Improvement and evaluation of a mathematical model for fertilization calcium waves in Caenorhabditis elegans

Ca²⁺ waves propagate through the oocyte during fertilization, activate the oocyte and induce embryonic development. Ca²⁺-induced Ca²⁺-release (CICR) is a mechanism of Ca²⁺ wave formation. We previously quantified the Ca²⁺ waves in the nematode Caenorhabditis elegans by using high-speed imaging and image analysis. We found that the waves consist of a rapid local rise at the point of sperm entry and a slow global wave. We demonstrated that the Nagumo model, which models the CICR by a reaction-diffusion equation, can produce a similar biphasic waveform. However, the model cannot represent the observed gradual decrease in maximum Ca²⁺ concentration with increasing distance from the point of sperm entry. In this study, we introduced a linear monotonically decreasing function into the reaction part of the Nagumo model. We demonstrated that our new model can produce the gradual decrease in maximum Ca²⁺ concentration with increasing distance from the point of sperm entry and a biphasic waveform simultaneously.