Functions (filled circles) and (open squares) from Eq. (10) have three intersections. The two interactions, one at 0.99 and one near 0.4, are clearly visible. To see the third one near 0.0004, we note that and the slopes of and at the origin are and , respectively. Thus there is a third intersection near the origin.
Two examples of as the solution to Eqs. (22a) and (22b). The lower line: , , , and ; therefore, . The top curve: , , , and ; therefore .
Molecular competition in homogeneous dual PdPC system according to Eqs. (19a) and (19b) or equivalently Eqs. (17a) and (17b) with and . Each panel has two nullclines whose intersections are the steady state of the system: open circles are unstable and filled circles are stable.
For and and with small and , the and nullclines are very near the , as they are left of the two vertical asymptotes shown in (a). The intersection between two nullclines, however, is near the if and near the if . Thus, as the function of defined in the text, there is an ultrasensitive transition between and shown in (b).
The dynamics of dual PdPC is confined on the triangle shown in (a): and . For heterogeneous dual PdPC operating under zeroth-order kinetics, the vector field in the triangle is a constant given in Eq. (33). The dynamics moves from the domain to its boundaries. When it is on the edge, the dynamics follows the direction shown in (b): if , the steady state is (0,0,1) and if , the steady state is (0,1,0). The same applies to the edge with . The dynamics confined on (near) the edge of is more delicate. See the text for more discussions.
A stochastic kinetic model for dual PdPC with kinase and phosphatase catalyzed reactions. One has the transition rates going right, left, up, and downward being , , , and , respectively. For heterogeneous dual PdPC with zeroth-order kinases and phosphatases, all are constant as in Eq. (33). However, for shared kinase and phosphatase, there are competitions for kinase the phosphatase when they are saturated. Hence the are no longer independent of . In fact, , , , and .
Steady state probability distribution of the stochastic model given in Fig. 6 for homogeneous dual PdPC with , , , and . The distribution is bimodal with peaks at upper-right and lower-left corners representing the and .
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