Analysis of energy state of a discrete fractionally damped spherical net of mouse Zona Pellucida before and after fertilization

Abstract

Zona pellucida (ZP) is a 3D matrix that surrounds mammalian oocytes and embryo until the stage of early blastocyst. This structure is important for fertilization, polyspermy block, integrity of the growing embryo, guiding the embryo through the oviduct. In the late blastocyst stage, this structure no longer exists. During oocyte maturation, fertilization and embryo development ZP dynamically changes its elasticity. To explain elasticity change of mouse ZP (mZP) during these processes, a discrete fractional order spherical net model of mZP is developed. Elements in the model correspond to ZP glycoproteins that are interconnected with standard light fractional order elements. This model is suitable for modelling different states of the ZP net. Using the part of the ZP spherical net model that still preserves the molar ratio of ZP glycoproteins, we determined characteristic eigen numbers of eigen fractional order oscillations and eigen modes for each chain. Analytical expressions for mechanical energy of the representative part of the ZP net before and after fertilization are defined. Generalised function of energy dissipation for the representative part of the ZP net as well as for the whole ZP net was created. We discussed dynamical change of elasticity of the mZP and dissipation of its energy after fertilisation in biological context.