where the subscripts 1 and 2 indicate the compartments separated by the membrane; L is the hydraulic conductivity coefficient; P1,2 are the hydrostatic pressures in the respective compartments; π1,2 are the respective osmotic pressures, and σ is the reflection coefficient, which is a measure of the impermeability of the membrane to the solute, ranging from 1 (fully impermeable membrane) to 0 (when the membrane does not discriminate between the solute and water). The osmotic pressure is given by the equation π = RTΣjCj(m) in which, according to Nobel (1974), Cj(m) = nj/(Vw*nw), where nj is the number of moles of osmotically active solute, nw is the number of moles of water, and Vw* is the partial molal volume of water (Vw*≅ 18 cm3 mol–1). The hydrostatic and osmotic pressures are the components of the water potential (Ψ), which is expressed as their difference (Nobel 1974):
Ψ= P –π.
The total flow(U) through a membrane of area A is
This equation is used to describe the influx of solutions from xylem and phloem into the fruit compartment. The fluxes are calculated as mass flow, in g h–1, but in the case of dilute solutions with density ≈ 1 g cm–3, they are numerically equal to the volume fluxes measured in cm3 h–1. Using subscripts x, p and f for xylem, phloem and fruit variables, respectively, and combining Eqns
This description of fluxes implies that different composite membranes separate the fruit compartment from phloem and from xylem, i.e. different kinetic parameters and different pathways may be involved. It is known that πx < < πf, so, in the following calculations, the approximations πx = 0 and σx = 1 are used, assuming that water entering from the xylem must cross the plasma membrane, which appears to have a reflection coefficient close to unity (Nobel 1974; Murphy & Smith 1994). The vascular network enters the fruit and enlarges as the fruit grows, with Ax and Ap increasing in parallel with fruit growth. It is convenient to relate this time dependence to a characteristic of the growing fruit having the dimension of area. Therefore, Ax and Ap are assumed to be proportional to the fruit surface area Af (calculated with Eqn
4), with constant non-dimensional coefficients of proportionality: Ax(t) = axAf(t) and Ap(t) = apAf(t).
If σp < 1, part of the sugar can be transported from the phloem to the fruit by mass flow (Up). The contribution of sugar to the mass flow is (1 –σp)CsUp, where Cs≅ (Cp + Cf)/2 is the mean concentration of the solute in the membrane, with Cp and Cf being the sugar concentrations in phloem and fruit, respectively, in accordance with non-equilibrium thermodynamics (Katchalsky & Curran 1965). In the following discussion, dimensionless (g g–1) values are used for these concentrations. This sugar contribution has to be subtracted from Up when the water inflow is calculated with Eqn
1, but the mass of water in the mass flow is much greater than that of the solute and so the water inflow may be approximated by Up. However, the transport of sugar by mass flow, (1 –σp)CsUp, must be taken into account when the sugar uptake is calculated. The saturating kinetics of the carbohydrate uptake observed in vitro for various fruits, as cited in the Introduction, points to the important role of active or facilitated transport (denoted by Ua). In addition, passive diffusion, driven by the difference in concentrations across the membrane, may take place. The total uptake of carbohydrates comprises these three constituents
where ps is the solute permeability coefficient. The saturating uptake rate, Ua, is assumed to be dependent on the phloem concentration according to the Michaelis-Menten equation. As observed by Johnson, Hall & Ho (1988), the uptake of sugars by fruit slices declines with increasing fruit age. The phenomenon of a reduction in the ability of the cells of senescing leaves to accumulate sugars was discussed by Milthorpe & Moorby (1969). This reduction may be of the same nature as that in fruit cells. To describe this phenomenon, a generalized form of the Michaelis-Menten equation has been applied. Generalization of the Michaelis-Menten theory, including the activity of a fully non-competitive inhibitor, leads to a speed reduction by a factor of 1 + CI/KI (Thornley & Johnson 1990), where CI is the inhibitor concentration and KI is the equilibrium constant for the formation of an inhibitor-carrier complex. The generalized Michaelis-Menten equation for fully non-competitive inhibition has the form
where vm is the maximum uptake rate per unit of dry mass and KM is the Michaelis constant. The rate Ua will decline with fruit age if the inhibitor accumulates in the growing fruit. We assumed that this accumulation proceeds exponentially, i.e. CI = CI*exp(t/τ), with CI* being the value of CI at the initial point t = 0. (If the time zero point is placed, for instance, at bloom termination, CI* must be recalculated accordingly.) It is more convenient to introduce another parameter: t* = –τ ln(CI*/KI), which has the dimension of time. Thus, Eqn
The generalized equation contains two additional parameters as compared with the basic Michaelis-Menten equation: t* and τ, describing the activity of an inhibitor. The exponential accumulation of the inhibitor is hypothesized for the model; we do not know what compound fulfils this role. The exponential accumulation characterizes an autocatalytic reaction. The concentration of ethylene has been observed to increase exponentially in the ripening peach fruit (Tonutti, Bonghi & Ramina 1996; Souty et al. 1997). A regulatory factor which inhibits sugar accumulation could be one of the co-products of ethylene biosynthesis.
The dry material loss through fruit respiration (Rf) comprises two components: that due to growth respiration, which is proportional to the rate of dry material intake, and that due to maintenance respiration, which is proportional to the dry mass (Thornley & Johnson 1990):
where qg and qm(T) are the coefficients for growth and maintenance respiration, respectively. The effect of temperature (T) on maintenance respiration is addressed by the Q10 concept (Penning de Vries & van Laar 1982; Pavel & DeJong 1993b): qm(T) = qm(293) Q10(T–293)/10.
Now all fluxes which contribute to the water and sugar balances are represented by equations containing the parameters and input functions, as well as osmotic and hydrostatic pressures in the fruit. The osmotic pressure can be calculated from the concentrations as described above. To calculate the hydrostatic pressure, the following procedure is performed. The relative rate of irreversible volume (V) growth of the fruit compartment is presented in the form of Lockhart’s equation (Lockhart 1965), yielding
where φ is the coefficient which describes extensibility of the cell walls and Y is the threshold value of the hydrostatic pressure in the fruit, above which irreversible expansion occurs. The change in fruit mass is calculated as the sum of Eqns
2. If each equation is divided by the appropriate density (Dw for water density and Ds for that of sugar), the rate of volume change may be obtained from
14 is small compared with the first and may be neglected. Under the condition of steady irreversible growth, the right-hand sides of Eqns
14 must be equal, which means that one of the variables on the right-hand sides of these equations is dependent and may be represented as a combination of other variables. This dependent variable is the hydrostatic pressure in the fruit. Setting Eqns
14 equal, inserting fluxes from Eqns
8, and solving the resulting equation for Pf, one obtains
where Δπ = πp–πf and, as mentioned above, σx = 1 and πx = 0. Equation
15 is valid under the condition Pf≥Y. The hydrostatic pressure Pf has the same physical meaning as the cell turgor discussed by Ray, Green & Cleland (1972) and represents ‘the value of turgor pressure that must prevail, during steady growth, to satisfy the requirement for simultaneous volume increase by water uptake and yield of the cell wall’. Equation
15 shows that Pf diminishes when the hydrostatic pressure in xylem and phloem decreases and when the transpiration rate rises. When Pf < Y, Eqns
15 are no longer valid. Because the cell walls are quite rigid, the volume will not change appreciably in response to the pressure changes, and the right-hand side of Eqn
14 becomes 0, which gives another equation for Pf:
If the environmental conditions lead to very low Pf values, the cell wall stresses are relieved and Pf is assumed to stay close to zero.
The water potentials in xylem and phloem, needed for calculation of Pf, are assumed to be equal to the measured water potential in the stem, Ψw. This gives Px = Ψw and Pp = Ψw + πp. Inserting Eqns
8 enables us to calculate the fluxes. After the fluxes have been calculated, integration of Eqns
2 over time yields the main state variables of the system:
where wo and so are initial masses of water and dry matter in the fruit flesh, respectively. After Eqns
18 have been solved, the fruit total mass and volume, sugar concentration, water content and other auxiliary variables which are important characteristics of the fruit may be calculated using the state variables w(t) and s(t). For osmotic pressure calculations, it is assumed that a fraction Z of the accumulated sugar remains in soluble form, and the rest is converted into structural material.