Fitting ecological and physiological data to rectangular hyperbolae: a comparison of methods using Monte Carlo simulations

Abstract

Recommendations were developed to help aquatic scientists select which curve fitting method should be used to fit data that are expected to follow rectangular hyperbolic relationships. Rectangular hyperbolae of the form V = ( V max S )/( K m + S ), where V is a biological rate and S is the concentration of some substrate, are widely used by researchers to model the kinetics of processes such as enzyme activity versus substrate concentration, nutrient uptake versus nutrient concentration, and grazing and growth rate versus prey concentration. A variety of procedures exist to estimate the parameters V max (the rate of the process at saturating substrate concentration) and K m , (the concentration of S at which half the saturated rate is achieved). There has been extensive discussion in the biochemical and ecological literature as to which fitting method is most appropriate, based largely on theoretical and statistical considerations. However, the assumptions inherent in these fitting procedures are typically violated by the data obtained in many field and laboratory studies, e.g. the measurement of S has an associated error, or error levels in the measurement of V may not be constant across S . Thus, there is a problem predicting a priori which fitting method should be used. In this study, this problem was approached using Monte Carlo simulations. Data sets with known V max and K m were constructed for 5 different data cases, ranging from data sets where saturation was not achieved to data sets where very few sub-saturated measurements were available. Random, normally distributed errors were assigned to each point based on a 10 %, 20 % or 50 % constant or variable error in the estimate of V , or 20 % error in both S and V Six fitting procedures were applied including linear methods (Lineweaver-Burk; Eadie-Hofstee; Hanes-Woolf), the median method (Eisenthal and Cornish-Bowden), and non-linear least-squares methods (Cleland-Wilkinson; Tseng-Hsu). Non-linear methods were generally superior, but for data sets with low error (10%) all methods gave almost equally accurate results. Data with constant error were more difficult to fit than those where error varied with V . Criteria for selecting a fitting method based on data characteristics are discussed and applied to actual data sets