A Beta Function-Based Model for Predicting Leaf Appearance and Expansion in Romaine Lettuce

A process-based developmental model was developed to predict leaf appearance and expansion in romaine lettuce (Lactuca sativa L. var. longifolia) using hourly air temperature and daily light integral as environmental inputs. The model consisted of two linked modules representing leaf appearance and individual leaf expansion as distinct biological processes. The leaf appearance module calculated the hourly leaf tip appearance rate as the product of a beta-type nonlinear temperature response function and a developmental stage weighting function based on growing degree days accumulated above a base temperature of 4 °C. The leaf expansion module estimated individual leaf expansion using photothermal age, a Gompertz growth function, and a leaf length–area allometric relationship, with potential leaf length determined by the mean growing temperature and leaf rank. The optimum temperatures for leaf appearance rate and potential leaf length differed by approximately 6 °C (26.7 °C vs. 20.4 °C), indicating distinct temperature response patterns between the two developmental processes. Model calibration was performed using datasets (n = 437) collected from a temperature-gradient greenhouse with a nutrient film technique hydroponic system across the spring, summer, and autumn growing seasons, yielding an overall model efficiency (EF) of 0.92 and a root mean square error (RMSE) of 4.26 leaves for leaf appearance, and an EF of 0.80 and an RMSE of 448.1 cm2 for leaf expansion. Independent model evaluation was also performed using datasets (n = 132) obtained from a commercial greenhouse with either a deep flow technique hydroponic system or a perlite-based substrate system across the same three growing seasons, yielding an EF of 0.96 and an RMSE of 2.24 leaves for leaf appearance, and an EF of 0.92 and an RMSE of 216.8 cm2 for leaf expansion. These results demonstrate that the model effectively described leaf appearance and expansion in romaine lettuce across the tested soilless culture systems under greenhouse conditions, highlighting its potential as a leaf development module for integration into canopy photosynthesis and biomass production models.