Supplementary File 1
Title: Functional contributions of the plasma membrane Ca2+ ATPase and the sodium
calcium exchanger at mouse parallel fibre to Purkinje neuron synapses
Authors: Chris J. Roome1, Thomas Knöpfel2 and Ruth M. Empson1
1Department
of Physiology, Brain Health Research Centre, University of Otago, Dunedin,
New Zealand.
2Laboratory
for Neuronal Circuit Dynamics, RIKEN Brain Science Institute, Wako-ishi,
Saitama, Japan
Description of the single compartment model for calcium dynamics
Due to the small volume of pre-synaptic terminals, spatial gradients in internal calcium
concentrations are assumed to dissipate very rapidly within the terminal, on much shorter
timescales than are relevant to the dynamics of residual pre-synaptic calcium [1]. Therefore, these
simulations consider only average homogenous internal calcium ([Ca2+]i) dynamics within a single
compartment. The compartment had a surface to volume ratio (G) of 3/0.5 m [2, 3] and boundaries
that maintained a constant resting membrane potential (Vm) at –70mV. Under resting conditions the
external calcium and sodium ionic concentrations ([Ca2+]e and [Na+]e) were held constant.
Additional calcium and sodium ion fluxes were included that exactly opposed calcium and sodium
flux via PMCA and NCX at rest. The time course of the opposing ion fluxes was set such that under
‘no-stimulus’ conditions basal internal calcium and sodium concentrations ([Ca2+]i and [Na+]i) were
held constant.
During ‘synaptic stimulation’ the membrane potential was briefly depolarised (to +20 mV for 1 ms)
using Gaussian functions to represent action potentials (APs) arriving at the pre-synaptic terminal.
The internal calcium and sodium ionic concentrations were also increased instantaneously for each
incoming AP to represent calcium and sodium influx through their voltage-gated channels. The presynaptic calcium dynamics that followed were then determined by the NCX, the PMCA, and
endogenous calcium buffers (see Fig. 4D in the main text for a model schematic).
The change in the free internal calcium and sodium concentrations for each time increment (t) was
described by ordinary differential equations [1] and [2], respectively:
∆[𝐶𝑎2+ ]𝑖
∆𝑡
∆[𝑁𝑎+ ]𝑖
∆𝑡
𝐺
1
= 𝑧𝐹 {𝐽𝑐𝑎𝑖𝑛 − 𝐽𝑛𝑐𝑥 − 𝐽𝑝𝑚𝑐𝑎 + 𝐽𝑐𝑎𝑙𝑒𝑎𝑘 } 1+𝑇
𝑒𝑛
𝐺
= 𝑧𝐹 {𝐽𝑛𝑎𝑖𝑛 + 3𝐽𝑛𝑐𝑥 + 𝐽𝑛𝑎𝑙𝑒𝑎𝑘 }
[1]
[2]
where [𝐶𝑎2+ ]𝑖 and [𝑁𝑎+ ]𝑖 are the changes in free internal ion concentrations, G is the single
compartment geometry factor (surface-to-volume ratio), and F is the Faraday constant. Jcain and Jnain
are calcium and sodium influx, respectively. Jncx and Jpmca are NCX- and PMCA-mediated calcium flux,
respectively. Assuming the rapid buffer approximation, endogenous calcium buffers were included
by means of a correction factor
1
1+𝑇𝑒𝑛
which depends only on calcium concentration, buffer
0
concentration (𝑏𝑒𝑛
), and the buffer dissociation constant (𝐾𝑒𝑛 ) [2] (Equation [3]).
𝑇𝑒𝑛 =
0 𝐾
𝑏𝑒𝑛
𝑒𝑛
(𝐾𝑒𝑛 + [𝐶𝑎 2+ ]𝑖 )2
𝑤𝑖𝑡ℎ 𝐾𝑒𝑛 =
−
𝑘𝑒𝑛
+
𝑘𝑒𝑛
[3]
+
−
where 𝑘𝑒𝑛
and 𝑘𝑒𝑛
are the calcium binding and unbinding rate constants.
Calcium removal via PMCA
Calcium efflux via PMCA was described by a simple Hill equation (4) [2, 4]:
̅
𝐽𝑝𝑚𝑐𝑎 = 𝐼𝑝𝑚𝑐𝑎
∙
[𝐶𝑎 2+ ]𝑖
𝑛
𝑛
[𝐶𝑎2+ ]𝑖 +(𝐾𝑝𝑚𝑐𝑎 )𝑛
∙ 𝜌𝑝𝑚𝑐𝑎
[4]
̅
where 𝐼𝑝𝑚𝑐𝑎
is the maximal calcium current via PMCA, 𝐾𝑝𝑚𝑐𝑎 is the half activation ion
concentration, 𝑛 is the Hill coefficient, and 𝜌𝑝𝑚𝑐𝑎 is the PMCA protein density.
Calcium and sodium exchange via NCX
The NCX current (𝐼𝑛𝑐𝑥 ) was described as the product of an electrochemical factor (ΔE; Equation [8])
and an allosteric factor (Allo; Equation [7]) [5], and was used to describe NCX-mediated calcium and
sodium influx/efflux by Equations [5] and [6], respectively.
̅ ∙ ∆𝐸 ∙ (𝐴𝑙𝑙𝑜) ∙ 𝜌𝑛𝑐𝑥
𝐶𝑎𝑙𝑐𝑖𝑢𝑚 𝑒𝑓𝑓𝑙𝑢𝑥 = −𝐽𝑛𝑐𝑥 = −𝐼𝑛𝑐𝑥
[5]
̅ ∙ ∆𝐸 ∙ (𝐴𝑙𝑙𝑜) ∙ 𝜌𝑛𝑐𝑥
𝑆𝑜𝑑𝑖𝑢𝑚 𝑖𝑛𝑓𝑙𝑢𝑥 = +3𝐽𝑛𝑐𝑥 = +3 ∙ 𝐼𝑛𝑐𝑥
[6]
𝐴𝑙𝑙𝑜 =
[𝐶𝑎2+ ]𝑖
𝑛
{[𝑁𝑎]𝑖 3 [𝐶𝑎]𝑜 𝑒
∆𝐸 =
[7]
𝑛
[𝐶𝑎2+ ]𝑖 +(𝐾𝑛𝑐𝑥 )𝑛
(𝜂−1)𝑉𝑚 𝐹
𝜂𝑉𝑚 𝐹
(
)
( 𝑅𝑇
)
𝑅𝑇
− [𝑁𝑎]𝑒 3 [𝐶𝑎]𝑖 𝑒
}
𝐾𝑚𝐶𝑎𝑒 [𝑁𝑎]3𝑖 +𝐾𝑚𝑁𝑎𝑒 [𝐶𝑎]𝑖 + 𝐾𝑚𝑁𝑎𝑖 3 [𝐶𝑎]𝑒 (1+
{
[𝐶𝑎]𝑖
)+
𝐾𝑚𝐶𝑎
𝑖
[𝑁𝑎]3
𝑖 )+ [𝑁𝑎]3 [𝐶𝑎] + [𝑁𝑎]3 [𝐶𝑎]
𝐾𝑚𝐶𝑎𝑖 [𝑁𝑎]3𝑒 (1+
𝑒
𝑖
𝑒
𝑖
𝐾𝑚𝑁𝑎 3
𝑖
[8]
{1+𝐾𝑠𝑎𝑡 𝑒
(
(𝜂−1)𝑉𝑚 𝐹
)
𝑅𝑇
}
}
̅ is the maximal calcium current via NCX, 𝐾𝑛𝑐𝑥 is the activation Ca2+ concentration, 𝑛 is the Hill
𝐼𝑛𝑐𝑥
coefficient, and 𝜌𝑛𝑐𝑥 is the NCX protein density. Km[ion]i/e values are dissociation constants for internal
(i) and external (e) sodium and calcium, η is the position of the energy barrier of NCX in the
membrane electric field, and Ksat is a factor controlling saturation of INCX at negative potentials. See
[5] for a complete description of the NCX model.
A complete list of universal constants and parameters used in the model for pre-synaptic calcium
dynamics is given in Table 1. All model simulations were implemented in MATLAB (code available on
request).
Table 1 Model parameters
Parameters
Symbol
Value
References
Faraday constant
F
96485 [C.mol-1]
Gas constant
R
8.315 [J.K-1.mol-1]
Temperature
T
300 [K]
Geometry factor
G
(3/0.5)x10-6 [m]
[3]
Resting membrane potential
Vm
-70 x10-3 [V]
[6]
Resting Ca2+ concentration
[Ca2+]i
0.05x10-6 [M]
[1]
Extracellular Ca2+ concentration
[Ca2+]e
2.4x10-3 [M]
[1]
Resting Na+ concentration
[Na+]i
10x10-3 [M]
[1]
Extracellular Na+ concentration
[Na+]e
150x10-3 [M]
[1]
Change in [Ca2+]i per action potential
[Ca2+]i
500x10-9M
[7]
Change in [Na+]i per action potential
[Na+]i
4x10-3M
adjusted
Endogenous buffer (based on calretinin)
Dissociation constant
Ken
1.5x10-6[M]
[8]
Buffer concentration
Ben
2000 x10-6[M],
[8]
Maximum activity rate
𝐼𝑝̅
2.7x10-17 [Cs-1]
[4]
PMCA density
𝜌𝑝
1x1013 [m-2]
adjusted
Half activation concentrations
𝐻𝑝
0.1x10-6 [M]
[4]
Maximum activity rate
𝐼𝑥̅
100*𝐼𝑝̅
[9]
Hill coefficient
𝑛𝑥
2
[2]
Activation Ca2+ concentration
𝐻𝑥
125x10-9 [M]
[5]
NCX density
𝜌𝑣
0.03 x 𝜌𝑝
[10]
1.3x10-3 [M]
[5]
PMCA
NCX
External Ca2+ dissociation const.
+
KmCao
-3
External Na dissociation const.
KmNao
87.5x10 [M]
[5]
Internal Ca2+ dissociation const.
KmCai
3.6x10-6 [M]
[5]
Internal Na+ dissociation const.
KmNai
12.3x10-3 [M]
[5]
Reverse mode saturation factor
Ksat
0.27
[5]
References:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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