Nonparametric Estimation of Rate Equations for Nutrient Uptake

Abstract

Knowledge of the rate of a biological process is important for characterizing the system and is necessary for gaining a deeper understanding of the process. Consider measurements, Y, made over time on a system following the model Y = f(t) + e, where f is a smooth, unknown function and e is measurement error. Although most statistical methodology has focused on estimating f(t) or f′(t), in some applications what is of real biological interest is the relationship between f and f′. One example is the study of nitrogen absorption by plant roots through a solution depletion experiment. In this case f(t) is the nitrate concentration of the solution surrounding the roots at time t and – f′(t) is the absorption rate of nitrate by plant roots at time t. One is interested in the rate of nitrate absorption as a function of concentration; that is, one is interested in Φ, where Φ(f) = – f′. Knowledge of Φ is important in quantifying the ability of a particular plant species to absorb nitrogen and in comparing the absorption ability of different crop varieties. A parametric model for f is usually not available, and thus a nonparametric estimate of Φ is particularly appropriate. This article proposes using spline-based curve estimates with the smoothing parameter chosen by cross-validation and suggests a method for obtaining confidence bands using a form of the parametric bootstrap. These methods are used to analyze a series of solution depletion experiments and are also examined by a simulation study designed to mimic the main features of such data. Although the true f is a monotonic function, simulation results indicate that for our specific application, constraining the estimate of f to be monotonic does not reduce the average squared error of the rate curve estimate, Φ. Although using a cross-validated estimate of the smoothing parameter tends to inflate the average squared error of the rate estimate, an analysis of a set of solution depletion experiments is still possible. Using the proposed methods, we are able to detect a difference in rate curves obtained under different experimental conditions. This is established by applying an analysis of variance (ANOVA)-like test to the estimated rate curves, where the critical value is determined by a parametric version of the bootstrap, and by examining confidence bands for the difference of two rate cures. This finding is important, because it suggests that the shape of Φ may not be constant under the experimental conditions examined.

Key Words:

• Confidence bands
• Cross-validation
• Michaelis-Menten rate equation
• Monotonicity constraints
• Parametric bootstrap
• Smoothing splines
• Solution depletion experiments