Figure 1.
The ratio of the concentration of kinases to the concentration of their substrates does not appear biased in S. cerevisiae.
(A) The distribution of kinase saturation levels, (the ratio of kinase concentration to substrate concentration), for the phosphorylation reactions in the BioGRID database [18] with levels of gene expression during log phase growth in rich media measured by Ghaemmaghami et al. [19]. The data contains 2850 phosphorylation reactions, comprising 98 unique kinases and 1136 unique substrate targets. (B) Equivalent distribution for phosphatases. The data contains 43 dephosphorylation reactions, comprising 16 unique phosphatases and 32 unique targets.
Figure 2.
A model of a phosphorylation-dephosphorylation cycle for an allosteric substrate with multiple phosphorylation sites has two “sink” states and multiple competitions between the modifying enzymes for the substrate.
For each level of phosphorylation, the free substrate switches conformations allosterically between (inactive, in grey) and
(active, in yellow). The rate of allosteric transitions from active states to inactive states is
and the reverse rate of allosteric transitions from inactive states to active states is
. The kinase (in red) binds to the active forms of the substrate; the phosphatase (in blue) binds to the inactive forms. The two sinks
and
, are shown with shadows. The rate of association of the kinase to the substrate is
; the rate of dissociation of the kinase-substrate complex is
. Phosphorylation of a phosphosite resulting in dissociation of the kinase from the substrate has rate
per phosphosite (distributive catalytic rate represented by the diagonal black arrows). Phosphorylation of a phosphosite whereby the enzyme remains bound to the substrate has a rate
(non-distributive catalytic rate represented by the straight grey arrows). The rate of association of the phosphatase to the substrate is
and its rate of dissociation is
. De-phosphorylation of a phosphosite resulting in dissociation of the phosphatase from the substrate has a distributive rate
per phosphosite. Dephosphorylation of a phosphosite whereby the enzyme remains bound to the substrate has a non-distributive rate
.
Figure 3.
Ultrasensitivity occurs because of sequestration of the substrate into one of two sinks (either fully phosphorylated and bound to kinase or fully dephosphorylated and bound to phosphatase) and is enhanced by the existence of an allosteric bias.
(A) Simulation of the response curve (in black) and the concentration of the states of the system (coloured bars) as a function of the signal, i.e., the ratio of phosphatase to kinase. Here we have ,
,
s−1,
s−1 (in units of the inverse total concentration of the substrate),
s−1,
s−1,
. The Hill number is approximately 3.5. (B) Contour plot of the Hill number as a function of the allosteric bias and the relative dissociation rate obtained from Eq. (17). In this panel
, the allosteric bias is given by
and the relative dissociation rate is
with
and
. (C) An allosteric bias allows non-distributive systems to become ultrasensitive. Contour plots of the Hill number as a function of the allosteric bias and the relative dissociation rate, obtained from solving Eq. (12) for
. Left: purely distributive case (
); centre: coexistence of distributivity and non-distributivity (
); right: non-distributive case (
). The relative dissociation rate is defined as in (B) and
, except for the right panel where the relative dissociation rate is defined as
and
. In (B) and (C), the solid black line marks the boundary between subsensitivity (above the line) and ultrasensitivity (below the line).
Figure 4.
Phosphorylation-dephosphorylation cycles can generate non-monotonic input-output relationships.
Here the allostery of the substrate inhibits further modifying reactions by disfavouring binding of the kinase when the substrate is phosphorylated and disfavouring binding of the phosphatase when the substrate is dephosphorylated. Simulation of the response curve (in black) and the concentration of the states of the system (coloured bars) as a function of the signal, i.e., the ratio of phosphatase to kinase, for a system where ,
,
s−1,
s−1,
s−1 (in units of the inverse total concentration of the substrate),
s−1,
s−1,
.
Figure 5.
Allostery is not required for ultrasensitivity and models with steric hindrance where the binding of one enzyme inhibits binding of the other can be highly ultrasensitive.
(A) A model with steric hindrance. The kinase (red) and the phosphatase (blue) bind either to the same docking site or to different docking sites and block the access of the other enzyme to its docking site. The two sinks, and
, are shown with shadows. The non-distributive catalytic rates (dashed grey arrows) were not allowed in this model. (B) Simulation of the response curve (in black) and the concentration of the states of the system (coloured bars) as a function of the signal, i.e., the ratio of phosphatase to kinase for a system, with
,
,
s−1,
s−1 (in units of the inverse total concentration of the substrate),
s−1,
s−1,
s−1. The Hill number is approximately 10.