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, 12 (11), e1005216
eCollection

Kinase Inhibition Leads to Hormesis in a Dual Phosphorylation-Dephosphorylation Cycle

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Kinase Inhibition Leads to Hormesis in a Dual Phosphorylation-Dephosphorylation Cycle

Peter Rashkov et al. PLoS Comput Biol.

Abstract

Many antimicrobial and anti-tumour drugs elicit hormetic responses characterised by low-dose stimulation and high-dose inhibition. While this can have profound consequences for human health, with low drug concentrations actually stimulating pathogen or tumour growth, the mechanistic understanding behind such responses is still lacking. We propose a novel, simple but general mechanism that could give rise to hormesis in systems where an inhibitor acts on an enzyme. At its core is one of the basic building blocks in intracellular signalling, the dual phosphorylation-dephosphorylation motif, found in diverse regulatory processes including control of cell proliferation and programmed cell death. Our analytically-derived conditions for observing hormesis provide clues as to why this mechanism has not been previously identified. Current mathematical models regularly make simplifying assumptions that lack empirical support but inadvertently preclude the observation of hormesis. In addition, due to the inherent population heterogeneities, the presence of hormesis is likely to be masked in empirical population-level studies. Therefore, examining hormetic responses at single-cell level coupled with improved mathematical models could substantially enhance detection and mechanistic understanding of hormesis.

Conflict of interest statement

IPB and CB were employees of AstraZeneca at time of writing of the manuscript. This does not alter the authors’ adherence to PLOS policies on sharing data and materials.

Figures

Fig 1
Fig 1. Dose response curve (log scale) for the double-phosphorylated substrate CPP in the presence of an inhibitor (inhtot), in the case when the motif Eq (1) is monostable.
In the absence of the inhibitor the stable steady state value of the double-phosphorylated substrate is denoted by CPP*. With the inhibitor present, the dose response exhibits hormetic properties whereby for sufficiently low inhibitor doses (inhtot<inh*) the computed steady state values of CPP increase monotonically, before monotonically decreasing for inhibitor concentrations inhtot>inh*. The magnitude of hormetic response is calculated as a difference between the CPP value at inh* and CPP*.
Fig 2
Fig 2. Dose response curve (log scale) for the double-phosphorylated substrate CPP in the presence of an inhibitor (inhtot), in the case when the motif Eq (1) is bistable.
In the absence of the inhibitor there are two stable steady states of the double-phosphorylated substrate (A) CPP,1* and (B) CPP,2*. Cell populations at these two steady state will react differently to the presence of an inhibitor: (A) cells at CPP,1* will exhibit a monotone dose-response while (B) cells at CPP,2* will exhibit a hormetic dose response whereby for sufficiently low inhibitor doses (inhtot<inh*) the computed steady state values of CPP increase monotonically, before monotonically decreasing for inhibitor concentrations inhtot>inh*. The dotted lines indicate a discontinuous jump in the steady state values of CPP in the presence of the inhibitor. The magnitude of hormetic response is calculated as a difference between the CPP value at inh* and CPP,2*.
Fig 3
Fig 3. Maximum hormetic response.
The maximum hormetic response is calculated as the maximal increase in CPP over all inhibitor doses, relative to the base line amount of CPP in the absence of inhibition. Since the baseline is represented at a level of 100%, hormesis is present if the maximum response strictly exceeds the baseline (> 100%). This maximum response is plotted (black dots) as a function of different substrate-kinase ratios achieved by varying the total mass of substrate and keeping the total mass of kinase constant.
Fig 4
Fig 4. Relationship between the maximum hormetic response and the flux C · kin · inh → C + inh · kin.
The maximum hormetic response is calculated as the maximal increase in CPP over all inhibitor doses, relative to the base line amount of CPP in the absence of inhibition. Since the baseline is represented at a level of 100%, hormesis is present if the maximum response strictly exceeds the baseline (> 100%). The flux is computed as e2[C · kin · inh] − e−2[C][kin · inh] using the steady state values [C](inh*), [kin · inh](inh*), [C · kin · inh](inh*) with inh* being the dose with the maximum hormetic response (see Fig 1). As e−2 increases, the flux decreases to 0.
Fig 5
Fig 5. Heterogeneous populations.
Dose response curves of type-1 (green line) and type-2 (blue line) cells in the presence of an inhibitor (inhtot), in the case when motif Eq (1) is bistable. A logistic curve (red line) is fitted to sample points (red dots) generated numerically from a population containing 88% type-1 cells and 12% type-2 cells. The logistic fit to data is used to estimate IC50 of the total population (see Supplementary Information). In the absence of the inhibitor type-1 cells are at CPP,1* stead state while type-2 cells are at CPP,2*. In the presence of the inhibitor at IC50, while the total population exhibits 50% inhibition, the same concentration has the opposing effects on the two sub-populations. In particular, type-1 cells are inhibited, which can be deduced from the observation that the steady state CPP,1** calculated at the population-level IC50 is lower than the steady state CPP,1* calculated in the absence of the inhibitor. Contrary to this type-2 cells are stimulated since the steady state CPP,2** calculated at the population-level IC50 is higher than the steady state CPP,2* calculated in the absence of the inhibitor. This stimulatory effect is amplified even further for IC*<IC50, as seen by comparing the relatively high values of the steady state CPP,2x at IC* to the relatively low values of the steady state CPP,2* in the absence of the inhibitor.

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Grant support

Funding bodies: BBSRC (BB/J010340/1); EPSRC (EP/I00503X/1); Wellcome Trust (ISSF to University of Exeter). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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