2.1.1 Network dynamics.

In order to characterize the dynamics of the signaling pathway related to the early capacitation response, we model each of the components and interactions in terms of a discrete dynamical network (DDN). The configuration of this kind of networks is given by a set of N discrete variables, {σ₁, σ₂, …, σ_N}, which represent the dynamical state of all the network nodes. The network configuration evolves in discrete time steps according to a set of dynamical rules that update the state of each node. One of most studied and used formalism within discrete dynamical networks is the Boolean network approximation. Such a network was proposed by Kauffman [25] as a modelling framework to study the dynamics of metabolic and genetic regulatory systems, with the underlying hypothesis that each particular temporal pattern can be associated to a cell phenotype. These type of models have been shown to be successful in the study of many biological networks, e. g. [26–30]. According to the Kauffman formalism, the dynamical state of a given node σ_i at time t + 1 is determined by the state of its own set of k regulators at time t as follows: (1) where F_i is a regulatory function that assigns a value from a discrete state repertoire to a node σ_i. Note that each node σ_i has its own regulatory function F_i, which is built to capture qualitatively the inhibitory/activating character of the interactions of its particular regulator set. This function can be encoded through a truth table or expressed in terms of Boolean algebra equations [31]. In the most simple variant, the dynamical state of all nodes is updated synchronously by applying their respective regulatory function (Eq 1) simultaneously at each iteration.

The Boolean model can be generalized to multi-state logical models, in which each node σ_i is allowed to take m states instead of only two: (2) where is the set of values σ_i can take, .

Most of the nodes in the capacitation network have two dynamical states. In the case of ion channels, the dynamical states portray: open (state 1) and closed (state 0). In the case of ion concentrations (Table 4), the dynamical states portray: high concentrations (state 1) and low concentrations (state 0). In the case of the activity of a given protein, the dynamical states portray: high activity (state 1) and low activity (state 0). In the case of current nodes, they have three dynamical states that represent the current direction: inward (state 1), null (state 0) and outward (state -1). In the case of membrane potential, the dynamical states portray: depolarized (state 1), resting (state 0) and hyperpolarized (state -1).

In general, there are many ways reported in the literature for building regulatory functions, e. g. truth tables with random outputs [25], Boolean threshold networks [32], majority rules [33], etc. The method for building regulatory functions in our model is ad hoc, related to thermodynamics and electrophysiological considerations explained in Section 4.2. The corresponding regulatory tables are shown in the S1 File. Furthermore, the dynamical rules for integration nodes of ion fluxes and membrane potential were built in terms of an innovative scheme inspired in neural nets, explained in Section 2.1.4.

2.1.2 Multitemporal updating DDN.

As pointed out in Section 1, it has been established experimentally that the time scales with which different nodes interact is diverse. In order to take this into account, in our model we define a time evolution equation that incorporates a stochastic interaction wherein each input of a given node is sampled at a different rate according to a biased Bernoulli process (i. e. unfair coin tossing). The mean value of this process is calibrated to qualitatively approximate the operation rates of each kind of regulatory nodes (Section 4.4.2). With the above, we generalize the Kauffman multi-state model as follows: (3a) (3b) where is a random variable from a biased Bernoulli process with mean , sampled at time t, and .

Given that PKA-dependent phosphorylation, cholesterol removal effects and calcium diffusion from the RNE to the principal piece show characteristic times greater than other network processes (e. g. ion transport) [34, 35], we introduce a persistence of their effects, reverting to earlier times steps in order to update, producing a memory effect in Eq 5; i. e., in our model, each of the nodes associated to these phenomena retain their previous value for a defined average number of steps (see Section 4.4 for more details). For these cases, instead of Eq 3b, the node evolution is given by:

for t = 0, (4) and for t > 0, (5) where , and j stands for nodes PKA, SPCA, IP3R and ChAcc. Notice that, though the network model evolution is synchronous, the biological asynchronicity is dealt with in the model through this particular stochastic updating at the regulator interaction level.

2.1.3 Ion flux dynamics: Neural network scheme.

In order to manage ion fluxes related to several types of transporters, we include ion flux integration nodes as auxiliary nodes that operate in a similar fashion as in neuron integrated signals coming from several inputs. Updating depends on whether a given threshold is surpassed. For theses cases, each total flux node σ_i evolves according to a regulatory function F_i that consists of a weighted sum of its regulators , as in the McCulloch&Pitts scheme [36]: (6) wherein the sgn function discretizes the sum of regulators weighted by , according to: (7) where x is a dummy variable. As in Section 2.1.2, we can generalize the neural network scheme to a multitemporal variant scheme as follows: (8) where the nodes σ_i now have a neuron-like regulatory function in which each regulator participates through a biased Bernoulli process. The values of the set of all weights of the network are based on biological evidence as it is shown in Section 4.4. We emphasize that Eq 8 is applicable only to ion flux integrating nodes.

2.1.4 Ion flux dynamics with heterogeneous ion transporter: Neural network scheme with weight variability at population level.

In order to introduce variability in the total and relative number of ion transporters, we can generate populations of networks by modifying the set of weights of each network replicate. Every network r from the constructed population will have a set of weights , where is a random number sampled from a truncated Gaussian probability distribution G(1, D) with mean 1 and standard deviation D, defined over an interval [0, 2] (see Section 4.4.1). In general, networks coming from such a population will have a different set of weights among them. It is at this stage that the second aforementioned stochasticity is included in the model. It acts on the values of the weights of regulators, on the links, not on the nodes themselves. Since, as shown in the methods section, each individual weight depends on the number of the corresponding ion transporter, the variability can be expressed in terms of number of ion transporters. We can extend this notion of variability to multitemporal-neuron-like regulatory functions F_i as follows: (9)

With all these extensions of the Kauffman multi-state model and McCulloch&Pitts scheme, we have constructed a novel general updating scheme for the modeling of the early mouse sperm capacitation.

2.1.5 Ion flux integration nodes.

Starting from Eq 9 and adapting it to the particular case of ion flux integration nodes (I_Ca, I_H, I_HCO3, I_Cl and I_Na), we introduce auxiliary variables , which integrate all the fluxes j of a particular ion i for a given sperm r. Each one of these auxiliary variables updates depending on the state of its particular ion transporter set at time t. In order to consider differences in operation rates among ion transporter , we multiply each of them by a random variable from a biased Bernoulli process with mean at each time step t. Notice that this is a particular case of the scheme shown in Section 2.1.4. The parameter controls the average operation rate of . The resulting dynamical rule is: (10) where is an aggregate variable that contains both the electromotive force (emf) of flux j and the gating variable (if there is any) of the corresponding transporter j. In those cases wherein no gating variable is modulating the flux, as in cotransporters, pumps and exchangers, we set this to 1, except for sNHE, which does have a gating variable since it is the only chimeric exchanger with a voltage-sensing gating domain similar to ion channels [37]). Ion transporter j will or will not contribute in the updating of at time t + 1, depending on the random variable , which can be either 0 or 1. Note that every spermatozoon r of the population has its own set of regulatory functions for the net ion fluxes which can differ from spermatozoon to spermatozoon. Also that regulator nodes represent the product of the emf associated with ion transporter j and the respective gating variable, whereas weights would be the analog of effective conductances. The product of these terms gives an ion current expression based on Ohm’s law. Regulator nodes have a state repertoire defined by , where -1 corresponds to an emf that generates outward ion flux, 0 corresponds to null force, and 1 corresponds to an emf favoring ion influx. In Fig 3 we can see how a change in ion concentration x_i occurs through the net ion flux described above. Weights and rates are reported in Section 4.4.

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Fig 3. Ion flux integration scheme.

Diagram of the mechanism (time line) by which ion concentration x_i (bottom line) changes via net ion flux integration (middle line) determined by the weighted sum of all its linked electromotive forces (top line).

https://doi.org/10.1371/journal.pone.0245816.g003

2.1.6 Membrane potential node.

The membrane potential is a heavily linked core component of the network that in our model can take values from {−1, 0, 1}, which correspond respectively to hyperpolarized, resting and depolarized. For its time evolution we build a discrete version of the Hodgkin&Huxley equation [38] (shown in Eq 23) through a discretizing partition defined by Eq 25, such that the resting state is robust under small perturbations. As with the ion flux integration nodes, V is determined by a weighted sum of multiple sources: CatSper, Na⁺-channel, Slo3, Cl^--channel, the cotransporter NBC, the exchanger SLC26 and the leak current I_L. The weight of the i-th contribution is given by its effective conductance. The last term of Eq 11 is a consequence of Kirchhoff’s law. Hence: (11) where z_j is the charge sign of the reference ion in the current carried by transporter j. The latter quantity ensures that ion fluxes that are capable of modifying the membrane potential follow the typical convention used in electrophysiology, i. e. negative ion currents correspond to the influx of positive charges (e. g. CatSper, Na⁺-channel), positive currents correspond to outward flux of positive charges (e. g. Slo3) or influx of negative charges (e. g. Cl^--channel, NBC). The random variable again is in charge of modeling different operation velocities of ion transporters. Weights and velocities ν_j are reported in Section 4.4.

2.1.7 Operational definition of capacitation at the single-spermatozoon level.

Most of the capacitation relevant variables are reported as isolated measurements of either population distributions or time series, i. e. there are very few available multivariate determinations in single spermatozoon [21, 39, 40]. This hinders the study of what changes occur simultaneously in a spermatozoon that reaches the capacitated state. Taking advantage of our modeling, we can set out explorations of the joint dynamics of changes involved in the early response. For this, we introduce an auxiliary node , that integrates, at the single-cell level, the physiological changes associated with early sperm capacitation extrapolated from what is observed in bulk-cell measurements. For such a joint node, we consider as a hallmark for early capacitation the coincidence over short time intervals of the following observable levels: hyperpolarized potential (state -1), high [Ca²⁺]_i (state 1), high pH_i (state 1) and high PKA activity (state 1). Though the levels of sodium, bicarbonate and chloride are also determinant for capacitation, they are not included in due to the relative scarcity of experimental single-cell measurements.

In order to overcome the lack of sustained simultaneity in the conditions of individual sperm realizations, inherent to the stochastic nature of the network dynamics, we measure the joint dynamics with an observation window W. Thus, for a given spermatozoon r, its joint node takes the state 1 at time t if the overlapping moving average with a window W of all its associated observables is above a threshold θ ∈ [0, 1]. This means that becomes active if along the preceding window W, each one of the above-mentioned variables is found in the state required for the local hallmark defined for capacitation more than a certain fraction of time θ. If any of the variables fails, the joint node takes the state 0. In other words, depends on the smoothed out values of our four selected variables, calculated at their respective window time-averages. The joint node threshold θ ∈ [0, 1] indicates what is the minimum percentage of time that the simultaneous conditions of the model defined capacitation needs to be fulfilled during an observation window. We define the dynamical rule of as follows: (12a) (12b) where s_i allocates the proper sign for the selected variable i. Summing up, the joint node tells us if, on average, the four selected variables were in their respective capacitation-associated states for a large enough fraction of time θ in an observation window W before the time t, i. e. at local temporal scale.

We will say that a single cell is early capacitated if the steady state time-average of is above another threshold θ_c. Hence, we propose a capacitation criterion in terms of the binary classifier Θ_c (Heaviside function), defined by: (13a) (13b) where is the effective output of our capacitation model, which is a discretization of the time average value of during a capacitation stimulus observation time L − t_r − 1, after discarding a transient t_r required to reach steady state dynamics, where L is the total duration of the capacitation stimulus. If , then the given sperm r will be considered capacitated by the end of the simulation. Under this operational definition, capacitation is irreversible.

2.1.8 Parameter summary.

For the calibration and analysis of the dynamics of our network model, we can distinguish three kinds of parameters:

• Those that affect the individual (spermatozoon) network dynamics: , a weight that determines the contribution (the influence) of regulator j on node i; , a kinetic parameter that sets a time scale that determines the influence rate of regulator j on node i.
• Those related to the network dynamics measurements that are instrumental for our capacitation operational definition (Section 2.1.7): W, an observation window of the moving average that sets the smoothing level of the joint node input variables; θ, a threshold of the observable average activity needed to turn on the joint node; θ_c, a threshold of the joint node average activity needed for a sperm to be classified as capacitated.
• Those that influence the spermatozoa network population: D, the standard deviation of the Gaussian distribution used to sample ion transporter weights.

2.2.1 Model validation.

To validate our model, we test the effect of external stimulation on wild-type sperm (WT) and several loss-of-function (LOF) single mutants. Comparison of reported experimental measurements with their respective model counterpart was performed by means of the time series averaged over a sperm population, shown in Supplementary material Section. Results in absence of variability in the determination of the weight sets () that define the interactions in our network, D = 0 (S1 Fig), coincide with the variability case with D = 0.25 (S2 Fig). A summary of the response (for both cases) under WT conditions, coming from several stimuli (separate and joint increase in external bicarbonate and cholesterol acceptor) is shown in Table 1. The response with LOF cases of the select nodes under the joint stimulus with variability D = 0.25 (see S3 to S7 Figs) is summarized in Table 2. All these tests contributed to the calibration of the model parameters.

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Table 1. Summary of model validation on selected observables in wild-type networks.

https://doi.org/10.1371/journal.pone.0245816.t001

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Table 2. Summary of model validation on selected LOF mutant networks.

https://doi.org/10.1371/journal.pone.0245816.t002

2.2.2 How ion transporter numbers in single sperm impinge on the fraction of the population capacitated.

Variability in the number of ion transporters is a known feature [56–58], here we explore the effect of modifying such variability on the capacitation response at the cell population level. For this, we calculate the probability density of time-averaged variables among those most relevant for capacitation (our selected set V, Ca_i, pH_i and PKA) during the capacitating stimulus for several variability distributions of weights parametrized by their standard deviation D, shown in Fig 4. Given the connection between weights and the number of each ion transporter (Section 4.4), by modifying D, we are looking into the effect of changes in the variability of these numbers among sperm from the cell population modeled by a Gaussian distribution (Section 2.1.5).

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Fig 4. Effect of variability in the probability density of time-averaged selected variables from stimulated 2 × 10³-sperm populations.

D is the numerical value of the standard deviation from the Gaussian distribution used to sample weight sets for each sperm r (Section 2.1.4).

https://doi.org/10.1371/journal.pone.0245816.g004

Notice the following from Fig 4: The membrane potential probability density (red) widens drastically as a function of D, and becomes slightly bimodal at higher D values, showing a minor peak in positive values (centered around 0.1). Similarly, the calcium probability density (purple) becomes bimodal at higher D levels, with the second peak located at lower average Ca_i values (centered around 0.37), however, its upper boundary does not change with function of variability. Also, the pH_i probability density (blue) widens drastically with the introduction of variability and changes from unimodal to bimodal. In contrast, PKA probability density (yellow) does not seem to be affected by the introduction of variability D (Fig 4).

2.2.3 Capacitation criteria parameters that reproduce response heterogeneity.

Here we explore the dependence of the capacitation percentage in population on the parameter values of the selected joint node time series: observation window W, capacitation time overlap θ, signaling time averaged threshold θ_c (see Eq 12), and the population variability distribution width D of the amount of ion transporters. The first three parameters are instrumental for the determination of our operational definition of capacitation, they are model data detection quantities used to analyze the time-series generated by each individual sperm realization and intervene in the capacitation selection criteria proposed in Section 2.1.7. On the other hand, D is, at population level, the standard deviation of the probability distribution of the set of weights (see Section 2.1.4), and at single cell network dynamics level it intervenes in the definition of the individual regulatory functions (Eq 9). It is noteworthy to mention that variability D acts as a population control parameter.

Notice from the above that our operational capacitation definition contributes to: 1) discern stochastic fluctuations relevant to capacitation from non-relevant noise (parameters θ and W), 2) find what is the joint activity time interval of our selected variables (parameter W), 3) measure the average joint activity strength of our selected variables along observation time (parameter θ_c), and ultimately 4) allow us to discriminate between capacitated sperm and non-capacitated sperm.

After an extensive parameter value exploration (Fig 5, S8 and S9 Figs), we often encountered that for fixed values of θ: i) as the activity threshold θ_c is increased the maximum reachable capacitation fraction decreases, as it should be expected, ii) there is global capacitation fraction maximum within a finite restricted range of W, iii) for values of θ_c > 0.125, there is a plateau of capacitation, i. e. as D increases, the capacitation fraction hits a “plateau” whose location depends on W. The last two conclusions also indicate what should be expected from a stochastic model. There will always be a fraction of the population that is not capacitated due to the probabilistic nature of the model parameters.

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Fig 5. Capacitated fraction (CF) in sperm populations, varying three parameters: Observation window size (W, in iteration units), standard deviation of Gaussian variability (D) and activity threshold of joint node (θ_c).

In all heatmaps, each data point comes from 2 × 10³-sperm sized populations with 2 × 10⁴ iterations long simulations (5 × 10³ iterations at resting state, 1.5 × 10⁴ post stimulus iterations) and joint threshold θ = 0.225. The classification of capacitated vs. non-capacitated sperm is applied on the last 10⁴ time iterations of each individual sperm. The color bar represents the level of capacitation fraction. Note that the white zone corresponds to a capacitation level of 30 ± 5%, which is close to the levels typically observed for in vitro capacitation in wild-type sperm. The chosen parameter set for the rest of the analyses (D = 0.25, W = 70, θ_c = 0.15, θ = 0.225) is indicated with a purple filled circle.

https://doi.org/10.1371/journal.pone.0245816.g005

In Fig 5, we show for θ = 0.225, variations of the capacitation fraction, as we modify D and W for several values of θ_c. When the latter parameter is 0.15, the figure exhibits a passage from non-capacitated population to capacitated, the peak of capacitation percentage is first attained around D = 0.25 with W ≈ 70 and a plateau sets in for higher values of D within a window ranges size (W ≈ 10 to W ≈ 200), and capacitation fraction levels around 1/3, typically reported in the literature [4]. This scenario is qualitatively retrieved in the S8 and S9 Figs for different sets of parameter values. By adjustment of the joint node (θ) and joint activity (θ_c) thresholds, again, a well delimited region (plateau) can be achieved wherein several combinations of W and D values reproduce a capacitation fraction similar to what is observed in bulk-cell experiments (≈ 30%). For the remaining this paper, we will be referring to the well delimited plateau region produced with parameters θ = 0.225 and θ_c = 0.15, from which we choose, as representative point, the parameter combination W = 70 and D = 0.25.

The point we want to highlight is the recurrence and robustness of this phenomenology. We have uncovered a procedure dependent on the degree of variability in the number of ion transporters, that leads to capacitation and that can regulate its fraction in the population. Experimental corroboration of this model scenario is essential. Furthermore, the understanding of the underlying processes requires both experimental and theoretical work. On the theoretical side, the incorporation for the analysis of this “functional” transition, of concepts, tools and methods for the study of physical and dynamical phase changes is enticing; on the experimental side, stoichiometric considerations come to mind.

2.2.4 Probability distributions of time-averaged selected variables in sorted sperm subpopulations.

After classifying sperm according to the above described criteria, we characterize for each sperm subpopulation (non-capacitated vs. capacitated) the distributions of time-averaged values of the selected variables: membrane potential (V, red), intracellular calcium (Ca_i, purple), intracellular protons (pH_i, blue) and protein kinase A activity (PKA, yellow) in Fig 6. In agreement with the literature, for Ca_i, pH_i and PKA distributions, a slight increment is observed in their respective median values of capacitated sperm with respect to that of non-capacitated sperm [22, 34, 42, 43], whereas the width of those three distributions does not seem to change substantially. In contrast, V shows a decrease in the capacitated sperm distribution median value and width with respect to non-capacitated sperm [45, 46], both lower and upper bounds move to more negative values. There is a clearer separation between capacitated and non-capacitated sperm distributions, as compared to the other three variables. In each case, the difference between the distributions of capacitated and non-capacitated sperm is statistically significant (p-value < 0.05). Interestingly, the introduction of variability induces bimodality in the distributions of membrane potential, calcium and pH_i (Fig 4).

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Fig 6. Probability distributions of time-averaged variables relevant to capacitation in sperm population simulations under capacitating conditions, here shown as normalized frequency histograms.

Sperm are sorted according to the classifier defined in Eq 13 (Section 2.1.7). For each plot, the distributions of those sperm that did not reach the capacitated state (NC) are colored in gray. The capacitated fraction of sperm populations are labeled as C. The parameters used for the classification of sperm are: joint threshold θ = 0.225, activity threshold θ_c = 0.15, observation window W = 70 and variability D = 0.25. The distributions are calculated over 2 × 10³-sperm populations. Each time series is averaged over a 1.5 × 10⁴-iteration long capacitating stimulus, after discarding a 5 × 10³-iteration long resting state. In order to compare the distributions, medians are indicated as dashed lines. These values are: V -0.040 (NC) and -0.122 (C), Ca_i 0.444 (NC) and 0.454 (C), pH_i 0.523 (NC) and 0.538 (C), finally PKA 0.434 (NC) and 0.440 (C). In all cases, the difference between distributions was analyzed using a Kolmogorov-Smirnov test (p-value <0.05).

https://doi.org/10.1371/journal.pone.0245816.g006

Summing up, the capacitated subpopulation consists of sperm with higher hyperpolarization and moderate increases in Ca_i, pH_i and PKA activity than in the non-capacitated subpopulation. Notice that the probability densities reported in Fig 4 are compatible with the proportions of the capacitated and the non-capacitated subpopulation in terms of the selected variable distributions shown in Fig 6 for D = 0.25.

2.2.5 Controlling capacitation fraction levels.

In order to attain some insight on the relative importance of some proteins previously reported as relevant to capacitation, we analyze the response to changes in single and double mutants generated from a list of nodes of interest (Fig 7). The analysis is constrained to the proteins that are known to be expressed more specifically in the sperm flagella (sNHE, Slo3, CatSper, PKA) and those directly modifying electrochemical variables related to capacitation response (Na and Cl channels). In addition to the previously presented LOF scheme (Table 2), we explore another perturbation scheme: gain-of-function mutants (GOF), in which the node of interest is fixed to the most active state. After a capacitating stimulus, the resulting capacitation fraction in each mutant is depicted with a colored box. As reference, a population of WT sperm (unperturbed control) reaches approximately 30% capacitation levels (green color delineates a 30 ± 5% region). The LOF-only and GOF-only scenarios are shown in the bottom-left and upper-right triangles, respectively, whereas mixed cases are contained in the upper-left quadrant. Notice that in the LOF-only mutant scenarios, the predominant effect is a general decrement of capacitation levels, except for individual ClC^LOF, NaC^LOF and their combinations.

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Fig 7. Fraction of capacitated sperm determined by the model when one or two chosen nodes are modified: Inactivated (LOF, state fixed to 0), over-activated (GOF, state fixed to 1).

Results correspond to average values of the last 10⁴ time steps of 1.5 × 10⁴-iteration long simulations, performed over populations of 2 × 10³ sperm, as in Fig 5. We used an observation window of size 70, Gaussian variability with D = 0.25, joint node threshold θ = 0.225 and activity threshold θ_c = 0.15. The green zone indicates capacitation levels that are within the range of those typically observed for in vitro capacitation in wild-type sperm (30 ± 5%).

https://doi.org/10.1371/journal.pone.0245816.g007

In the triangle of GOF-only network variants, the predominant effect is an increase of the capacitation levels, except for CatSper^GOF, ClC^GOF and NaC^GOF combinations.

The mixed LOF and GOF scenarios display a more complex pattern than the two above-mentioned types of mutants. More specifically, this kind of scenarios allow us to assess possible recovery or interference between nodes. For example, the model predicts that the over-activation of any of the selected nodes is not sufficient to recover from the capacitation fraction decrease produced by the blockage of either Slo3, CatSper or PKA. On the other hand, in a sNHE^LOF background, Slo3 can totally recover the capacitation response, and even surpass the WT capacitation levels.

We remark that since the over stimulation of either Slo3 or PKA can maximize the capacitation levels in most of the combinations, they are predicted to be the strongest enhancers of capacitation. This is in agreement with the fact that they are strongly affecting membrane potential, a variable that is markedly different between capacitated and non-capacitated subpopulations (Fig 6).

4.2.1 Intracellular and extracellular ion concentration nodes.

Ion concentration nodes only consider two states: low concentration (state 0) and high concentration (state 1). Also, they have two regulators: the total flux of their corresponding ion and the intracellular concentration itself. Depending on the direction of the flux and intracellular concentration, there may be an increase, no change or a decrease in the concentration. For intracellular pH, sodium and bicarbonate, there is an extra regulatory node related to basal state recovery. Table 4 summarizes numeric values of voltage, extracellular and intracellular concentrations, used for the calculation of ion transporter flux direction and magnitude.

4.2.2 Gating nodes for ion channels.

4.2.2.1 CatSper For the modeling of CatSper gating, we considered four regulators: voltage, intracellular calcium, pH_i and PKA. The following equations determine the effect of the first three regulators, as in [69]: (14a) (14b) (14c) where P_V is the opening probability of CatSper channels by influence of membrane potential and protons. Eq 4a is a function used to model the sigmoid (conductance vs potential) G/V curve, as reported in [70]. Eq 14b (V_pH) models the pH_i-dependent shift of the half-activation voltage of Eq 14a. P_Ca is the opening probability of CatSper channels by influence of calcium, where K_pH and K_Ca are empirical fitted parameters. Also, n_Ca and n_pH are cooperativity coefficients of calcium and pH_i, respectively. Finally, we multiply Eq 14a and 14c in order to obtain the opening probability of CatSper channels, P_CatSper, by influence of the first three regulators. We discretize this final probability by a threshold value max(P_V P_Ca)/2, where function max reaches the greatest possible value for the given regulator combination. We associate the state 0 (basal gating) to probabilities below this threshold value and state 1 (more permeable gating) to probabilities above of this threshold value. The only action of PKA in the gating consists in shifting the permeability to state 1 in spite of resting pH_i and resting membrane potential. Table 5 contains a summary of numerical values used in the gating equations.

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Table 5. Numerical values of the constants used in CatSper gating equations.

https://doi.org/10.1371/journal.pone.0245816.t005

4.2.2.2 Na⁺-channel. The sodium conductance increases (state 1) with respect to basal levels (state 0) when its only regulator ClC is closed (state 0).

4.2.2.3 Cl^--channel. The chloride conductance increases (state 1) with respect to basal levels (state 0) when its only regulator PKA is active (state 1).

4.2.2.4 Slo3. In order to choose whether the node will change from basal level (state 0) to a more permeable state (state 1), we used the gating equation of the supplementary material in [71] with the respective numerical constants. The discretization was done under the same criteria mentioned previously in the case of CatSper. Under depolarized potential and high pH_i, Slo3 takes the state 1 if ChAcc or PKA or both are in state 1. Under high pH_i, if PKA and ChAcc are both in state 1, Slo3 can take the state 1 even in resting potential. If any of the above mentioned conditions is not fulfilled, then the conductance will return to its basal value 0.

4.2.3 Current nodes of ion channels.

4.2.3.1 CatSper current. The magnitude and direction for this Ca²⁺ flux was estimated only considering the calcium gradient by means of the Nernst potential and Ohm’s equation as follows: (15a) (15b) (15c) where R is the Molar gas constant, T is the temperature, F is the Faraday constant, and the 2 multiplying F in the denominator stands for the charge number of calcium ions, P_CatSper is the output value of the gating node of CatSper, defined in Section 4.2.2, N_CatSper is the number of channels and g_CatSper is the CatSper unitary conductance. According to the combination of its regulator states (Ca_e, Ca_i, P_CatSper, V) based on Eq 15a to 15c, we determined if the CatSper current, I_CatSper node will have an outward calcium current (I_CatSper = -1), a null current (I_CatSper = 0) or an inward calcium current (I_CatSper = 1). Table 6 shows the numerical values of constants used in Eq 15 through Eq 18.

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Table 6. Summary of the parameters used in current and energy equations of this section.

https://doi.org/10.1371/journal.pone.0245816.t006

4.2.3.2 Na⁺-channel current. To estimate the magnitude and direction of this ion flux, we considered the sodium gradient and we used the Na⁺ Nernst potential and the Ohm equation as follows: (16a) (16b) (16c) where P_NaC is the Na-channel gating node output value, N_NaC is the number of Na channels and g_NaC is the unitary conductance of Na channels. According to the combination of regulator states (Na_e, Na_i, P_NaC, V) we decided with the help of Eq 16a to 16c if the Na-channel current will have an outward sodium current (state -1), a null current (state 0) or an inward sodium current (state 1) in the same way as in the previous case of CatSper current.

4.2.3.3 Cl^--channel current. In order to obtain some notion of magnitude and direction of this ion flux, we considered a chloride/bicarbonate mixed current as in CFTR channel, and we used the Goldman-Hodgkin-Katz (GHK) voltage equation and the Ohm equation as follows: (17a) (17b) (17c) where P_ClC is the Cl-channel gating node output value, N_ClC is the number of Cl channels and g_ClC is the unitary conductance of Cl channels, P_HCO3Cl is the relative permeability HCO3 to Cl. According to the combination of regulatory states (HCO_3e, Cl_e, HCO_3i, Cl_i, P_ClC, V), based on Eq 17a to 17c, we determined if current of ClC will have an outward chloride/bicarbonate current (state -1), a null current (state 0) or an inward chloride/bicarbonate current (state 1) in the same way as in the previous case of CatSper current.

4.2.3.4 KSper. It is known that Slo3 is mainly responsible for the sperm potassium current named KSper [73]. However, unlike Slo1 channels, Slo3 selectivity for potassium is weak, given by a K⁺ to Na⁺ relative permeability of ∼ 5, this measured in heterologously expressed channels [72], which would result in a reversal potential close to the resting potential. Even though it has been reported that the blockage or deletion of Slo3 abolishes the typically observed hyperpolarization in capacitated sperm [50, 51], the opening of a channel with those characteristics would never be able to generate hyperpolarizing currents. To overcome this apparent contradiction, we decided to model this current as a purely K⁺-current from possible additional K⁺ channels downstream of Slo3 activity and with higher selectivity. In order to obtain some notion of magnitude and direction for this current, we considered a potassium flux according to the K⁺ Nernst potential and Ohm equation as follows: (18a) (18b) (18c) where P_Slo3 is the Slo3 gating node output value, N_Slo3 is the number of Slo3 channels and g_Slo3 is the unitary conductance of Slo3 channels. Taking into account the combination of regulator states (K_e, K_i, P_Slo3, V), for the KSper electromotive force we discretized Eq 18b. The state of this node will be -1 if the gradient generates inward flux, 0 when there is no net flux, and 1 if the gradient causes outward flux.

Since the expected concentration changes of K_i due to ion channels opening, typically observed in excitable cells, are negligible (<10% of the intracellular potassium concentration) [79], we took K_i as a constant during all the simulations. Note that K_i is not shown in Fig 2 and Table 4 because those only include nodes that vary with time.

4.2.4 Cotransporter and exchanger nodes.

4.2.4.1 Electrogenic sodium/calcium exchanger. For the flux direction of sodium/calcium exchanger (NCX), we used an equation from appendix in [8] with the respective reported constants. We chose the state -1 for outward flux of sodium and influx of calcium, state 0 for null flux, state 1 for influx of sodium and outward flux of calcium.

4.2.4.2 Electroneutral sodium/proton exchanger. Estimation of the flux direction of the sperm-specific electroneutral sodium/proton exchanger (sNHE) was performed following energetic consideration for determining the most probable state of sNHE according to the regulator chemical gradients as follows: (19) where H_i and H_e (intra- and extracellular proton concentrations) are calculated as and , respectively. We assigned state -1 to the sodium outward flux coupled with protons influx, state 0 for null flux, state 1 for sodium influx coupled with protons outward flux. This node has a gating regulator that depends on voltage and cAMP. If the cytosolic cAMP concentration increases and the voltage is hyperpolarized then the flux is enhanced. If only one of the two regulators is present in the right state, the flux is permitted. If none of these regulators are in the right state, the flux is forbidden.

4.2.4.3 Electrogenic sodium/bicarbonate cotransporter. We calculated the flux direction of sodium/ bicarbonate cotransporter (NBC), by means of the reversal potential and the following energetic consideration for determining the most probable state of NBC according to the regulator gradients and membrane potential: (20a) (20b)

We assigned the state -1 to outward flux, state 0 for null flux and state 1 for influx of sodium and bicarbonate. The sodium/bicarbonate stoichiometry per individual transport event by this cotransporter, n_NBC, is reported in Table 6.

4.2.4.4 Electroneutral chloride/bicarbonate exchanger. In order to obtain the ion flux direction of electroneutral exchanger (SLCAE), we used the following energetic consideration for determining the most probable state of SLCAE according to the regulator gradients: (21)

We assigned the state -1 to influx of chloride and efflux of bicarbonate, state 0 for null flux, state 1 for efflux of chloride and influx of bicarbonate.

4.2.4.5 Electrogenic chloride/bicarbonate exchanger. The flux direction of electrogenic exchanger (SLC26A3) was estimated using the following energetic consideration for determining the most probable state of SLC26 according to the regulator gradients and membrane potential: (22)

We assigned state -1 to influx of chloride coupled with extrusion of bicarbonate, state 0 for null flux, state 1 for extrusion of chloride coupled with influx of bicarbonate. The chloride to bicarbonate stoichiometry of this exchanger, n_SLC26, is reported in Table 6.

4.2.5 Early phosphorylation nodes.

The soluble adenylate cyclase (sAC) only has two regulators: intracellular calcium (Ca_i) and intracellular bicarbonate (HCO_3i). The sAC will activate (state 1) if Ca_i or HCO_3i is in high concentration in the cytosol (state 1), otherwise it will inactivate (state 0).

Phosphodiesterase (PDE) will increase its activity from basal (state 0) to high (state 1) if there is adenosine monophosphate in cytosol (state 1).

Cyclic Adenosine monophosphate (cAMP) has three regulators: itself, sAC and PDE. The cAMP will remain in its previous state if both sAC and PDE remain inactive. There will be production of cAMP (state 1) as long as sAC is activated (state 1). There will be a decrease of cAMP (state 0) only when PDE is activated (state 1). There will be a decrease of cAMP (state 0) if PDE and sAC are both activated (state 1).

Protein kinase A (PKA) will remain activate (state 1) while there is cAMP in the cytosol (state 1) otherwise PKA will remain inactivated (state 0).

4.2.6 Auxiliary recovery nodes.

When the recovery nodes of (Δμ_pHR, Δμ_NaR and Δμ_HCO3R) turn on, they set their respective ion concentration (pH_i, Na_i and HCO3_i) from the high (state 1) to their resting levels.

The leak current node (I_L) is a nonspecific counterbalance mechanism to voltage changes, i. e. it tends to depolarize if V is hyperpolarized (state -1) and to hyperpolarize if V is depolarized (state 1), leading in both cases to resting values.

4.2.7 Miscellany of nodes.

The inositol trisphosphate (IP3) node will increase its concentration in cytosol (state 1), provided that both Ca_i and cAMP remain high (state 1). We discretized the continuous model proposed by [80], in which it is argued that IP3 is synthesized near the midpiece through a PLC isoform and can be regulated by calcium and cAMP.

IP3 sensitive calcium channels from calcium reservoirs (IP3R) node will promote a calcium current into the cytosol (state 1) as long as IP3 remains high (state 1) and Ca_i remains low (state 0) in the cytosol.

Calcium pump from calcium reservoirs (SPCA) node will be activated (state 1) if there is a high intracellular calcium (state 1), promoting a calcium current from the cytosol to the interior of calcium reservoirs.

Calcium pump from the main piece of flagellum (PMCA) node will be activated (state 1) whenever intracellular calcium concentration increases (state 1), promoting a current from the cytosol to the outside of the cell [8].

4.4.1 Variability in the amount of ion transporters.

In order to introduce variability in the amount of ion transporters present in the sperm populations, we multiply each of the by a random variable that samples a normal distribution with average 1 and deviation D. This distribution was truncated over the interval because negative conductances have no meaning in our modelling; furthermore, given that the center of the distribution is 1, in order to avoid any sampling bias while keeping symmetry, we fixed the interval up to 2. Thus, a random set of weights is assigned to each single sperm r at the beginning of the simulations. Note that the probabilistic element here introduced acts directly on network links through the stochastic modulation of the integrator weights, furthermore, the distribution of the numbers of ionic transporters per cell in a sperm population, considered in our model by means of Eq 26, is experimentally challenging to determine.

4.4.2 Ion transporter operation rates.

We consider differences in operation rates of various ion transporters, by using a random variable , introduced in Section 2.1.2, that samples a biased Bernoulli distribution at each iteration t, where is the distribution mean. If , the ion transporter i_j will participate in the weighted sum of its respective ion flux output value at time t, otherwise, it will not participate. The participation probability given by the average of the biased Bernoulli distribution of each ion transporter i_j is related to an operation velocity as follows: (27)

In our simulations, the constant 0.25 is a free parameter that changes the time-scale of the dynamics, modifying the length of the transients but leaving steady state values (which are our main concern) unaltered. If the value of the constant increases, the length of the transient decreases, while if it decreases we observe the converse. During the validation process, we found the value 0.25 is appropriate for the model calibration. For example, if in S2 Fig we had chosen a constant value of 1, the transient kinetics would have been so fast that they would approach a step function, making calibration difficult. The kinetic parameters influence the flux given by the total number of each type of ion transporter in the flagellum. Unlike weight parameters, due to the scarcity of their numerical values in the scientific literature, we have opted for an educated guess approach, taking into account the compatibility of our simulations with reported experiments. Within our manual adjustment, we assigned to the fast ion channels and used this as a reference value to express other velocities in relative terms, namely, corresponds to exchangers and co-transporters; , to flagella calcium pumps; , to phosphorylation regulator nodes; , to calcium diffusion from RNE to the principal piece (calcium pumps from the principal piece); and , to ChAcc regulator node.

4.5.1 Chloride fluxes.

We determine the chloride flux integrator node regulation with: (28) where the chloride transporter modifier functions P^t and Q^t are defined as follows: (29) (30)

The signs preceding the terms related to the bicarbonate/chloride exchangers reflect the stoichiometries of their corresponding ion transport events, e. g. the electroneutral exchanger SLCAE allows the exit of 1 Cl^- ion coupled to the entry of 1 HCO₃^- ion. Based on experiments reported in [43], the functions P^t and Q^t model the potentiating role of cholesterol removal on membrane potential hyperpolarization by ion transporters. Unlike Q^t, P^t considers an intermediate activation state, 2, that depends solely on PKA and ClC; in this kind of regulation, we model the potentiation that results from the interaction between PKA-dependent phosphorylated forms of both SLC26 and CFTR [67].

4.5.2 Bicarbonate fluxes.

We determine the bicarbonate flux integrator node regulation: (31) where the bicarbonate transporter modifiers P^t and Q^t are defined in the same way of chloride.

4.5.3 Sodium fluxes.

We determine the sodium flux integrator node regulation: (32) where the bicarbonate transporter modifier Q^t is defined in the same way of chloride.