Supplemental data
Supplemental text S1. Microscale gas exchange model
Model equations
Since the observed diameter of the intercellular pores was typically above 10µm, we assumed that gas transport through the pores and the (lumped) cytoplasm was governed by Fickian diffusion instead of Knudsen diffusion
(1)
where (mol m-3) is the O2 concentration, is the gradient operator (m-1) and (m2 s-1) the O2 diffusivity. (mol m-3 s-1) is the source term representing respiration. In the intercellular space this term is zero while in the cytoplasm it is the consumption rate of O2 due to respiration. Michaelis-Menten kinetics were used as a semi-empirical model to describe the oxygen consumption rate of cells (Lammertyn et al., 2001)
(2)
with (mol m-3 s-1), the O2 consumption rate;(mol m-3 s-1) the maximal O2 consumption rate; and (mol m-3) the Michaelis constant for O2 consumption.
Like O2, CO2 was assumed to diffuse through the pores accordingly to the following equation:
(3)
where (mol m-3)is the CO2 concentration in a certain phase, (m2 s-1) is the CO2 diffusivity. The model of CO2 transport in the liquid phase was
(4)
(5)
with (mol m-3), (m2 s-1) the cytoplasmic concentration and diffusivity of HCO-3 , respectively, (mol m-3 s-1) the cytoplasmic CO2 production rate, k1 (s-1) and k2 (s-1) the CO2 hydration rate constant and H2CO3 dehydration rate constant, respectively. [H]+ (mol L-1) and K (mol L-1) are the concentration of protons H+ and the acid dissociation constant for H2CO3. The latter two terms of Eq. 4and 5represent the forward and backward conversion rate of CO2 to HCO3-, respectively. The equation for production rate of CO2 in the cytoplasm accounts for both oxidative and fermentative respiration.
(6)
The first term on the right hand side indicates the oxidative CO2 production rate due to consumption of O2; the second term represents anoxic conditions in the cell where the oxidative respiration process is inhibited and replaced by a fermentation pathway.
The O2 and CO2 transport model of the cells was written in term of equivalent concentration in the gas phase according Henry's law:
(7)
with R (8.314 J mol-1 K-1) the universal gas constant, T (K) the temperature and H (mol m-3 kPa-1) Henry's constant for O2 and CO2, respectively. The indicesp and c represent the intercellular (pore) space and cell, respectively.
The gas transport properties of the plasma membrane and cell wall were incorporated as a resistance term at the interface between intercellular space and cell. The flux J(mol m-2 s-1) through the intercellular space and cell was written as
(8)
where C* (mol m-3) is the equilibrium gas concentration in the liquid phase of the outer cell wall. Peff(ms-1) is the effective permeability of the membrane and cell wall.
Physical properties and respiration parameters
The values and sources of the material properties and respiration parameters are listed in Table S1. The diffusivity of O2 in air at 20°C was taken from Lide (1999) and was equal to 1.6×10-5 m2 s-1 while the O2 diffusivity in the intracellular liquid was equal to 2.01×10-9 m2 s-1 (Lide, 1999). Respiration parameters were measured forapple fruit. The measured of tissuewas divided the porosity of the tissue samples. (mol m-3) was set equal to 3µM (3×10-3 mol m-3) (Lammertyn et al., 2001).at the cellular scale was not found in the literature. However, this fermentative process is completely inhibited when the oxidative respiration is at saturation (Peppelenbos et al., 1996; Hertog et al., 1998). The was assumed to be equal to 1, representing fully oxidative respiration of hexoses (Andrich et al., 2006).
The cell wall diffusivity of O2 and CO2was assumed to be equal to that of water; the cell wall thickness of apple tissuewas about 1.67±0.71 µm (Mebatsion et al., 2009). Uchida et al. (1992) found that the diffusivity of the plasma membrane was equal to 2.91×10-9 m2 s-1. The plasma membrane has been found to be only about 8 nm thick (Gunning and Steer 1996). The membrane permeability for O2 was calculated as . The CO2 membrane permeability was 3.5×10-3 ms-1 (Gutknecht et al., 1977). The resistance of the cell wall and membrane for a gas i (O2 or CO2) was taken into account at the interface between intercellular space and cell through an effective permeability value calculated from (Nobel, 1991)
(9)
Supplemental text S2. Macroscale gas exchange model
Model equations
The permeation-diffusion-reaction of Ho et al. (2008) is based on the following transport equation:
(10)
and at the boundary
(11)
with αithe gas capacity of component i (O2, CO2 and N2) of the tissue (Ho et al., 2006b),Di (m2 s-1) the macroscopic apparent diffusion coefficient, u (m s-1) the apparent velocity vector, Ri (mol m-3 s-1) the tissue respiration(related to O2 consumption or CO2 production). The index refers to the gas concentration in the ambient atmosphere.Differences in diffusion rates of the different gasses lead to total pressure gradients that caused convective exchange as described by Darcy's law.
(12)
with K (m2) the permeation coefficient; P (Pa) the pressure and µ (Pa.s) the viscosity of the gas. The relation between gas concentration and pressure was assumed to follow the ideal gas law ().
The non-competitive inhibition Michaelis Menten model was used to describe O2 consumption and CO2 production of fruit tissue (Ho et al., 2010).The gas transport properties of the fruit tissue were measured using fluorescent optical probes. More details of procedure can be found in Ho et al. (2006a,b).The respiratory activity of apple fruit depends on its maturity (Veltman et al., 1999). Since the respiration rate was assumed to be determined by one rate limiting enzymatic reaction (Chevillotte, 1973), the dependence of the Michaelis Menten constantsKm for O2 and CO2, which are ratios of rate constants, on temperature was assumed to be negligible (Hertog et al., 1998). The maximal O2 consumption rate Vm,O2 and maximal CO2 production rate Vm,f,CO2 are a function of the initially available enzyme concentration (Hertog et al., 1998) and depend on fruit maturity and season. For the model validation the values of Vm,O2 and Vm,f,CO2 were therefore set equal to these of intact fruit measured from the batch used in the validation experiments.
Respiration measurement
Apple tissue samples were placed in airtight glass jars and the respiration rate was measured at different initial gas concentrations with three repetitions for each gas conditions according to the methodology described by Ho et al., (2008).
The respiration rate of apple tissue samples was measured at 20°C at 0, 0.5, 3, 5, 20 kPa O2; the partial pressure of CO2 was 0 kPa. To study the inhibitory effect of CO2, respiration measurements were carried out at 0, 5 and 20 kPa O2 in combination with 10 kPa CO2. For quantifying the effect of temperature on the respiration rate, measurements were carried out at 5, 10 and 20°C at 0 and 20 kPa O2 in combination with 0 kPa CO2. The results are shown in Supplemental Table S2 and Figure S1.
Supplemental text S3. Monte Carlosimulations
In order to evaluate the effect of sample size on the apparent diffusivities, a Monte Carlo analysis was carried out. A series of 500 brick-like computational domains was constructed by assembling 18 (332), 64 (444) and 216 (666) cubic microscale zones. The O2 and CO2 diffusivity of each microscale zone was randomly selected from a set of diffusivity values computed based on 16 different cubical cortex tissue samples with an edge length of 1.28 mm. The overall apparent diffusivity of each of the 500 computational domains was subsequently calculated. A normal distributionwas fitted to these values (Supplemental Figure S4).
Supplemental text S4. Richardson extrapolation
Richardson extrapolation was applied to evaluate the effect of the size of the computational domain on the apparent O2 and CO2 diffusivity (Kiusalaas,2005). The results are shown in Supplemental Figure S5. In the limit for with x the size of the computation domain, the apparent O2 and CO2 diffusivity converge to a value of 1.26×10-8 m2 s-1 for O2 and 3.19×10-8 m2 s-1 for CO2, respectively.
Supplemental text S5. Cellular conductance
The mean flux J of the gas exchange through the interface between cells and intercellular space was computed from
(13)
where (mol m-3 s-1) is the mean respiration rate in the cellular compartment of the computational domain, Vcell (m3) is the total volume of the cellular compartment of the computational domain and As (m2) is the total exchange surface between cells and intercellular space. The cellular conductance gc was computed from the mean flux J as follows
(14)
with (kPa) and (kPa), the mean concentration in the intercellular space and the equivalent concentration in the cellular compartment, respectively.
Supplemental text S6. Local gas concentration gradient
The mean local gas gradient in phase i, with i denoting either the cellular compartment or the intercellular space, was calculated as follows:
(15)
where Vi(m3) is the total volume of phase i.
Supplemental text S7. Sensitivity analysis
A sensitivity analysis was performed to study how sensitive the computed cellular conductance was with respect to small changes in model parameters. A high value of the relative sensitivity of a parameter indicates that the particular predicted model solution is highly influenced by a small change in that parameter value. The relative sensitivity of the predicted cellular conductance gc with respect to parameter P was defined as follows
(16)
The perturbation of the parameters was taken as 10% of the nominal value of P which was used for simulation. The results are shown in Supplemental Table S4.
Supplemental text S8. Effect of cell pH on CO2 exchange
CO2 transport in the cells is more complex than O2 because, depending on the pH, in water it is in equilibrium with HCO3-. Apple cells contain large amounts of organic acids in their (large) vacuole that typically has a low pH (Smith and Raven, 1979). As the vacuole volume can be up to 90% of the total volume of a mature plant cell (Nobel, 1991), it seemed reasonable to assume in the simulations that the overall pH of the cell is 4 obtained from the measurement of tissue juice. Simulations at this pH showed that there was almost no difference between model predictions obtained with or without the transport equation for HCO3- (Table 2). This is due to the fact that H2CO3, which is much less soluble in water than HCO3-, isalmost completely undissociatedat pH 4. For completeness we also carried out simulations at pH 7 which is the normal pH of the cytoplasm (Kurkdjian et al., 1978; Roberts et al. 1981 and 1982). The CO2 conductance increased by 7.8% .
Supplemental text S9. Effect of compartmentalization on intracellular gas transport
Plant mitochondria appear as spherical or roded-shaped entities 0.5 to 1 µm in diameter and 1 to 3 µm long.The number of mitochondria per plant cell varies and is related primarily to the metabolic activity of the cells. In a very active cell, a large fraction (up to 20%) of the volume of the cytoplasm may be occupied by mitochondria. The vacuoles usually occupy more than 30% of the cell volume and can amount to 90% of the cell volume in a large mature cells (Buchanan et al., 2000). In the microscale model we assumed that the cytoplasm can be considered as a lumped homogeneous liquid; we neglected the compartmentalization of the cell. In order to investigate the consequences of this assumption we also implemented a model which included both the cytoplasm and vacuole. Because of the limitations of the computational resources we had access to we implemented this in 2-D only. We extended the 2-D model developed by Ho et al. (2009) to include mitochondria as well.
As the vacuole volume is very variable, we created vacuoles by shrinking the cell geometry until the ratio of the resulting vacuole area to original cell area was around 0.6. The mitochondria were created as spheres with a diameter of 3µm distributed near the cell membranes; a smaller and a more realistic rod-like shape would require excessive computer time and memory requirements. The mitochondria occupied 23% of the volume of the cytoplasm.
Simulations showed that the O2 concentration was not different between the model with and without compartmentalization (Figure S8). The CO2 transport inside the cell is more complex due to the difference in pH between the vacuole and the other compartments. Different values of HCO3- membrane permeability () of the membrane have been reported in literature, ranging from 4.3×10-8 ms-1 for phospholipid vesicles (Norris Powell, 1992) to 5.6×10-6 ms-1from human erythrocytic membrane(Sieger et al., 1994). Gas transport simulations were carried out with both values and by assuming a pH equal to 4 for the vacuole and 7 for the cytoplasm and mitochondria (Figure S9B and S9C). There was clearly a much larger CO2 gradient inside the cell compared to that predicted by the lumped model in which compartmentalization was not modeled (Figure S9A).
The transport of ions into the vacuole is controlled by the tonoplast. When tonoplast was assumed to be impermeable, the CO2 concentration was very similar to that obtained with the lumped CO2 transport model. More research is required to understand the role of the tonoplast on CO2 transport and the resulting CO2 distribution in the cell.
Supplemental Table S1 Physical parameters of the microscale gas transport model
Model parameters / O2 / CO2Intercellular space / Cell / Cell wall / Intercellular space / Cell / Cell wall
Diffusivity (m2 s-1) / 1.6×10-5 (1) / 2.01×10-9(1) / 2.01×10-9(1) / 1.6×10-5(1) / 1.67×10-9(1) / 1.67×10-9(1)
Thickness (µm) / - / - / 1.67 / - / - / 1.67
Henry’s constant (molm-3 kPa-1) / 1.37×10-2(1) / 0.3876 (1)
Cell membrane thickness (nm) / 6-10 (2) / 6-10 (2)
Membrane permeability (ms-1) / 3.63×10-2* / 3.5×10-3(4)
Respiration
- (mol m-3 s-1) / -1.29×10-4**
-(mol m-3 s-1) / 1.35×10-4**
- (mol m-3) / 3×10-3(3)
- rq,ox / 1
(1) Lide (1999), (2) Gunning and Steer (1996), (3) Lammertyn et al. (2001), (4) Gutknecht et al.(1977).
O2 diffusivity of cell membrane was 2.91×10-9 m2s-1 (Uchida et al., 1992). *The value of O2 membrane permeabilitywas calculated as .
**Values were expressed in term of tissue respiration.
Supplemental Table S2. Gas transport properties and respiration kinetics parameters of macroscale model
Parameters / Unit / Skin / TissueDO2 / 10-9 m2 s-1 / 0.19±0.15 / 10.10±6.21
DCO2 / 10-9 m2 s-1 / 0.31±0.23 / 35.10±12.3
DN2 / 10-9 m2 s-1 / 0.3±0.09 / 18.1±7.9
K / 10-17 m2 / 0.59 / 92.3
Vm,O2* / 10-5 mol m-3 s-1 / 4.91±0.27
Ea,VmO2 / (kJ mol-1) / 62.5±7.3
Km,O2 / (kPa) / 1.64±0.16
Kmn,CO2 / (kPa) / 163 ± 74
Vm,f,CO2* / 10-5 mol m-3 s-1 / 4.44±0.56
Ea,Vm,f,CO2 / (kJ mol-1) / 81.7±7.7
Km,f,O2 / (kPa) / 0.89±0.18
rq,ox / 1.02±0.04
*Value for intact fruit, expressed at 283°K, ± Standard error. Definition of symbols is indicated in Table S1. (Ho et al., 2010)
Supplemental Table S3. Effect of cell wall diffusivity (Dw) on equivalent diffusivity of a cell cluster (Dcluster).
O2 / CO2Dw (m2s-1) / Dcluster (m2s-1) / / Dw (m2s-1) / Dcluster (m2s-1) /
2DO2,c / 1.98×10-9 / 0.984 / 2DCO2,c / 1.63×10-9 / 0.976
DO2,c / 1.95×10-9 / 0.97 / DCO2,c / 1.61×10-9 / 0.96
0.5DO2,c / 1.89×10-9 / 0.94 / 0.5DCO2,c / 1.56×10-9 / 0.93
0.25DO2,c / 1.79×10-9 / 0.89 / 0.25DCO2,c / 1.47×10-9 / 0.88
Dc is the diffusivity of the gas in the cytoplasm.
Supplemental TableS4. Relative sensitivity of cellular conductance of O2 and CO2 with respect to the microscale model parameters. Symbols are defined in Supplemental Text S1.
Parameters / / Parameters // 1.6×10-5 m2s-1 / -3.55×10-4 / / 1.6×10-5 m2s-1 / -9.17×10-3
/ 2.01×10-9 m2s-1 / 9.53×10-1 / / 1.67×10-9 m2s-1 / 9.29×10-1
/ 2.01×10-9 m2s-1 / 4.3×10-2 / / 1.67×10-9 m2 s-1 / 4.2×10-2
/ 3.63×10-2 ms-1 / 1.45×10-2 / / 3.5×10-3 ms-1 / 1.19×10-2
/ 1.37×10-2 molm-3 kPa-1 / 1 / / 0.3876 molm-3 kPa-1 / 1.006
/ 1.17×10-9 m2s-1 / 1.3×10-3
k1 / 0.039 s-1 / 2.6×10-2
k2 / 23 s-1 / 1.3×10-3
K / 2.5×10-4 molL-1 / 3.8×10-2
Suplemental Figure S1(a) Typical O2 and CO2 gas partial pressure as a function of time in the gas diffusion measurement setup;(b) Arrhenius plot of maximal O2 consumption and CO2 production rate of tissue at different temperatures;(c) and (d) O2 consumption and CO2 production rate in pear tissue disks at 20°C. Symbols denote measurement and the lines denote fitted model equation. (Adapted from Ho et al. 2010).
Suplemental Figure S2.O2(A)and CO2(B)partial pressure distribution at 20 kPa O2, 0 kPa CO2 at 20°C of the intact fruit.
(A) / (B)SupplementalFigure S3Computational domain for Monte Carlo analyses. The brick like geometry is constructed by assembling different zones with diffusivities which were randomly selected from the set of apparent diffusivity values calculated using the microscale model.
SupplementalFigureS4 Probability density distribution of the overallapparent diffusivity of O2 and CO2 based on 500 Monte Carlo simulations for computational domains of 18 (A, B), 64 (C, D) and 216 (E, F) microscale zones. In (A) and (B) a β-distribution was fitted while normal distribution was fitted in (C), (D), (E) and (F).
(A) / (B)(C) / (D)
(E) / (F)
SupplementalFigure S5.Effect of size of the computational domain (x) on apparent O2 (A) and CO2 (B) diffusivities calculated by means of Monte Carlo simulation. The bars indicate standard deviation around the means. The line indicates the Richardson extrapolation for .
(A) / (B)Supplemental Figure S6 (A)O2 distribution of the intact fruit at typical commercial storage conditions (1 kPa O2, 2.5 kPa CO2 and 1°C). Color bar indicates O2 partial pressure (kPa). (B)Simulated intra-cellular O2 concentration of microscopic tissuecomputed from the lowest O2 concentration near the core of the macroscale simulation. Color bar indicates O2 concentration in cytoplasm (µM).
(A) / (B)SupplementalFigure S7Normalised oxygen consumption () of intact apple fruit as a function of the ambient O2 partial pressure at 10°C with 3 different apple radii. is the value of O2 partial pressure where =0.5.
SupplementalFigure S8Simulated intracellular O2 concentration in cortex tissue of Jonagold apple near the core. At the top of the computational domain, a partial pressure of 6.72 kPa O2computed using the macroscale model at the corresponding position was applied. The other sides of the sample were assumed to be impermeable. The color bars expressequivalent gas concentrations (kPa). (A)and (B) model without and with compartments of mitochondria and vacuoles, respectively.
(A) / (B)SupplementalFigure S9Simulated intracellular CO2concentration in cortex tissue of Jonagold apple near the core. At the top of the computational domain, a partial pressure equal to 8.39 kPa CO2computed using the macroscale model at the corresponding position was applied. The other sides of the sample were assumed to be impermeable. Lumped model (A); compartmentalized models with value taken from Sieger et al. (1994) (B) and Norris Powell (1992) (C); compartmentalized model with no flux of HCO3- through the vacuole membrane (D).
(A) / (B)(C) / (D)
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