi-solid layers to the interior (Woods
The existence of a tradeoff suggests an optimality
et al., 2005). Consequently, in insects the ratio of eggshell
question: what conductance should an eggshell have? An
conductances to oxygen and water vapor will reflect the
interesting analysis of this problem has been developed in
diffusivity ratio in liquids and solids—with possible further
the context of bird eggshells, by Alexander (1996). He
modification by differential chemical interactions between
asserts, reasonably, that an egg is harmed either by losing a
the diffusing gases and their medium.
large fraction (a) of its initial mass to evaporation or by

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ARTICLE IN PRESS
490
H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
experiencing large partial pressure gradients of oxygen
Now, noting that kmA$K5 and substituting Eq. 9 for the
ðDPO Þ from the environment to the embryo (correspond-
total developmental time, t
2
D (and neglecting K3, which is
ing to low internal PO ). If so, natural selection is likely to
51), total water loss is given by
2
minimize the function
ÀÀ
Á
Á
K5K1 K2=PO
þ j
TWL$
2
.
(19)
F ¼ a þ KDPO ,
(17)
2
j
where K (representing a weighting parameter that describes
The dimensionless (fractional) concentration of oxygen
the relative importance of oxygen and water) is unknown
inside the egg, j, can also be rewritten as a function of K5
but assumed not to vary among species. By replacing the
by substituting K4 ¼ ðK1K5ÞÀ1 into Eq. (13), giving
ÀÀ
Á À
Á
Á qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ÀÀ
Á À
Á
Á2
À
Á
À
K2=PO
þ 1=P
K
À 1 þ
K
þ 1=P
K
À 1
þ 4 K
2
O2
1K 5
2=PO2
O2
1K 5
2=PO2
j ¼
,
(20)
2
right-hand terms with specific descriptions of how they
Finally, we can transform j into the egg’s internal
depend on egg mass, metabolism, incubation time, and
partial pressure of oxygen (in kPa) by multiplying it by
eggshell conductance, it is possible to differentiate and find
atmospheric PO (assumed to be 21 kPa).
2
a simple equation for the conductance (Gopt) that
A plot of TWL and internal PO (denoted hereafter as
2
minimizes F. Because K is unknown, one cannot calculate
Pegg) together as functions of K
O
5 (Fig. 6) supports an
2
Gopt for any particular egg. But it is straightforward to
interesting view of how insect eggs balance oxygen gain and
show (Alexander, 1996) that Gopt should scale as mass0.78—
water loss. We describe this view in the section below.
an exponent indistinguishable from the one (0.81) deter-
First, though, we describe how different features of the
mined empirically by Ar and Rahn (1978).
model lead to this broader view. Note that the values of
Below we develop an alternative model, which differs
TWL are reported on an arbitrary scale since the actual
from Alexander’s in that it is derived explicitly from
constants for the equality in Eq. (19) are not known.
parameterized equations describing how development
Comparison of relative values of TWL for different values
depends on mass transfer and consumption of oxygen
of K5 is the most important function of this plot.
(Eqs. (9)–(14)).
The model indicates that the equivalent partial pressure
Because the fluxes of water and oxygen across the insect
of oxygen inside the egg ðPegg ¼ HC
O
I Þ increases mono-
2
eggshell are controlled by the same sets of structures,
tonically from 0 to 21 kPa (assumed atmospheric PO )
2
total water lost (TWL) will be proportional to the mass-
with increasing K5. The S-shape of these curves reflects
transfer coefficient multiplied by the total developmental
coupling of reaction kinetics and diffusive transport of
time, or
oxygen. At large K5 (rapid mass transfer), oxygen moves
easily into the egg and metabolic reactions are saturated
TWL$kmAtD.
(18)
(and total development time invariant) regardless of
This formula assumes that eggshell conductance is
variation in K5. However, as K5 declines below $0.0015,
constant over development. We know that for Manduca
metabolic reactions increasingly draw down internal
this is not so—both developmental stage and current
oxygen stores, because oxygen moves less easily into the
environmental conditions affect eggshell conductance
egg. Finally, at K5 below $0.005 (depending on tempera-
(Woods et al., 2005; Zrubek and Woods, 2006). Eq. 18
ture), metabolic reactions draw oxygen stores to very low
therefore equates an integrated parameter (TWL) with a
levels—as K5 goes to zero, so too do Pegg and metabolic
O2
nominally instantaneous one (km). We still think, however,
rate.
that TWL is a reasonable ‘controlling’ parameter. First,
The TWL curves, by contrast, transition from a flat
whether or not the egg produces a viable hatchling will
plateau at low K5 to linearly increasing TWL at large K5. In
depend on water reserves left near the end of development,
the linearly increasing phase metabolic reactions are
not the instantaneous rate of water loss at any moment.
saturated, and, therefore, increasing K5 leads to higher
That is, TWL, is likely Author's personal copy
to be more closely related to fitness
rates of water loss without also shortening total develop-
(and thus under selection) than is the mass-transfer
ment time ðtDÞ. The plateau at low K5 arises from interplay
coefficient, km. Second, km at any one point may indeed
of eggshell conductance and total development time. As K5
constitute a good statistical predictor of TWL, because km
declines, water loss per unit time also declines. But the rate
in general rises over development (if km is high now, it will
of oxygen influx per unit time also decreases, leading to
go only higher) and because environmental conditions
longer development times over which water loss occurs. A
often are autocorrelated over time scales relevant to egg
priori we had expected that TWL would increase at low K5,
development (days). Third, the fitted value of km (deter-
because development would stall completely while water
mined from developmental data) is, in fact, time-averaged,
would continue to be lost. This result did not occur because
as development occurs over several days.
the fitted values of K3 (representing oxygen consumption

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ARTICLE IN PRESS
H. Arthur Woods, R.T. Bonnecaze / Journal of Theoretical Biology 243 (2006) 483–492
491
devoted only to maintenance, not development) were
marginal value to be gained from still higher conductance
negligibly small.
becomes increasingly small.
Temperature affects both Pegg and TWL. Increasing
The fitted model parameters and graphical analysis
O2
temperature depresses the curves of Pegg, because the
shown in Fig. 6 are specific to M. sexta. However, the
O2
activation energy of metabolism is larger than that of mass
general conclusions should be applicable to eggs of most
transport—i.e., as temperature rises, metabolic consump-
terrestrial organisms. All terrestrial eggs face the twin
tion of oxygen increasingly outstrips diffusive delivery of
pressures of having to limit water loss while still developing
oxygen. However, this effect is mostly offset by increasing
rapidly, and this situation will lead to the evolution of
eggshell conductance (K5) at higher temperatures. The net
eggshell conductances that place embryos at risk for both
effect (seen where the vertical lines, representing fitted
desiccation and oxygen starvation. We note that two
values of K5, cross the curves of Pegg) is that Pegg is
additional factors likely also shape eggshell function: the
O2
O2
predicted not to change much at different temperatures.
drying power of the environment (microsite temperature
Using an oxygen microelectrode, Woods et al. (2005)
and humidity together) and the predation risk. The latter is
measured internal Pegg in ambient air at 24 and 37 1C,
particularly interesting. High predation risk should favor
O2
finding that it was slightly but significantly lower at the
the evolution of higher conductances (increasing total
higher temperature. We note, however, that the theoretical
water expenditure) as a means of minimizing total
analysis extends only to 32 1C whereas the electrode data
development time.
were from 37 1C. Also, the model also assumes an average
oxygen demand throughout development. Indeed the
Acknowledgments
demand for oxygen increases with time as the number of
living cells increases over development. This would lead to
A special thanks to Terry Watts for building the
a greater-than-predicted decrease in the oxygen concentra-
aluminum thermal gradient bar used in this project. This
tion toward the end of development.
work was supported by the NSF (IBN-0213087 to HAW)
Counter-intuitively, the model also predicts that at a
and the University of Texas at Austin.
given K5 increasing temperature depresses TWL. This
phenomenon occurs because temperature’s effect on short-
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O
5, the
2

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