Probability fluxes and transition paths in a Markovian model describing complex subunit cooperativity in HCN2 channels

Abstract

Hyperpolarization-activated cyclic nucleotide-modulated (HCN) channels are voltage-gated tetrameric cation channels that generate electrical rhythmicity in neurons and cardiomyocytes. Activation can be enhanced by the binding of adenosine-3',5'-cyclic monophosphate (cAMP) to an intracellular cyclic nucleotide binding domain. Based on previously determined rate constants for a complex Markovian model describing the gating of homotetrameric HCN2 channels, we analyzed probability fluxes within this model, including unidirectional probability fluxes and the probability flux along transition paths. The time-dependent probability fluxes quantify the contributions of all 13 transitions of the model to channel activation. The binding of the first, third and fourth ligand evoked robust channel opening whereas the binding of the second ligand obstructed channel opening similar to the empty channel. Analysis of the net probability fluxes in terms of the transition path theory revealed pronounced hysteresis for channel activation and deactivation. These results provide quantitative insight into the complex interaction of the four structurally equal subunits, leading to non-equality in their function.

Figure 1. The C4L-O4L model.

(A) The C4L-O4L model is composed of five closed (C _x) and five open states (O _x; x = 0…4). Ligand (L) binding is possible in both the closed and the open channel. The closed-open isomerization can proceed at each degree of liganding. The rate constants and their errors were determined previously . k _O3C3 was set equal to k _O4C4. Since for these rate constants only a lower border could be estimated, they were set to 3 s⁻¹ herein if not otherwise noted. This results in k _C3O3 = k _C4O4 = 2.0×10² s^.1. (B) Time-dependent probability to stay in a closed state, P _C,x; (x = 0…4) for a ligand jump (green bar) from zero to 7.5 µM and back to zero. P _c denotes the sum of all P _C,x at each time. (C) Time-dependent probability to stay in an open state, P _O,x; (x = 0…4) for a ligand jump (green bar) from zero to 7.5 µM and back to zero. P _o, the sum of all P _O,x at each time, indicates the measured time course of the total open probability. (D) Scheme of the C4L-O4L model with means of rate constants provided by Table S1.

Figure 2. Time courses of net probability flux density in the C4L-O4L model.

The time courses for the net probability flux density, f _XY, in the C4L-O4L model were computed for the event of applying the ligand fcAMP (7.5 µM; left diagrams) and removing it again (right diagrams). The inset diagrams show the original diagram with an either higher amplitude or time resolution. (A) Net probability flux densities between closed states following ligand application (left) and removal (right) obtained by equation (1). (B) Net probability flux densities between open states following ligand application (left) and removal (right) obtained by equation (2). (C) Net probability flux density of the closed-open isomerizations following ligand application (left) and removal (right) obtained by equation (3).

Figure 3. Total net probability fluxes in the individual transitions of the C4L-O4L model.

The total net probability fluxes per transition, F _XY (given as fraction of unity), were obtained by integrating (equation (4)) the time courses of the net probability flux densities obtained by equation (1) to (3). The data after stepping to 7.5 µM fcAMP and after its removal are shown on the left and right side, respectively. The dotted bars indicate the effect of fitting with k _O3C3 to 30, instead of 3, on F _C3C4, F _O3O4, F _C3O3, and F _C4C4. The sum of F _C3O3 and F _C4C4 is practically unchanged. The other total net probability fluxes were unchanged. (A) Total net probability fluxes for the transitions between neighbored closed states. (B) Total net probability fluxes for the transitions between neighbored open states. (C) Total net probability fluxes for the five closed-open transitions.

Figure 4. Transition pathways in the C4L-O4L model at saturating fcAMP.

The weight of the pathway net probability fluxes, obtained by equation (5), is given by the numbers besides the colored graphs (as fraction of unity) and, for values >0.030, also encoded by the thickness of the arrows. Fluxes from 0.002 to 0.030 are illustrated by equally thick dotted arrows. All other fluxes are only very small (<4.5×10⁻⁴) and therefore not represented by arrows. For further explanation see text. (A) Pathway net probability fluxes along the pathways C ₀ to O ₄ (red) and O ₀ to O ₄ (blue) after switching to 7.5 µM fcAMP. (B) Pathway net probability fluxes along the pathways O ₄ to C ₀ (red) and O ₄ to O ₀ (blue) after removing fcAMP.

Figure 5. Transition pathways in the C4L-O4L model at subsaturating fcAMP.

The illustration is analogous to Fig. 4. The two main net probability fluxes at each concentration are shown. Pathway net probability fluxes associated with activation (C ₀→O _x, O ₀→O _x) and deactivation (C _x→O ₀, O _x→O ₀) are shown in red and blue color, respectively. (A) 0.75 µM fcAMP, C ₀↔O ₄. (B) 0.75 µM fcAMP, net probability flux along C ₀↔O ₂ in addition to that included in C ₀↔O ₄. (C) 0.075 µM fcAMP, C ₀↔O ₂. (D) 0.075 µM fcAMP, net probability flux along C ₀↔O ₁ in addition to that included in C ₀↔O ₂.

Figure 6. Unidirectional probability flux densities in the closed-open isomerizations.

The time courses of the unidirectional probability flux densities, f _U,CxOx and f _U,OxCx (x = 0…4) for the closed-open isomerizations are plotted together with the respective net probability flux densities, f _CxOx, when applying 7.5 µM fcAMP (left) and removing it (right). The arrows indicate the time point of concentration change. For explanation see text.