Fusion and fission events regulate endosome maturation and viral escape

Many intra-cellular processes rely on transport by endosomes. Recent experimental techniques have provided insights into organelle maturation and its specific role in, for instance, the ability of a virus to escape an endosome and release its genetic material in the cytoplasm. Endosome maturation and dynamics depend on GTPases called Rabs, found on their membrane. Here, we introduce a mathematical framework, combining coagulation and fragmentation of endosomes with two variables internal to each organelle, to model endosomes as intra-cellular compartments characterised by their levels of (active) Rab5 and Rab7. The key element in our framework is the "per-cell endosomal distribution" and its its dynamical equation or Boltzmann equation. The Boltzmann equation, then, allows one to deduce simple equations for the total number of endosomes in a cell, and for the mean and standard deviation of the Rab5 and Rab7 levels. We compare our solutions with experimental data sets of Dengue viral escape from endosomes. The relationship between endosomal Rab levels and pH suggests a mechanism which can account for the observed variability in viral escape times, which in turn regulate the viability of a viral intra-cellular infection. Author summaryEndosomes are intra-cellular receptacle-like organelles, which transport endocytosed cargo upon internalisation from the plasma membrane. These early endosomes, also known as sorting endosomes, mature to late endosomes, with a lower pH than early ones, as a consequence of the intricate dynamics of a family of molecules called Rabs. Viruses exploit this endosomal pH drop to their advantage. Here we bring together experimental data on Dengue viral escape times from endosomes and a novel mathematical framework inspired by the theory of droplet coalescence, to improve our understanding of endosome maturation, and in turn to quantify the large variability of viral escape times. This mathematical framework can easily be generalised to model the dynamics of other intra-cellular organelles, such as mitochondria or the endoplasmic reticulum.