Experimental design and inverse problems in plant biological modeling

We develop a mathematical and statistical framework to model the actions of underlying metabolites for carbon dioxide assimilation in photosynthesis. This study was motivated by a challenge posed by Syngenta Biotechnology to use modeling to better characterize photosynthesis in plants and subsequently better understand growth and crop yield. We use a dynamical system model proposed by Zhu., et al. [16], which describes the Calvin Cycle in spinach plants through changes in the concentrations of 38 metabolites (state variables) using non-linear enzyme kinetic ordinary differential equations and mass-balance laws that contain a total of 165 parameters. In our study of the CO2 assimilation rate, we pose our research questions with this dynamical system mathematical model and a statistical model to describe the observation process. In particular, we address the research question "Once a subset of parameters is fixed and the times at which data is collected are determined, can we identify which metabolites we should measure in order to optimize the confidence in our parameter estimates?" This problem of choosing the best subset of metabolites to measure is not well-explored in the existing literature. Here, we propose two methods. The first, the ad hoc Simple Linear Regression (SLR) method, models carbon assimilation as a linear function of the metabolites. Using simulated data, we implement a step-wise selection algorithm to determine the best subset of metabolites at each of 10 fixed time points. The second method, the Optimal Design Criterion method, fixes the number of metabolites to be observed and searches for the best combination by calculating a measure of the size of the Fisher Information Matrix (FIM) associated with measuring only the selected metabolites. Both methods suggest promise in determining appropriate sets of metabolites to observe for successful implementation of inverse problems. Our conclusions represent a paradigm for estimating parameters and designing experiments more efficiently in plant biological modeling. Ultimately, our results may be used to help engineer seeds that maximize carbon dioxide assimilation in photosynthesis and hence promote plant growth.