Mineral Nutrition of Plants

One of the more interesting methods of generating nutrient uptake data is the nutrient draw down method of Claasen and Barber (1973). By this method, growing plants are placed in a limited volume of nutrient solution and are allowed to draw down the nutrient concentrations over a short period of time. Aliquots are removed and analyzed for concentrations of the nutrient or nutrients of interest. In a simple case, volume changes due to evapotranspiration and quantities of nutrients withdrawn with the aliquots are small to the point of being negligible. The slope of concentration (or quantity) with respect to time is the rate of nutrient uptake, dC/dt, and the rate of nutrient uptake can be measured for a range of concentrations in a single experiment as the nutrient concentration is drawn down by plant uptake.

Assuming Michaelis-Menten kinetics, dC/dt = (1/Vol) v_maxC/(K_m+C), to be generally applicable to this problem, the issue divides into two problems: 1) expressing Michaelis-Menten kinetics, a derivative equation, in integral form as a function of time, and 2) parameter optimization of v_max and K_m to generate the best fit of the kinetic model to the experimental data.

1) Integration: The Michaelis-Menten equation may be analytically integrated to give an expression explicit in C but not in t. This means that if C is measured, then with values of v_max, K_m, and C₀, the time t predicted by the Michaelis-Menten kinetic model to draw down concentrations to that value of C can be calculated. In practice, one typically measures a value of C at an interval of t, not the other way around, so this information is not as useful as it may seem.

Alternatively, the Michaelis-Menten equation can be numerically integrated. The simplest approach uses Euler's method: C_i+1=C_i + dC_i/dt t. Inspection shows that this approach to numerical integration simply ties together a number of straight line segments. The slope at the beginning of each segment is correct, but that slope is not altered until the end of the segment. Error is cumulative and the only recourse is to reduce the size of t and recalculate dC_i/dt more frequently. How often is enough? Hard to say in advance. Fortunately there are better, if slightly less intuitive, methods.

The Runge-Kutta algorithm makes four evaluations of dC/dt between each time step, which are combined to create a single step. The overall accuracy is much better than taking four smaller steps. The first evaluation is identical to Euler's method. The second and third use the previous step to evaluate dC/dt at the middle of the time step, and the fourth evaluation uses the third at the end of the time step. The four evaluations are weighted and used to create a single step from i to i+1. The equations are:

k₁ = t dC_i/dt
k₂ = t d(C+k₁/2)/dt
k₃ = t d(C+k₂/2)/dt
k₄ = t d(C+k₃)/dt

C_i+1 = C_i + k₁/6 + k₂/3+ k₃/3 +k₄/6

These equations may rather simply incorporated into spreadsheet format, permitting evaluation of C as a function of time, given a set of values for v_max, K_m, and C₀. Additional time steps may be required on occasion if the behavior of the function is too abrupt, although this is not likely the case for Michaelis-Menton kinetics.

2) Parameter Optimization: Obviously, there are too many choices for v_max, K_m, and C₀ to sort through manually when trying to fit a model to real data. An obvious choice is to decide upon a figure-of-merit, a criterion of goodness of fit, and then seek those values of the parameters that minimize that figure-of-merit. Least squares of the residuals, or minimizing the sum of squared differences between predicted and measured values, is long established as acceptable and is often termed SSE, sum of the squares of the errors.

Numerous parameter optimization schemes are known and computer code is available from Numerical Recipes. Alternatively, spreadsheets contain such code in the "Optimizer" feature of QuattroPro and the "Solver" add-in of Excel, and the details of algorithm are not important. Once the Michaelis-Menten model has been written with a given set of vmax, Km, and C0 values, predicted values may be compared to measured values to calculate SSE, and then Optimizer/Solver may be used to search for and find values of vmax, Km, and C0 that minimize SSE. Parameter optimization typically proceeds until either a preset number of iterations has been completed or continued iterations do not improve the overall fit by a preset value. The beauty of parameter optimization is that the initial guesses for the parameter values affect the path of the parameter search but not the outcome (unless there is a degenerate situation in which the parameter makes no difference at all in the model.) Sometimes there is need to constrain values of the parameters to ensure their logical meaning; for example, Km and C0 must be positive.

There are yet more wrinkles to this story. One can use much the same techniques to write spreadsheet entries that include removal of aliquots and changes in volume due to evapotranspiration. Calculations can be written that include C_min, a concentration below which no nutrient uptake can occur. Other kinetic models can be evaluated numerically as well, including first and zero order kinetics.

A sample MS Excel spreadsheets can be downloaded: MS Excel v5/v7: mmdemo.xls (uses Solver add-in; corrected and amended, Oct 2002)

References:

A method for characterizing the relation between nutrient concentration and flux into roots of intact plants. N. Classen and S.A. Barber. 1974. Plant Physiol. 54:564-568.

Chap 16: Integration of Ordinary Differential Equations: 16.1 Runge-Kutta Method (p 704; see also 16.2 Adaptive Stepsize Control for Runge-Kutta p. 708); Chap 15: Modeling of Data: 15.1 Least Squares as a Maximum Likelihood Estimator; 15.5 Nonlinear Models; 15.6 Confidence Limits on Estimated Model Parameters. In: Numerical Recipes in Fortran 77 (2nd ed.) or Numerical Recipes in C (2nd ed.)
(see also Numerical Recipes homepage)

Leonor Michaelis (b. Berlin, 1875; d. New York City, 1949). Dictionary of Scientific Biography. Vol. 18 (Suppl. II):620-625.
A wonderful scientific biography of a scientist who worked out kinetic laws of enzymes when the pH scale was less than 10 years old and when enzymes would not be recognized as proteins for another 15 years! Maude Leonora Menten, a visitor to his lab during the period that led to the 1913 paper that established Michaelis-Menten kinetics, went on to become a distinguished pathologist.

This page was last modified by Phillip Barak, Univ. of Wisconsin, on 1 Oct 2002. All rights reserved. This material may not be reposted on any server without express permission of the author.