R1. To see this, consider the limiting case r2 = 0. As we saw above,

R1. To see this, consider the limiting case r2 = 0. As we saw above, at long times all finite holoenzymes evolve to aPhys Biol. Author manuscript; available in PMC 2013 June 08.watermark-text watermark-text watermark-textMichalski and LoewPagesteady state where only a fraction of subunits are phosphorylated. This fraction is 1/2 for the dimer, 2/3 for the trimer, and approaches 0.62 for large holoenzymes. This effect is a direct manifestation of spatial correlations within the holoenzyme. In the ISHA all spatial information is lost and the autophosphorylation reaction never stops. Thus, the infinite holoenzyme evolves towards a state with all subunits phosphorylated, even in the limit r2 = 0. As the ISHA is a good approximation when r2 = r1, there must be some critical value of r2/r1 below which the ISHA fails. In Fig. S1 we show that the ISHA is an acceptable approximation to the hexamer for r2/r1 0.4. We conclude that the ISHA is an excellent approximation to a finite holoenzyme provided that (1) the concentration of CaM4 is not too small and (2) r2 is not significantly less than r1. The first condition can be made more explicit by noticing that, with our scaling, CaM4 is short for kon,uCaM4/r1. Examining Fig. 7 then shows that the ISHA is an excellent approximation provided kon,uCaM4/r1 0.5. In the real system, kon,u 30 M-1 s-1 and r1 1?0 s-1. Therefore, we expect the ISHA to be accurate for CaM4 50 nM, depending on the value of r1. 3.2. Six State Model Here we consider a more detailed six state model (as described in the methods) to demonstrate that the conclusions of the last section do not depend on the use of the simplified three state model. Exact models of finite holoenzymes showed only weak dependence on the size of the holoenzyme, in line with the differences in the three state model (data not shown). For convenience we compare the ISHA to the trimer, as larger holoenzymes generated extremely large and computationally taxing networks. In Fig. 8 we compare the ISHA to the trimer by plotting the fraction of I-CBP112 biological activity phospho-T286 subunits (a ) and the fraction of phospho-T305 subunits (c ) as a function of Ca2+ (assumed buffered) after a 10 second autophosphorylation reaction for a system with 0.005 M CaMKII and 1.0 M total CaM. Figures 8(a) and (c) show the ISHA is an excellent approximation to the trimer in the absence of PP1. Notably, both models predict that the amount of phospho-T305 is a non-monotonic function of Ca2+. This behavior is easy to understand: at low calcium the curve increases because adding more calcium generates more CaM-bound CaMKII, which generates more phospho-T286 and phospho-T305. At high calcium the curve decreases with increasing calcium because CaM gets “trapped” on CaMKII, which protects T305 from phosphorylation. Figures 8(b) and (d) show that the ISHA overestimates equilibrium phosphorylation levels PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/21114274 in the presence of PP1, as expected from the previous section, but that the deviations at 1 M PP1 minimal. Again, both models capture the non-monotonic behavior of phospho-T305. All of the systems considered so far have CaMKIItot = 0.005 M. The results of the three state model with CaM4 buffered are independent of total CaMKII, but with a fixed amount of CaM the accuracy of the ISHA may depend on the total CaMKII. In Figs. S2 and S3 we compare the ISHA to the trimer model for varying amounts of CaMKII both with and without phosphatases. We find that the ISHA is best if CaMKIItot < CaMtot, but that the ap.