Optimization modeling of plant root architecture for water and phosphorus acquisition

An optimization model is presented that examines the relationship between root architecture and multiple resource acquisition, specifically water and phosphorus in spatially heterogeneous environments. The basal root growth angle of an individual common bean plant, which determines the orientation and localization of the bulk of the root system, was modeled as the decision variable. The total payoff to the plant, the benefit obtained from water and phosphorus acquisition, minus the costs of spatial competition between roots, is given as a function of the (x,y) coordinates of the basal root in two-dimensional Cartesian space. We obtained a general solution and applied it to four unique environmental cases which are as follows: (1) the case of uniformly distributed water and phosphorus; (2) the case of localized shallow phosphorus; (3) the case of localized deep water; and (4) the case of shallow phosphorus and deep water. The general solution states that the optimal basal root growth angle will occur at the point where the total rate of change in the value of the resources acquired equals the total rate of change in cost that results from locating the root deeper in the soil. An optimizing plant locates its roots deeper in the soil profile until the marginal benefit exactly equals the marginal cost. The model predicts that the basal root angle of an optimizing plant will be shallower for Case 2 and deeper for Case 3, relative to the basal root angle obtained in the case of uniformly distributed water and phosphorus. The optimal basal root angle for Case 4 will depend on the marginal rate of substitution of water availability for phosphorus availability that occurs with depth. Empirical observations of bean root architecture in the greenhouse and in the field confirm model results and are discussed. In addition, the potential importance of phenotypic plasticity and phenotypic variation are discussed in relation to optimization of traits and adaptation to spatially heterogeneous environments.