APPENDIX 1EQUATIONS AND PARAMETERS OF THE THREE-DIMENSIONAL COMPUTER MODELA1. Principles of Model Construction, Selection of Initial and Boundary Conditions, and Parameter Fitting

The three-dimensional model was created using VCell software (http://www.vcell.org/). The corresponding Virtual Cell model, MouseSpermCalcium, is available publicly on http://www.vcell.org/ under the username “Juliajessica”. Details of the model geometry are presented in Table A1. The geometry of the three-dimensional model includes the cytosol (56 µm3), divided into the flagellum of the spermatozoon (0.4 µm in diameter, 115 µm in length [46]), the neck (0.8 µm in diameter [47], 20 µm in length [48]), and the head of the spermatozoon (8 × 3 × 2.5 µm [48]), the acrosome of the spermatozoon, serving as a calcium store (1.3 µm3) [17], located in the head of the spermatozoon (Fig. A1, Table A1), as well as the cell membrane and the acrosome membrane. Second-type boundary conditions were used for all substances. Data on the localization of substances and consideration of diffusion are given in Table A2. All substances in the acrosome were considered well-mixed. To convert volumetric concentrations and dissociation constants to surface ones, the coefficient \(_} = _}\frac_}}}_}}}\), was used, where \(_}\) is Avogadro’s number, \(_}\) is the volume of compartment q, \(_}}\) is the surface area of compartment q. \(_}}}}\) and \(_}}}}\) correspond to the acrosome and cytosol, respectively.

Table A1.   Model geometryFig. A1.

General view of the model geometry. Species localization. Spatial restriction of certain reactions. The blue area represents the cytosol of the spermatozoon. The red area represents the acrosome of the spermatozoon. The yellow area represents the membrane of the spermatozoon.

Table A2.   Initial values of variables (surface concentrations for membrane-bound substances, volume concentrations for soluble substances; all concentrations correspond to basal levels except for calcium) and diffusion coefficients (for the three-dimensional model)A2. Model Equations

Model variable denotations are given in Table A2.

Model parameters are given in Table A4.

First module consists of 11 equations:

$$\begin \frac_}}}}} \right]}}} = _}}}}(\left[ \right] \\ - \,\,[Progesteron_}}}}]) + _}}}}\Delta \left[ _}}}}} \right], \\ \end $$

(1)

$$\begin \frac \right]}}} = \frac_}}\left[ \right]\left[ ^} \right]}}_}}}}_}} + \left[ \right]}} \\ - \,\,_}}\left[ \right] - _}}_}}\left[ \right] + _}}\Delta \left[ \right], \\ \end $$

(2)

$$\begin \frac \textPGE2]}}} = _}}\left[ \right]} \mathord_}}\left[ \right]} _ }}}}}}} \right. \kern-0em} _ }}}}}} \\ _}}_}}\left[ \right]} \mathord_}}_}}\left[ \right]} _ }}}}}}} \right. \kern-0em} _ }}}}}} \\ - \,\,_}}\left[ \right] \right]} \mathord \right]} _ }}}}_}})}}} \right. \kern-0em} _ }}}}_}})}} \\ - \,\,_}}\left[ \right] + _}}\Delta \left[ \right], \\ \end $$

(3)

$$\begin \frac^} \right]}}} = _}}\left[ \right]\left[ _}}}}} \right] \\ - \,\,_}}\left[ ^} \right] + _}}\Delta \left[ ^} \right], \\ \end $$

(4)

$$\begin \frac \right]}}} = - \frac_}}\left[ \right]\left[ ^} \right]}}_}}}}_}} + \left[ \right]}} \\ + \,\,_}}\left[ \right] + _}}\Delta \left[ \right], \\ \end $$

(5)

$$\begin \frac \right]}}} = _}} \right]\left[ \right]} \mathord \right]\left[ \right]} _}}}}} \right. \kern-0em} _}}}} \\ - \,\,_}}\left[ \right] - _}}}}\left[ \right] \\ + \,\,_}}\Delta \left[ \right], \\ \end $$

(6)

$$\begin \frac \right]}}} = - _}} \right]\left[ \right]} \mathord \right]\left[ \right]} _}}}}} \right. \kern-0em} _}}}} \\ + \,\,_}}\left[ \right] + _}}\Delta \left[ \right], \\ \end $$

(7)

$$\begin \frac \right]}}} = _}}}}\left[ \right]\frac_}}}}M - \left[ \right]}}_}}}}_}} + \left( _}}}}M - \left[ \right]} \right)}} \\ - \,\,_}}\left[ \right]\left( \right] - [G^}}]} \right) + _}}\Delta \left[ \right], \\ \end $$

(8)

$$\begin \frac \textG^}}]}}} = _}}}}\left[ \right] \\ \times \,\,\frac_}}}}M - \left[ \right]}}_}}}}_}} + \left( _}}}}M - \left[ \right]} \right)}} \\ - \,\,_}}}}_}}}}\left[ \right]\frac^}}]}}_}}}}_}} + [G^}}]}} + _}}\Delta [G^}}], \\ \end $$

(9)

$$\begin \frac \right]}}} = - \left[ \right]_}} \\ + \,\,_}[G^}}]\left[ \right] + _}}\Delta \left[ \right], \\ \end $$

(10)

$$\begin \frac \right]}}} = \left[ \right]_}} \\ - \,\,_}[G^}}]\left[ \right] + _}}\Delta \left[ \right]. \\ \end $$

(11)

Rate sum in right sides of Eqs. (10) and (11) equals to zero, therefore for the point model one of the equations can be discarded.

Fig. A2.

Dynamics of the amount of activated phospholipase for different parameters of the first module of the model. (a) Mode 4 from [34], (b) Mode 2 from [34]. (c) Comparison of the full model and the model with the quasi-equilibrium approximation applied to ABHD2.

The obtained system of equations was simplified to 9 equations by studying the hierarchy of characteristic times and reduction using Tikhonov’s theorem. Initially, dimensionless scaling of the variables was performed:

$$\begin PLC_} = \frac \right]}}_}}}}}},\,\,\,\,PLC_}} = \frac \right]}}_}}}}}}, \\ \beta _} = \frac \right]}}_}}}}}},\,\,\,\,G\alpha _^} = \frac^}}]}}}}}^}}}, \\ GPCRac_} = \frac \right]}}_}}}}}},\,\,\,\,GPC_} = \frac \right]}}_}}}}}}, \\ A_} = \frac \right]}}_}}}}}},\,\,\,\,A_} = \frac \right]}}_}}}}}},\,\,\,\,PGE_} = \frac \right]}}_}}}}}}, \\ ABHD_} = \frac \right]}}_}}}}}},\,\,\,\,ABHD2_^ = \frac^} \right]}}}}}^}}. \\ \end $$

Values with the subscript \(}\) correspond to the average values of variables over the period of activation. Numerical values are given in Table A3.

Table A3.   Time-averaged values of variables upon activation with 50 µM progesterone

Numerical values of the constants were put into the dimensionless system, and the equation was rewritten in the form \(\frac_}}}} = ^}O\left( 1 \right)\). The obtained values for the coefficients \(^}}\) were of following magnitude: \(~^}} \ll ^}} \sim ^^}}}}} \ll ^}}\) \( \ll ^}} \ll ^}}\). When applying the quasi-steady-state approximation for ABHD2, the model completely retained its behavior (Fig. A2b). In the final calculations, the concentrations of ABHD2 and ABHD2* were given by the following expressions:

$$\begin \left[ ^} \right] = \right]}_} - \left[ \right], \\ \left[ \right] = _}} \right]}}_}} \right)} \mathord \right]}}_}} \right)} }} \right. \kern-0em} } \\ \left( _}}\left[ _}}}}} \right] + _}}} \right). \\ \end $$

In the second module, the activity of PLC was input using the following expression:

$$ \right]\,\, = \,\,8^ - 600)}}^}}}}^}}}}}}}} \mathord \right]\,\, = \,\,8^ - 600)}}^}}}}^}}}}}}}} }}^}}}} \right. \kern-0em} }}^}}}$$

The activity of PLCδ4 was described as a calci-um‑dependent flow of IP3 into the cytosol – see equation (27). The second module consists of 8 differential equations. The variable denotations of the model are listed in Table A2. The parameters of the model are listed in Table A4.

$$\begin \frac \text}_}}}^}]}}} = _},}}}} - _}} - _}} \\ + \,\,_}} + _}}}} + _}} + _}}}}\Delta \left[ }_}}}^}} \right], \\ \end $$

(12)

$$\frac [}_}}}^}]}}} = _}} - _}}}} - _}} + _}},$$

(13)

$$\begin _}} \\ = \left\ \frac_}}}_}}}^}]}}^}}}}^ + }_}}}^}]}}^}}},\,\,\,\,\vec \in }\,\,} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,}} \hfill \\ \end \right. \\ \end $$

(14)

$$_}} = \left\ \frac_}}}_}}}^}]}}^}}}}^ + }_}}}^}]}}^}}},\,\,\,\,~\vec \in }\,\,} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,}} \hfill \\ \end \right.$$

(15)

$$_},}}}} = \left\ _}}}}\left( }_}}}^}] - [}_}}}^}]} \right),\,\,\,\,\vec \in }\,\,} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,}} \hfill \\ \end \right.$$

(16)

$$_}}}} = \left\ _}PP}}},~\,\,\,\,\vec \in }\,\,} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,}} \hfill \\ \end \right.$$

(17)

$$_}} = \left\ _}}_}}_}\left( }_}}}^}] - [}_}}}^}]} \right),\,\,\,\,~\vec \in }\,\,} \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,}} \hfill \\ \end \right.$$

(18)

where \(_} = m_^^}\) is the probability of IP3R remaining in the open state, as determined according to the Lee-Rinzel model [52], h is the proportion of channels not inactivated by calcium.

$$\frac}} = \frac_} - h}}_}}},$$

(19)

where \(_} = \frac_}\left( _} + [}_}}}^}]} \right)}}\), \(_} = \left( _}} \right]}}_}} \right] + _}}}} \right)\) \(\left( }_}}}^}]}}}_}}}^}] + _}}}} \right)\), \(_} = \frac_}}}_} + [}_}}}^}]}}\), \(_} = _}\frac_}} \right] + _}}}_}} \right] + _}}}\)

$$\begin _}} = - \left[ \right][}_}}}^}] k_}^ \\ + \,\,\left[ \right]k_}^ \\ - \,\,\left[ \right][}_}}}^}]k_}^ + \left[ \right]k_}^, \\ \end $$

(20)

$$\begin _}} = - \left[ \right][}_}}}^}] k_}^ \\ + \,\,\left[ \right]k_}^ \\ - \,\,\left[ \right][}_}}}^}] k_}^ + \left[ \right]k_}^, \\ \end $$

(21)

where \(_}}\) is SPCA-generated calcium flux, \(_}}\) is PMCA-generated calcium flux, \(_}}\) is IP3R-generated calcium flux, \(_}}}}\) and \(_}}}}}\) are outside and store calcium leaks, respectively.

$$\begin \frac \right]}}} = - \left[ \right][}_}}}^}] _}} \\ + \,\,\left[ \right]\frac}^}}}^}}, \\ \end $$

(22)

$$\begin \frac \right]}}} = - \left[ \right][}_}}}^}] _}} \\ + \,\,\left[ \right]\frac_}}}}_}}}}, \\ \end $$

(23)

$$\begin \frac \right]}}} = - \left[ \right][}_}}}^}] _}} \\ + \,\,\left[ \right]_}}, \\ \end $$

(24)

$$\begin \frac \right]}}} = - \left[ \right][}_}}}^}] _}} \\ + \,\,\left[ \right]\frac_}}}}_}}}}, \\ \end $$

(25)

$$\frac \right]}}} = _}} - _}} + _}} + ~_}}\Delta \left[ \right],$$

(26)

$$_}} = \left\ \frac_}} _}}[}_}}}^}]}}_}_}}}}\left( _}} + [}_}}}^}]} \right)}},\,\,\,\,~\vec \in }\,\,} + } \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,} + }} \hfill \\ \end \right.$$

(27)

$$_}} = \left\ \left[ \right]_}},\,\,\,\,\vec \in }\,\,} + } \hfill \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec \notin }\,\,} + }} \hfill \\ \end \right.$$

(28)

$$_}} = \frac_}}_}}[IP3]}}_}_}}}}(_}} + [IP3])}},$$

(29)

where \(_}}\) is IP3 degradation rate by inositol-5-phosphatase (IP5pp), \(_}}\) is IP3 production rate by PLCδ; \(_}}\) is IP3 production rate by PLC\(\beta \).

The obtained system of 8 equations was simplified to 7 equations by studying the hierarchy of characteristic times and reduction using Tikhonov’s theorem. Initially, dimensionless scaling of the variables was performed:

$$\begin IP_} = \frac \right]}}_}}}}}},\,\,\,\,}_}}}_}}}^} = \frac}_}}}^}]}}}_}}}_}}}}}}^}}}, \\ }_}}}_}}}^} = \frac}_}}}^}]}}}_}}}_}}}}}}^}}},\,\,\,\,CaRetSlo_} = \frac \right]}}_}}}}}}, \\ \end $$

$$\begin CaRetFas_} = \frac \right]}}_}}}}}}, \\ CaMSlo_} = \frac \right]}}_}}}}}}, \\ CaMFas_} = \frac \right]}}_}}}}}}. \\ \end $$

Values with the subscript \(}\) correspond to the average values of variables over the period of activation. Numerical values are given in Table A3.

In the dimensionless system, numerical values for constants were applied, and the equation was rewritten in the form \(\frac_}}}} = ^}O\left( 1 \right)\). The obtained values for the coefficients \(^}}\) were of following magnitude:

$$\begin ^}} \sim ^}} \sim ^}} \\ \ll ^}} \ll ^}} \sim ^}} \ll ^}}. \\ \end $$

When applying the quasi-stationary approximation for \(}_}}}^}\) (Fig. A3a), the model diverged. When applying the quasi-stationary approximation for \(}_}}}^}\), (Fig. A3b), there was no calcium response to progesterone. A similar result was obtained for the use of the stationary approximation for CaRetFast and CaRetSlow (Fig. A3b). In the case of using the stationary approximation for IP3, the response was observed in full, but the model was not able to reproduce calcium oscillations (Fig. A3b). The use of the quasi-stationary approximation for the fast calmodulin site did not change the nature of the calcium response, and subsequently, this approximation was used both for the point and the three-dimensional model.

The quasi-stationary approximation for the fast calmodulin site was defined as follows:

$$\begin \left[ \right] = \right]}_} - \left[ \right], \\ \left[ \right] = _}} \right]}}_}} \mathord \right]}}_}} }} \right. \kern-0em} } \\ \left( }_}}}^}] _}} + _}}} \right). \\ \end $$

Fig. A3.

Use of quasi-stationary and quasi-equilibrium approximations for the second module of the model. (a) When using the quasi-stationary approximation for the concentration of the fast site of calmodulin, the calcium response does not change (black curve). For store calcium, using the quasi-stationary approximation, the calcium response to progesterone is absent. (b) When using the quasi-equilibrium approximation for calreticulin, the calcium response is absent. When using the quasi-stationary approximation for IP3, the calcium response is preserved, but oscillations in the model are absent at any IP3 concentration values.

Table A4.   Model parameters