Figure 1.
CaMKII Autophosphorylation and PP1 Saturation Lead to Bistability in the Phosphorylation States of CaMKII at Resting Ca2+
(A) Phosphorylation of the first subunit is slow at resting Ca2+ because of the requirement for two Ca2+/calmodulin molecules (see text). Subsequent phosphorylation is faster because only one Ca2+/calmodulin is required.
(B) Schematic figure indicating how bistability arises from the dependence of phosphorylation and dephosphorylation rates on the number of subunits phosphorylated. Stable states are at the left (DOWN) and right (UP) intersections of the two curves. The middle crossing is unstable. The greater the area of the shaded region between a stable steady state and the unstable steady state, the harder it is for fluctuations to destabilize that stable steady state (the larger their basins of attraction).
(C) A 2-s pulse of high Ca2+ switches the system (with 16 holoenzymes) from a low state of phosphorylation to a higher state within the basin of attraction of the UP state (see [D]). Phosphorylation fraction is Sptot/12NCaMK.
(D) After the end of the Ca2+ pulse, it can take tens of minutes for the system to reach the dynamic equilibrium in the UP state.
(E) The UP state is stable for many years in a system with 16 holoenzymes.
Table 1.
Parameters Used in the Model
Figure 2.
Switch Stability Is a Trade-Off between Lifetimes of UP and DOWN States
(A) Spontaneous switching between UP and DOWN states in a system with eight CaMKII holoenzymes.
(B) The distribution of lifetimes between switching events is exponential, as demonstrated by the straight line fit for lifetimes of the UP state on a semi-logarithmic scale.
(C and D) Removing one PP1 enzyme (seven instead of eight) (C) leads to a longer lifetime for the UP state but shorter lifetime for the DOWN state; whereas adding one PP1 enzyme (nine instead of eight) (D) yields the opposite effect.
(E) Dependence of average lifetimes of the UP state (squares) and DOWN state (circles) as a function of the number of PP1 enzymes (with eight CaMKII holoenzymes). Filled blue symbols correspond to data points from (A), (C), and (D). Lines are approximate, analytic results (based on Materials and Methods and [51]). The optimal lifetime of the switch is defined by the intersection point of the two curves, at which the lifetimes of the UP and DOWN states are equal. Blue indicates reference parameters. Red indicates k1 = 0.75 s−1, one-half of the standard value. The lifetime does depend on kinetic parameters, but maximum stability is approximately the same, albeit with a different number of PP1 molecules.
Figure 3.
Stability of the Switch Increases Exponentially with System Size (Number of CaMKII Holoenzymes and PP1 Molecules)
(A) Spontaneous transitions in a system with four CaMKII holoenzymes and four PP1 enzymes. Phosphorylation fraction is Sptot/12NCaMK.
(B) System with 16 holoenzymes and 16 PP1 enzymes, with four times the volume of (A). Note different timescales between (A) and (B). Phosphorylation fraction is Sptot/12NCaMK.
(C) The switch's lifetime increases exponentially with system size. Numbers of all species scale together with system volume. Circles are data points, line is a linear fit, indicating an exponential dependence, because the ordinate is in logarithmic scale. The red asterisks indicate data points where we included the PP1–I1P fluctuations explicitly.
Figure 4.
The Rate of Protein Turnover Limits the Maximal Lifetime of the System and Leads to a Minimal Rate of Energy-Consuming Activity
(A and B) Stochastic changes in total phosphorylation during a transition from the UP to DOWN state, with turnover events marked by arrows, in a system with (A) four CaMKII holoenzymes and (B) 16 CaMKII holoenzymes. Red arrows indicate turnover events, which cause an abrupt drop in the level of phosphorylation.
(C) Log–log plot of lifetime of the switch as a function of turnover rate for the system with eight CaMKII holoenzymes (the red asterisk marks 30-h turnover, used as standard in this paper).
(D) The rate for an individual ring of subunits to switch off as a function of the total number of rings that are on (shown here for a system with eight CaMKII holoenzymes). As more rings are turned on, the phosphatase activity saturates and the equilibrium level of phosphorylation per ring increases. As a result, the switching-off rate for rings in the UP state for the system approaches the turnover rate (dashed line), because the probability of total ring dephosphorylation by PP1 becomes small.
(E) Dynamic equilibrium between turnover (vertical solid black lines) and switching on of rings (colored step-like lines) when the system is in the UP state. At the time of the first turnover, five rings are already unphosphorylated by prior turnover. The system is stable because the rate of rings switching on matches the rate of turnover of phosphorylated rings (each turnover event can result in the loss of zero, one, or two phosphorylated rings). A system with 20 holoenzymes is shown.
Figure 5.
Switch Stability in the Presence of Spontaneous Fluctuations in Free Calcium Concentration and the Total Number of Enzymes
(A) Effect of Ca2+ fluctuations on stability (lifetime) as a function of number of CaMKII holoenzymes. Circles indicate the original system without Ca2+ fluctuations. Squares indicate the original system with free Ca2+ fluctuations of amplitude 0.1 μM and baseline 0.07 μM. Triangles indicate adjusted system with free Ca2+ fluctuations of amplitude 1.0 μM and baseline 0.1 μM. The adjusted system has alternative parameters, such that NPP1 = NCaMK/2, k1 = 6 s−1, k2 = 7 s−1, and KH1 = 4.0 μM. The ordinate is in logarithmic scale.
(B) Effect of fluctuations in the number of PP1 molecules on the lifetime of UP states (squares/solid line) and DOWN states (circles/dashed line) (16 holoenzymes). The timescale on the abcissa is the average time for the number of PP1 molecules to increase or decrease by one. Red indicates 12 < NPP1 < 20. Blue indicates 8 < NPP1 < 24. The ordinate is in logarithmic scale.
(C) Lifetime of UP states (squares/solid line) and DOWN states (circles/dashed line) when the number of holoenzymes and PP1 molecules fluctuate in the respective ranges 14 < NCaMK < 18 and 8 < NPP1 < 24. The timescale for PP1 fluctuations varies along the abcissa. The timescale for CaMKII fluctuations is fixed by the turnover rate (30 h per holoenzyme). The ordinate is in logarithmic scale.