2.1 A modular signalling network

The potential signalling network illustrated in Fig 1 is organised in three modules: an Upstream module, which includes the ligation of the receptor by SAP and the early transient drop of membrane potential V generated by the cGMP response, and two alternative downstream modules that can generate Ca²⁺ oscillatory responses. The Ca_V+BK module includes voltage-dependent Ca²⁺-channels and Ca²⁺-dependent K⁺-channel BK; while the CatSper+NHE module is composed of the voltage-, pH_i- and Ca²⁺-dependent channel CatSper, the voltage-dependent sodium-proton exchanger and the intraflagellar proton concentration. The variables, parameters and differential equations used to model the components of this network are described in Sec. 4.1 and Sec. 4.2.

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Fig 1. The modular organisation of the signalling network transducing SAP signals to Ca²⁺-spike trains.

The structure is separated into 3 modules coupled by the membrane potential variable (V). In the Upstream module (described by the variables S, R_F, R_H, R_L, G, , , and ) free receptors R_F bind SAP molecules and transit irreversibly through three receptor forms, each with less guanylate cyclase activity: High R_H, low R_L and inactive R_I; the cGMP synthesised by these receptors binds and opens KCNG channels, which conduct a hyperpolarising potassium current. The membrane hyperpolarisation promotes the opening of spHCN channels, which exert the opposite action on V when conducting a cationic inward current. Two candidate modules are presented to explain the calcium spike trains: one that includes classic voltage-dependent Ca²⁺ channels and BK channels (described by variables , , and ), and another that considers CatSper, NHE exchanger and proton concentration (described by , and H). Note in the Ca_V+BK module, that the calcium channel is purely voltage dependent, whereas CatSper in the other module has threefold regulation, namely Ca²⁺, H⁺ and V. The complexity of the CatSper channel is illustrated with two gates. Voltage-regulated channel transitions are depicted with blue arrows, where the darker or lighter tones indicate whether the transition is promoted by membrane hyperpolarisation or depolarisation respectively. The blue dotted lines linking some of these arrows to the membrane potential box top or bottom provide the same information. The components in the middle dashed boxes, namely cGMP (G), Ca²⁺ (C), proton (H) and membrane potential (V) are the main experimental observables. The ions K⁺ and Na⁺ are depicted in grey to indicate that the model describes only the currents while neglecting concentration changes.

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The modular structure facilitated modelling and analysis. As a strategic simplification, we hypothesised that most of the components of the Upstream module were independent of the variables downstream and studied it isolated from the remaining dynamics. We tested the validity of this simplifying assumption by showing that this Upstream module accounted quantitatively for experimental time series of cGMP and membrane potential in response to SAP (Sec. 2.2). Once the values of the parameters of this module were obtained, they remained fixed in the subsequent analyses of the dynamics of the downstream components. In these analyses, the modules featuring Ca²⁺-channels were constrained such that given the input of the Upstream module, they generated an output that was consistent with the observed Ca²⁺-spike trains, i.e. these two putative modules were constrained by independent data sets at the both input and output levels. Hence, by coupling the downstream Ca_V+BK and CatSper+NHE modules to this Upstream module, we show in Sec. 2.3 that either module can independently account for the observed Ca²⁺-spike trains patterns.

2.2 The Upstream module of the SAP-activated response explains the long term kinetics of cGMP independently of the downstream signals

cGMP is the core component of the Upstream module (Fig 1). cGMP levels rise in response to SAP-induced guanylate cyclase activity and activates KCNG channels that in turn lead to hyperpolarising currents. A. punctulata sperm stimulated with the SAP Resact exhibit a rapid cGMP response that peaks before ∼400 ms and rapidly drops to a pseudo plateau that slowly decays over several seconds, as observed by stopped-flow kinetics [22]. This biphasic dynamics prompted us to postulate that SAP receptors transit irreversibly through three forms with different levels of associated guanylate cyclase activity, and that the guanylate cyclase and phosphodiesterases activities are insensitive to downstream signals (Fig 1; Sec. 4.2.1 and Sec. 4.1.2). We are thus neglecting putative regulation of phosphodiasterases activity by cGMP itself and by other components of the signalling network, such as pH_i or [Ca²⁺]_i [23]. In order to assess these assumptions we fitted a model of the Upstream module to cGMP measurements in bulk cell assays wherein sperm were stimulated with pulses of uncaged SAP along a concentration range spanning several orders of magnitude [22]. The good quality of the fitting (S1 Fig), involving appropriate scaling factors, supported quantitatively the assumptions and provided estimates of kinetic rate constants for receptor state transition, cGMP synthesis and decay (kinetics illustrated in Fig 2A for a SAP concentration that saturates the response of the membrane potential V [10]). The long term dynamics of the average cGMP level in the sperm population is described well by our single cell model, which as a first approximation neglected any potential feedbacks from downstream processes. This is our first modelling result.

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Fig 2. Numerical solution of the Upstream module overlaid with experimental data obtained in bulk sperm populations.

The model with state space {S, R_F, R_H, R_L, G, ,, V}) was fitted to the cGMP concentration data measured on sperm populations stimulated at t = 0 with a SAP pulse S₀ = 25 nM (data points read from the graphs in publications [22] and [10] for cGMP and V, respectively; see also S5 Fig). In the top graph, receptor activation dynamics and SAP consumption are shown. The middle graph displays cGMP concentration dynamics in the model (G, line) and the respective experimental measurements (dots) in nM. The bottom graph, shows two separate model solutions with KCNG+spHCN (solid black) and KCNG only (dotted; obtained by setting g_hc=0). See main text for further details.

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By adjusting the density of KCNG and spHCN channels to membrane potential from sperm population data [10], we found values that reproduce the early transient drop in membrane potential and the early recovery (Fig 2). To illustrate the separate contribution of each channel, we showed a reference case where KCNG is the only channel present (dotted line in Fig 2) and, as expected, its opening alone drives membrane potential towards E_K ≈ −80 mV (potassium Nernst potential). The latter reference value sets a lower limit to the V response that is well below the value reached at the minimum V produced by the KCNG+spHCN case. It is worth noticing that under this scenario, in which only these two channels are assumed to be present, it is not possible to find a given combination of channel densities such that the spHCN current alone counterbalances the hyperpolarising current carried by KCNG, so as to bring membrane potential back to resting levels without abrogating the hyperpolarisation pulse. Therefore, full recovery of voltage requires the additional activity of other ion channels with depolarising currents after these initial events (namely the calcium channels).

2.3 Two alternative signalling modules can explain semiquantitatively the structure of the Ca²⁺-spike trains

The [Ca²⁺]_i spike trains observed in individual sperm cells responding to the SAP stimulus are noisy but show some conspicuous trends, as illustrated by those of S. purpuratus sperm (Fig 3A). The regression coefficients of the relative spike amplitude b_A and the relative interspike interval b_T over the spike index (illustrated on Fig 3A right) provide numeric measurements of these trends in the individual responses (Sec. 4.4). The amplitude of [Ca²⁺]-spikes declines progressively in 87% of the responses analysed (Fig 3B top), whereas the interspike interval tends to increase in 65% of the cases (Fig 3B bottom). The bivariate plot of b_A vs b_T (Fig 3C) shows that a majority of 56% of the [Ca²⁺]_i responses to SAP, as well as the median coefficients, fall within the bottom right quadrant. The convex hull defined by the experimental values that fall within this quadrant (the dashed polygon in Fig 3C) offers a first selection criterium to assess how close the dynamics predicted by the model are to the observations. The complementary criteria are the mean interspike interval and the steady state [Ca²⁺]_i value before the stimulus (Sec. 4.4). The rationale for the first selection criterium is twofold. First, it is the most representative pattern of the experimental spike trains analysed here as well as of those reported in other species (e.g. see fig. 3 in reference [9] for A. punctulata sperm). Second, it is the trend that was most difficult to obtain in the model under either the Ca_V+BK or the CatSper+NHE modules, therefore being most suitable to challenge network composition. The spike trains falling in the bottom left quadrant, besides being less frequent experimentally, were easily obtained with the model and therefore were less challenging.

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Fig 3. [Ca²⁺]_i-spike trains produced by the sperm activating peptides in sea urchin sperm flagellum and its modelling.

A) On the left, two examples are shown of [Ca²⁺]_i time series measured with fluorescent probes in flagella of S. purpuratus individual sperm bound to a coverslip following uncaging of Speract analogue by UV light. The raw measurements (grey) were smoothed using cubic spline (black). The Ca-spikes (vertical grey dashed lines) were calculated as the maxima in the smooth curve higher than 3x the standard deviation of the measurements pre-uncaging (horizontal gray dashed line). For each spike indexed k, the amplitude A_k and the interspike interval T_k = t_k+1 − t_k were calculated. The shaded region indicates the UV light pulse. On the right, the relative spike amplitude A_k/A₁ (left axis; black) and relative interspike interval T_k/T₁ (right axis; grey) are plotted as a function the spike index k for the two examples. The lines are linear regressions of the relative spike amplitude (black) and relative interspike interval (grey) over the spike index k from which the respective regression coefficients b_A and b_T were obtained. B) Cumulative frequency distributions of the spike-amplitude regression coefficient b_A and interspike interval regression coefficient b_T for all experimental time series analysed. C) Bivariate plot of the spike amplitude coefficient b_A versus spike amplitude coefficient b_A in experimental time series (gray dots). The cross indicates the median values. The numbers indicate the percentages of time series falling in the four quadrants. The polygon in the bottom-right quadrant represents the convex hull of the experimental points in the quadrant where most the time series fall in. The pink and green dots represent the numerical solution of the model represented in D and E. D, E) [Ca²⁺]_i dynamics as predicted by the models featuring either the Ca_V+BK module (green) (D) or the CatSper+NHE module (E), using the reference parameters (Table 2). On top, the curves represent the numerical solutions of the variable C as a function of time. The Ca-spikes are identified (vertical dashed lines) and indexed by k (labels). The relative spike amplitude and relative interspike interval are plotted as function of spike index k on the bottom graph together with the regression lines used to obtain the regression coefficients b_A and b_T for the model, which were represented in C.

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We found that the two alternative signalling modules, Ca_V+BK and CatSper+NHE (Fig 1), can explain the characteristic envelope and increasing delay of the Ca²⁺-spikes when coupled alone to the Upstream module. These modules are composed of specific combinations of ion channels and transporters. Each of these combinations has the capacity to produce, given the input of the cGMP kinetics used to calibrate the Upstream module (Fig 2), a series of Ca²⁺-spikes that are coherent with the predominantly observed ones (Sec. 4.4). The corresponding numerical solutions (Fig 3D and 3E top) are characterised by amplitude and interspike interval regression coefficients (Fig 3D and 3E bottom) that fall within the values of the most likely observed responses in individual sperm cells of S. purpuratus (represented by the labelled circles in Fig 3C).

The two modules and their properties are analysed separately in the following sections. For each module, a first section characterises the Ca²⁺ oscillations and explains how they are generated (Sec. 2.4, Sec. 2.6), and a second focuses on explaining the progressive amplitude decay and delay between spikes (Sec. 2.5, Sec. 2.7). We will build on this knowledge to analyse and understand mixed scenarios (Sec. 2.8). The Ca_V channel was the first Ca²⁺ channel to be implicated in SAP signalling in sea urchin sperm literature and following this chronology we will start the analysis of its module.

2.4 Characteristics and mechanism of the [Ca²⁺]_i oscillations generated by the Ca_V+BK module

The module features the Ca_V channels and the Ca²⁺-dependent BK potassium channels (Fig 1). The Ca_V channels transit irreversibly through three forms—inactive, closed and open—defining an ordered cycle, in which the transition rates are controlled by voltage. Only in the open form Ca_Vs allow inward Ca²⁺ currents that tend to depolarise the membrane. Provided that the transition rates between these forms are of similar order of magnitude, these channels alone give rise to a limit cycle, which as we will see has a constant period that is unrealistically short (Sec. 2.5). This disagreement with the observations called into action the other channel in the module, BK, which once coupled with the Ca_V results in larger progressively increasing intervals between the peaks and also increases the peak amplitude.

The numerical solution of the variables of the model including the Ca_V+BK module under the reference parameters is shown in Fig 4. The uprise and slow decay of cGMP produced by the Upstream module leads to Ca²⁺-spike trains with an envelope that follows closely the cGMP values (top) and increasing interspike intervals. The initial transient hyperpolarisation (bottom) driven by the opening and currents of KCNG ( and I_kn) precedes and triggers the first peak of [Ca²⁺]_i (C). The nadirs of the [Ca²⁺]_i oscillations are well above the basal concentration of this cation and this minimal plateau of about 500 nM is maintained until the Ca²⁺ activity vanishes at about 17 s. A similar oscillatory behaviour with similar period but different phases is observed on the fractions of open channels (second row) and the respective currents (third row), as well as the membrane potential (bottom).

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Fig 4. Dynamics of the model featuring the Ca_V+BK module.

The state space is {S, R_F, R_H, R_L, G, , , V, , , , C}. As in Fig 2, the initial stimulus is S(0) = 25 nM is used. From top to bottom, the numerical solutions obtained with the parameters listed in Table 2 are shown for: [Ca²⁺]_i (C) together with cGMP concentration (G), open channel fractions, ionic currents and membrane potential. The ionic currents are displayed in current densities (left axis) and as current normalised by capacitance (right axis).

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The temporal dynamics of the [Ca²⁺]_i fluctuations predicted by this module can be easily understood qualitatively. The hyperpolarisation induced by KCNG enables the transition from inactive to closed forms of the Ca_V and the spHCN-mediated increase in membrane potential opens these closed channels. The open Ca_V channels will overcome the hyperpolarising current driven by KCNG and thus depolarise the membrane as Ca²⁺ ions flow in. The depolarisation of the membrane leads to the inactivation of the Ca_V channels with the hyperpolarisation currents predominating again. At the hyperpolarised membrane potentials the inactive channels will again transit to closed form, which eventually will open as described above, reinitiating the cycle. Intraflagellar calcium, by activating the BK channels, adds a strong hyperpolarising current to that of KCNG. This couples the net hyperpolarising currents to the amplitude of the previous Ca²⁺-spike, in such a way the higher the [Ca²⁺]_i peak, the higher the BK-driven hyperpolarising current will be.

2.5 Control of [Ca²⁺]_i-spike amplitude and interspike interval in the Ca_V+BK module

The envelope of the [Ca²⁺]_i spikes follows closely that of the cGMP levels upstream. This led us to speculate that the time course of cGMP might offer a way to understand the fluctuations envelope. To gain a better quantitative insight, a variant of the model was made wherein cGMP concentration, G, instead of a variable became a constant parameter, thus ignoring the upstream variables and reducing the state space (Sec. 4.3). In this simplified model, the qualitative changes of the [Ca^{2 +}]_i and its period at the stationary states were analysed as a function of the G parameter. This is known as bifurcation analysis in dynamical systems theory, which is useful for characterising the global properties of a system of differential equations, in particular the dependence of asymptotic behaviours on a given parameter of interest (bifurcation parameter). These behaviours can correspond to attractors of fixed point (invariant in time) or limit cycles (stationary periodic), which in turn can be stable or unstable, among others. Wherever there is a qualitative alteration of these behaviours by changing the parameter of interest, it is said that a bifurcation has occurred.

When cGMP concentration is varied in this subsystem, the bifurcation diagram shows 5 bifurcation points (Fig 5A, labeled from I to V). In I, by means of a Hopf-type bifurcation, as cGMP concentration increases above the basal level, the system moves from a [Ca²⁺]_i stable stationary state to a limit cycle. When it reaches II, the limit cycle disappears giving rise again to a stable stationary equilibrium by means of an inverse Hopf-type bifurcation. Further on in III, the system undergoes an inverse saddle-node bifurcation, defined over cGMP concentration increments, in which a stable equilibrium state coalesces with an unstable one. In this way, in traversing the branch over the stable equilibrium state, the system goes to an unstable state. Now, when advancing on this last unstable equilibrium, this turns to a stable one by means of the direct saddle-node bifurcation (IV). The presence of points III and IV suggest the possible existence of a cusp bifurcation defined in a space of higher dimension, which is corroborated in panel B of the same figure. Finally, the last stable equilibrium state, again by means of a direct Hopf-type bifurcation, gives rise to another limit cycle.

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Fig 5. Bifurcation analysis of the model featuring the Ca_V+BK module.

In this analysis, cGMP concentration is a constant input, i.e. G is the bifurcation parameter and the upstream variables are ignored, thus reducing the state space to ,, V, ,, f_bk, C}. A) Bifurcation diagrams calculated with XPPAUT [24] using the reference parameters (Table 2). Top: the graphs show the C variable ([Ca²⁺]_i) as a function of the input parameter G (cGMP concentration). The thick continuous lines are stable equilibria, the dashed ones indicate unstable equilibria, the thin continuous lines are the maxima and minima of stable limit cycles. The bifurcation points are marked with Roman numerals (I to V). The numbered grey circles indicate the peak value for consecutive spikes obtained by numerically solving the complete model (see Fig 4). Bottom: the solid lines represent the period of stable limit cycles as a function of the constant input G. The numbered gray circles correspond to the interspike interval in the numerical solutions, as in the top graph. B) Bifurcation diagrams parameterised by maximal BK conductance density (g_bk). Top and bottom: the lines are as in the graphs in (A); for clarity, the minima and maxima of the limit cycles are omitted. The numbers in the range of 0.0 to 1.1 are the fold change factors multiplying the reference value of g_bk (e.g. the curves labelled 1.0 coincide with those of (A), whereas the curves marked 0.0 correspond to a cell without BK channels).

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The period of the limit cycle in the upper branch increases as the cGMP concentration decreases (Fig 5A, bottom), but this is not the case for the limit cycle in the lower branch, in which the period shows an opposite yet less pronounced trend. In addition, there is a considerable difference between the two limit cycles in terms of their amplitude, demarcated by the minima and maxima of calcium levels in the bifurcation diagram.

To better understand the role of the BK channel on calcium dynamics in a Ca_V-dependent scenario, bifurcation diagrams were calculated within the cGMP concentration range considered in Fig 5A, for different density values of BK, represented by g_bk (Fig 5B). In B, unlike the upper panel of the Fig 5A, the lower and upper lines representing the limit cycles were omitted for clarity; their respective unstable steady state values are shown instead (dashed lines). In the bifurcation analysis, the extreme case where g_bk = 0, that is, the variant of the model that only presents Ca_V, predicts the existence of a single branch for the calcium concentration with a limit cycle whose envelope is controlled by cGMP levels but has a constant period of approximately 0.2 s (see the lines labeled 0.0, in both panels of Fig 5). The gradual addition of BK reveals how bistability is generated (Fig 5B), and exhibits the appearance of an upper limit cycle within the range of cGMP concentration values observed in Fig 5A. The behaviour of the periods associated with the different values of g_bk is shown in the lower panel of the Fig 5B. In a three-dimensional representation, with g_bk as one of the axes, the bifurcation pattern is a cusp with mounted limit cycles. The presence of the new upper periodic attractor allows the nadir of the [Ca²⁺]_i fluctuations to be placed on a plateau above the basal level, and both their amplitude and increasing interspike interval to approach the physiological response. This implies that the aforementioned properties arise from the coupling of the Ca_V and BK channels.

The parameters have been chosen such that as the cGMP concentration values transiently exceed the bifurcation point, the system is forced to oscillate in the upper limit cycle and fall into the lower limit cycle after the decrease in cGMP causes the system to pass through the saddle-node bifurcation (IV). This explains the abrupt [Ca²⁺]_i drop and the low amplitude oscillations that persist afterwards, which are particularly visible in the numerical solution of V (Fig 4, bottom). If the value of g_bk were increased above the reference value used in panel A (e.g. by multiplying it by a factor of 1.1), the saddle-node bifurcation would occur at a lower cGMP concentration value and these low amplitude fluctuations would not be revealed (see Fig 5B).

Considering the bifurcations of the model obtained with cGMP as control constant, the behaviour of the full model is easy to understand: as SAP binds to its receptor, a sustained yet decreasing cGMP activity leads the system to approach asymptotically the attractors of the model with constant cGMP concentration, such that the system displays fluctuations whose spike amplitude and interspike interval tend to the amplitude and the period of the attractors (compare the numbered gray circles corresponding to the solution of the full system, which takes into account the temporal evolution of cGMP concentration as in Fig 4, to the asymptotic limit cycles for different fixed values in Fig 5A).

It is important to note that at intermediate levels of cGMP, there is a range of values where limit cycles coexist. From what has been reported in the literature, only the dynamics compatible with the upper limit cycle has been experimentally observed. Presumably, even if the small oscillations of the lower limit cycle were present, they would be masked by measurement errors.

2.6 Characteristics and mechanism of the [Ca²⁺]_i oscillations generated by the CatSper+NHE module

We now consider the alternative module whose central component is CatSper, the Ca²⁺ channel that has recently been implicated in SAP signalling in sea urchin sperm. We assumed here, that CatSper opening relies on two independent gates: one modulated by both pH_i and voltage, and the other one responsive to intracellular calcium. The fraction of open channels is given by the product of the fractions of these two open gates (Eq 44). The module includes also as variables the fraction of active forms of the voltage-sensitive Na⁺/H⁺ exchangers, , and the intraflagellar concentration of protons, H. The numerical solutions obtained under the same conditions as in the alternative model are shown in Fig 6.

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Fig 6. Dynamics of the model featuring the CatSper+NHE module.

The state space is {S, R_F, R_H, R_L, G,,, V, ,, C, H}. As in Fig 4, the initial stimulus is S(0) = 25 nM, and the graphs are the numerical solutions for the indicated variables and intermediate quantities, obtained with the parameters listed in Table 2.

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The rise and fall of the cGMP produced by the Upstream module leads to a Ca²⁺-spike train with a slow decaying amplitude and an increasing interspike interval (Fig 6, top). The nadir of the Ca²⁺-oscillations is close or even below the resting level, in contrast with the oscillations predicted by the alternative model (Fig 4). A similar oscillatory behaviour with similar period but different phases is observed on the fractions of open channels (Fig 6, second row), the respective currents (third row), as well as the membrane potential and proton concentration (bottom).

Compared to the Ca_V+BK module, the pattern of the Ca²⁺ oscillations is differently shaped by the CatSper+NHE module. The individual Ca²⁺-spikes are more prolonged and the corresponding nadir is relatively shorter. The decay of the spike train envelope is less pronounced for the reference parameters in the CatSper+NHE module than in the Ca_V+BK module, while the increase in the interspike interval is more marked in the former than in the latter. It is remarkable that the CatSper+NHE module does not predict the sustained plateau of the Ca²⁺-oscillation nadirs present in the Ca_V+BK module. Finally, another interesting difference between the two modules is the envelope of the voltage spikes whose amplitudes decrease progressively in the Ca_V+BK module and increase in the CatSper+NHE module (bottom in Figs 4 and 6).

The basis for the limit cycle in CatSper+NHE module is qualitatively straightforward. The hyperpolarisation increases the activity of the Na⁺/H⁺ exchanger, this triggers the extrusion of protons raising pH_i. As a consequence of this alkalinisation, the voltage sensitivity of CatSper shifts to lower values (see S3D Fig) and some of these channels open. The inward Ca²⁺ currents raise the membrane potential which in turn tends to increase the fraction of open channels, in a positive feedback loop. However, the progressive depolarisation of the membrane inhibits the exchanger and the pH_i tends to return to its basal level, shifting the CatSper voltage sensitivity back to higher values. This allows the hyperpolarising current to surpass the depolarising current of CatSper (see Fig 6, third row). The cycle restarts when the membrane becomes again hyperpolarised by the still ongoing KCNG current (Fig 6, second and third row), leading to a new round of enhanced exchanger activity, a transient rise in pH_i, and the recovery of CatSper currents that eventually overcome the hyperpolarising currents. Although the feedback inhibition of CatSper by Ca²⁺-inactivation could have a potential role in the limit cycle, its contribution is actually negligible under the reference parameter regime.

2.7 Control of [Ca²⁺]_i-spike amplitude and interspike interval in the CatSper+NHE module

As in the case of the Ca_V+BK module, the bifurcation analysis of the model featuring the CatSper+NHE module with constant input cGMP reveals a rather complex dynamical structure. The system displays a cusp involving two saddle-node bifurcations, which, in contrast to the module Ca_V + BK, is in lower cGMP concentration ranges, such that two [Ca²⁺]_i stable equilibria coexist close to the resting state. At higher cGMP values, there is a regime where the stable equilibrium solution coexists with a stable limit cycle, whose maxima and minima are indicated by a succession of black circles in the top graph in Fig 7. It is worth noticing that the coexistence of stable attractors within a given range of cGMP concentrations suggests the presence of bifurcation points that determine the transition of one behaviour to the other, at the boundaries of that range. However, for this model, with the available bifurcation analysis software (e.g. XPPAUT [24]), it was not possible to discover these points or the limit cycle. In fact, the stable limit cycle was detected by solving the model numerically, exploring the phase space by randomisation of initial conditions of the system and allowing it to evolve for long times (>5 × 10⁵ iterations). Also an exploration of initial conditions between the two attractors, suggests that a possible presence of highly interlaced attraction basins, which makes numerical calculations difficult.

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Fig 7. Bifurcation analysis of the model featuring the CatSper+NHE module.

The analysis was done under conditions wherein cGMP is defined as a constant input (G = constant) and upstream variables ignored reducing the state space to ,, V, , m_cs, h_cs, C, H}. Top: The graphs depict the variable C ([Ca²⁺]_i) as a function of the input parameter G (cGMP concentration). The thick continuous lines are the stable equilibria and the dashed lines indicate unstable equilibria, as obtained by XPPAUT [24]. The thick dotted lines are the maxima and minima of the stable limit cycle obtained by numerical solutions of the system under random initial conditions. The numbered gray dots indicate the maxima and minima of the consecutive spikes obtained by solving numerically the full model, corresponding to those depicted in Fig 4. Bottom: The dotted lines represent the period of the stable limit cycles as a function of the constant input G obtained by numerical solutions. The numbered grey dots correspond to the interspike interval in the numerical solutions, as in the top graph.

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For this scenario, as was the case with the Ca_V+BK module, the amplitude decreases and the period of the limit cycle increases (Fig 7, bottom) as the constant input of cGMP returns to basal levels after having reached high values.

The patterns of the [Ca²⁺]_i fluctuations obtained by solving the full model with the CatSper+NHE module (Fig 6) are now easy to understand. As the cGMP concentration increases rapidly from the resting state, the system falls within the basin of attraction of the limit cycle determined for constant G values; and as the cGMP returns slowly to its basal level, the time required to approach those limit cycles decreases. This can be seen in Fig 7 by means of the empty circles superimposed on the bifurcation diagram, which show how the system tends towards this limit cycle with a spike amplitude reduction and an increase in the interspike intervals. The differences between the asymptotic values of amplitude and periods predicted by this variant of the model with fixed values of G (black circles), and those of the actual peaks obtained when G varies in time (numbered empty circles) are more marked in this module than in the case of the Ca_V+BK module.

2.8 The decaying envelope and temporal organisation of the Ca-spike trains can be obtained in mixed scenarios involving components of the Ca_V+BK and CatSper+NHE modules

In the previous sections, we have shown that both alternative modules, once coupled with the Upstream module, can account for the observed data patterns. These two modules share some features and their components could potentially coexist in the flagellum being coupled through the membrane potential, pH_i, and [Ca²⁺]_i. As a first approach to investigate how the coupling of the modules would affect the dynamics, we considered a scenario in which the actual composition of the system would be a weighted combination of the two modules by means of weight parameters given by θ^cs = θ and θ^cs = (1 − θ) (implementation details in Sec. 4.5). For θ = 0, one recovers the submodel with CatSper module alone, for θ = 1 one recovers the submodel featuring Ca_V+BK module only, and when θ = 0.5 the total conductance of the components in each module are half of their respective reference values. Taking the CatSper+NHE module we titrated in the components of the Ca_V+BK module under several combination schemes. First, we explored the addition of Ca_V and BK simultaneously maintaining their conductance ratio as in the original Ca_V+BK module (S4 Fig), then we added either channel alone (S5 Fig). This simple analysis showed that the modules cannot be combined freely without severely compromising the dynamics. Only in situations where one of the original modules largely predominates over the other (either θ ∼1 or θ ∼0) the mixed models predict the characteristic decay of the amplitude and the increase interspike interval. BK alone added to the CatSper+NHE module will cancel the oscillations at values as small as 10% of its reference value. At even lower densities it is functionally negligible, modulating the average interspike interval or the duration of the whole spike train without modifying the characteristics of the oscillations. These results are not entirely unexpected considering that in each oscillation of the Ca_V+BK module the system spends most of the time hyperpolarised, while in the CatSper+NHE it spends a significant fraction of the cycle depolarised, which is an absolute prerequisite for pH_i to return to lower levels closing CatSper gate.

These considerations notwithstanding, the initial analysis of the mixed scenarios was done under a single parameter regime in all the remaining parameters, and it is possible that this may have constrained results. To investigate this possibility we explored the parameter space in search for mixed scenarios that would be coherent with the experimental Ca-spike trains based on the criteria defined in Sec. 4.4. The purpose was not to identify which parameters have the greatest impact on the models’ quantitative output, for which we could have resorted to Latin hypercube sampling and quantitative sensitivity analysis as done by others (e.g. [25, 26]). Our more immediate goal was to relax the parameter settings in order to detect potential mixed scenarios. Because our assessment of the model output is essentially a qualitative (binary) criterium of coherence with observations, the quantitative sensitivity analysis via partial rank correlation coefficients (as in [25]) or other methods (as in [26]) was not feasible. Based on these considerations, we devised a simple algorithm inspired on adaptive evolutionary dynamics to explore the parameter space. This algorithm (Sec. 4.5 and S1 Algorithm) evolved a seeding population of parameter vectors in a regular grid first neutrally and subsequently under stringent binary selection criteria. Variants of the parameter vectors were generated by a stochastic process, based on a random walk in logarithmic parameter space. The initial grid ensured that the abundances of the components of the Ca_V+BK and CatSper+NHE modules were systematically explored in broad ranges, between negligible and excessive conductances. The generation of parameter vector variants ensured a random exploration of the parameter space centred on the realistic estimates of the other parameters, to avoid diversions into spurious parameter regimes. The final ensemble of parameter vectors produced by the algorithm contains both a population that was selected for being coherent with experimental data and an unselected population derived from and closely located to the former (Sec. 4.5 and S1 Algorithm). The projection of the parameter vectors in these two populations on the plane (g_cv, g_cs) shows 6 clusters, labelled a-f, depicted in Fig 8A. It is rather conspicuous that the cluster including the reference parameter set for the CatSper+NHE module (cluster b) is narrower than the other clusters. The cluster seems to be more sensitive to parameter changes as the within-cluster fraction of selected parameters is 8%. which is about half the average fraction of selected parameter vectors in the ensemble of the other clusters (16%). The mapping to the plane of the predicted spike amplitude and interspike interval regression coefficients indicates a narrow range of spike amplitude decay (Fig 8B). The other five clusters, including all mixed scenarios and the cluster centred on the reference Ca_V+BK module, are more spread both in parameter space (g_cv, g_cs) and in the plane (b_T, b_A) summarising the predicted output. To understand the mixed scenarios we asked whether the presence of all the channels Ca_V, BK or CatSper was essential to produce the predicted dynamics. To this end, we compared the numerical solution obtained with a given parameter set with the solutions obtained by specifically cancelling the conductance of Ca_V and BK on the one hand, or that of CatSper, on the other hand (examples of these numerical solutions are shown in Fig 8 for parameters singled out from each cluster). These comparisons revealed that CatSper has no functional consequences in the parameter vectors in clusters a, c and f whereas Ca_V and BK are irrelevant in those of cluster b. In clusters d and e, which include conductance densities of CatSper higher than the reference ones, the characteristic dynamics were severely distorted if either CatSper or Ca_V and BK were eliminated. In other words, the reference parameter set for the Ca_V+BK module captures the essential qualitative features of the dynamics associated with clusters a, c and f and the reference CatSper+BK module represents well those of cluster b. However, the dynamics corresponding to clusters d and e may not be fully captured by the reference modules.

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Fig 8. Exploration of mixed scenarios involving the components of the Ca_V+BK and CatSper+NHE modules.

The state space of the model was extended to {S, R_F, R_H, R_L, G, ,, V,,,,,, C, H}. A large number of parameter vectors resulting from a search of parameter space for coherence with experimental time series is illustrated in this figure according to clustering in the projection plane (g_cv, g_cs), predicted dynamics, and predicted functionality of the components of the two modules. During the exploration, coherence with experimental time series required that the resting C(0) is about 100nM, the average spike interval is in the range 0.37s-1.38 s, the predicted (b_T, b_A) point falls within the convex hull of most representative experimental values in the bottom right quadrant (polygon in A, right). A) On the left, the clustering of the parameter vectors in projection in the (g_cv, g_cs) plane is shown. On the right, the envelope and temporal organisation of Ca-spike trains predicted by each of these parameter vectors is represented as the bivariate plot of the spike-amplitude coefficient b_A vs the interspike interval coefficient b_T. In both graphs the dots represent individual parameter vectors which are coloured according to the cluster if they are fully coherent with experimental time series (selected) and grey otherwise (unselected). For each cluster in the plane (g_cv, g_cs) a parameter vector has been singled out (smaller filled circles). The two larger filled circles represent the reference parameter vectors for Ca_V+BK and CatSper+NHE modules. B) Illustration of the dynamics corresponding to the parameters vectors singled out in clusters c (top left), d (top right), e (bottom left) and f (bottom right) in A. The curves are the numerical solutions of the variable C in the model under the full parameter vector (coloured according to the cluster), setting g_cs = 0 (black) and setting g_cv = g_bk = 0 (grey). Notice the overlap between coloured and black curves on the graphs on top.

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It is worth noticing that the numerical solutions obtained with the parameter vectors in the CatSper+NHE module cluster (cluster b) show in each oscillation cycle the typical preponderance of depolarisation, with blunt Ca-spikes and marked pH_i oscillations. The parameter vectors in the other clusters display very transient depolarisation times, with sharp Ca-spikes (Fig 8B) and pH_i oscillations with shallow amplitude around a relatively high value, qualitatively similar to those obtained with the reference Ca_V+BK module (Fig 4). As we will show in the next section, this unique characteristic of the CatSper+NHE module cluster offers an avenue for model selection.

2.9 The CatSper+NHE module, but not the Ca_V+BK module or any mixed scenario, predicts the cancelling of oscillations and sustained [Ca²⁺]_i following controlled alkalinisation

The two candidate modules for the signalling network activated by SAPs are, by construction, differently coupled to the pH_i dynamics. The NHE and [H⁺]_i can be coupled to the Ca_V+BK module in the extended model by making θ^cv = 1 and θ^cs = 0 in Eqs 48–52. Under these conditions the exchanger is activated by hyperpolarisation and the proton concentration becomes an output variable, in the sense that pH_i does not feedback on any of the processes that determine Ca²⁺ fluctuations (proton concentration H does not appear in any of the equations that govern the variables of that model variant). Therefore, this module would predict that changes in the pH_i should not affect the Ca²⁺-spike trains. In contrast, in the CatSper+NHE module, the proton dynamics is tightly coupled to the Ca²⁺ oscillations and is an essential part of the mechanisms underlying the oscillatory behaviour. In our analysis of the parameter dependence of the CatSper+NHE module, we systematically found oscillations in [Ca²⁺]_i that were concomitant with pH_i oscillations having the same period, albeit phase differences. Parameters that lead to slower proton dynamics such that the pH_i oscillations vanish, resulted in cancellation of the oscillation in [Ca²⁺]_i.

The characteristics of the two modules imply distinctive responses to a manipulation that would artificially raise and fix the pH_i inside the flagellum. In this situation, the Ca_V+BK module would predict that the Ca²⁺-fluctuations would be unaffected, while the CatSper+NHE module predicts that sustained pH_i increase would cancel [Ca²⁺]_i fluctuations and this cation would be sustained at higher concentration. These effects are illustrated in Fig 9A and 9B, where we also overlay the results of registering Ca²⁺ dynamics in the presence of ammonium chloride (NH₄Cl), an agent that arguably raises the pH_i without affecting any of the other channels modelled here. As observed in the experimental trace, the exogenous alkalinisation gives rise to a dynamics that is qualitatively similar to the one predicted by the pH_i-dependent CatSper+NHE module. The conspicuous effect of alkalinisation on [Ca²⁺]_i dynamics is not predicted by any of the mixed scenarios that were otherwise coherent with the experimental time series studied the previous Sec. 2.8 (Fig 9C). Although this result might appear surprising it has a simple rational. The oscillations in the CatSper+NHE module (as well as those obtained under all parameter vectors in cluster b) are characterised by and are strictly dependent on large amplitude alkalinisation and acidification cycles, which in turn require sufficiently long depolarisation in each cycle (Fig 9D). In contrast, the oscillations in the Ca_V+BK module as well as those of the candidate mixed scenarios are characterised by very transient depolarisations resulting in sustained alkalinisation with shallow pH_i oscillations. In these cases, the concomitant oscillations in the fraction of open CatSper are not driven by alkalinisation-acidification cycles and therefore manipulations that keep the cytosol in an alkaline state do not block the oscillations (Fig 9D).

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Fig 9. Responses to manipulation of pH_i in the reference Ca_V+BK and CatSper+NHE modules, and in mixed scenarios.

In A and B, the graphs show the numerical solutions of the indicated variables under the reference parameters of the Ca_V+BK (as in Fig 4) and the CatSper+NHE (as in Fig 6) modules. In both cases, the normal response to SAP release at time 0 s (vertical dashed line) develops until it is perturbed by an artificial raise in pH_i (indicated by the second vertical dashed line) to a constant value maintained thereafter. Variables representing pH_i and NHE in the model were coupled to those of the Ca_V+BK module with the same reference parameters used in the CatSper module. The grey dots are a rescaled trace of the relative intensity of a Ca-sensitive fluorescent probe in S. purpuratus individual sperm cell bound to a coverslip obtained by adding first 100 nM Speract (vertical dashed line) and subsequently 10 mM NH₄Cl (second vertical line). C) Parameter vectors projected onto the plane (g_cv, g_cs) (as in Fig 8A) were coloured according to predicted model responsiveness during identical pH_i manipulation protocol: if alkalinisation results in sustained, slow decaying C dynamics the parameter set is representing according to the cluster’s colour, otherwise it is depicted in grey. D) Plot of the fraction of open CatSper channels vs pH_i in the numerical solutions of the model parameterised under the reference CatSper+NHE module, the reference Ca_V+BK module, and with the parameter vectors singled out from the clusters c, d, e and f illustrated in Fig 8. The trajectories are displayed for the interval 2-12 s to avoid transients.

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2.10 Coexistence of two stable states in unstimulated sperm is predicted by the CatSper+NHE module and may be revealed by transient membrane depolarisation

The CatSper+NHE module is overall the most coherent with the experimental observations. Another of its most distinctive features is that it predicts the coexistence of two possible stable equilibria characterised by [Ca²⁺]_i at either basal levels or above basal levels. This latter state is further characterised for being more acidic and with higher membrane potential than the basal state. The coexistence of two states is best seen in the bifurcation diagrams in the top graph of Fig 7. According to the CatSper+NHE module, a sufficiently strong perturbation of the membrane potential may force a switch from the lower basal steady state to the state characterised by higher [Ca²⁺]_i, where it will remain (Fig 10B). This behaviour contrasts with any model with a single resting stable state, such as the one with Ca_V+BK module, in which the rise in membrane potential has, at best, transient effects on membrane potential or pH_i but not on [Ca²⁺]_i, as illustrated in (Fig 10B). As a corollary of these properties, the Ca²⁺-spike train will always terminate in the basal level according to the Ca_V+BK module, while according to the CatSper+NHE module, the [Ca²⁺]_i may remain at high values after the spike trains vanish under particular dynamics that favour depolarised membrane potential, as is the case of the time course depicted in Fig 6.

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Fig 10. Ca_V+BK (A) and CatSper+NHE (B) modules predict distinctive responses to manipulation of membrane potential.

The graphs show the numerical solutions of the indicated variables under the reference parameters as in Figs 4 and 6. The normal resting state is perturbed (in the absence of SAP, S = 0 nM) by setting the membrane potential to a higher value (V = 0 mV) at time t = 0 for a short period of 0.25 s (indicated by the two vertical dashed lines).

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4.1 The general modelling framework

Our strategy is to describe the dynamics of the SAP-activated signalling pathway in order to understand how the [Ca²⁺]_i fluctuations are controlled by different molecular components. A system of ordinary differential equations (ODE) describes the dynamics of the membrane potential and of the intraflagellar concentrations of cGMP, protons and Ca²⁺. These four essential variables depend on the activities of different flagellar channels and enzymes, the dynamics of which are also described by ODE (Sec. 4.2). The four variables are the main observables in the model and offer the means to assess its predictions by comparison with experimental data.

4.2 Signalling components and channels

In the following section, we will present the molecular components and channels that were implicated in SAP signal transduction and that are relevant for our study. For each of these components we will describe: i) the evidence for their role in SAP-signalling, ii) a mathematical model for its dynamics, iii) the reference parameter values and how they were obtained.

4.3 Computational implementation and parameter estimations

The systems of differential equations were solved numerically with Mathematica 11 (Wolfram Research, Inc., Mathematica, Version 11.0, Champaign, IL), XPPAUT 8 [24] or R (using the package deSolve [75]). XPPAUT 8 was used for the bifurcation analysis (see below). Mathematica was further used to fit the model to experimental data using the Nonlinearmodelfit function with the optimisation method Differential Evolution with constraints. In the analysis, we defined several model variants based on subsets of the variables and equations presented above. Some of these defined modules were used to estimate the parameters and some were used to analyse specific properties.

The Upstream module (illustrated in Fig 1) was fitted to data in two phases:

• The state space comprised by the variables {S, R_F, R_H, R_L, G}, which includes the reactions involving the SAP-receptor (Eqs 8–11) and the cGMP dynamics (Eq 4), was fitted to kinetic data of cGMP measured in bulk cell populations [22]. Data points were read from the indicated publication and are expressed in units of pmol per 10⁸ cells. The cGMP kinetics obtained with the different SAP concentrations used in [22] were fitted simultaneously with the model (S1 Fig). During this fitting the dimension cGMP was pmol per 10⁸ cells instead of molar concentration, such that the scaling factor θ_G (described in the Sec. 4.1.2) was not called in. The parameter value of σ_G obtained in this phase, being measured in pmol per 10⁸ cells per s, was subsequently rescaled to molar concentrations per s, taking into account the flagellar volume.
• The remaining variables of the module (, , V), which involve the early electrophysiological changes caused by SAP (i.e. hyperpolarisation and repolarisation due to the opening of KCNG and spHCN, respectively), were used to fit the gating parameters of the ion channels. The spHCN parameters were either extracted or calculated from [62, 63, 61, 76], while those of KCNG were based on the values reported in [10, 59, 48]. Fixing these values, the conductance density parameters of these channels were calibrated together with B_G (included as a free parameter in the scaling factor θ_G), by fitting the dynamics of V with the kinetic data from [10], which comprise the early response of the membrane potential change. Only the first second of the time series was used in the fitting, taking into consideration that, by itself, this module is expected to fit only the early transient behaviour of V before other downstream channels with strong effect come into play.

In order to assess and calibrate this module, we assumed that the overall dynamics under both single-cell and population regimes can be compared using a scaling factor; this depends on the flagellar volume term to convert the cGMP signal into the effective concentration capable of exerting a physiological effect at the single cell level (see parameter θ_G, Sec. 4.1.2).

To study the dynamics of intraflagellar calcium, we set out to explore the possible contribution of different channels under two main scenarios.

• In one scenario, the model features the Ca_V+BK module and has the state space {S, R_F, R_H, R_L, G, , , V, , , , C}. The values of the Ca_V channel parameters were adjusted based on the characteristics of low-voltage-activated T-type Ca²⁺ channels [77], for which there is evidence suggesting their presence in sperm [14, 10] (Table 1).
• In the alternative scenario, the model features the CatSper+NHE module and has the following state space {S, R_F, R_H, R_L, G, , , V, m_cs, h_cs, , C, H}. The CatSper gating parameters were set such that its [Ca²⁺]_i and pH_i sensitivity range is placed within the physiologically expected values in reported natural SAP responses (S3 Fig).

The channel parameters were tuned such that the numerical solution of the variable C representing [Ca²⁺]_i in the model were comparable to those of the experimental time series of Ca²⁺-sensitive fluorescent probes using the criteria defined in the next Subsection 4.4.

Finally, to understand how the Ca²⁺-spike train patterns are controlled by the Ca_V+BK and the CatSper+NHE module, we performed bifurcation analyses of two simpler models, in which G was input parameter and state spaces were respectively , , V, , , , C} and , , V, m_cs, h_cs, , C, H}.

4.4 Comparison of the structure of the Ca-spike trains in experiments and models

The [Ca²⁺]_i dynamics in response to SAP was imaged in individual S. purpuratus sperm immobilised on a cover slip using intracellular Ca²⁺-sensitive fluorescent probe following the uncaging of a photoactivatable Speract analogue by a 0.3s pulse of UV light (experimental details in [78]). By measuring the average fluorescence of the flagellum of each cell at each time point we obtained a time series of this proxy of [Ca²⁺]_i). Each time series was smoothed using a cubic spline (as implemented by the R function smooth.spline) and the maxima of the smooth curve used to identify Ca-spike trains that were characterised as (t_k, A_k, T_k), where k is the ordered spike index (k = 1, 2, 3, …) and t_k, A_k, and T_k = t_k+1 − t_k are the time, the amplitude and interspike interval. To summarise the structure of each individual time series we computed the average interspike time T_k, and the regression coefficients of the relative amplitude A_k/A₁ and relative interspike interval T_k/T₁ over the spike index, denoted b_A and b_T, respectively. To compare the [Ca²⁺]_i dynamics in the model the same procedure was applied to the numerical solutions of C. The structure of the spike trains in the model was considered to be coherent with those experimentally observed according to the following selection criteria: the average spike interval is in the range 0.37 s-1.38 s, the predicted (b_T, b_A) point falls within the convex hull defined by experimental values in the bottom right quadrant, i.e. the spike amplitude decrease whereas the interspike interval increase in time b_A < 0 and b_T > 0. As mentioned in the Sec. 2.3, this pattern of the spike trains is the most representative of the experimental time series and the most challenging for the models. In addition, to filter out spontaneous oscillations and alternative equilibria in multistable regimes, we imposed that the resting steady state value of [Ca²⁺]_i, , obtained following numerical solutions with S = 0, is about 100nM. It is important to notice that the fluorescent probe intensities were not calibrated and therefore mean flagellar intensities are not accurate [Ca²⁺]_i measurements. We assumed that the fluorescence intensity was proportional to the actual concentration of [Ca²⁺]_i, focusing our analyses on relative differences and values. This assumption underlies the co-plot time series and model solutions presented in the figures and also the direct comparison of the theoretical and experimental b_A values. We believe this assumption is reasonable, considering that all Ca-fluorescence intensity spikes analysed were below the control intensities elicited by ionomycin.

4.5 Mixed scenarios parametrisation and parameter sensitivity

We analysed mixtures of Ca_V+BK and CatSper+NHE modules by assembling the most extended model with state space {S, R_H, R_L, G, , , V, , , , m_cs, h_cs, , C, H}. The parameters of this model are the union of all the parameters listed in Table 2.

Any mixed scenario involving the two modules can be parameterised by setting: (48) (49) (50) (51) (52) with the superscripts cv and cs denoting the reference values in the Ca_V+BK and the CatSper+NHE modules, respectively, and all the remaining parameters taking the values in Table 2. θ^cv and θ^cs are non-dimensional scaling factors that allow to weigh the contribution of each module. With this formulation, the original Ca_V+BK module dynamics can be recovered in this extended model by setting θ^cv = 1 and θ^cs = 0, whereas the CatSper+NHE module can be obtained with θ^cv = 0 and θ^cs = 1.

From the above equations, we can express a ratio of the maximal conductance density of CatSper with respect to that of Ca_V: (53)

In a first approach to mixed scenarios we interpolated between the two modules. Starting with the CatSper+NHE module we added fractions of Ca_V and BK channels in three ways: a) adding Ca_V + BK while decreasing proportionally CatSper by setting θ^cv = θ and θ^cs = (1 − θ) with θ ∈ [0, 1] (S4 Fig); b) adding only Ca_V by setting θ^cv = θ and θ^cs = (1 − θ) while forcing g_bk = 0 (S5A Fig); and c) adding only BK by defining and setting θ^cv = 0 and θ^cs = 1 (S5B Fig).

A more systematic analysis of mixed scenarios contemplated the conductance of channels that compose the two modules but also variants of the values of all the other parameters, except for those representing the Upstream Module (which were fixed at their point estimates in Table 2). The exploration of this parameter hyperspace was done using an algorithm inspired on adaptive evolutionary dynamics, specified in S1 Algorithm. Briefly, the algorithm comprehends two phases, one of neutral exploration of the parameter space by evolving a seed population of parameter vectors in a regular grid, followed by further exploitation of parameter vectors selected according to the criteria described above (Sec. 4.4). First, each parameter vector in the nodes of the grid was obtained by taking θ^cv, θ^cs ∈ {0.0001, 0.1, 0.2, 0.3, …, 2.0}. Each vector in this population was used randomly as a seed to generate a daughter parameter vector obtained by multiplying each parameter value by an independent lognormal distributed random variable (specified as e^x with x being normally distributed with mean 0 and standard deviation 0.04). Each daughter parameter vector was appended to the evolving population, irrespective of the output behaviour, serving as random seed to generate new daughters. The method to generate parameter vector variants favoured values close to the reference ones (Table 2), reducing the probability that the random exploration would spread to unrealistic regions of the parameter space (e.g. leading to inactivation, opening and closing of Ca_V within incongruous membrane potential intervals). The dynamics predicted by each parameter vector , seeds and daughters alike, was obtained by solving numerically the extended model. The process of generation of new parameter vectors was repeated until the clouds of points, spreading from the nodes of the grid, once projected in the plane (g_cv, g_cs), started to visibly coalesce. At this stage, several parameter vectors satisfied the selection criteria for coherence with experimental observations and were selected for further exploration in the second phase of the algorithm. In each iteration, a vector in the selected population was randomly picked to generate a daughter vector by the same procedure as above. This daughter vector was analysed and added to the evolving selected population, serving as seed for further exploration, if it met the selection criteria; otherwise it was stored in a list of unselected parameter vectors. The latter vectors are by construction closely located in the parameter space to those of the evolving selected population from which they derived but, being incoherent with observations, they were not further explored. The selected and unselected vector populations were used to build clusters based on the proximity in the projection plane (g_cv, g_cs). The iterative generation of daughter parameters from the selected population was repeated until the clustering in the plane (g_cv, g_cs) was stable (i.e. no new clusters were observed after a large number of iterations). Within each final cluster, the fraction of number of selected parameter vectors over the total number of vectors is a kind of measure of the parameter sensitivity. The smaller this fraction the more sensitive the model is within the cluster.

4.6 Resting state and initial conditions

Initial conditions of the gating variables of KCNG, spHCN, Ca_V, BK and NHE, as well as parameters σ_C, E_L and δ_H, were all calculated by solving the system at equilibrium in the absence of SAP, that is, setting each differential equation to 0 under the condition S = 0. Physiological observables G, C, H and V were initialised to their respective resting values, as reported in the literature (Table 2).

The parameters governing NHE activation were chosen such that approximately 40% of exchangers were active at the resting state, as inferred from experimental data in activated sperm without any chemoattractant stimulation [35, 79], whereas the maximum effective proton flux by NHE, J_max, was fitted to limit the pH_i up to ∼7.7, after physiological stimulation by SAP [80, 81].