High probability neurotransmitter release sites represent an energy efficient design

Abstract

Nerve terminals contain multiple sites specialized for the release of neurotransmitters. Release usually occurs with low probability, a design thought to confer many advantages. High probability release sites are not uncommon but their advantages are not well understood. Here we test the hypothesis that high probability release sites represent an energy efficient design. We examined release site probabilities and energy efficiency at the terminals of two glutamatergic motor neurons synapsing on the same muscle fiber in Drosophila larvae. Through electrophysiological and ultrastructural measurements we calculated release site probabilities to differ considerably between terminals (0.33 vs. 0.11). We estimated the energy required to release and recycle glutamate from the same measurements. The energy required to remove calcium and sodium ions subsequent to nerve excitation was estimated through microfluorimetric and morphological measurements. We calculated energy efficiency as the number of glutamate molecules released per ATP molecule hydrolyzed, and high probability release site terminals were found to be more efficient (0.13 vs. 0.06). Our analytical model indicates that energy efficiency is optimal (~0.15) at high release site probabilities (~0.76). As limitations in energy supply constrain neural function, high probability release sites might ameliorate such constraints by demanding less energy. Energy efficiency can be viewed as one aspect of nerve terminal function, in balance with others, because high efficiency terminals depress significantly during episodic bursts of activity.

Introduction

The strength of a synaptic connection is a function of three parameters: the number (N) of release sites, or active zones (AZs), at which neurotransmitter is released; the average probability of release from each AZ (P_AZ); and the average amplitude of the postsynaptic response to each packet of neurotransmitter [1]. It is not known why AZs with either a low or high P_AZ exist at any particular terminal. A low P_AZ confers advantages such as a high capacity for information storage [2], resistance to depression [3, 4], and energy efficient information transfer at convergent synaptic inputs [5, 6]. Insofar as the advantages of a low P_AZ design confer organisms with selective fitness, selection pressures might promote a common low P_AZ. However, P_AZ varies greatly between presynaptic terminals [7-14]. While reasons for a non-zero P_AZ are readily evident, the occurrence of high P_AZ synapses causes us to question what advantages these might confer to offset those of low P_AZ.

We propose that presynaptic energy efficiency, defined as the number of glutamate molecules released for each ATP molecule hydrolyzed, is one of the advantages inherent in high P_AZ release sites. Multiple calcium ions (Ca²⁺) are required to trigger the release of neurotransmitters and this generates a steep dependency of P_AZ on Ca²⁺ entry, described as a sigmoid function [15]. As the cost of Ca²⁺ removal is one of the primary costs of the presynaptic terminal [16, 17] we might expect that energy efficiency will be optimized when neurotransmitter release is maximized relative to Ca²⁺ entry. This point does not occur until gains in neurotransmitter release become marginal relative to Ca²⁺ entry, i.e. high on the sigmoid curve, synonymous with high P_AZ.

Here we used direct measurements to make the first bottom-up estimates of presynaptic energy efficiency at individually identified neurons. We took advantage of two motor neurons that stereotypically innervate a single muscle fiber in Drosophila larvae. One neuron forms a small terminal with few AZs while the other forms a larger terminal with more AZs. We established that P_AZ differs considerably between the two terminal types. We then proceeded to test the hypothesis that high P_AZ confers high energy efficiency, estimating energy consumption through direct measurements of neurotransmitter release and Ca²⁺ entry, and by estimating sodium ion (Na⁺) entry theoretically. We found that in response to an isolated AP the small terminal with high P_AZ was significantly more efficient, consistent with our hypothesis. Data collected during AP bursts supported the same hypothesis, but revealed that a high efficiency design based on high P_AZ will not sustain high output.

Results

QUANTIFICATION OF RELEASE SITE PROBABILITY (P_AZ)

We set out to measure the average probability of release from AZs (P_AZ) at each of two glutamatergic motor neuron (MN) terminals that synapse on muscle fiber #6 (Figure 1A). Average P_AZ can be determined using Equation 1:

Figure 1. Small motor neuron terminals (type-Is) have fewer active zones than large terminals (type-Ib).

A, A confocal micrograph of fluorescence from terminals of two different motor neurons (MNSNb/d-Is and MN6/7-Ib) innervating muscle fibers #6 and #7. Both MNs express GFP, but the terminal of MNSNb/d-Is is filled with a greater concentration of AF647-dextran (magenta). B-D, Total terminal surface area and volume quantified on fiber #6 quantified from confocal microscopy. B, Fixed preparations immunolabelled to reveal the neuronal plasmamembrane [PM, (Horseradish Peroxidase, HRP)], the postsynaptic sub-synaptic reticulum [SSR, (Discs Large, DLG)] and active zones [AZs, (Bruchpilot, nc82)]. A series of images collected while advancing through the entire depth of the terminals, then collapsed into a maximal intensity z-projection stack viewed as an inverted grayscale. C and D, Plots of the total terminal surface area and volume for 6 different pairs of terminals. Averages shown as open circles (*, P<0.001, paired Student’s t-test). SD shown in C and D. E, Transmission electron micrograph of a 100nm thick section through a type-Ib bouton on fiber #6 collected at 11,500X. AZs are evident with (arrow) and without (arrowhead) accompanying T-bars. F, The number of AZs per unit volume was plotted for each of three pairs of terminals on fiber #6 (3 separate larvae). SD shown. G, Estimates of the average number of AZs (with or without T-bars) for the different terminal types. Estimates were made by combining data in D (N=6) and F (N=3). SEM shown in G. Error bars in G calculated according to propagation of uncertainty theory.

$${\mathrm{P}}_{\text{AZ}}=\text{QC}∕{\text{N}}_{\text{AZ}}$$ (1)

where the number of packages of glutamate released [quantal content (QC)] by each terminal is determined electrophysiologically, and the number of AZs (N_AZ) determined by microscopy.

Type-Is terminals have fewer active zones than type-Ib

Muscle fiber #6 in Drosophila larvae is innervated by type-Is “small” bouton terminals of MNSNb/d-Is [18], and type-Ib “big” bouton terminals of MN6/7-Ib [19, 20]. The distinct identities of the live MN terminals are illustrated by differentially filling them with two fluorophores (Figure 1A). In fixed tissue, anti-horseradish peroxidase (HRP; [21]) was used to define neuronal membrane, anti-Bruchpilot (Brp; nc82; [22]) to identify AZs, and anti-Discs Large (DLG; [23]) to label the sub-synaptic reticulum (SSR) in the muscle and facilitate discrimination between boutons of different terminal types (Figure 1B and Figures S1A and S1B). Type-Is terminals on muscle 6 have a smaller surface area (Is: 234±110; Ib: 568±70 µm²; N=6 pairs, SD, P<0.001; Figure 1C; Table S1) and volume (Is: 90±49; Ib: 310±49 µm³; N=6 pairs, SD, P<0.001; Figure 1D; Table S1) than type-Ib terminals.

The number of AZs per unit volume was determined by counting AZ profiles in a series of transmission electron micrographs of 100 nm sections (Figure 1E and 1F). The entire extent of each terminal was too large to reconstruct and we relied instead on synaptic vesicle (SV) sizes rather than morphologies to identify terminals because SV diameters differ between terminal types [24] (Figure S1C). Counts of AZ profiles per unit volume revealed a similar AZ density in the two terminal types (Figure 1F; Table S1). We estimated N_AZ for each of 6 terminals examined by light microscopy, multiplying its volume by the average number of AZs per unit volume. Average N_AZ was lower for type-Is terminals (Is: 223±66; Ib: 747±114 AZs; SEM; Figure 1G; Table S1).

Type-Is terminals release more neurotransmitter than type-Ib terminals

Quantal content (QC) can be determined for all AZs of each terminal type using an electrophysiological recording protocol that takes advantage of our knowledge of the stereotypical innervation of the body wall muscle fibers (see Experimental Procedures and Figure 2A). The dependence of SV release on extracellular Ca²⁺ concentration ([Ca²⁺]_o) was initially quantified in a low range of [Ca²⁺]_o (0.3-0.5 mM) in Hemolymph-Like solution #6 (HL6) [25] and type-Is terminals were found to have a much larger QC, i.e. they release more SVs across this range of [Ca²⁺]_o (Figure S2A-S2D). However, to quantify SV release under physiological conditions, [Ca²⁺]_o was raised to 2 mM [Ca²⁺]_o and synaptic events recorded under two-electrode voltage clamp (Figure 2B). Type-Is Excitatory Junction Currents (EJCs) were significantly larger than type-Ib EJCs (Is: 80.8±4.0, N=17; Ib: 59.9±3.1 nA; N=15, SEM, P<0.001; Figure 2C), but QC estimates were similar (QC: Is: 72.9±4.8; Ib: 80.6±4.5; SEM, P=0.25; Figure 2D; Table S1) after adjustments were made for the larger miniature EJPs (mEJCs) that originate from type-Is terminals (~50% larger, see Experimental Procedures). QC determined in another commonly used saline [Hemolymph-Like solution #3 (HL3)] [26] using physiological [Ca²⁺]_o gave similar results (Figure S2E). The capacity of each terminal to release glutamate beyond amounts recorded at physiological [Ca²⁺]_o was demonstrated by decreasing [Mg²⁺]_o in HL6 from 15mM to 10mM; EJC amplitude increased by 15.3% at both terminals (Is: 93.2±6.01; Ib: 69.8±4.18 nA; N=7, SEM, P<0.01).

Figure 2. Type-Is terminals release more neurotransmitter than type-Ib during a single action potential.

A, Cross-section of muscle fibers and their innervating MNs. MNs are color-coded. MNSNb/d-II and MN12-III and other terminals of MNSNb/d-Is (*) are omitted for clarity. The current-clamp recording configuration is shown with micropipette (electrode) tips in adjacent muscle fibers (A). Two-electrode voltage clamp (TEVC) was achieved after withdrawing a micropipette from muscle 13 and placing it in muscle 6 (grey outline) (data in B-E). B, Excitatory Junction Currents (EJCs) recorded in muscle 6 in two separate preparations, and miniature EJCs (mEJCs) from one of those preparations. C, Average EJC amplitudes (Is: N=17; Ib: N=15; Student’s t-tests, *P<0.01). Miniature EJCs (mEJCs) recorded from the two terminals at the same time as the EJCs had similar amplitudes (Is: 0.919±0.040 nA, N=17; Ib: 0.916±0.046 nA, N=15, SEM, P=0.96). D, Quantal content shown after correcting mEJC amplitudes using scaling factors (see Experimental Procedures for explanation). E, Estimates of average release site probability (P_AZ), calculated by dividing corrected quantal content (QC, Is: N=17; Ib: N=15) (D and Table S1) by total AZ number (N_AZ, N=6, Figure 1G) for each terminal type. Error bars show SEM in C, D and E. Error bars calculated according to propagation of uncertainty theory in E. All recordings performed in TEVC in 2mM [Ca²⁺]_o 15mM [Mg²⁺]_o HL6.

Type-Is terminals have a higher P_AZ than type-Ib

P_AZ was estimated for each terminal using equation 1 and the estimates of QC made in 2 mM [Ca²⁺]_o HL6. P_AZ was 3-fold higher for type-Is terminals than for type-Ib terminals (Is: P_AZ = QC / N_AZ = 72.9/223 = 0.33±0.10; Ib: P_AZ = 80.6/747 = 0.11±0.02; SEM) (Figure 2E; Table S1). The assumption that all AZs identified from EM are functional, is favored over the alternative that only AZs with T-bars (Figure S1D) are functional, because the latter assumption generates P_AZ estimates > 1 in type-Is terminals (Is: P_AZ = 72.9/59 = 1.24; Ib: P_AZ = 80.6/421 = 0.19).

QUANTIFICATION OF PRESYNAPTIC ENERGY EFFICIENCY

Our conception of energy efficiency is based on Maxwell’s definition [27], based on output relative to input (see Experimental Procedures). To test our hypothesis that high P_AZ terminals are more energy efficient we must estimate the number of glutamate molecules released from each terminal relative to the amount of energy required to release and recycle the glutamate and to remove Ca²⁺ and Na⁺. The QC estimates above contribute to an estimation of both the number of glutamate molecules released and the number of ATP molecules needed to recycle the glutamate and SVs. Next, microfluorimetric measurements in conjunction with our morphological data were used to quantify Ca²⁺ number, and a theoretical approach based on those data used to quantify Na⁺ number. These estimates for Ca²⁺ number and Na⁺ number were then used to estimate the numbers of ATP molecules required to power Ca²⁺ and Na⁺ removal.

Glutamate Handling

Type-Is terminals release the most glutamate during an AP

The number of glutamate molecules released in response to a single AP (Glu) can be determined using equation 2:

$$\text{Glu}=\text{QC}\times \epsilon$$ (2)

where QC is the number of SVs released, determined above (Figure 2D), and ε is the number of glutamate molecules in a SV. Values for ε were estimated for SVs in each terminal type by combining a nominal value of ε generated in a biophysical study at the Drosophila larval NMJ [28] and scaling factors derived from amplitude differences in uni-quantal events produced by each terminal (ε; Is: 9600; Ib: 6400 molecules). Using Equation 2, we calculated that type-Is terminals release more glutamate per AP [Is: (7.00±0.46) × 10⁵; Ib: (5.16±0.29) × 10⁵ glutamate molecules; SEM] (Table S1).

The number of ATP molecules required to recycle and refill SVs with glutamate after an AP (E_Glu) can be determined using equation 3:

$${\mathrm{E}}_{\text{Glu.}}=\text{QC}\times 410.5+\text{Glu}\times 2.67$$ (3)

where the estimated cost of exocytosis and endocytosis of a single SV of measured diameter is 410.5 ATP molecules [29], and it costs 2.67 ATP molecules to load a glutamate molecule into a SV. The value of 410.5 has been adopted for SVs of both terminals. For a single AP, type-Is terminals spend more ATP recycling and refilling SVs with glutamate [Is: (1.90±0.12) × 10⁶; Ib: (1.41±0.08) × 10⁶ ATP molecules; SEM] (Table S1).

Calcium Handling

The number of calcium ions (Ca²⁺_total) that enter each terminal during an AP can be determined using equation 4:

$${{\text{Ca}}^{2+}}_{\text{total}}=\Delta {\left[{\text{Ca}}^{2+}\right]}_{\text{total}}\times \text{vol}\times {\mathrm{A}}_{\text{N}}$$ (4)

where Δ[Ca²⁺]_total is the change in total Ca²⁺ concentration (free+bound) (moles / L), vol is the volume of the terminal (L) which has been determined (Figure 1D), and A_N is Avogadro’s constant, but Δ[Ca²⁺]_total must be calculated using equation 5 [(equation 4) of [30]]:

$$\Delta {\left[{\text{Ca}}^{2+}\right]}_{\text{total}}=\Delta {\left[{\text{Ca}}^{2+}\right]}_{\text{AP}}\times (1+{\mathrm{K}}_{\text{S}}+\mathrm{K}{’}_{\text{B}})$$ (5)

where Δ[Ca²⁺]_AP is the change in cytosolic free Ca²⁺ concentration ([Ca²⁺]_i) in response to a single AP, K_S is the endogenous Ca²⁺ binding ratio, and K’_B the incremental Ca²⁺ binding ratio for the exogenous Ca²⁺ buffer, Oregon-Green BAPTA-1 (OGB-1).

Type-Is terminals display the largest transients in cytosolic free Ca²⁺ concentration

We determined changes in [Ca²⁺]_i by forward-filling terminals with a mixture of a Ca²⁺ sensitive fluorescent dye and a Ca²⁺-insensitive fluorescent dye, in a constant ratio (~15:1; OGB-1 dextran : AF647 dextran) (Figure 3). A single AP evoked a greater increase in OGB-1 fluorescence, and thus [Ca²⁺]_i, in type-Is terminals than type-Ib (Is: 412.3±59.9; Ib, 292.1±34.2 nM; N=12 pairs, SEM, P<0.05; Figure 3B and 3C). See Experimental Procedures for calibration of the ratio against [Ca²⁺]_i. The time course of decay (τ) of [Ca²⁺]_i was similar between terminal types (Is: 75.9±4.7; Ib: 75.3±6.5 ms; SEM; Figure S3A and S3B), and the time integral of the [Ca²⁺]_i transient (Δ[Ca²⁺]_AP·τ), was greater in type-Is terminals (Is: 28.9±3.2; Ib: 20.2±1.7 nM·s; SEM, P<0.05; Figure S3C). The endogenous Ca²⁺ binding ratio (K_S) was calculated as described in the Experimental Procedures (K_S: Is: 82, Ib: 49; Figure S3D and S3F). Using equation 5 we calculated the change in total [Ca²⁺]_i (Δ[Ca²⁺]_total) to be significantly greater in type-Is terminals (Is: 40.1±5.5; Ib: 20.6±2.2 µM; SEM, P<0.05; Figure 3D).

Figure 3. Type-Is terminals display larger [Ca²⁺]_i transients during single action potentials, but less total Ca²⁺ enters these terminals.

A, Inverted grayscale images of type-Ib and -Is bouton terminals filled with AF647-dextran (top; average of 20 frames collected at 20 fps) and of Oregon-Green BAPTA-1 (OGB-1) dextran (middle & bottom; average of 10 frames at 100 fps). The middle and bottom images show an average of OGB-1 fluorescence in the 100 ms period prior, and subsequent to, each of 10 stimuli delivered at 1 Hz, respectively. B, Average [Ca²⁺]_i for a single AP, calculated from 10 synchronized stimuli, estimated from the ratio of OGB-1 to AF647 fluorescence calibrated relative to [Ca²⁺]_i (see Experimental Procedures). C, Plots of the average maximal change in [Ca²⁺]_i (Δ[Ca²⁺]_AP) for 12 different pairs of terminals filled with OGB-1 and AF647 on muscle 6. Averages shown as open symbols (*, P<0.01). D, Plots of the change in total [Ca²⁺]_i (free plus bound), calculated using equation 5 : Δ[Ca²⁺]_total = Δ[Ca²⁺]_AP × (1 + K_S + K’_B), (*, P<0.05). E, A plot of the average number of Ca²⁺ ions that enter each terminal in response to a single AP (calculated using equation 4). Paired Student's t-tests were conducted in C and D, and SEM is shown. Error bars calculated according to propagation of uncertainty theory in E.

Type-Is terminals admit the least Ca²⁺

The number of calcium ions that enter the entire terminal during an AP (Ca²⁺ total) was calculated using equation 4. This analysis revealed that total Ca²⁺ entry is least for type-Is terminals [Is: (2.18±0.57) × 10⁶; Ib: (3.86±0.49) × 10⁶ ions; SEM; Figure 3E] consistent with their 3.5 fold smaller volume, despite a 2-fold greater Δ[Ca²⁺]_total. Significantly, Ca²⁺ entry per-unit-volume was much higher in type-Is terminals (Is: 2.42 × 10⁴; Ib: 1.24 × 10⁴ ions / μm³; SEM), consistent with their larger [Ca²⁺]_i transients (Figure 3B and 3C), despite their higher estimated K_S. We also estimated Ca²⁺ entry per AZ, as this is most likely to vary in proportion with microdomain Ca²⁺ concentration which is most relevant to the Ca²⁺ sensor at the AZ. Ca²⁺ entry per AZ is greater for type Is terminals [Is: (9.77±3.87) × 10³; Ib: (5.17±1.02) × 10³ ions; SEM; Table S1] and this provides a simple explanation for the higher P_AZ of type-Is terminals.

The number of ATP molecules required to extrude the Ca²⁺ that enters during an AP (E_Ca²⁺) can be calculated using equation 6:

$${\text{E}}_{{\text{Ca}}^{2+}}={{\text{Ca}}^{2+}}_{\text{total}}\times 1\text{ATP}∕1{\text{Ca}}^{2+}$$ (6)

where 1 ATP molecule is required to extrude 1 calcium ion regardless of its route of exit [29]. Type-Is terminals were found to spend far less ATP on Ca²⁺ extrusion per AP [Is: (2.18±0.57) × 10⁶; Ib: (3.86±0.49) × 10⁶ molecules; SEM] (Figure 4A and 4B; Table S1).

Figure 4. In response to a single action potential, type-Is terminals spend less ATP to release more glutamate.

A, Pie chart representations of relative ATP demands for glutamate release and recycling [equation 3 (E_Glu)], Ca²⁺ extrusion [equation 6 (E_Ca2+)], and Na⁺ extrusion [equation 8 (E_Na+)] for each terminal type. The area of each sector is proportional to the number of ATP molecules used for each activity (Table S1). B, Plot of the average number of glutamate molecules released from each terminal type (Glu), adjacent to a stacked bar plot of the number of ATP molecules required for each activity (E_Glu, E_Ca2+ and E_Na+). C, A plot of the energy efficiency of both terminals calculated using equation 11. Error bars in B-C calculated according to propagation of uncertainty theory.

Sodium Handling

Type-Is terminals require the least Na⁺ current to depolarize

An indirect estimate of the minimum Na⁺ entering the presynaptic terminal during an AP can be calculated using equation 7:

$$\text{q}=\text{V}\times \text{C}$$ (7)

where a charge (q; Coulombs) is needed to change the voltage (V; volts) across the capacitance (C; uF) of presynaptic membrane by the amplitude of the AP [29, 31, 32]. The AP amplitude was taken as a nominal 100 mV, specific membrane capacitance as 1 µF/cm², and the area was estimated from confocal microscopic examination (Is: 234±110; Ib: 568±70 μm²; SD; Figure 1C). The resulting estimate of Na⁺ charge entry (Na⁺_total) had to be multiplied by an "overlap factor” representing the degree to which K⁺ entry works against Na⁺ entry during the rising phase of the AP [29, 31-33]. Patch recordings have not been made from Drosophila glutamatergic MN terminals, but an estimate of 3.05 comes from Kenyon cell axons of the honeybee, Apis mellifera [34]. The resulting estimates of Na⁺ entry showed that considerably more Na⁺ was expected to enter the larger type-Ib terminals [Is: (0.45±0.09) × 10⁷; Ib: (1.08±0.06) × 10⁷ Na⁺ ions; SEM; Table S1].

The number of ATP molecules required to extrude Na⁺ (E_Na⁺) can be calculated using equation 8:

$${\text{E}}_{{\text{Na}}^{+}}={{\text{Na}}^{+}}_{\text{total}}∕3\text{ATP}$$ (8)

where 1 ATP molecule can extrude 3 sodium ions [29]. Type-Is terminals were found to spend far less ATP on Na⁺ extrusion per AP [Is: (1.48±0.28) × 10⁶; Ib: (3.60±0.18) × 10⁶ ATP molecules; SEM; Figure 4A and 4B; Table S1].

Type-Is terminals, with the highest P_AZ, release neurotransmitter with the highest efficiency

The total number of ATP molecules (E_total) required to release and recycle glutamate and to remove Ca²⁺ and Na⁺ after a single AP can be calculated using equation 9:

$${\mathrm{E}}_{\text{total}}={\text{E}}_{\text{Glu}}+{\mathrm{E}}_{{\text{Ca}}^{2+}}+{\text{E}}_{{\text{Na}}^{+}}$$ (9)

where E_Glu, E_Ca2+ and E_Na+ were determined above. Although type-Is terminals release more glutamate per AP [Glu: Is: (7.00±0.46) × 10⁵; Ib: (5.16±0.29) × 10⁵ glutamate molecules; SEM] (Figure 4A and 4B; Table S1), they expend less ATP in doing so [E_total: Is: (5.56±0.65) × 10⁶; Ib: (8.88±0.52) × 10⁶ ATP molecules; SEM] (Figure 4A and 4B; Table S1), primarily because they need to spend less ATP extruding Ca²⁺ and Na⁺. Energy efficiency (E.E.) can be calculated using equation 10:

$$\text{E.E.}=\text{Glu}∕{\text{E}}_{\text{total}}$$ (10)

where, Glu and E_total have already been determined. Expressed in terms of the number of glutamate molecules released for each ATP molecule hydrolyzed, type-Is terminals are twice as efficient as type-Ib (Is: 0.126±0.017; Ib: 0.058±0.005; SEM; Figure 4C; Table S1). The hydrolysis of ATP to ADP under conditions that might be found within the cells of an invertebrate yields ~56.1 kJoules / mol [35] and so the efficiency values above might be re-expressed in terms of Joules consumed per glutamate molecule released (Is: 0.74 × 10⁻¹⁸; Ib: 1.61 × 10⁻¹⁸) or mole of glutamate released (Is: 4.45 × 10⁵; Ib: 9.67 × 10⁵).

A test of the “relative” energy efficiency of nerve terminals, one which does not rely on the accuracy of estimates for the amount of ATP, but rather on parameters measured directly in this study, is the number of SVs released relative to the amount of Ca²⁺ pumped [Is: (3.35±0.90) × 10⁻⁵; Ib: (2.09±0.29) × 10⁻⁵; SEM; Table S1], or the number of glutamate molecules released per number of Ca²⁺ ions that enter (Is: 0.321±0.087; Ib: 0.134±0.018; SEM; Table S1). In both cases, far less Ca²⁺ is required for type-Is terminals, reaffirming our conclusion that type-Is terminals with a high P_AZ embody a more energy efficient design for releasing glutamate.

SIMULATION OF PRESYNAPTIC ENERGY EFFICIENCY

Although our data support the hypothesis that high P_AZ confers greater energy efficiency, it relies on measurements from the terminals of just two MNs and is therefore not a powerful test. To further explore the influences on presynaptic energy efficiency we built an analytical model based on the data collected from the two MN terminals (Appendix I). This model allowed us to examine the influence of systematically altering various aspects of presynaptic morphology, physiology and biochemistry on presynaptic energy efficiency during a single AP.

P_AZ itself can be expressed as a function of 3 parameters (Appendix I): Ca²⁺ entry per AZ; Ca²⁺ sensitivity of the trigger for exocytosis (S); and cooperativity between Ca²⁺ binding sites on the trigger (n_H; [15, 36]). Simulation of changes in the magnitude of each parameter, while holding the other parameters constant, demonstrates that P_AZ is highly responsive to changes in Ca²⁺ entry per AZ and even more responsive to n_H (Figure 5A). Simulating the influence of S, n_H, N_AZ, Ca²⁺ entry per AZ and SV size on glutamate release we see that Ca²⁺ entry per AZ and n_H are again highly influential (S, n_H & Ca²⁺/AZ: Figure 5B; N_AZ, SV size: Figure S4A). Total Ca²⁺ influx was sensitive to changes in Ca²⁺ entry per AZ and N_AZ but not n_H, S or SV size (Figure S4B). Total Na⁺ entry was sensitive to a different set of parameters, such as surface area and a number of parameters for which no direct data are available for Drosophila terminals; AP overlap factor, unit capacitance and AP voltage change (Figure S4C).

Figure 5. Output plots from the analytical model (Appendix I) that explores the influences on presynaptic energy efficiency during a single action potential (AP).

A, Simulation of the influence of changes in the magnitude of each parameter that contributes to P_AZ [Ca²⁺ entry per AZ (Ca²⁺/AZ), sensitivity (S) and cooperativity (n_H)] on P_AZ itself. Type-Is and -Ib terminals are labeled in red and green, respectively. B, Simulation of the influence of changes in parameters that contribute to P_AZ on the amount of glutamate released by each terminal. C, Simulation of the influence of parameters that contribute to P_AZ on the energy efficiency of each terminal. D, Simulation of the influence of P_AZ (magnitude adjusted by changing Ca²⁺ entry per AZ alone) on the energy efficiency of each terminal. Inset, Plots duplicated from B, showing that maximum efficiency does not occur until high on the sigmoid curves describing the dependence of glutamate release on Ca²⁺ entry per AZ, that part of the curves synonymous with high P_AZ. In each plot, curves were normalized to physiological levels along the abscissa (a nominal range of 0.2 to 5), by dividing by the physiological value of the dependent (ordinate) variable.

We simulated the influence of each parameter on energy efficiency in Figure 5C and S4D. Glutamate release was unconstrained (as in Figure 5B) and each parameter systematically changed relative to measured values. Terminal size influences efficiency not through volume, but rather, through surface area, because it influences capacitance and the amount of Na⁺ current required for depolarization. An increase in N_AZ leads to a greater increase in efficiency only because more glutamate will be released while Na⁺ entry remains fixed (as terminal surface area is fixed). Also in Figure 5C, we see that parameters of n_H and Ca²⁺ entry per AZ, as well as S, are again highly influential.

It is unclear how variable the parameters underlying P_AZ may be in the program of any MN, but certainly Drosophila MN terminals have the capacity to change Ca²⁺ entry within minutes in response to an unconditioned AP [37]. Simulation of the effect of increasing Ca²⁺ entry per AZ to increase P_AZ shows that both terminals can improve their efficiency in the short term by as much as 25% for type-Is terminals and 150% for type-Ib (Figure 5D). The corresponding P_AZ for energy efficiency maximization is high in both terminals (Is: 0.76; Ib: 0.73). Energy efficiency will be maximized immediately before the point at which the incremental cost of releasing one more glutamate molecule exceeds the average cost of glutamate already released, a point high on the sigmoid curve that represents the dependence of glutamate release on Ca²⁺ entry per AZ (Figure 5D, inset, arrowheads).

QUANTIFICATION OF PRESYNAPTIC ENERGY EFFICIENCY DURING MOTOR PATTERNS

While a single MN AP is sufficient to elicit twitch in the muscle fiber, MNs cause muscle contractions relevant to locomotion through bursts of APs called motor patterns [38]. Nerve terminals containing high P_AZ release sites commonly show a depressing release profile during trains of APs, while those with low P_AZ release sites show a facilitating profile [3]. It seems possible then that if the high P_AZ terminals depress sufficiently then differences in efficiency between terminals may disappear. Energy efficiency during bursts of activity (E.E._bursts) can be calculated using equation 11:

$${\text{E.E.}}_{\text{bursts}}={\text{Glu}}_{\text{bursts}}∕{\text{E}}_{\text{total-bursts}}$$ (11)

where, Glu_bursts is quantified as Glu for an AP in the middle of a 2 s train of stimuli delivered at the endogenous firing rate (EFR), and E_total-bursts is the total number of ATP molecules required to “support” neurotransmitter release and recycling, and Ca²⁺ and Na⁺ regulation.

High-efficiency type-Is terminals fire at the lowest rate

To determine the endogenous firing rate (EFR) of each MN during motor patterns we recorded synaptic activity from adjacent muscle fibers (Figure 6A) while the central pattern generator (CPG) drove activity (Figure 6B), an experimental condition referred to as fictive locomotion. Using a previously described method [39] we were able to determine the EFR for each of the two MNs (see also Figures S5A and S5B). The EFR was significantly lower for MNSNb/d-Is than for MN6/7-Ib (Is: 7.8±0.7Hz; Ib: 20.7±0.8Hz; SEM, P<0.001; 2mM [Ca²⁺]_o) (Figure S5C).

Figure 6. Motor neurons innervating muscle fiber 6 both fire and depress at different rates.

A, Cross section of muscle fibers, their innervating MNs, and the current clamp recording configuration used to record EJPs during fictive locomotion. B, Sample trace of EJPs recorded in muscle fiber #6 in 0.8mM [Ca²⁺]_o, while a separate electrode recorded simultaneously in muscle 13 (trace not shown) enabling identification of the MNs contributing EJPs – in this case all were from MN6/7-Ib. C, Profiles of EJCs recorded in TEVC in muscle 6 in separate preparations evoked through impulses delivered to the segment nerve in 2 mM [Ca²⁺]_o. EJCs in each trace represent the current that flows across the plasmamembrane of muscle 6 when release is evoked exclusively from a MNSNb/d-Is (red) or MN6/7-Ib terminal (green). D, EJC profile summary; each symbol represents the average amplitude of EJCs for ≥6 preparations. EJC amplitude is measured as vertical displacement from the baseline immediately before each event. SD shown. E, Plot of the average number of glutamate molecules released in response to an AP, calculated from EJCs depressed to mid-train levels (Glu_bursts). F, Plot of the average number of glutamate molecules released per unit time (s) calculated from mid-train QC and the endogenous firing frequency (EFR) of each terminal type. Error bars in E and F calculated according to propagation of uncertainty theory. G, Pie chart representations of relative ATP demands / s for glutamate release and recycling, Ca²⁺ extrusion, and Na⁺ extrusion for each terminal type when firing at their endogenous firing frequencies. * indicates abbreviations of E_Glu-bursts / s, E_Ca2+-bursts / s and E_Na+-bursts / s, as E_Glu*, E_Ca2+* and E_Na+* respectively. Total pie chart areas are proportional to E_total-bursts / s.

High-efficiency terminals show the greatest frequency depression

When firing was driven at 10Hz (Figure 6C-6D), close to the MNSNb/d-Is EFR of 7.8Hz, type-Is terminals show greater frequency depression of EJCs than type-Ib (Is: 37.9±1.5% mid-train, N=27; Ib: 8.0±2.2% mid-train, N=8; SEM, P<0.001; Figure S5E). When driven at 22Hz, close to the MN6/7-Ib EFR of 20.7Hz, frequency depression was not significantly greater in type-Ib terminals (Figure 6D and Figures S5D and S5E; 10.5±6.0% mid-train; N=6, SEM), demonstrating their robust capacity to sustain neurotransmitter release. After one second of activity (mid-train) at their respective EFRs, EJC amplitudes were similar between terminal types (Is: 48.5±10.8, N=20; Ib, 47.4±6.0 nA, N=6; SD; Figures 6D and S5F), yielding the following QC estimates (Is: 42.3±9.4; Ib, 62.1±7.9; SD, P<0.05), after adjusting for differences in mEJC amplitudes. Similar amounts of glutamate were released by the two terminals during a mid-train AP (Glu_bursts) (Figure 6E; Table S2) but type-Ib released much more glutamate per unit time (s) [calculated as the product of Glu_bursts and the EFR (Figure 6F; Table S2)]. The ATP required to support presynaptic activity (E_total-bursts) per unit time (E_total-bursts / s) may be represented as the sum of E_Glu-bursts / s, E_Ca2+-bursts / s and E_Na+-bursts / s, as shown in Figure 6G.

High efficiency terminals show the greatest deterioration in efficiency during bursts

With estimates of Glu_bursts and E_total-bursts, equation 11 was used to calculate energy efficiency during bursts (E.E._bursts). Efficiency diminished, from single AP levels of 0.126±0.017 and 0.058±0.005 for terminal types-Is and -Ib respectively, to 0.085±0.020 and 0.046±0.009, respectively. The most significant decrease occurred in type-Is terminals (Is: 32.5% decline; Ib: 20.7% decline), resulting from their greater frequency depression. Unlike glutamate release from one AP to the next, we have assumed that there is no change in Ca²⁺ or Na⁺ entry from one AP to the next. Despite greater frequency depression in type-Is terminals, the difference in efficiency between terminals does not readily diminish during bursts of activity.

High-efficiency terminals fire only for short durations during fictive locomotion

To estimate each terminal’s neurotransmitter output rate in a freely moving larva, MN duty cycle estimates are required. Peristaltic body wall contractions occur at a frequency of ~1 Hz in freely moving larvae [40], but the duty cycle of MN6/7-Ib and MNSNb/d-Is during locomotion is not known. Electrophysiological recordings during fictive locomotion reveal that MN6/7-Ib is active 5/6 of the contraction cycle [40], but no estimates are available for MNSNb/d-Is. To estimate the relative duty cycle of the terminals we adopted an optical approach monitoring changes in cytosolic GCaMP5 florescence as a proxy for electrical activity (Figure 7A and 7B). Muscle contractions were blocked by adding 7mM glutamate to the saline [41]. Type-Ib terminals were active for a greater proportion of the time than type-Is terminals (Is: 0.074±0.024; Ib: 0.289±0.098; N=6, SD, P<0.001; Figure 7B).

Figure 7. Low-efficiency type-Ib terminals fire longest during fictive locomotion.

A, An activity plot for both terminal types on muscle fiber #6, collected simultaneously; GCaMP5 fluorescence intensity is proportional to the most recent instantaneous firing frequency. GCaMP5 fluorescence divided by the fluorescence of co-expressed DsRed, and the ratio for each terminal normalized between 0 (rest) and 1 (maximum). DsRed is insensitive to physiological ranges of Ca²⁺ and pH. B, Plots of the proportion of time during which the activity trace exceeded 20% (dotted line) for each terminal in a pair (relative duty cycle, N=6). Averages shown as open circles. SD shown (*, P<0.01, paired Student’s t-test). C, Plot of the average number of glutamate molecules released per unit time (Glu / s _loco.) from each terminal type on muscle fiber #6 (see Equation 12). Estimates of the summed output from all terminals of the same axon are represented by the arrows. D, Plot of the number of ATP molecules required to support the activity calculated in C (including Ca²⁺ and Na⁺ handling) (E_total / s _loco., see Equation 13), calculated to mimic presynaptic energy demands during body wall peristalsis in a freely moving animal. Estimates of the summed ATP demands of all terminals are represented by the arrows. E, Pie chart representations of the relative ATP demands per unit time (s) for glutamate release and recycling, Ca²⁺ and Na⁺ extrusion for both terminal types. Estimates of the summed ATP demands of all terminals of each MN are represented by the areas within the dashed circumferences. Error bars in C and D calculated according to propagation of uncertainty theory.

High efficiency terminals likely have a low output in freely moving larvae

Glutamate release per unit time (Glu / s_loco.) and total ATP consumption per unit time (E_total / s_loco.) can be estimated in freely moving larvae using Equations 12 and 13:

$$\text{Glu}∕\text{s}{}_{\text{loco.}}={\text{Glu}}_{\text{bursts}}\times \text{EFR}\times \text{D.C.}$$ (12)

$${\text{E}}_{\text{total}}∕\text{s}{}_{\text{loco.}}={\text{E}}_{\text{total-bursts}}\times \text{EFR}\times \text{D.C.}$$ (13)

where D.C. represents Duty Cycle (Is: 5/6 s × 0.074/0.289 = 0.213 s; Ib: 5/6 × 1 = 0.833 s, Table S2). Type-Ib terminals release over 10 fold more neurotransmitter per unit time (Glu / s _loco.; Is: (6.8±1.1) × 10⁵; Ib: (68.5±10.4) × 10⁵ glutamate molecules / s; SEM; Figure 7C) but require almost 18 fold more ATP to do so (E_total / s _loco.; Is: (7.9±1.3) × 10⁶; Ib: (147.4±16.1) × 10⁶ ATP/s; SEM; Figure 7D). This difference during endogenous activity is stark and to be able to sustain release at such high levels type-Ib terminals appear to pay a premium in having a low efficiency (Figures 6G and 7E).

Given the large difference in the rate of output for each terminal, the question arises as to the ATP demand per unit volume to sustain these relative rates. Despite being much larger, ATP is required at a rate 5 times faster per unit volume in type-Ib terminals [Is: (8.8±2.4) × 10⁴; Ib: (47.5±6.1) × 10⁴ ATP molecules/μm³/s; SEM]. The implication is that terminal volume is itself not rate limiting for neurotransmitter release from type-Is terminals.

In this study we only quantified the output of each MN on a single muscle fiber, but MN6/7-Ib innervates another muscle fiber and MNSNb/d-Is innervates seven other muscle fibers [18]. As neurotransmitter output on the other fibers is similar (data not shown), and terminals from the same axon all fire with the same frequency and duty cycle, we can estimate the summed glutamate for each MN. The glutamate output from all eight MNSNb/d-Is terminals is still less than the output from the two MN6/7-Ib terminals (60% less; Figure 7C arrows), and total ATP consumption would be considerably less (Figure 7D arrows).

Discussion

We report the relationship between P_AZ and energy efficiency in terminals of two glutamatergic MNs innervating the same target muscle fiber. P_AZ was three times higher in one of the terminals and this terminal was also twice as efficient, indicating that a high probability release site is more energy efficient. Given that the brain’s energy demands are high [29], selection away from low P_AZ synapses because of their low efficiency might be expected to favor the adoption of a uniform high P_AZ design. However, we found that P_AZ values in situ fell short of the high P_AZ values predicted for optimal energy efficiency. Terminals with the lowest P_AZ and lowest energy efficiency depressed the least at endogenous release rates and performed most of the work during fictive locomotion. Our interpretation of these data is that selection away from energy inefficiency has favored high P_AZ but that increased P_AZ is held in check by selection away from an inability to sustain release.

Energy efficiency optimization as a selective pressure for high P_AZ

Selection away from energy inefficiency is thought to have influenced the size and other properties of synapses [42, 43], and in turn limited neural computational power [5, 44]. Here we suggest that selection away from energy inefficiency has selected for high P_AZ synapses, and we have demonstrated a positive correlation between P_AZ and energy efficiency under physiological conditions. High P_AZ can result from high Ca²⁺ entry/AZ, sensitivity (S) or cooperativity (n_H) of release [15, 36]. At first glance, these parameters seem to have qualitatively different influences on energy efficiency. High S or n_H lead to higher P_AZ, and, in turn, higher P_AZ leads to higher energy efficiency. More Ca²⁺ entry/AZ also results in higher P_AZ, but simulation of how changes in Ca²⁺ entry/AZ affect energy efficiency produced a curve with a distinct optimum. Further analysis revealed that energy efficiency would be optimized if P_AZ were elevated to 0.76 by an increase in Ca²⁺ entry/AZ. Taken together these simulation results show that a nerve terminal needs a relatively high P_AZ for optimal energy efficiency, but that efficiency diminishes at the highest P_AZ values because the purchase of more Ca²⁺ yields no more neurotransmitter release in return.

Trade-offs between presynaptic energy efficiency and function determine P_AZ

Trade-offs between energy efficiency and function have been observed previously [42, 45-47]. For relay synapses such as the NMJ, the trade-off appears to be between presynaptic energy efficiency and the capacity to sustain neurotransmitter release. Some mammalian central synapses show a capacity for sustained release and are exclusively low P_AZ synapses [48-51]. Here, we find that low P_AZ terminals depressed less than high P_AZ terminals, consistent with previous studies showing that low P_AZ synapses are likely to facilitate, whereas high P_AZ synapses are likely to depress [3]. Type-Ib terminals with an endogenous firing rate of about 20 Hz offset the risk of depletion with a low P_AZ, but this measure is attended by low efficiency. The selection potential for low P_AZ becomes apparent when, in combination with large N_AZ, it confers a capacity to sustain high levels of release, which may translate to sustaining organismal locomotion without fatigue.

Presynaptic energy consumption in the context of postsynaptic energy consumption

While there is a clear rationale for selection against presynaptic design unable to sustain release at a NMJ, the rationale for selection against presynaptic energy inefficiency at a NMJ assumes that the presynaptic terminal consumes (or once consumed) a non-negligible proportion of the NMJ energy budget. Postsynaptic energy consumption can be estimated from EJC measurements that allow calculation of the amount of charge crossing the postsynaptic plasmamembrane in response to neurotransmitter released during a single presynaptic AP and the amount of ATP then needed to remove those ions (see Supplemental Information). Several assumptions that are difficult to defend have to be made to assess postsynaptic energy consumption (Is: 8.16 × 10⁸, and, Ib: 5.18 × 10⁸ ATP molecules). Yet, even if correct only in order of magnitude, we would conclude that the presynaptic terminal consumes ~1% of the NMJ energy budget. This proportion contrasts starkly with estimates at mammalian central synapses, at which presynaptic energy demands are ~30% of the total synaptic signaling cost [16, 17]. If presynaptic terminals only consume 1% of the NMJ energy budget we suggest that the ability to locomote without fatigue ultimately conferred a greater selection advantage than the energy saved by an efficient terminal, and that as a result low P_AZ release sites persisted at the expense of high P_AZ sites.

Highlights.

• - We estimated the energy efficiency of individual motor nerve terminals in situ.

• - Energy efficiency was calculated as glutamate release relative to ATP hydrolysis.

• - Terminals with high probability neurotransmitter release sites are most efficient.

• - Simulations indicate that most release sites operate well below optimal efficiency.

eTOC blurb.

High probability release sites are not uncommon, but what are their advantages? Lu et al. show that energy efficiency is one of their advantages. However, the probability value at any particular synapse might be seen as a trade-off between energy efficiency and other functional properties such as a capacity for sustained release.

Appendix I.

An analytical model that simulates the influence of aspects of presynaptic morphology and function on presynaptic energy efficiency.

Part A: The relationship between P_AZ and Ca²⁺ influx per active zone (Ca)

P_AZ is defined as the average likelihood of a single quantum of neurotransmitter being released per active zone (AZ) in response to a presynaptic action potential:

$${\text{P}}_{\text{AZ}}=\frac{\text{QC}}{{\text{N}}_{\text{AZ}}}$$ (A. 1)

where QC = total number of synaptic vesicles (SVs) exocytosed;

N_AZ = total number of AZs

Numerous studies have demonstrated that the dependence of neurotransmitter release on Ca²⁺ influx can be described by the Hill equation. The relationship between QC and Ca is described as the following equation:

$$\text{QC}=\frac{{\mathrm{N}}_{\text{AZ}}}{1+{\text{S}}^{-1}\times {\text{Ca}}^{-{\text{n}}_{\text{H}}}}$$ (A. 2)

where S is the sensitivity to release neurotransmitter and n_H is the Hill coefficient [15, 36]. Combining equations (A.1) and (A.2), yields:

$${\mathrm{P}}_{\text{AZ}}=\frac{1}{1+{\text{S}}^{-1}\times {\text{Ca}}^{-{\text{n}}_{\text{H}}}}$$ (A. 3)

From equation (A.3), it is evident that P_AZ is a compound parameter, determined by Ca, S and n_H, but independent of N_AZ. Plotting P_AZ against Ca, S or n_H (Figure 5A), shows that greater Ca²⁺ influx, S or n_H gives rise to higher P_AZ. For both terminals, physiological values of n_H were fixed at 3.

Part B: The relationship between Ca, and the relationship between P_AZ, and energy efficiency: a model of presynaptic energetics

Presynaptic energy efficiency is defined here as:

$$\text{E.E.}=\frac{\text{Glu}}{{\text{E}}_{\text{total}}}$$ (A. 4)

$$\text{Glu}=\text{QC}\times \epsilon$$ (A. 5)

where E.E. represents energy efficiency; ε = number of glutamate molecules in a SV, and ε is proportional to SV volume.

Together with equation (A.2), the number of glutamate molecules released is seen as a function of Ca, S, N_AZ, vesicle size and n_H (Figure 5B; Figure S4A).

Initially, total energy demand can be approximated as the sum of the cost of SV release and recycling, and the cost on Ca²⁺ extrusion [17, 29]:

$${\mathrm{E}}_{\text{total}}={\mathrm{E}}_{\text{Glu}}+{\mathrm{E}}_{{\text{Ca}}^{2+}}$$ (A. 6)

where E_Glu represents the cost of releasing glutamate, which combines the costs of loading SVs with glutamate and recycling SVs. E_Ca²⁺ represents the cost of Ca²⁺ extrusion.

As Na⁺ entry also contributes to action potential propagation in the nerve terminal, it is necessary to involve the cost of Na⁺ extrusion in the nerve terminal. Taken together, equation (A.6) is now rewritten as:

$${\text{E}}_{\text{total}}={\text{E}}_{\text{Glu}}+{\text{E}}_{{\text{Ca}}^{2+}}+{\text{E}}_{{\text{Na}}^{+}}$$ (A. 7)

where E_Na⁺ is the cost of the Na⁺ load.

Each term in equation (A.7) is dealt with separately below.

E_Glu is calculated as the product of the number of SVs released (QC) and the cost per SV (ATP_SV):

$${\text{E}}_{\text{Glu}}=\text{QC}\times {\text{ATP}}_{\text{SV}}=\text{QC}\times (\beta +\epsilon \times \theta )$$ (A. 8)

where β = number of ATP molecules spent to recycle a SV;

ε = number of glutamate molecules in a SV, proportional to SV size;

θ = number of ATP molecules needed to load a glutamate molecule

β = 410.5 and θ = 2.67 [29]. For the following simulation, these parameters are treated as constants while SV size changes along with Ca, N_AZ, S and n_H.

E_Ca²⁺ is calculated as the product of the total number of Ca²⁺ ions that enter the terminal (Ca_total, the same number that must be extruded) and the cost to extrude each Ca²⁺ (1 ATP molecule):

$${\text{E}}_{{\text{Ca}}^{2+}}={\text{Ca}}_{\text{total}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}=\text{Ca}\times {\text{N}}_{\text{AZ}}\times 1\text{ATP}$$ (A. 9)

Therefore, E_Ca²⁺ is a function of Ca and N_AZ.

Similarly, E_Na⁺ is calculated as:

$${\text{E}}_{{\text{Na}}^{+}}={\text{Na}}_{\text{total}}\times \frac{1}{3}\times 1\text{ATP}$$ (A. 10)

Na_total (Na⁺ load) can be calculated using the following formula:

$${\text{Na}}_{\text{total}}=\text{C}\times \text{V}\times \text{O}$$ (A. 11)

where C represents the capacitance and can be calculated as the product of capacitance per surface area (C_m= 1 μF/μm²) and surface area of the terminal; V represents the voltage change of an AP; O represents overlap factor.

We measured the surface area of type-Ib and type-Is terminals (Table S1). The overlap factor refers to the ratio of the total integrated Na⁺ current during the action potential to the minimum charge transfer necessary for action potential depolarization [32]. The overlap factor for honeybee neurons is 3.05 [34] and we assume that Drosophila nerve terminals have the same overlap factor. Taken together, we find that, under physiological conditions, E_Na⁺ for type-Ib terminals is 3.60 × 10⁶ ATP molecules whileE_Na⁺ for type-Is nerve terminals is 1.48 × 10⁶ molecules.

Simulating the effect of changing variables in equation (A.11), it is clear that [like total Ca²⁺ influx (Figure S4B)] the Na⁺ load changes linearly with alterations in these parameters (Figure S4C). For subsequent analysis it is assumed that E_Na⁺ is independent of other parameters such as Ca, S, N_AZ, vesicle size and n_H. In other words, partial derivatives of variables such as Ca, E_Na⁺ will be constant.

Combining equations (A.5), (A.7), (A.8), (A.9) and (A.10) yields:

$$\text{E.E.}=\frac{\text{QC}\times \epsilon }{\text{QC}\times (\beta +\epsilon \times \theta )+\text{Ca}\times {\text{N}}_{\text{AZ}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}+{\text{E}}_{{\text{Na}}^{+}}}$$ (A. 12)

Based on equation (A.12), we can conclude that energy efficiency is a function of Ca as well as other parameters such as S, N_AZ, vesicle size and n_H (Figure 5C; Figure S4D). Parameters such as surface area and capacitance per surface area also affect energy efficiency through increase of E_Na⁺ [equation (A.11)], but unlike parameters discussed above, such parameters should be as small as possible to reduce the total energy demand. In other words, when E_Na⁺ is negligible, it is beneficial for increasing energy efficiency of a nerve terminal.

Furthermore, based on equations (A.1) and (A.3), equation (A.12) can be rewritten as:

$$\text{E.E.}=\frac{{\mathrm{P}}_{\text{AZ}}\times \epsilon }{{\text{P}}_{\text{AZ}}\times (\beta +\epsilon \times \theta )+{((\frac{1}{{\mathrm{P}}_{\text{AZ}}}-1)\times \text{S})}^{{\mathrm{n}}_{\text{H}}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}+\frac{{\text{E}}_{{\text{Na}}^{+}}}{{\text{N}}_{\text{AZ}}}}$$ (A. 13)

From the simulation data (Figure 5D), maximal energy efficiency is achieved when P_AZ is 0.76 in type-Is nerve terminals (physiological P_AZ in Is nerve terminals is 0.33); P_AZ is 0.73 in type-Ib nerve terminals (physiological P_AZ in Ib nerve terminals is 0.11). Therefore, the simulation indicates that a relatively high P_AZ confers high energy efficiency.

Part C, Proof: Optimization of energy efficiency drives towards high P_AZ

Ca is a unique variable since it is subject to changes within milliseconds [52]. In addition, the bell shaped plot of energy efficiency versus Ca is unique relative to the simulated dependency on other parameters in Figure 5C. For these reasons we elaborated our model to determine whether a relatively high P_AZ is required for maximization of energy efficiency where Ca is the only independent variable.

Taking the derivative of equation (A.12) where Ca is the only independent variable (partial derivative):

$$\begin{array}{cc}\hfill & \frac{\text{d}\left(\text{E.E.}\right)}{\text{d}\left(\text{Ca}\right)}\hfill \\ \hfill & =-\left({\text{N}}_{\text{AZ}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}\times \frac{1}{\text{QC}}-{\text{N}}_{\text{AZ}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}\times \frac{\text{Ca}}{{\text{QC}}^2}\frac{\mathrm{d}\left(\text{QC}\right)}{\mathrm{d}\left(\text{Ca}\right)}-{\mathrm{E}}_{{\text{Na}}^{+}}\times \frac{1}{{\text{QC}}^2}\frac{\mathrm{d}\left(\text{QC}\right)}{\mathrm{d}\left(\text{Ca}\right)}\right)\hfill \\ \hfill & \times \frac{\epsilon }{{(\beta +\epsilon \times \theta +\text{Ca}\times {\text{N}}_{\text{AZ}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}÷\text{QC}+{\text{E}}_{{\text{Na}}^{+}}÷\text{QC})}^2}\hfill \end{array}$$ (A. 14)

When $\frac{\mathrm{d}\left(\text{E.E.}\right)}{\mathrm{d}\left(\mathrm{Ca}\right)}=0$, energy efficiency could be maximal, which is :

$$\frac{\mathrm{d}\left(\text{QC}\right)}{\text{d}\left(\text{Ca}\right)}=\frac{\text{QC}}{\text{Ca}+\frac{{\text{E}}_{{\text{Na}}^{+}}}{{\text{N}}_{\text{AZ}}\times {\text{ATP}}_{{\text{Ca}}^{2+}}}}$$ (A. 15)

A general solution for equation (A.15) is:

$$\text{QC}=\phi \times \text{Ca}+\omega$$ (A. 16)

with φ and ω as non-zero constants.

The range of Ca for optimal energy efficiency can be further refined as follows. From equation (A.2), if Z = 1 + S⁻¹ × Ca^−n_H then:

$$\begin{array}{c}\hfill \frac{\mathrm{d}\left(\text{QC}\right)}{\mathrm{d}\left(\text{Ca}\right)}={\text{N}}_{\text{AZ}}\times \left(\frac{-{\text{Z}}^{\prime }}{{\text{Z}}^2}\right)=-\frac{{\text{N}}_{\text{AZ}}}{{\text{Z}}^2}\times {\text{S}}^{-1}\times (-{\mathrm{n}}_{\text{H}})\times {\text{Ca}}^{-{\text{n}}_{\text{H}}-1}\hfill \\ \hfill =\frac{{\mathrm{N}}_{\text{AZ}}\times {\mathrm{S}}^{-1}\times {\mathrm{n}}_{\text{H}}}{{\text{Z}}^2}\times {\text{Ca}}^{-{\text{n}}_{\text{H}}-1}\hfill \end{array}$$ (A. 17)

Combining equation (A.17) with equations (A.2) and (A.14), we get:

$${\text{S}}^{-1}\times ({\text{n}}_{\text{H}}-1)\times \text{Ca}+{\text{E}}_{{\text{Na}}^{+}}\times {\text{S}}^{-1}\times {\text{n}}_{\text{H}}÷({\text{N}}_{\text{AZ}}\times 1\text{ATP})={\text{Ca}}^{{\text{n}}_{{\text{H}}^{+1}}}$$ (A. 18)

After rearranging equation (A.18);

$${\text{Ca}}^{{\text{n}}_{{\text{H}}^{+1}}}-{\text{S}}^{-1}\times ({\text{n}}_{\text{H}}-1)\times \text{Ca}={\text{E}}_{{\text{Na}}^{+}}\times {\text{S}}^{-1}\times {\text{n}}_{\text{H}}÷({\text{N}}_{\text{AZ}}\times 1\text{ATP})$$

As the term on the right must be a positive value;

$${\text{Ca}}^{{\text{n}}_{{\text{H}}^{+1}}}-{\text{S}}^{-1}\times ({\text{n}}_{\text{H}}-1)\times \text{Ca}>0$$

After rearranging:

$$\text{Ca}>{\left(\frac{{\text{n}}_{\text{H}}-1}{\text{S}}\right)}^{\left(\frac{1}{{\text{n}}_{\text{H}}}\right)}$$ (A. 19)

This indicates that Ca needs to exceed a certain value for optimal energy efficiency. According to equation (A.3), high Ca gives rise to high P_AZ. Similarly, optimization of energy efficiency requires high S and n_H (Figure 5C); and high S or n_H yields high P_AZ (Figure 5A). To conclude, a high P_AZ value exceeding a certain value is required for maximization of energy efficiency. The actual value is determined as follows:

In a special case, if E_Na⁺ is negligible compared with E_Glu and E_Ca²⁺, then equation (18) can be rewritten as follows:

$${\text{S}}^{-1}\times ({\text{n}}_{\text{H}}-1)\times \text{Ca}={\text{Ca}}^{{\text{n}}_{{\text{H}}^{+1}}}$$ (A. 20)

After rearranging:

$$\text{Ca}={\left(\frac{{\text{n}}_{\text{H}}-1}{\text{S}}\right)}^{\left(\frac{1}{{\text{n}}_{\text{H}}}\right)}$$ (A. 21)

Therefore, there is a single solution for Ca where energy efficiency is maximized, and from our data we determined the values to be: Is, 18,100 ions; Ib: 14,650 ions.

Combining equation (A.21) and equation (A.3), P_AZ can be calculated under conditions where E_Na⁺ is negligible, and energy efficiency is optimal:

$${\text{P}}_{\text{AZ}}=\frac{1}{1+{\text{S}}^{-1}\times {\text{Ca}}^{-{\text{n}}_{\text{H}}}}=\frac{1}{1+{\text{S}}^{-1}\times {\left({\left(\frac{{\text{n}}_{{\text{H}}^{-1}}}{\text{S}}\right)}^{\left(\frac{1}{{\text{n}}_{\text{H}}}\right)}\right)}^{-{\text{n}}_{\text{H}}}}=\frac{1}{1+{\text{S}}^{-1}\times {\left(\frac{{\text{n}}_{{\text{H}}^{-1}}}{\text{S}}\right)}^{-1}}=\frac{{\text{n}}_{\text{H}}-1}{{\text{n}}_{\text{H}}}$$ (A. 22)

As n_H = 3 in both type-Is and type-Ib nerve terminals, from equation (A.22), P_AZ=0.67. If E_Na⁺ is not negligible, efficiency maximization can only be reached when P_AZ>0.67, consistent with the plots in figure 5D.

Therefore, our analysis suggests that optimization of energy efficiency drives towards high P_AZ, a conclusion that should be generalizable to all presynaptic terminals where neurotransmitter release has a steep dependence on Ca²⁺ entry.