Understanding how genes regulate each other and how gene expression is controlled in living cells is crucial to cure genetic diseases such as cancer and represents a fundamental step towards personalised medicine. The complexity and the high concurrency of gene regulatory networks require the use of formal techniques to analyse the dynamical properties that control cell proliferation and differentiation. However, for these techniques to be used and be useful, they must be accessible to biologists, who are currently not trained to operate with abstract formal models of concurrency. Petri nets, owing to their appealing graphical representation, have proved to be able to bridge this interdisciplinary gap and provide an accessible framework for the construction and execution of biological networks. In this paper, we propose a novel Petri net representation, tightly designed around the classic basic definition of the formalism by introducing only a small number of extensions while making the framework intuitively accessible to a biology-trained audience with no expertise in concurrency theory. Finally, we show how this Petri net framework has been successfully applied in practice to capture haematopoietic stem cell differentiation, and the value of this approach in understanding the heterogeneity of a stem cell population.