A mathematical model captures the role of adenyl cyclase Cyr1 and guanidine exchange factor Ira2 in creating a growth-to-hyphal bistable switch in Candida albicans

Abstract

Recent biochemical experiments have indicated that in Candida albicans, a commensal fungal pathogen, the Ras signaling pathway plays a significant role in the yeast-to-hyphal transition; specifically, two enzymes in this pathway, Adenyl Cyclase Cyr1 and GTPase activating protein Ira2, facilitate this transition, in the presence of energy sensor ATP. However, the precise mechanism by which protein interactions between Ira2 and Cyr1 and the energy sensor ATP result in the yeast-to-hyphal transition and create a switch-like process are unknown. We propose a new set of biochemical reaction steps that captures all the essential interactions between Ira2, Cyr1, and ATP in the Ras pathway. With the help of chemical reaction network theory, we demonstrate that this set of biochemical reaction steps results in bistability. Further, bifurcation analysis of the differential equations based on this set of reaction steps supports the existence of a bistable switch, and this switch may act as a checkpoint mechanism for the promotion of growth-to-hyphal transition in C. albicans.

Keywords: Candida albicans; ODE; Ras signaling pathway; bistable; positive feedback; yeast-hyphal switching.

Fig. 1

Biological network of Ras‐GTP pathway. Yellow boxes with names are protein molecules. We first start on the top left of the circuit inside the dotted box with RD, a short form for Ras‐GDP, and RT for Ras‐GTP. The interactions inside the dotted box show the new reaction for the interaction between Ira2 and Cyr1, ATP and Cyr1, which plays a vital role in forming cAMP. Subsequently, the other module outside the box is the derepression of PKA by cAMP that releases the catalytic C part of the PKA and downregulates cAMP through phosphodiesteRases PDE‐1 and 2 to form a negative feedback loop. The ri's represent the rates of the reactions based on mass action kinetic laws. In detail, Ras‐GDP is converted to RT, the Ras‐GTP, by the enzyme cdc25. The enzyme Ira2 carries out the reverse reaction. Cyr1, the enzyme adenyl‐cyclase, binds to Ira2 to form a complex C5 to arrest the conversion of Ras‐GTP to Ras‐GDP, and thereby reduce its activity. Cyr1 also binds to ATP, which forms an inactive complex C6. This inactive complex C6 binds to RT to form a trimer C7 that facilitates cAMP formation. The competition between complexes C5 and C6 decides the RT activity and cAMP formation. This tug‐of‐war for RTP by C5 and C6 leads to a positive feedback loop that generates bistability for the choice of parameters. Both cAMP and ATP undergo first‐order exponential degradation. ri are the rates of the reaction. The only role of cAMP is to derepress the catalytic part of PKA from the regulatory part by a sequence of steps that forms complexes C13–C16. The final complex C16 dissociates PKA and ejects out cAMP. The catalytic part of PKA involves cAMP degradation through negative feedback that involves low‐affinity phosphodiesteRases‐1 (PDE1) and gets converted to AMP. The high‐affinity phosphodiester Rase‐2 (PDE2) also converts cAMP to AMP to maintain the cAMP homeostasis. Note that we do not provide the details of the enzymatic conversion of PDE1 to PDE1P and have given only the rates ri's.

Fig. 2

Bifurcation diagram: One parameter bifurcation diagram with Cyr1T as the bifurcation parameter and RTP (A), ATP (B), and CAMP (C) as steady‐state dynamical variables. Blue and broken red lines indicate the stable node and unstable saddle steady‐states. The meeting point of these lines is a saddle‐node bifurcation, where the switch from the yeast‐to‐hyphae (Y‐H) transition or another way (H‐Y) takes place. (D) Bistability is lost in the absence of ATP production (kf12 = 0), although it is biphasic. ATP and CAMP values are zero. Table 1 provides the value of kinetic constants. The other total concentration fixed in constructing the bifurcation parameters are Ira2T = 50, CD25T = 4, RasT = 100, PDE1T = 5, PDE2T = 5, CT = 50, RT = 50, PPAT = 5. We have provided the xppaut file in Appendix S5.

Fig. 3

Bifurcation diagram: One parameter bifurcation diagram with Ira2T as the bifurcation parameter and RTP (A), ATP (B), and CAMP (C) as steady‐state dynamical variables. (D) Two‐parameter bifurcation diagram with total Ira2 as the second parameter. Two cusp bifurcations are seen. The red dot is inside the boundary of the cusp where both normal yeast and hyphal morphology coexist. Outside the left boundary, predominantly the hyphae form exists, while on the right boundary with low Cyr1 total, the yeast form exists. Table 1 provides the values of the kinetic constant. The other total concentration fixed in constructing the bifurcation parameters are Cyr1T = 500, CD25T = 4, RasT = 100, PDE1T = 5, PDE2T = 5, CT = 50, RT = 50, PPAT = 5.

Fig. 4

Bifurcation diagrams on the effect of ATP rate constant kf12 on the dynamics. (A) One parameter bifurcation diagram indicates a rapid change in ATP and cAMP with kf12 as a bifurcation diagram keeping Cyr1T = 500 and Ira2T = 50. (B–D) Delay in the saddle‐node bifurcation with the increase of kf12 for RTP, ATP, and CAMP with Cyr1T as the bifurcation parameter. Three kf12 values are chosen and they are kf12 = 4 (left), kf12 = 9.5940921 (middle, taken as standard), and kf12 = 9.5940921.

Fig. 5

(A) Two‐parameter bifurcation diagram in kf12‐Ira2T and kf12‐Cyr1T parameter plane that exhibits cusp bifurcation. To simulate, we took Ira2T = 15 and Cyr1T = 500, keeping the other parameters constant. (B) In the top subplot, Ira2T = 0 and Cyr1T = 5. Bistability is absent. RTP and ATP are at high levels. CAMP is zero. In the bottom subplot, CYR1T = 0, IRA2T = 50. Here also bistability is absent. However, when both RTP and cAMP are zero, ATP remains high. (C) In the top subplot, both Ira2T = Cyr1T = 50. Again, bistability is absent. In the bottom subplot, Cyr1T = 150 = 3 × Ira2T (= 50). Bistability is present. The inset plot shows the magnification of the ATP and cAMP bifurcation diagram. The condition Cyr1T ≫ Ira2T is important for the presence of bistability by keeping the other parameters constant. See main text for explanation. (D) Increase in PDE2T, a high‐affinity phosphodiesterase‐2 from 5 to 500, results in a lowering of the cAMP level. There is no change seen in the bifurcation points. However, RTP and ATP steady‐state values remained the same (not shown) since PDE2 is a downstream effector of only cAMP.

Fig. 6

Biological circuit of Ras‐GTP pathway for model‐2. Yellow boxes denote the proteins with their names. We first start from the top left of the circuit inside the dotted box with RD, a short form for Ras‐GDP and RT for Ras‐GTP. The circuits inside the dotted box show the newly proposed reaction for the interaction between Ira2 and Cyr1, ATP, and Cyr1, which plays an essential role in forming cAMP. Subsequently, the other module outside the box is the derepression of PKA by cAMP that releases the catalytic C part of the PKA and downregulates cAMP through phosphodiester Rases PDE‐1 and 2 to form a negative feedback loop. The ri's are the corresponding rates of the reaction based on mass action kinetic laws. In this network, Ras‐GDP is converted to RT, the Ras‐GTP, by the enzyme cdc25. The enzyme Ira2 carries out the reverse reaction. Cyr1, the enzyme adenyl‐cyclase, binds to Ira2 to form a complex C5 to arrest the conversion of Ras‐GTP to Ras‐GDP, and thereby reduce its activity. Cyr1 also binds to RTP, which forms an inactive complex C8. This inactive complex binds to ATP to form a trimer C9 to activate Cyr1 and facilitates cAMP formation. The rest of the details about PKA interaction with CAMP are the same as provided in the caption of Fig. 1.

Fig. 7

Bifurcation diagram for model‐2: One‐parameter bifurcation diagram with Cyr1T as the bifurcation parameter and RTP (A), ATP (B), and CAMP (C) as steady‐state dynamical variables. (D) Two‐parameter bifurcation diagram with total Ira2 as the second parameter. Two cusp bifurcation is seen. The total conserved species are Ira2T = 15, CD25T = 10, RasT = 150, PDE1T = 5, PDE2T = 5, CT = 5, RT = 5, PPAT = 5. The xppaut file is given in Appendix S8.