Supporting Material Concise whole-cell modeling of BK Ca CaV activity controlled by local coupling and stoichiometry.

Supporting Material Concise whole-cell modeling of BK Ca CaV activity controlled by local coupling and stoichiometry F Montefusco, A Tagliavini, M Ferrante, M Pedersen List of contents. Model of the : BK Ca -CaV complex.. Monte Carlo Simulations.. Time to first opening and phase-type distributions. Time-scale analysis and model simplifications.. ODE model of the : BK Ca -CaV complex.. Model simplification.3. Time scale analysis.4. Responses of the : BK Ca -CaV complex to voltage steps and AP 3. Model for BK Ca activation in complexes with k non-inactivated CaVs and its approximation 4. Whole-cell models 4.. Hypothalamic neuronal model 4.. Human β-cell model 4.3. Pituitary lactotroph model Supporting Figures Figure S...... p. 3 Figure S...... p. 6 Figure S3...... p. 8 Figure S4...... p. 9 Figure S5...... p. Figure S6...... p. 5 Figure S7...... p. 7 Supporting Tables Table S...... p. 4 Table S...... p. 4 Table S3...... p. 9 Supporting References

Model of the : BK Ca -CaV complex First, we justify the product formulation for the voltage and Ca + -dependent rate constants k and k + (Eqs. and 3 in the main text). We assume (i) that, for fixed Ca + concentration (Ca), BK Ca activity is described by a Boltzmann function, and (ii) that the slope parameter of the Boltzmann function is independent of Ca, a reasonable assumption for Ca + concentrations above µm ( 3) as expected in BK Ca -CaV complexes (, 4). Based on assumption (i), we express the rate constants by the standard expressions k (V, Ca) = w (Ca)e wyx(ca)v, k + (V, Ca) = w + (Ca)e wxy(ca)v. The open fraction of BK Ca channels is then p Y = k + (V, Ca) k (V, Ca) + k + (V, Ca) = e V V (Ca) S (Ca) where we have highlighted that in principal V and S depend on Ca. In particular the slope parameter is given by S (Ca) = w yx (Ca) w xy (Ca), and from assumption (ii) we obtain that w xy (Ca) is equal to w yx (Ca) except from a constant independent of Ca. We make the simplifying assumption that also w yx and w xy are independent of Ca. Finally, writing, w (Ca) = w f (Ca), w + (Ca) = w + f + (Ca) we obtain Eqs. -5 in the main text. We couple the two state (closed and open) model for the BK Ca channel (see Methods, Table S reporting the optimal model parameters, and Figure S showing a representation of the model and the fits to the data) with the three state (closed, open and inactivated or blocked) model for the CaV channel (see Methods and Table S). Figure A in the main text shows a cartoon of the model of the : BK Ca -CaV complex, where CX, OX and BX correspond to the closed state for the BK Ca channel (X) coupled with the closed (C), open (O) and inactivated (B) states for the CaV, respectively, and CY, OY and BY correspond to the open state for the BK Ca channel (Y ) coupled with the closed (C), open (O) and inactivated (B) states for the CaV, respectively. The parameters kc and ko (k c + and k o + ) are defined by Eq. (Eq. 3) of the main text with Ca equal to Ca c and Ca o, respectively. Ca c is the concentration at the BK Ca channel when the associated CaV is closed (or inactivated, i.e. Ca c = Ca b ) and is set equal to. µm (background Ca + concentration). Ca o is the concentration at the BK Ca channel when the CaV is open and is given by i Ca Ca o = 8πrD Ca F exp r, (S) DCa k + B [B total] where i Ca = ḡ Ca (V V Ca ) is the single-channel Ca + current, and r = 3 nm the distance between CaV and BK Ca channels (, 4). At V = mv, Ca o 9 µm. Other parameters are given in Table S (). Note that for the parameters used here (Table S), k c + (i.e., the probability of BK Ca opening when the CaV is closed is practically zero).

A Ca + X w + w - f - f + Y Ca + B C Y p.8.6 + Ca [μμ] 8.4 7.8..4.9 p Y τ.5.5 D E τp Y F.5.5 τp Y G.5.5 τp Y 6 4 τp Y 6 4 Figure S: BK Ca channel model: fit to the data. (A) Schematic representation of the model, where X and Y indicate the closed and open states of the BK Ca channel. (B) Steadystate BK Ca open probabilities vs. voltage at different Ca + concentrations (different markers for different Ca + levels as indicated in the legend) and the corresponding fit obtained by the model (dashed lines). (C)-(G) Time constants (in ms) vs. voltage data at given Ca + concentrations (see legend in panel B), and the corresponding model fits (dashed lines). 3

Table S: Optimal model parameters for the BK Ca -CaV complex. BK Ca channel model described by Eq. of the main text Parameter Value Unit w 3.3 ms w +. ms w yx. mv w xy.36 mv K yx. µm K xy 6.6 µm n yx.46 - n xy.33 - CaV activation described by Eqs. 6 and 7 of the main text Parameter Value Unit α.979 ms α.639 mv β.665 ms β.73 mv ρ.39 - Table S: Parameters for calculating Ca + at the BK Ca channel taken from (). Parameter Value Unit r 3 nm D Ca 5 µm s F 9.6485 C mol k B 5 µm s B total 3 µm V Ca 6 mv ḡ Ca.8 ps 4

. Monte Carlo Simulations We perform Monte Carlo simulations for the devised complex in order to model its stochastic gating. Initially the CaV and BK Ca channels are closed. Then, for both the channels, the transitions from one state (closed, open or inactivated for CaV, closed or open for BK Ca ) to the others are given by the procedure explained below. First, we define the transition matrices for the CaV, P [C, t + t C, t] P [C, t + t O, t] Q CaV = P [O, t + t C, t] P [O, t + t O, t] P [O, t + t B, t] P [B, t + t O, t] P [B, t + t B, t] α t β t (S) = α t β t δ t γ t, δ t γ t and for the BK channel, [ ] P [X, t + t X, t] P [X, t + t Y, t] Q BKCa = P [Y, t + t X, t] P [Y, t + t Y, t] [ k = + t k ] (S3) t k + t k t where the elements correspond to the transition probabilities between the indicated states in the time interval [t, t + t], provided that t is small. At any time point, we compute the state of the BK Ca -CaV complex according the following procedure. We choose a random number ξ uniformly distributed on the interval [, ] for the CaV channel, and make a transition based upon the subinterval in which ξ falls. For example, if the CaV channel is open (O) (see the second column of Q CaV defined by Eq. S), it remains open if ξ < β t δ t, while a transition to the inactivated (B) state occurs if ξ β t δ t and ξ < β t, otherwise (ξ β t and ξ ) a transition to the closed (C) state occurs. Similarly, we choose a random number η uniformly distributed on the interval [, ] for the BK Ca channel, and make a transition based upon the subinterval in which η falls. Note that the Ca + concentration for the BK Ca channel at any time point is determined by the state of the CaV channel. This procedure is repeated for every time point to achieve the Monte Carlo simulations. We used t =. ms. Smaller time steps were checked and gave identical results.. Time to first opening and phase-type distributions The continuous-time Markov chain Z in Figure A in the main text takes values in the state space S = {CX, OX, BX, BY, CY, OY }. We denote its initial distribution by λ and its generating matrix by Q, where α k c + α k + c β β δ k o + δ k + o Q = γ γ k c + k c + kc γ kc γ. kc α kc α ko δ β β δ ko 5

CaV BK Ca.5 5.5 5.5 5.5 5 5.5.5 5 Fraction of BK Ca opening at least once.8.6.4. 5 Figure S: Time to first opening for the : BK Ca -CaV complex. Left: Three Markov chain realizations for the : BK Ca -CaV complex. Right: The empirical fraction of BK Ca channels that exhibit the first channel opening before t (red) and the theoretical expression P (T CX,Y < t) (black) from Eq. of the main text. Note that in the second and third examples of Markov chain realizations, the CaV inactivates early before the BK Ca channel opens, and only later when the CaV reactivates (at 5 ms and 7 ms) does the BK Ca open for the first time. Such cases cause the late, slowly rising phase in the right panel, whereas the more typical case where the BK Ca channel opens early (as in the first example shown on the left) creates the rapid early rise in the right panel. Assuming that k c + =, the generating matrix Q become α α β β δ k o + δ k + o Q = γ γ kc γ kc γ. (S4) k c α kc α ko δ β β δ ko Define the random variable H = inf{t : Z t {CY, OY, BY }}, (S5) i.e., the (random) time to first opening, which can be described with phase-type distribution theory (5). We are interested in evaluating its conditional distribution and expectation, given that Z = CX, i.e., with the initial distribution λ = (,,,,, ). Defined S = {CX, OX, BX, BY, CY } and the matrix Q = (q i,j ) i,j S, a simple application of phase-type distribution theory yields Eqs. and of the main text, as shown in Time to first opening in the Results section. Figure S shows that the simulated first opening times are well described by Eq. of the main text. 6

Time-scale analysis and model simplifications. ODE model of the : BK Ca -CaV complex The deterministic description of the complex, corresponding to the Markov Chain described above, is given by the following ODE system dp CX dp CY dp OX dp OY dp BX dp BY = k c p CY + βp OX (k + c + α)p CX, (S6) = k + c p CX + βp OY (k c + α)p CY, (S7) = k o p OY + αp CX + γp BX (k + o + β + δ)p OX, = k + o p OX + αp CY + γp BY (k o + β + δ)p OY, (S8) (S9) = k c p BY + δp OX (k + c + γ)p BX, (S) = k + c p BX + δp OY (k c + γ)p BY, (S) where p Z, with Z {CX, CY, OX, OY, BX, BY }, represents the probability of the complex to be in one of the six states of the model. Then p Y = p CY + p OY + p BY. In the following, we assume k c + =.. Model simplification As explained in the Results section, since re- and inactivation of CaVs are slower than (de- )activation, the ODE model defined by Eqs. S6 S can be split into two submodels with respectively 4 and states (the green and blue boxes in Figure A in the main text). Then, by considering only the activation of the BK Ca -CaV complexes with non-inactivated CaV, and exploiting the relations p CY +p OY = m BK, p OX +p OY = m CaV, and p CX +p CY = m CaV, where m BK and m CaV are the BK Ca and the CaV activation variables, respectively, BK Ca activation can be modeled by the two following ODEs, dp CY dm BK = βm BK (k c + α + β)p CY, (S) = m CaV k + o + ( k + o + k o + α + β ) p CY (k + o + k o + β)m BK, (S3) with m CaV modeled by Eq. 8 of the main text. Assuming quasi-steady state for p CY ( dp CY ) then yields p CY = β k c + α + β m BK. (S4) By substituting Eq. S4 into Eq. S3, we achieve a single ODE describing the dynamics of BK Ca activation in complexes with non-inactivated CaV, dm BK = m CaV k + o (k+ o + k o )(k c + α) + βk c α + β + k c m BK. (S5) 7

8 6 4 k o + k o k c α β t pcy /t mbk 5 5 Figure S3: Time scale analysis for the : BK Ca -CaV complex. Model parameters as function of voltage, showing that the transitions described by α, β and kc are the fastest. This fact leads to a separation of time scales, as shown by the black curve illustrating t pcy /t mbk in Eq. S8, which is always less than.33, and for most voltages considerably smaller. Therefore, the equilibrium open fraction of the BK Ca channels in complexes with non-inactivated CaV, m BK,, and the corresponding time constant, τ BK, are given as in Eq. 4 of the main text. Coupling the one state model for the BK Ca activation (Eq. S5) with the inactivation of the CaV channel allows us to reproduce the dynamics of the BK Ca -CaV complex described by Eqs. S6 S and the corresponding Monte Carlo simulations (see Figure in the main text). In particular, the open probability of the BK Ca channel, p Y = p CY + p OY + p BY, can be described by p Y m BK h, (S6) where h is the fraction of non-inactivated CaV (see Eqs. 5 and of the main text)..3 Time scale analysis For a more formal analysis, we follow (6). The quasi-steady state approximation for p CY is valid when the time scales of the two variables in question differ, i.e., t pcy t mbk, where t mbk is the time scale of changes m BK after an initial (fast) transient, i.e., when the approximation is consistent (6). The (fast) time scale for p CY is given from Eq. S, assuming that m BK is constant, as t pcy = α + β + kc. (S7) The (slow) time scale for m BK is given as the time scale of m BK assuming that the quasi-steady state approximation holds, i.e., t mbk = τ BK in Eq. 4 of the main text. Then which is small (Figure S3). t pcy t mbk = (k o + k + o )(k c + α) + βk c (α + β + k c ), (S8) 8

A B C D E 4 4 6 8 6 I CaV [pa] 5 5 5 5 F G H 6 6 5 5 6 I BK [na] 4 4 4 4 Figure S4: Model responses to different voltage steps for the : BK Ca -CaV complex. (A) Voltage steps from -8 mv to from -4 mv to 4 mv in mv increments for ms and then back to -8 mv. Simulated CaV currents in response to the different voltage steps obtained from (B) the 7-state Markov chain model (see (); gray), (C) the ODE model corresponding to the 3- state Markov chain model (Eqs. 3-5 of the main text; blue), and (D) the corresponding model assuming instantaneous activation m CaV = m CaV, (Eq. of the main text; green). Simulated BK Ca currents in response to the different voltage steps obtained from (E) the original 7-state Markov chain model (see (); gray), (F) the ODE model corresponding to the 6-state Markov chain model (Eqs. S9 S4; blue), (G) the simplified Hodgkin-Huxley-type model (Eq. 5 of the main text; red), and (H) the corresponding model assuming instantaneous activation m CaV = m CaV, (green; see main text). Each trace is the sum of single complex responses. 9

A 4 B C 4 6 I CaV [pa] 5 I BK [na].5.5 8 4 5 4 4 Figure S5: Model responses to a simulated AP generated by a model of hypothalamic neurosecretory cells (7). (A) Simulated AP. (B) Simulated CaV currents in response to the simulated AP obtained from the 7-state Markov chain model ((); dotted gray), the ODE model corresponding to the 3-state Markov chain model (Eqs. 3-5 of the main text; solid blue), and the corresponding model assuming instantaneous activation m CaV = m CaV, (Eq. of the main text; dash-dotted green). (C) Simulated BK Ca currents in response to the simulated AP obtained from the original 7-state Markov chain model ((); gray), the ODE model corresponding to the 6- state Markov chain model (Eqs. S9 S4; solid blue), the simplified Hodgkin-Huxley-type model (Eq. 5 of the main text; dashed red), and the corresponding model assuming instantaneous activation m CaV = m CaV, (dash-dotted green; see main text). Each trace is the sum of single complex responses..4 Responses of the : BK Ca -CaV complex to voltage steps and AP Figure S4 shows the current response of the : BK Ca -CaV complex to different step voltages (Figure S4A). The whole-cell CaV current for the 3-state ODE model (I CaV = N CaV ḡ Ca m CaV h, where N CaV = is the number of CaV channels, ḡ Ca is defined in Table S and m CaV and h are given by Eqs. 3-8 of the main text) approximates the 7-state Markov chain model () current very well for each step voltage (Figure S4BC). Also, the further simplification for the 3-state ODE model assuming instantaneous activation of the CaV currents (m CaV = m CaV,, green plots in Figure S4D) provides a good approximation of the Monte Carlo simulations. The whole-cell BK Ca current for our simplified 6-state Markov chain model (I BK = N BK ḡ BK p Y, where N BK = is the number of BK Ca channels, ḡ BK = pf is the single BK Ca channel conductance (8) and p Y is given by Eqs. S9-S4) approximates the 7-state Markov chain model () current very well for each step voltage (Figure S4EF). Moreover, our simplified Hodgkin-Huxley-type model current for the BK Ca channel (Eq. 5 of the main text; red plots in Figure S4G), and the corresponding model assuming instantaneous activation of the CaV currents (green plots in Figure S4H) also works very well. Figure S5 shows the current response of the : BK Ca -CaV complex to a simulated action potential (AP) generated by a model of hypothalamic neurosecretory cells (7) (Figure S5A). As for the voltage step response case, the whole-cell CaV current for the 3-state ODE model approximates the 7-state Markov chain model current very well (see solid blue and dotted gray plots in Figure S5B). However, the further simplification for the 3-state ODE model assuming instantaneous activation of the CaV currents (dash-dotted green plot in Figure S5B) determines an additional early peak of current and a faster decay. For the whole-cell BK Ca current, the peak obtained from the 7-state Markov chain model is lower than that obtained from our simplified

models (Figure S5C). This discrepancy is likely due to an overestimation of the BK Ca activation time constant of BK Ca model devised by Cox () at specific Ca o (compare the left lower panel of Figure D in () with Figure SD, where Ca o = µm). The slower time constant prevents complete activation of the BK Ca channels for Cox 7-state Markov chain model during the imposed AP. 3 Model for BK Ca activation in complexes with k non-inactivated CaVs and its approximation For the case of :n BK Ca -CaV stoichiometry, we split the system according to the number k of non-inactivated CaV channels. Then the equivalent :k BK Ca -CaV complex can be described by the following ODE system: dp Ck X = βp Ck OX + kc p Ck Y (kα + k c + )p Ck X (S9) dp Ck Y = βp Cn OY + k c + p CnX (kα + kc )p CnY (S) dp Ck i O i X = (i + )βp Ck (i+) O i+ X + (k (i ))αp Ck (i ) O i X + koi p C k i O i Y ((k i)α + iβ + k oi + )p C k i O i X, i =,..., k (S) dp Ck i O i Y = (i + )βp Cn (i+) O i+ Y + (k (i ))αp Ck (i ) O i Y + k oi + p C k i O i X ((k i)α + iβ + koi )p C k i O i Y, i =,..., k (S) dp Ok X = αp COk X + k ok p O k Y (kβ + k + ok )p O k X (S3) dp Ok Y = αp COk Y + k + ok p O k X (kβ + k ok )p O k Y (S4) where, e.g., p Ck i O i X and p Ck i O i Y correspond to the probability of having k i closed and i open CaVs coupled with the closed (X) and open (Y ) BK Ca channel, respectively. The activation of the BK Ca surrounded by k non-inactivated CaVs, m (k) BK, is then BK = p k C k Y + p Ck i O i Y + p Ok Y. m (k) i= (S5) By taking into account that p Ck X = ( m CaV ) k p Ck Y (S6) ( ) k p Ck i O i X = ( m CaV ) k i m i CaV p Ck i O i i Y, i =,..., k (S7) p Ok X = m k CaV p Ok Y (S8)

and renaming the state variables as follows p Y = p Ck Y (S9) p Y = p Ck O Y + p Y (S3). p Yi = p Ck i O i Y + p Yi (S3). m (k) BK = p Y k = p Ok Y + p Yk (S3) we can reduce the ODE system from (k + ) to (k + ) equations. Moreover, assuming the quasi-steady state approximation for p Yi, with i =,..., k, then dp Yi =, p Y = A p Y + D. p Yi = A i+ p Yi+ + D i+ (S33) (S34) where (for brevity, m = m CaV ). p Yk = A k m (k) BK + D n β β A = = kα + β + kc + k c + kα + β + kc D = k+ c ( m) k = kα + β + kc + k c + A i = iβ, i =,..., k B i i ( ) ( D i = k i ) j ( m) k j m j kc D + D j A l /B i j j= j= l= i ( k oj oj) + + i l k ( A j ) D j+ + A m D j /B i ( + j= ( (k (i )) α + k + o(i ) + k o(i ) l=j+ D l m=j+ ) D i ) /B i, i =,..., k (S35) with ( ) B i = (k (i )) α + k + o(i ) + k o(i ) ( A i ) + iβ + kc i ( ) + k oj + + k oj ( A j ) j= i l=j+ A l, i =,..., k. i j= A j

Then k p Yi = j= k p Ck Y = j= A j A j m (k) BK + D i+ + p Ck i O i Y = ( A i ) m (k) BK + D + k j=i+ A j D i, i =,..., k p Ok Y = ( A k ) m (k) BK D k. k j=i+ ( j ) D j A l, i =,..., k l=i ( k j ) D j A l j= l= m (k) BK + ( A i) D i+ + k j=i+ ( j D j l=i+ A l ) Finally, we achieve one ODE for describing the dynamics of the BK Ca activation with k noninactivated CaVs: dm (k) BK k ( ) k k = ( m) k i m i k oi + i k c p Ck Y i= i= { k ( ) ( k k j = ( m) k i m i k oi + i k c D + D j i= j= l= k ( k oj oj) k + + k ( A i ) D i+ + i= + ( } k + ok + ) k ok Dn ( k + ok + ) k ok ( Ak ) + kc j= ( ) k oj + + k oj p Ck i O i Y j=i+ i= ( j D j A l ) l=i+ k k ( ) A j + k oj + + k oj ( A i ) ) A l D i k j=i+ A j m (k) BK. (S36) Therefore, the activation time constant of the BK Ca channels in complexes with k non-inactivated CaVs, τ (k) BK, is the inverse of the expression in square brackets in Eq. S36, whereas the BK Ca equilibrium open fraction, m (k) (k) BK,, is equal to the product of τ BK and the expression in curly brackets. Note that τ (k) BK depends only on parameters, whereas m(k) BK depends also on m CaV via D i. If assuming instantaneous activation of CaVs, we have ( ) k p Ck i O i Y = ( m CaV, ) k i m i i CaV, p Y, i =,..., k, 3

and (cf. Eq. 9 of the main text) dm (k) BK = = k i= ( ) k ( m CaV, ) n i m i i CaV, k io + k c p Ck Y k ( k i ( i= k i= ( ) k oj + + k oj p Ck i O i Y ) ( m CaV, ) k i m i CaV, k + oi (S37) ( m CaV, ) k k c + 4 Whole-cell models k i= 4. Hypothalamic neuronal model ( ) k ( m CaV, ) k i m i ( CaV, k + i oi + koi) ) m BK. We extended a model of electrical activity in hypothalamic neurosecretory cells (7) with our devised BK Ca -CaV model. The model is described by dv = C (I BK + I SK + I K + I A + I Na + I Ca + I leak ). (S38) where I BK and I SK represent the calcium- and voltage-dependent K + currents carried by BK and SK channels, respectively; I K and I A denote the delayed rectifier and the A-current respectively; I Na represents the sodium current and I Ca the calcium current. C is the membrane capacitance and is equal to µf cm. We modified the calcium current and inserted our whole-cell BK Ca model in place of the original representation of BK Ca currents. In particular, according to experimental data (9, ), we introduced CaV activation dynamics, whereas the original model assumed instantaneous activation of CaVs, and modified the equilibrium voltage-dependent activation. Then I Ca = g Ca m CaV (V V Ca ) (S39) where g Ca and V Ca are the maximal whole-cell conductance and the Ca + reverse potential, respectively (7). m CaV is defined by Eq. 8 of the main text, with τ CaV =.5 ms. For the equilibrium voltage-dependent activation, given in (7) and defined by m CaV, = ) V Vm, ( + e km we modified the V m parameter of the Boltzmann function (V m = 5 mv). The I BK current is modeled by Eq. 8 of the main text, where the BK Ca activation, m (n) BK, is given by Eq. 6. Moreover, we also considered the case where the activation of the BK Ca channel is given by the complete ODE model described by Eqs. S9 S5, and the case where the BK Ca activation is simplified by assuming instantaneous activation of CaVs (see Eq. S37 and Eq. 9 of the main text). The other currents are expressed as (7) I SK = g SK q (V V K ), I K = g K m 3 K(V V K ), I A = g A m 4 Ah A (V V K ), I Na = g Na [m Na, (V )] 3 h Na (V V Na ), I leak = g leak (V V leak ), (S4) (S4) (S4) (S43) (S44) 4

n= n= n=4 4 4 4 4 4 4 6 6 6 8 5 5 8 5 5 8 5 5 Figure S6: Whole-cell simulations for the neuronal model (7) with the devised BK Ca - CaV model. Simulated APs with the BK Ca channels coupled in complexes with n noninactivating CaVs (n =, left panel; n =, middle; n = 4, right). In each panel, the whole-cell BK Ca current is described by Eq. 8 of the main text (i.e. BK Ca coupled with non-inactivating CaVs), where the BK Ca activation, m (n) BK, is modeled by the full ODE model described by Eqs. S9 S5 (blue curve), by Eq. 6 (see Eq. S36 for the details) (red curve), and by Eq. 9 (green curve), and g BK = ms cm. The black curve shows the case of BK Ca block (g BK = ms cm ). where the voltage-dependent activation variables, m X (and similarly inactivation variabels, h X, where X denotes the type of current), follow dm X = m X, (V ) m X τ mx, (S45) where τ mx (respectively τ hx ) is the time-constant of activation (respectively inactivation for h X ), and m X, (V ) (respectively h X, (V )) is the steady-state voltage-dependent activation (respectively inactivation) of the current, described with Boltzmann function. g X represents the whole-cell conductance of the channel X and V X the reverse potential. q is the calciumdependent activation function for I SK (see Appendix in (7) for parameter values). Figure S6 shows the simulated AP in this neuronal model with :n stoichiometry BK Ca -CaV complexes, where n =, or 4. The BK Ca activation is modeled by the complete ODE system described by Eqs. S9 S5, by Eq. 6 of the main text, or by Eq. 9. The simplified model (Eq. 9) is not able to reproduce the fast after-hyperpolarization (fahp), whereas the other two models, which take CaV activation dynamics into account, show how increasing the number of CaVs coupled with BK Ca helps to generate fahp. The difference associated with the choice of the BK Ca model suggests that the neuronal model is sensitive to the kinetics of BK Ca activation. 5

4. Human β-cell model The human β-cell model () is as follows. The membrane potential is described by dv = (I BK + I Kv + I HERG + I Na + I CaL + I CaP Q + I CaT + I KAT P + I leak ), (S46) where the BK Ca current I BK is modeled assuming that BK Ca channels are located in complexes with either T-type, L-type or P/Q-type Ca + channels. In particular, I BK is described by Eq. 7 of the main text (with inactivating T- and L-type CaVs) or Eq. 8 (with non-inactivating P/Q-type CaVs), where the BK Ca activation, m (n) BK, is modeled by Eqs. S9 S5 (full model), Eq. 6, or Eq. 9 (instantaneous CaV activation). As shown in Figure S7, the model is quite robust to the choice of BK Ca -CaV model. The other currents are I Kv = g Kv m Kv (V V K ), I HERG = g HERG m HERG h HERG (V V K ), I Na = g Na m Na, (V )h Na (V V Na ), I CaL = g CaL m CaL, (V )h CaL (V V Ca ), I CaP Q = g CaP Q m CaP Q, (V )(V V Ca ), I CaT = g CaT m CaT, (V )h CaT (V V Ca ), I K(AT P ) = g K(AT P ) (V V K ), I leak = g leak (V V leak ), (S47) (S48) (S49) (S5) (S5) (S5) (S53) (S54) where activation variables (and similarly inactivations variabels, h X, where X denotes the type of current) follow dm X = m X, (V ) m X τ mx, (S55) where τ mx (respectively τ hx ) is the time-constant of activation (respectively inactivation for h X ), and m X, (V ) (respectively h X, (V )) is the steady-state voltage-dependent activation (respectively inactivation) of the current. The steady-state activation (and inactivation) functions are described with Boltzmann functions, except m X, (V ) = + exp((v V mx )/n mx ), (S56) h CaL, (V ) = max (, min {, + [m CaL, (V )(V V Ca )]/57mV} ), (S57) for Ca + -dependent inactivation of L-type Ca + channels. The time-constant for activation of Kv-channes is assumed to be voltage-dependent (, ), { τ mkv, + exp ( ) mv V τ mkv = 6mV ms, for V 6.6 mv, (S58) τ mkv, + 3 ms, for V < 6.6 mv. We refer to the original paper () for details regarding modeling of the different currents. 6

n= 4 I BK [pa/pf] n= I BK [pa/pf] n=4 I BK [pa/pf] 4 4 BK coupled with n T-type Ca + 5 5 4 6 4 6 4 6 5 n= I BK [pa/pf] 4 n= 4 I [pa/pf] BK n=4 I BK [pa/pf] 4 BK coupled with n L-type Ca + 5 5 4 6 4 6 4 6 5 n= I BK [pa/pf] n= I BK [pa/pf] n=4 I BK [pa/pf] 4 4 4 BK coupled with n P/Q-type Ca + 5 5 4 6 4 6 4 6 5 Figure S7: Whole-cell simulations for the human β-cell model () with the devised BK Ca -CaV model. Simulated APs with BK Ca channels located in complexes with n T-type (left panels), L-type (middle), or P/Q-type (right) CaVs (n =, upper panels; n =, middle; n = 4, lower). The β-cell model shows little sensitivity to the choice of BK Ca -CaV model. In each panel, the whole-cell BK Ca current is described by Eq. 7 of the main text (with inactivating T- and L-type CaVs) or Eq. 8 (with non-inactivating P/Q-type CaVs), where the BK Ca activation,, is modeled by the full ODE model described by Eqs. S9 S5 (blue curve), by Eq. 6 (see Eq. S36 for the details) (red curve), and by Eq. 9 (green curve). Left subpanels show fit to data from (), right subpanels report model simulations. The black traces show simulations with BK Ca block. m (n) BK 7

4.3 Pituitary lactotroph model We extended a pituitary lactotroph model (3) with our concise Hodgkin-Huxley-type model of the BK Ca -CaV complex (Eq. 8 of the main text). The original model includes a single Ca + current (I Ca ), a delayed-rectifier K + current (I K ), a Ca + -gated SK current (I SK ), and a leak current (I leak ), in addition to the BK current. The membrane potential V evolves in time according to C dv = (I Ca + I K + I SK + I BK + I leak ), (S59) where C is the membrane capacitance. The currents are modeled as I Ca = g Ca m CaV, (V )(V V Ca ), (S6) I K = g K n(v V K ), (S6) I SK = g SK s ([Ca])(V V K ), (S6) I leak = g l (V V l ), (S63) I BK = g BK m (n) BK (V V K), where the activation variable of the delayed rectifier is given by The equilibrium functions are described as (S64) dn = n (V ) n. (S65) τ n m CaV, (V ) = [ + exp((v m V )/s m )], (S66) n (V ) = [ + exp((v n V )/s n )], (S67) s ([Ca]) = [Ca] [Ca] + ks. (S68) The differential equation for the cytosolic Ca + concentration is d[ca] = f c (αi Ca + k c [Ca]). (S69) Table S3 reports the parameter values of the pituitary model (3). 8

Table S3: Parameter values of the pituitary model (3). Parameter Value Unit C pf g Ca ns V Ca 6 mv v m - mv s m mv g k 3 ns V k -75 mv v n -5 mv s n mv g SK. ns k s.4 µm g BK ns g l. ns V l -5 mv f c. - α.5 µm fc k c. ms 9

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