10. The data from the packaging experiment were analysed together in one run, simultaneously
using temperature, film area, p
O
2
air
and p
CO
2
air
as independent variables and r
O
2
, r
CO
2
, p
O
2
cav
and p
CO
2
cav
as dependent variables using the model formula- tion from Eqs. 14, 15, 8 and 9 with the
temperature dependence according Arrhenius’ law Eq. 10 applied to r
O
2
max
, r
CO
2
max
, P
O
2
film
and P
CO
2
film
multi-response, multivariate, non-linear regres- sion analysis. The data on temperature depen-
dence of the additional respiration measurements were analysed using the temperature dependence
according to Arrhenius’ law Eq. 10. The refer- ence temperature for Arrhenius’s law was in all
cases fixed at 15°C 288.15 K. The non-linear equations were applied directly, without transfor-
mation to data or equations.
3. Model development
3
.
1
. Gas exchange The O
2
uptake of the packed fruit was modelled as a function of p
O
2
cav
applying Michaelis – Menten kinetics as introduced by Chevillotte 1973:
r
O
2
= r
O
2
max
· p
O
2
cav
Km
O
2
+ p
O
2
cav
5 where r
O
2
max
is the maximum O
2
consumption rate mol·kg
− 1
·s
− 1
unconstrained by O
2
availability, Km
O
2
is the Michaelis constant for O
2
consump- tion Pa and p
O
2
cav
is the O
2
partial pressure Pa in the cavity of the capsicum.
The CO
2
production of the fruit was modelled as described by Peppelenbos et al. 1996, taking
into account CO
2
coming from both oxidative and fermentative processes:
r
CO
2
= RQ
ox
· r
O
2
+ r
CO
2
max
1 + p
O
2
cav
km
O
2
f
6 where RQ
ox
represents the respiration quotient ratio of CO
2
production to O
2
consumption for oxidative respiration, r
CO
2
max
is the maximum fer- mentative CO
2
production rate mol·kg
− 1
·s
− 1
and Km
O
2
f
is the Michaelis constant for the inhibition of this fermentative CO
2
production by O
2
Pa. As the Michaelis – Menten approach is a sim-
plified representation of a more complex biochem- ical and physiological process including several
diffusion steps, the constants Km
O
2
and Km
O
2
f
are in fact apparent Kms instead of pure Michaelis constants.
3
.
2
. Package conditions At steady state, the rate of O
2
diffusion through the packing film equals the rate of O
2
diffusion into the fruit, which equals the rate of O
2
con- sumption due to respiration, resulting in:
r
O
2
max
· p
O
2
cav
Km
O
2
+ p
O
2
cav
· M = P
O
2
film
· A
film
D X
film
· p
O
2
atm
− p
O
2
pkg
= P
O
2
fruit
· A
fruit
· p
O
2
pkg
− p
O
2
cav
7 with
P
O
2
fruit
the fruit
permeance for
O
2
mol·s
− 1
·m
− 2
·Pa
− 1
and A
fruit
, the diffusion area m
2
. Using these relationships, p
O
2
cav
can be ex- pressed as a function of p
O
2
atm
: p
O
2
cav
= 1
2 · a +
1 2
a
2
+ 4 · Km
O
2
· p
O
2
atm
8 with: a = p
O
2
atm
− Km
O
2
− r
O
2
max
· M · 1
P
O
2
fruit
· A
fruit
+ D
X
film
P
O
2
film
· A
film
A comparable approach is taken to express p
CO
2
cav
as a function of p
CO
2
atm
and p
CO
2
cav
at steady state, resulting in:
p
CO
2
cav
= p
CO
2
atm
+ RQ
ox
· r
O
2
· M + M · r
CO2
max
· Km
O
2
f
Km
O
2
f
+ p
O
2
cav
· 1
P
CO
2
fruit
· A
fruit
+ D
X
film
P
CO
2
film
· A
film
9 with
P
CO
2
fruit
the fruit
permeance for
CO
2
mol·s
− 1
·m
− 2
·Pa
− 1
. Equivalent approaches were applied by Dadzie
et al. 1996 describing internal atmospheres of unpacked apples, Banks et al. 1993 describing
internal atmospheres
in waxed
apples, and
Cameron et al. 1995 MA packaging of mini- mally processed produce.
3
.
3
. Temperature dependence Temperature influences the system through
both its effects on gas exchange and film perme- ability. In both cases the effect of temperature can
be described according to Arrhenius’ law:
k = k
ref
· e
Ea R
1 T
ref
− 1
T
10 where the energy of activation Ea J·mol
− 1
ex- presses the dependence of a given rate k exact
unit depends on the rate on temperature T K. The parameter k
ref
exact unit depends on the rate is the rate at an arbitrarily chosen reference
temperature T
ref
K. In accordance with Hertog et al. 1998 it was assumed that the temperature
effect on respiration can be accurately described by solely applying the Arrhenius equation to the
parameters r
O
2
max
and r
CO
2
max
, assuming Km
O
2
and Km
O
2
f
being relatively temperature independent. For packaging films it is also generally accepted
to express permeability as a function of tempera- ture according to Arrhenius’ law Beaudry et al.,
1992; Exama et al., 1993; Cameron et al., 1994. The permeability of the fruit was considered rea-
sonably temperature independent as, with cap- sicums, most of the diffusion takes place in air
through gaps underneath the stem plate into the central cavity De Vries et al., 1996; personal
communications N.H. Banks and J.P. Bower.
3
.
4
. Fruit-to-fruit 6ariation As all the experimental work was done on
individually packed fruit, large variations were expected due to fruit-to-fruit variation in both gas
exchange and fruit permeability. Mechanistic models
enable discrimination
between those
parameters likely to vary between individual fruit and those that would be likely to be constant for
all fruit. Variation in experimental data has previ- ously been handled by relating it to specific
parameters in models that include cultivar, batch and individual fruit effects Hertog et al., 1997b;
Tijskens et al., 1997; Hertog et al., 1999. Fruit permeance P
O
2
fruit
would probably vary substantially between fruit. In addition to this, the
diffusion area A
fruit
would vary between the fruit. As these features are quite hard to measure
directly, the initial measurements in air at 20°C made before packing, were used to determine the
combination of these two parameters for each of the individual fruit. This was done by analogy to
Eqs. 3 and 4 assuming a steady state between respiration and diffusion resulting in:
P
O
2
fruit
· A
fruit
= r
O
2
· M p
O
2
atm
− p
O
2
cav
11 P
CO
2
fruit
· A
fruit
= r
CO
2
· M p
CO
2
cav
− p
CO
2
atm
12 The meaning of the parameters from Eqs. 5 and
6 describing respiration, provides useful insight into the likely sources of variation in fruit gas
exchange. The Michaelis constants Km
O
2
and Km
O
2
f
are based on enzymatic reaction rate con- stants while the maximum rates r
O
2
max
and r
CO
2
max
are also related to the amount of enzymes present Her-
tog et al., 1998. Assuming that pure rate constants are intrinsic attributes of enzymes, and at the same
time realising that the amount of enzyme present would vary between fruit, supports the conclusion
that differences between individual capsicums would most likely result from differences in r
O
2
max
and r
CO
2
max
. The rate constants Km
O
2
and Km
O
2
f
can be safely assumed constant for a given capsicum
cultivar. Based on this reasoning, the relative respi- ration of the individual fruit rr
fruit
was determined relative to the mean respiration r
O
2
, using the initial measurements in air at 20°C before packing as:
rr
fruit
= r
O
2
fruit
r
O
2
13 This relative respiration was subsequently intro-
duced into the formulation of gas exchange Eqs. 5 and 6, to take into account the variation
between fruit, resulting in:
r
O
2
= rr
fruit
· r
O
2
max
· p
O
2
cav
Km
O
2
+ p
O
2
cav
14 r
CO
2
= RQ
ox
· r
O
2
+ rr
fruit
· r
CO
2
max
1 + p
O
2
cav
Km
O
2
f
15
4. Results and discussion
