dataesr/scientific-paragraphs-categorization huggingface dataset:
4 ). A similar oscillatory behaviour with similar period but different phases is observed on the fractions of open channels (Fig. 4 , 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 Ca 2+ oscillations is differently shaped by the CatSper module. The individual Ca 2+ -spikes are more prolonged and the corresponding nadir is relatively shorter. The decay of the spike train envelop is less pronounced for the reference parameters in the CatSper 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 module does not predict the sustained plateau of the Ca 2+ -oscillation nadirs present in the Ca V +BK module.
Finally, another interesting difference between the two modules is the envelop of the voltage spikes whose amplitudes decrease progressively in the Ca V +BK module and increase in the CatSper module (bottom in Figs. 4  and 6 ). The basis for the limit cycle with CatSper is qualitatively straightforward. The hyperpolarisation increases the activity of the Na + /H + exchanger, this triggers the extrusion of protons and accordingly raises pH i. As a consequence of this alkalinisation, the voltage sensitivity of CatSper shifts to lower values (see Fig. S3D ) and some of these channels open. The inward Ca 2+ currents rise 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 reduces the activity of the exchanger and the pH i tends to return to its basal level, shifting the CatSper voltage sensitivity back to higher values.
This, together with Ca 2+ -inactivation of CatSper, 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 raise in pH i , and the recovery of CatSper currents that eventually overcome the hyperpolarising currents. As in the case of the Ca V +BK module, the bifurcation analysis of the model 15/43 featuring the CatSper module but with constant input G 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 G ranges, such that two [Ca 2+ ] 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 G values 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  [25] ), 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 × 10 5 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 (data not shown). 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  Ca-spike amplitude (nM) q q q q q q q q q q q q q q q q 1 3 5 7 9 11  13   Ca-spike interval (s) q q q q q q q q q q q q q q q 1 3 5 7 9 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 {f o kn ,f o hc ,V , f a nh ,m cs ,h cs ,C,H}. Top: The graphs depict the variable C as a function of the input parameter G. The thick continuous lines are the stable equilibria and the dashed lines indicate unstable equilibria, as obtained by XPPAUT  [25].
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. 6. 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 gray dots correspond to the interspike interval in the numerical solutions, as in the top graph. CatSper module (Fig. 6 ) are now amenable to a quantitative interpretation. 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 (empty circles numbered) are more marked in this module than in the case of the Ca V +BK module. 17/43 3. 4 Ca V +BK and CatSper modules predict distinctive responses to controlled manipulations of intracellular pH and of membrane potential One of the most direct ways of distinguishing which of the two modules is the most adequate to describe the essential structure of the pathway activated by SAPs is to test their coupling with the pH i dynamics.
If sNHE and [H + ] i are coupled with the Ca V + BK module, the former would be activated by transient hyperpolarisation, and the proton concentration would turn out to be just an output variable, in the sense that pH i does not feedback on any of the processes that determine Ca 2+ fluctuations; in other words, the concentration of protons 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 entail major changes in the Ca 2+ -spike trains. In contrast, in the CatSper module, the proton dynamics is tightly coupled to the Ca 2+ oscillations and is an essential part of the mechanisms underlying the oscillatory behaviour. In our analysis of the parameter dependence of the CatSper module, we systematically found oscillations in [Ca 2+ ] 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 [Ca 2+ ] i. It is worth noticing that attempts to detect periodic pH i oscillations in sea urchin sperm have failed hitherto (see  [23] ; unpublished observations), which might be explained by experimental limitations or simply because periodic pH i oscillations do not happen in vivo. Nevertheless, the different dependence of the two modules on pH i offers a means to distinguish their possible role experimentally. Suppose that one could artificially raise and fix the pH i inside the flagellum. In this situation, the Ca V +BK module would predict that the Ca 2+ -fluctuations would not be affected, while the CatSper module predicts that sustained pH i increase would cancel [Ca 2+ ] i fluctuations and this cation would be maintained at higher concentration.
This effect is illustrated in Fig. 8A  and B , where we also overlay the results of registering Ca 2+ dynamics in the presence of ammonium chloride (NH 4 Cl), an agent that arguably raises the intracellular pH 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 module. Another distinct feature of the two candidate modules is that at resting state the Ca V +BK module has a single stable equilibrium, while the alternative module with CatSper predicts the coexistence of two possible stable equilibria, characterised by [Ca 2+ ] i at either basal levels or above basal levels. This is best seen in the bifurcation diagrams in the top graph of Fig.
7. A temporal raise in membrane potential, according to the Ca V +BK module, may lead to an increase in [Ca 2+ ] i that would rapidly return to its unique steady state (Fig. 8C ). In contrast, according to the CatSper 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 [Ca 2+ ] i , where it will remain (Fig. 8D ). As a corollary of these properties, the Ca 2+ -spike train will always terminate in the basal level according to the Ca V +BK module, while according to the CatSper module, the [Ca 2+ ] 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.
The output of the CatSper module is tuned by the Ca V + BK channels if present in low proportions In the previous sections, we have shown that both alternative modules, once coupled with the upstream module, can semi-quantitatively describe the observed data. These 18/43 ). The graphs show the numerical solutions of the indicated variables under the reference parameters as in Fig. 4  and 6. In A and B, a normal response to SAP develops during the first 5 s, and is then perturbed by an artificial raise in pH i to a constant value maintained thereafter. Variables related to pH i and sNHE are coupled to Ca V +BK module with the same reference parameters used in the CatSper module. The noisy gray line is a rescaled trace of the intensity of a pH-sensitive fluorescent probe in S. purpuratus sperm cells (obtained as described in  [23] ).
5 the total conductance of the components in each module are half of their respective reference values. To ensure the recovery ot the aforementioned extreme scenarios, in our new weighted scheme, the parameters that regulate the resting membrane potential and ion concentrations were weighted as well 19/43 (see equation 9). The importance of how the pH i dynamics influences the [Ca 2+ ] i response was illustrated in the previous section, so we now take the CatSper module as our reference scenario since it contains the necessary components to account for this regulatory mechanism. We titrate in the components of the Ca V +BK module under three combination schemes. First, we explore the addition of Ca V and BK separately (Fig. 9 ), then we add both components maintaining their conductance ratio as in the original module Ca V +BK (Fig.
10 ). This simple analysis shows that the modules cannot be combined freely without compromising the dynamics. In A and B, we show numerical solutions for calcium under different values of the weighting parameter θ. As reference, the original scenario of only CatSper, i. e. θ = 0, is shown in the top row, while the subsequent rows correspond to the gradual increase of θ. In A, the reference value of the Ca V conductance density, g cv , is multiplied directly by θ, whereas the reference value of CatSper conductance density is multiplied by (1 -θ); the resulting ratio of CatSper/Ca V conductance densities, r g , is shown for each θ value. The higher the value of r g , the greater the predominance of CatSper with respect to Ca V. In B, θ sets the percentage of the reference value of g bk , the conductance density of BK; unlike scenario A, the conductance density of CatSper is not weighted by θ.
For each titration, the corresponding modified g bk value is shown The most noticeable effect of adding relatively small titers of Ca V to the CatSper module, depicted in Fig. 9A , is the overall response acceleration without qualitatively modifying the envelop or the shape of the peaks. For the first two lower titers of Ca V (second and third rows), the frequency increases but the total duration of the spike train is shortened compared to the original CatSper scenario (first row). For the third titer, the response is both accelerated an extended. However, by the fourth titration (last row), the response lasts for tens of seconds and tends to a high calcium level instead of approaching the resting state, these features being incompatible with what is observed 20/43 experimentally. The second combination scheme in which BK is added to the CatSper module (Fig.
9B ), shows how extremely sensitive this module is when coupled with a channel that has the characteristics of BK. From very small values of conductance density, g bk , the oscillations disappear while leaving only a tonic calcium increase that reaches intermediate values above the basal level (rows 3 and onwards). Nonetheless, in considerably small values of θ, the capacity of the system to oscillate is able to cope with the addition of BK to some extent; this is seen by a change in the time scale of the response without altering the qualitative form of the peaks (second row). In the latter case, there is only a subtle increase in frequency, perceivable by considering that in the first 15 s there are 13 fluctuations without BK, whereas with BK there are 14. Regarding the total total length of the spike train, with BK the response ends just 2 seconds later compared to the CatSper module alone.
Starting from the model that includes the CatSper module, numerical solutions for calcium are shown under different values of the weighting parameter θ, which controls the percentage of module Ca V + BK that is being added. θ multiplies the reference conductance g cv and g bk , and correspondingly the conductance density of CatSper decreases by the factor (1 -θ). For reference, the original CatSper-only scenario is shown first, i. e. θ = 0, and the subsequent rows correspond to the gradual increase of θ. The effect of this parameter on the channel densities is reported in the coefficient r g , which measures the total conductance ratio CatSper/Ca V , as well as the new values g bk. As in the figure  9 , the greater the value of r g , the greater the predominance of CatSper on the total calcium conductance with respect to Ca V.
When analysing the most complex scenario, consisting of the combination of the two modules, similar effects to the observed in the titration of only Ca V were retrieved, but at different values of θ (Fig. 9A ), i. e. the increase in the frequency of oscillations without introducing significant changes to the shape of the peaks or to their envelop, as well as the response lengthening. On the other hand, when comparing the two previous schemes with the mixed one, it is suggested that only when BK is added together with Ca V , the CatSper-dependent response is more tolerant to the presence of that channel at higher conductance densities (compare values of g bk in figures 9B and 10); moreover, BK is also responsible for extending the duration of the spike train (e. g. compare the third row of figures 9A and 10).
The upstream module alone was able to explain the early hyperpolarisation dynamics based on the joint action of KCNG and spHCN channels (Fig. 2 , bottom). The parameters were set such that, in the absence of any other channels, the repolarisation initiated by spHCN following the hyperpolarising current of KCNG, leads to a partial recovery of the membrane potential that remains below the resting value. The partial recovery implies that spHCN effective conductance is sufficient to set into motion the downstream modules (e. g. the opening of closed Ca V channels or shifting to higher opening rate of the voltage-gate of CatSper) but not too strong to prevent the oscillatory [Ca 2+ ] i dynamics (results not shown). This property of the signalling network is amenable to experimental corroboration by assessing whether the membrane potential recovery is only partial under a setting in which all calcium entry is prevented by using zero external concentrations of this cation.
In this module, the oscillations are strongly modulated in amplitude and tempo by the Ca 2+ -dependent channel BK to produce the experimentally observed progressive increase in the intervals between [Ca 2+ ] i spikes. The oscillations predicted by this module vanish when cGMP falls below a critical value involving a Hopf-type bifurcation (Fig. 5A ). The second module capable of explaining the organisation of the [Ca 2+ ] i fluctuations features a single channel, CatSper, that is voltage-and pH-dependent and is reversibly inactivated by intracellular Ca 2+ ions. In contrast with the previous module, which is set in motion directly by the effect of transient hyperpolarisation that relieves Ca V channels from inactivation, the initial opening of CatSper channels is triggered indirectly by alkalinisation of the cytoplasm driven by the hyperpolarisation-dependent Na + /H + -exchanger activity (Fig.
S3 ). The cycling behaviour is produced by the negative feedback that depolarisation driven by CatSper activity has on inactivating the Na + /H + -exchanger, reducing pH i and decreasing CatSper activity. Another feedback arising from the inactivation of CatSper by [Ca 2+ ] i plays a more preponderant role in controlling the length of the Ca 2+ -spike train and in increasing the interspike intervals. As in the case of the first module, the oscillations generated by the module stop when the cGMP activity falls below a critical value (Fig. 7 ). It is noteworthy that, under the reference parameters used in this article, the overall temporal scale of the calcium response produced by the CatSper module is slower compared to the experimental tracings, in terms of real time units; however, if the relative progressive increase of interspike intervals is contrasted instead, this property becomes more marked in this model than in the one featuring the Ca V + BK module, and more closely related to the experimentally observed (Fig.
3C ). In the numerical solutions with the former module the interval between the last spikes can be two or three fold larger than the interval between the first spikes in the train, as observed in the experiments in single cells. A comparable increase in interspike intervals can be obtained with the Ca V +BK module under different parameter regimes (not shown). Conversely, the progressive decrease in the amplitude observed experimentally is better accounted for by the Ca V +BK module than by the CatSper module under the reference parameters used here. A more evident decline of the envelop of the Ca 2+ -spikes can be obtained using the CatSper module under other parameter regimes in which the spikes became less realistically interspersed in time. These two downstream modules confer some distinguishable properties to the signalling network when coupled to the upstream module.
One of those properties is the envelop of the membrane potential oscillations. In the module featuring the Ca V + BK channels pair. This envelop has a shape similar to that of the [Ca 2+ ] i spikes. In contrast, the CatSper module predicts that the amplitude in the membrane potential peaks should increase until it suddenly ceases (Fig. 6 , bottom), thus establishing an opposite trend to the calcium-spike train envelop. Interestingly, the ascending V envelop is more similar to the observed at prolonged times in membrane potential measurements of sperm populations stimulated with high doses of SAP  [8]. Another good example of distinctive features between the alternative scenarios is the permissive G range that allows for oscillatory solutions. At too low (close to resting) or too high intracellular GMP concentrations, the model with CatSper predicts a sustained rise [Ca 2+ ] i without noticeable oscillations (Fig.
Albeit these two candidate modules predict qualitatively the observed calcium responses under particular parameters regimes, we found that the three channels, Ca V + BK on the one hand and CatSper on the other, interfere with each other leading to unrealistic behaviours when the effective calcium currents are of the same order of magnitude (bottom in Fig. 9  and 10 ). This modelling result suggested that the two modules could be mutually exclusive raising the question of which one is predominating in sea urchin spermatozoa. The present analysis indicates that the two modules predict very distinct responses to the artificial induction of sustained alkalinisation. Hence, the CatSper module, but not the one featuring Ca V and BK channels, predicts that transient or sustained alkalinisation should result in sustained [Ca 2+ ] i as observed experimentally following the addition of NH 4 Cl to S.
purpuratus spermatozoa (Fig. 8A ). The confrontation of these two modules with the experimental measurements suggests that the channel responsible for the [Ca 2+ ] i responses to SAP is the pH-dependent CatSper instead of Ca V. Further considerations on the above matter are called for. It is worth emphasising some additional implications of the CatSper module, as these are also amenable to experimentation. Under the reference parameters, the oscillatory regime in the CatSper requires oscillations of proton fluxes and intracellular pH. Rather than inactivation of CatSper by [Ca 2+ ] i , the effective feedback loop explaining the intracellular Ca 2+ oscillations is the retroinhibition of the sNHE activity by depolarising currents. In other words, observing [Ca 2+ ] i oscillations dependent on CatSper implies necessarily the concomitant observations on pH i oscillations with the same period and different phase.
The observation that a rise in pH i precedes the first Ca 2+ spike in sea urchin sperm stimulated by SAP has been interpreted as an indication that Ca 2+ currents are a consequence of this early alkalinisation  [14, 22, 23]. This interpretation has become a major cornerstone of the hypothesis that pH-sensitive CatSper channel is responsible for Ca 2+ fluctuations. It is worth calling attention to the early time courses of pH i and [Ca 2+ ] i obtained by the numerical solution of the model featuring the Ca V +BK module (Fig. 8A ). In this model, where Ca 2+ currents are, by construction, independent and unaffected by pH i one can nevertheless observe an onset of alkalinisation that precedes the first [Ca 2+ ] i spike. This illustrates the frailty of inferring causation from time series data in general and raises a cautionary note on the interpretation that a pH-sensitive 24/43 Ca 2+ channel must be involved in sea urchin sperm responses to SAP.
Another testable prediction of the CatSper module is the existence of bistability at the resting values of cGMP. This implies that perturbations exist that may lead to sustained [Ca 2+ ] i value above the normal resting one, in the complete absence of SAP signals. We illustrated this by a transient rise in membrane potential that locks the system in the stable state characterised by higher, tonic-like [Ca 2+ ] i (Fig. 8D ). Another perturbation with comparable effects would be a transient yet complete depletion of cGMP that would push the system across a bifurcation point in which the lower equilibrium vanishes. Following such perturbation, hysteresis is predicted such that the system could remain in the higher [Ca 2+ ] i state, after restoration of the basal cGMP levels. The CatSper module bistability also implies that after a [Ca 2+ ] i -spike train terminates, the system may or may not return to the basal level.