Mathematical description for nodule size distribution, and number, weight, and diameter of nodules deduced from a theoretical model of photosynthate competition among nodules

Abstract

To describe the nodule size distribution as well as the total number, weight, and diameter, and average weight and diameter of nodules using a set of input parameters, mathematical equations were derived from a model in which photosynthate allocation determined the nodule size after competition for photosynthates among nodules. This model contains four input parameters, fC, N, a, and b, representing the rate of photosynthate supply to the root system, distribution density of nodulation sites, a partition coefficient of photosynthates in the root system, and a threshold of photosynthate supply to maintain a visible nodule, respectively. However, the mathematical equations finally required only a set of three values, namely N / S ( = N / afC), N / a, and b for the description. The equation of the nodule size distribution was applied to the nodulation of soybeans (Glycine max L. Merr. cvs. Kitamusume and Toyosuzu) grown under different conditions and for various periods of time. A set of N / S, N / a, and b values was determined by simultaneously solving the equation of the nodule size distribution at a peak and another point. Consequently, whole theoretical nodule size distribution, and theoretical values of nodule number, size, and diameter were calculated from the obtained values of N / S, N / a, and b. They were compared with the actual distribution and values to evaluate the accuracy of the equations. They showed a good consistency with the actual ones. Averages of relative differences between the calculated and measured values were 21.9, 17.2, 43.2, 10.2, and 35.5% for the total number, diameter, and weight of the nodules, and average diameter and weight of a nodule, respectively. Simulation demonstrated the role of the parameters in the model and proved similarity between C and photosynthesis ability and between N and inoculation dose in their behaviour.