A Mathematical Model of Sap Exudation in Maple Trees Governed by Ice Melting, Gas Dissolution, and Osmosis

Abstract

We develop a mathematical model for sap exudation in a maple tree that is based on a purely physical mechanism for internal pressure generation in trees in the leafless state. There has been a long-standing controversy in the tree physiology literature over precisely what mechanism drives sap exudation, and we aim to cast light on this issue. Our model is based on the work of Milburn and O'Malley [Can. J. Bot., 62(10):2101-2106, 1984] who hypothesized that elevated sap pressures derive from compressed gas that is trapped within certain wood cells and subsequently released when frozen sap thaws in the spring. We also incorporate the extension of Tyree [in Tree Sap, pp. 37-45, eds. M. Terazawa et al., Hokkaido Univ. Press, 1995] who argued that gas bubbles are prevented from dissolving because of osmotic pressure that derives from differences in sap sugar concentrations and the selective permeability of cell walls. We derive a system of differential-algebraic equations based on conservation principles that is used to test the validity of the Milburn-O'Malley hypothesis and also to determine the extent to which osmosis is required. This work represents the first attempt to derive a detailed mathematical model of sap exudation at the micro-scale.