Stability analysis of a signaling circuit with dual species of GTPase switches

GTPases are molecular switches that regulate a wide range of cellular processes, such as organelle biogenesis, position, shape, and function, vesicular transport between organelles, and signal transduction. These hydrolase enzymes operate by toggling between an active ("ON") guanosine triphosphate (GTP)-bound state and an inactive ("OFF") guanosine diphosphate (GDP)-bound state; such a toggle is regulated by GEFs (guanine nucleotide exchange factors) and GAPs (GTPase activating proteins). Here we dissect a network motif between monomeric (m) and trimeric (t) GTPases assembled exclusively in eukaryotic cells of multicellular organisms. To this end, we develop a system of ordinary differential equations in which these two classes of GTPases are interlinked conditional to their ON/OFF states within a motif through feedforward and feedback loops. We provide explicit formulas for the steady states of the system and perform classical local stability analysis to systematically investigate the role of the different connections between the GTPase switches. Interestingly, a feedforward from the active mGTPase to the GEF of the tGTPase was sufficient to provide two locally stable states: one where both active/inactive forms of the mGTPase can be interpreted as having low concentrations and the other where both m- and tGTPase have high concentrations. Moreover, when a feedback loop from the GEF of the tGTPase to the GAP of the mGTPase was added to the feedforward system, two other locally stable states emerged, both having the tGTPase inactivated and being interpreted as having low active tGTPase concentrations. Finally, the addition of a second feedback loop, from the active tGTPase to the GAP of the mGTPase, gives rise to a family of steady states that can be parametrized by a range of inactive tGTPase concentrations. Our findings reveal that the coupling of these two different GTPase motifs can dramatically change their steady state behaviors and shed light on how such coupling may impact information processing in eukaryotic cells.