This model describes a core process during endocytosis. Intracellular vesicles called early endosomes contain the endocytosed cargo, e.g. signaling components like growth factors and RTKs, pathogens like viruses and nutrients like iron in transferrin. Early endosomes form an interacting pool of thousands of vesicles and jointly constitute the sorting and transport machinery in the endocytic pathway. Together with the cargo, membrane components travel to other compartments of the pathway which perform distinct functions. Hence, the functionality of the membrane components needs to be controlled dynamically in response to their instantaneous position within the pathway. This key role is taken by Rab GTPases. Early endosomes are controlled by the membrane organizer Rab5:GTP. Rab5:GTP recruits specific effector proteins which perform the specific functions of early endosomes. Here we model the recruitment, activation, hydrolysis and solubilization of Rab5:GTP. This model includes the spatial interaction terms. The results are reported in

L. Brusch, P. Del Conte-Zerial, Y. Kalaidzidis, J. Rink, B. Habermann, M. Zerial, and A. Deutsch, Protein Domains of GTPases on Membranes: Do They Rely on Turings Mechanism? In A. Deutsch, L. Brusch, H. Byrne, G. de Vries, H. Herzel (Eds): Mathematical Modeling of Biological Systems, pp. 35-48 (Birkhaeuser, Boston, 2007).

s kh kh s

s p ke EM ks ke s EM p ks p

s km1 km1 s

K1 K1

10 \frac{\partial^{2} u}{\partial x^{2}}

1 \frac{\partial^{2} v}{\partial x^{2}} + 1 p (((\frac{3 v}{2} + \frac{1}{2}) {(\frac{\partial^{2} v}{\partial x^{2}})}^{2} + (4 v - \frac{2}{3}) \frac{\partial v}{\partial x} \frac{\partial^{3} v}{\partial x^{3}} + (\frac{5 v2}{4} - \frac{7 v}{6}) \frac{\partial^{4} v}{\partial x^{4}}) Δ^{2} + (6 v - 2) {(\frac{\partial v}{\partial x})}^{2} + (3 v2 - 2 v) \frac{\partial^{2} v}{\partial x^{2}}) - Dv q (((2 - \frac{3 v}{2}) {(\frac{\partial^{2} v}{\partial x^{2}})}^{2} + (\frac{10}{3} - 4 v) \frac{\partial v}{\partial x} \frac{\partial^{3} v}{\partial x^{3}} + (\frac{4 v}{3} - \frac{5 v2}{4}) \frac{\partial^{4} v}{\partial x^{4}}) Δ^{2} + (4 - 6 v) {(\frac{\partial v}{\partial x})}^{2} + (4 v - 3 v2) \frac{\partial^{2} v}{\partial x^{2}})

0 \frac{\partial^{2} u}{\partial x^{2}}

recruitment K1

dissociation species_1 km1

compartment_0 activation species_1 species_2 ke EM ks

compartment_0 hydrolysis species_2 kh