2 Classical model
2.8 Conclusions
The classical model, based on the analytical solution, has been analysed and applied to raw CPP data obtained from Dr. Henzler and Mr. Ye. The analytical solution was validated, and the necessary simplifying assumptions were shown to be numerically justified. The model was implemented in Excel, and Excel Solver used as a tool to estimate membrane parameters from observed P-t data. Membrane parameters were estimated using 2 methods: (a) the Classical method, which uses curve characteristics to fit the data, and (b) the RMS method, which optimizes the parameters such that the overall RMS error is minimized.
Results show that quite a good fit to observed P-t data can be obtained for the classical model with both fitting methods, for both HPP and OPP experiments. Although the methods are not mutually exclusive, they emphasise fitting different regions of the curve
and give different estimated parameters. The Classical method gives more weight to certain characteristics of the curve. Drawbacks are that its accuracy depends on how well curve characteristics from the data can be determined, and it does not fit noisy data well. The RMS method gives equal weight to all data points. A potential drawback is that regions where the cell dynamics are changing rapidly may not be fit well. However, this problem can be overcome by differentially weighting points in the minimized function (the RMS value) to improve the fit to the less well-fitted regions. The RMS method, therefore, will be used for subsequent data fits in this study. For comments on further use of curve characteristics, see §A.1 in appendix. While a low RMS error will be used as a guide to “best” fit, the RMS error does not uniquely determine a best fit, and the final decision will be made by making an analysis of the overall residuals between the model and data.
Fits with the Classical method confirmed that only a short initial period of the semilog plot should be used to calculate τw for the HPP curve (0-3 s for the data set used). This is
due to the non-linearity in the semilog plot. Although Ye et al. (2006) state that this non- linearity is “an artefact” arising from measurement errors, using 2 additional values of PE
(0.001 MPa above and 0.001 MPa below the calculated PE of the observed data) to
calculate the slope of the semilog plot still revealed a slight nonlinearity (R2=0.997). Although this is statistically very close to linear, the semilog plot slopes for the 3 different values of PE still differed by about 10%, and the fact remains that Lp is very
sensitive to the slope of the semilog plot. The results in §2.5.1 gave halftimes of 1.62s and 1.99s for semilog plot slopes which differed by 23%.
The method by which Lp from OPP experiments is estimated in other studies has not been
described in the literature. In this study, Lp was determined numerically using an equation
from the analytical solution to the KK equations. Using this method, it was found that the average Lp for the HPP experiments was 17% higher than Lp for the OPP experiment (Lp
=2.99 ± 0.05 m s-1 MPa -1 and Lp = 2.57 ± 0.21 m s-1 MPa -1 respectively). A higher Lp
for HPP experiments agrees with results from Steudle and Tyerman (1983), who suggest that this behaviour is due to an external unstirred layer influencing the pressure dynamics
in OPP experiments. Estimated parameters for positive and negative pulses for both the HPP and OPP experiments did not differ significantly, although differences in the estimated Lp between positive and negative pulses have been observed in the literature
(ibid. 1983).
The classical model was found to predict the cell dynamics very well, despite the
simplifying assumptions in the theory. The main drawbacks of the model are its inability to properly fit the shoulder of HPP relaxation curves, and the initial curvature and time delay in OPP relaxation curves. It was found that the first could be solved by fitting the HPP data with a double exponential, and the second could be solved by assuming a gradual rather than an instantaneous change in the external concentration.
These results suggest that a single exponential does not accurately represent the cell dynamics in a HPP experiment. A likely explanation is that the influence of the tonoplast on cell dynamics is being ignored, thus illustrating the limitations to viewing the cell as a single membrane rather than a composite membrane. Models of HPP pressure relations in wheat root cells (Zhang and Tyerman, 1999) revealed that a double exponential function fit the data better when aquaporins were blocked, showing the inadequacy of using a single exponential function when the influence of the tonoplast and plasma membrane are both significant. The blocking of aquaporins may impact the hydraulic conductivity of the tonoplast and plasma membrane differently depending on the amount of aquaporins in each.
A double exponential representation would mean that the expression for the hydraulic conductivity Lp in Eq. (2.13) no longer applies.This will not be explored here, but merely
pointed out that the expression for Lp used in current practice may be incorrect, and
impact on the accuracy to which Lp can be determined by current means.
Although the ramp change in external concentration assumed in an OPP is unrealistic, the resulting improvement to the classical model shows that the time and form in which the external perturbation impacts on the cell is an important consideration. If ramping is not
included, t0 must be adjusted or optimized to obtain a good fit to the data. Lp in particular
is very sensitive to the value of t0.
The time-delay observed in the OPP data may, however, be attributed to a combination of ramping in the external concentration (see §1.2), and effects of an external unstirred layer which would delay the external solute from reaching the membrane. The classical model may be made more realistic by the incorporation of ULs, which would impact on the parameter estimation. (It may also be made more realistic by including the effects of the tonoplast, but that is beyond the scope of this study.) In Chapter 3 we will incorporate UL effects into the models, and explore their impact on the model fits and parameter