A model for the expression of gap genes based on the Jeffreys-type equation.

Abstract

We propose the third-order model equation of the Jeffreys type for concentrations of gap gene proteins in order to take into account particle inertia. Gap genes are responsible for formation of body segments in Drosophila melanogaster embryo during its early development. Usually the expression of the genes is described by the model of protein transport based on conventional diffusion equation. However, the model is known to govern the Brownian (non-inertial) motion of particles; hence, it is hardly applicable to the description of protein transport. Analysis of the Jeffreys-type equation results in the necessary condition for the problem to be well-posed. Application of the Jeffreys-type equation with non-linear terms to description of the dynamics of gap gene network demonstrates better fitting to experimental data than the conventional model. Implementation of solver algorithms and the software are freely available from: https://github.com/wswgG/solver-for-the-Jeffreys-type-equations-system .