Table 1.
Summary of experimental evidence on the main calcium channels proposed in the pathway activated by SAPs.
aIt is worthwhile to mention that out of the pharmacological blockers listed, none of them are 100% specific to the associated channels. bAntibody raised against a polypeptide of the α subunit in rat, which also recognises the homologous version in mouse and human sperm. cAntibody raised against pore-forming subunit polypeptides of A. punctulata [12] and S. purpuratus [18]. d It is of note that direct electrophysiological recordings for this ion channel have not been achieved in mature sea urchin spermatozoa compared with the homologous complex in mouse, human and macaque spermatozoa [19].
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, RF, RH, RL, G, ,
,
and
) free receptors RF bind SAP molecules and transit irreversibly through three receptor forms, each with less guanylate cyclase activity: High RH, low RL and inactive RI; 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 CaV+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.
Fig 2.
Numerical solution of the Upstream module overlaid with experimental data obtained in bulk sperm populations.
The model with state space {S, RF, RH, RL, 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 ghc=0). See main text for further details.
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 Ak and the interspike interval Tk = tk+1 − tk were calculated. The shaded region indicates the UV light pulse. On the right, the relative spike amplitude Ak/A1 (left axis; black) and relative interspike interval Tk/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 bA and bT were obtained. B) Cumulative frequency distributions of the spike-amplitude regression coefficient bA and interspike interval regression coefficient bT for all experimental time series analysed. C) Bivariate plot of the spike amplitude coefficient bA versus spike amplitude coefficient bA 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 CaV+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 bA and bT for the model, which were represented in C.
Fig 4.
Dynamics of the model featuring the CaV+BK module.
The state space is {S, RF, RH, RL, 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).
Fig 5.
Bifurcation analysis of the model featuring the CaV+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,
,
, fbk, 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 (gbk). 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 gbk (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).
Fig 6.
Dynamics of the model featuring the CatSper+NHE module.
The state space is {S, RF, RH, RL, 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.
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,
, mcs, hcs, 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.
Fig 8.
Exploration of mixed scenarios involving the components of the CaV+BK and CatSper+NHE modules.
The state space of the model was extended to {S, RF, RH, RL, 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 (gcv, gcs), 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 (bT, bA) 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 (gcv, gcs) 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 bA vs the interspike interval coefficient bT. 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 (gcv, gcs) a parameter vector has been singled out (smaller filled circles). The two larger filled circles represent the reference parameter vectors for CaV+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 gcs = 0 (black) and setting gcv = gbk = 0 (grey). Notice the overlap between coloured and black curves on the graphs on top.
Fig 9.
Responses to manipulation of pHi in the reference CaV+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 CaV+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 pHi (indicated by the second vertical dashed line) to a constant value maintained thereafter. Variables representing pHi and NHE in the model were coupled to those of the CaV+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 (gcv, gcs) (as in Fig 8A) were coloured according to predicted model responsiveness during identical pHi 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 pHi in the numerical solutions of the model parameterised under the reference CatSper+NHE module, the reference CaV+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.
Fig 10.
CaV+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).
Table 2.
Variables, intermediate functions, and reference parameters values.