Table 2. Variables, intermediate functions, and reference parameters values.

Symbol	Description	Units	Reference value	Source
Upstream module: SAP Receptor and cGMP signalling
SAP and receptor
S	Extracellular SAP concentration	nM	Variable a	Eqs 8–11 b
RF	Number of free SAP receptors per cell (flagellum)	cell−1	Variable	Eqs 8–11
RH	Number of high activity SAP-receptor complexes per cell (flagellum)	cell−1	Variable	Eqs 8–11
RL	Number of low activity SAP-receptor complexes per cell (flagellum)	cell−1	Variable	Eqs 8–11
RT	Total number of SAP receptors per cell (flagellum)	cell−1	7.36 × 105	F [22], S1 Fig
θR	Receptor-bound to soluble SAP concentration conversion factor	nM cell	6.14 × 10−5	D (sec. 4.2.1)
s	Sperm density in measurement chamber	cell mL−1	3.706 × 107	F [22], S1 Fig
r1	SAP-receptor association rate constant	nM−1 s−1	2.70 × 10−2	[50]
r2	RH to RL conversion rate constant	s−1	8.70	F [22], S1 Fig
r3	RL inactivation rate constant	s−1	5.70 × 10−2	F [22], S1 Fig
cGMP
G	Intraflagellar cGMP concentration	nM	Variable	Eq 4
Gr	Resting cGMP concentration	nM	1.25	F [22]
σG	Receptor-independent cGMP synthesis rate	nM s−1	23.65	F [22], S1 Fig
δG	cGMP turnover rate constant	s−1	18.92	F [22], S1 Fig
ρG	Receptor-dependent cGMP rate	cell−1 s−1	Function	Eq 14
kL	cGMP synthesis rate constant per RL	s−1	3.25	F [22], S1 Fig
kH	cGMP synthesis rate constant per RH	s−1	40.20	F [22], S1 Fig
BG	Flagellum cGMP buffering power	cell−1	5.49 × 103	F [10]
θG	Conversion factor to turn cGMP molecule number into effective intraflagellar concentration	nmcell	1.89 × 10−4	D (sec. 4.1.2)
KCNG channels
$f_{\text{kn}}^o$	Fraction of open channels		Variable	Eq 15
αkn	Opening rate constant	nM−1 s−1	10	F [10]
βkn	Closing rate constant	s−1	257	[48]
gkn	Maximal conductance density	pS μm−2	135.30	Ac [10]
Ikn	Current density	fA μm−2	Function	Eq 18
spHCN channels
mhc	Fraction of open gates		Variable	Eq 19
$f_{\text{hc}}^o$	Fraction of open channels		Function	$m_{\text{hc}}^3$
${\widehat{m}}_{\text{hc}}$	Steady state fraction of open channels		Function	Eq 20
τhc	Characteristic time of gating	s	Function	Eq 21
h1	Half-activation voltage	mV	-50.80	[62, 76]
h2	Voltage sensitivity of activation	mV	6.60	[62, 76]
h3	τhc basal value	s	9.18 × 10−2	F [62], S2 Fig
h4	τhc amplitude	s	1.10	F [62], S2 Fig
h5	Voltage of maximum τhc	mV	-42	F [62], S2 Fig
h6	Characteristic width of τhc	mV	18.50	F [62], S2 Fig
ghc	Maximal conductance density	pS μm−2	193.50	Ad [10]
Ihc	Current density	fA μm−2	Function	Eq 22
CaV + BK module
Equilibrium parameters for V and C
gL	Leakage conductance density	pS μm−2	1.94	F [10]e
EL	Reversal potential of leakage current	mV	80.05	D (Eq 3)
σC	CaV-independent Ca2+ source rate constant	nm s−1	1.14 × 108	D (sec. 4.6)
δC	Ca2+ extrusion rate constant	s−1	1.16 × 106	MF
CaV channels
$f_{\text{cv}}^o$	Fraction of open channels		Variable	Eq 30
$f_{\text{cv}}^c$	Fraction of closed channels		Variable	Eq 31
αcv	Inactivation recovery rate	s−1	Function	Eq 27
βcv	Opening rate	s−1	Function	Eq 28
γcv	Inactivation rate	s−1	Function	Eq 29
v1	Inactivation recovery rate constant	mV−1 s−1	10.10	MF
v2	inactivation recovery rate voltage inflexion point	mV	-55	MF
v3	Inactivation recovery rate voltage sensitivity	mV	1.42	MF
v4	Opening rate constant	mV−1 s−1	33.80	MF
v5	Opening rate voltage inflexion point	mV	-39	MF
v6	Opening rate voltage sensitivity	mV	2.10	MF
v7	Inactivation rate constant	mV−1 s−1	8.10	MF
v8	Inactivation rate voltage inflexion point	mV	-18	MF
v9	Inactivation rate voltage sensitivity	mV	7.50	MF
gcv	Maximal conductance density	pS μm−2	185.16	MFf
Icv	Current density	fA μm−2	Function	Eq 33
BK channels
$f_{\text{bk}}^o$	Fraction of open channels		Variable	Eq 45
αbk	Opening rate	s−1	Function	Eq 46
βbk	Closing rate constant	s−1	97	MF
b1	Opening rate constant	nM−1 s−1	0.97	MF
b2	Opening rate [Ca2+]i inflexion point αbk	nM	316	MF
b3	Opening rate [Ca2+]i sensitivity	nM	30	MF
gbk	Maximal conductance density	pS μm−2	214.50	MFg
Ibk	Current density	fA μm−2	Function	Eq 47
CatSper + NHE module
Equilibrium parameters for V and C
gL	Leakage conductance density	pS μm−2	9	MFh
EL	Reversal potential of leakage current	mV	-20.4	D (Eq 3)
σC	CatSper-independent Ca2+ source rate constant	nM s−1	3.72 × 107	D (sec. 4.6)
δC	Ca2+ extrusion rate constant	s−1	4.27 × 105	MF
NHE exchangers
$f_{\text{nh}}^a$	Fraction of active exchangers		Variable	Eq 23
αnh	Activation rate	s−1	Function	Eq 24
βnh	Inactivation rate	s−1	Function	Eq 25
n1	Activation rate constant	mV−1 s−1	1.51	MF
n2	activation rate voltage inflexion point	mV	-65	MF
n3	activation rate voltage sensitivity	mV	7	MF
n4	inactivation rate constant	mV−1 s−1	0.90	MF
n5	inactivation rate voltage inflexion point	mV	-30	MF
n6	inactivation rate voltage sensitivity	mV	7	MF
Jmax	Maximal H+ flux by total active exchangers	fmol s−1	3 × 10−7	MF
KNa	Na+ dissociation constant	mM	25	MF [69]i
CatSper channels
$f_{\text{cs}}^o$	Fraction of open channels		Function	Eq 34
mcs	Fraction of channels with voltage- and pH-dependent gate open		Variable	Eq 35
hcs	Fraction of channels with Ca2+-dependent gate not inactivated		Variable	Eq 36
vcs	pH-dependent voltage leading to half-maximal gate opening at equilibrium	mV	Function	Eq 40
s1	Voltage-dependent gate opening rate constant	s−1	270	MF
s2	mcs voltage sensitivity	mV	10	MF [12, 83], S3 Fig
s3	Amplitude of vcs response to pHi	mV	79.70	F [72], S3 Fig
s4	pHi leading to half-maximal vcs value		7.20	MF [72], S3 Fig
s5	Hill-coefficient of vcs		35.52	F [72]
s6	pHi-independent, minimal vcs value	mV	-41	MF, S3 Fig
s7	Scaling factor	nM	1	MF
s8	[Ca2+]i leading to half-maximal hcs inactivation (IC50)	nM	700	MF
s9	Hill coefficient of Ca2+-inactivation curve		2.50	MF
s10	Ca2+-inactivation rate constant	nM−1 s−1	1.40 × 10−4	MF
gcs	Maximal conductance density	pS μm−2	39.90	MF j [72]
Ics	Current density	fA μm−2	Function	Eq 44
Common variables and parameters
V	Membrane potential	mV	Variable	Eq 1
Em	Resting membrane potential	mV	-40	[80, 84, 85, 10]
Cm	Specific capacitance of flagellum membrane	pF μm−2	1 × 10−2	[86] k
vf	Intraflagellar volume	fL	1.60	[48]
sf	Surface area of flagellar membrane	μm2	30	D l
[K+]o	Extracellular potassium concentration	mM	9	[87] m
[K+]i	Resting intraflagellar potassium concentration	mM	219	[66]
[Na+]o	Extracellular sodium concentration	mM	423	[87] m
[Na+]i	Resting intraflagellar sodium concentration	mM	20	[81]
pHo	Extracellular pH		8	[87] m
pHr	Resting intraflagellar pH		7.10	[81]
[Ca2+]o	Extracellular calcium concentration	mM	10	[87] m
Cr	Resting intraflagellar calcium concentration	nM	100	[81, 14]
C	Intraflagellar calcium concentration	nM	Variable	Eq 6
H	Intraflagellar proton concentration	M	Variable	Eq 5
σH	NHE-independent proton source rate constant	M s−1	1.50 × 10−7	D (sec. 4.6)
δH	Proton extrusion/consumption rate constant	s−1	1.28	D (Eq 5)
EK	K+ equilibrium potential	mV	-80	Dn
Ehc	Reversal potential of spHCN current	mV	-18	[61, 62, 63]
ECa	Ca2+ equilibrium potential	mV	144	Do

^a The column entries are a numeric value if the quantity is a parameter or a label whether it is a variable or a function

^b The column entries: are a bibliographic reference unlabelled if the values were used as reported in the original publication or with labels F and MF if further processed; a derivation labelled D followed by a reference to the section where its derivation is explained; or a reference to the defining equation(s) defining a variable or function.

^c Taking into account the membrane surface of the flagellum s_f, and KCNG unitary conductance (110 pS [59]), this implies ∼37 channels per flagellum

^d Taking into account the membrane surface of the flagellum s_f, and spHCN unitary conductance (43 pS [63]), this would imply ∼135 channels per flagellum

^e Fitted in the Upstream module

^f Taking into account the membrane surface of the flagellum s_f, and Ca_V unitary conductance (4.70 pS [82]), this implies ∼1182 channels per flagellum

^g Taking into account the membrane surface of the flagellum s_f, and BK unitary conductance (70 pS [42]), this implies ∼92 channels per flagellum

^h Fitted in the complete CatSper + NHE module to conserve the initial drop of membrane potential, whereas allowing calcium signal to return to levels close to resting value after spike-train finishes

ⁱ Based on figure 1 of [69]

^j Taking into account the membrane surface of the flagellum s_f, and assuming that the Ca²⁺ unitary conductance of CatSper might be below 1 pS, this implies ∼1200 channels per flagellum

^k Conventional value for eukaryotic cytoplasmic membranes

^l Calculated as the lateral surface of a cylinder with length 40 μm and diameter 0.240 μm

^m Typical concentration in artificial seawater used for embryology experiments on marine organisms

ⁿ Calculated with the Nernst equation, $E=\frac{RT}{zF}ln\frac{{\left[X\right]}_o}{{\left[X\right]}_i}$, having temperature T = 17°C

^o Calculated with the Nernst equation, setting temperature T = 17°C, and using the C_r parameter as internal concentration value