Dynamical robustness of a Boolean model for the human gonadal sex determination

Gonadal sex determination (GSD) is a complex but poorly understood process in the early stages of embryonic development. This process determines whether the bipotential gonadal primordium (BGP) will differentiate into testes or ovaries through the activation of genetic factors related to Sertoli or Granulosa cells, respectively. The study of this developmental process remains challenging due to experimental limitations and the complexity of the underlying genetic interactions. Boolean Networks (BNs) are binary networks that simulate genetic behavior and are commonly used for modeling gene regulatory networks (GRNs) due to their simplicity when dealing with a high number of gene interactions. Reported BNs usually use a synchronous (parallel) update scheme, which means that all the nodes (representing genes) update their values simultaneously. However, the use of this update scheme has been criticized because it cannot represent biological systems that are highly regulated at a temporal scale. Asynchronous and block-sequential updating schemes appear as an alternative to tackle this issue. In the first case, the updating scheme follows a random behavior while, in the second case, the set of network nodes is partitioned into blocks such that the nodes within a block are updated simultaneously, and the blocks are considered in a specific order sequence. To assess the impact of different updating approaches in a GRN associated to GSD we first made a node reduction without losing the main dynamics of the original network which are related to the formation of testes and ovaries. Then, we tested the effect of perturbations given by the inactivation of genes on the network attractors, specifically the SRY and WNT4 genes, since the former is only present in the Y chromosome and the latter is of importance in early embryo development. We found that both genes were crucial, but WNT4 alone showed a higher percentage of attractors towards a phenotype than the SRY alone. Finally, we found that using asynchronous and block-sequential updating schemes, the attraction basins - i.e., the set of configurations that reach an attractor - remain with similar percentages to those of the original network, which supports the robustness of the model.