article: A biophysical model of receptor potential generation in the male moth olfactory receptor neuron is presented. It takes into account all pre-effector processes—the translocation of pheromone molecules from air to sensillum lymph, their deactivation and interaction with the receptors, and the G-protein and effector enzyme activation—and focuses on the main post-effector processes. These processes involve the production and degradation of second messengers (IP3 and DAG), the opening and closing of a series of ionic channels (IP3-gated Ca2+ channel, DAG-gated cationic channel, Ca2+-gated Cl− channel, and Ca2+- and voltage-gated K+ channel), and Ca2+ extrusion mechanisms. The whole network is regulated by modulators (protein kinase C and Ca2+-calmodulin) that exert feedback inhibition on the effector and channels. The evolution in time of these linked chemical species and currents and the resulting membrane potentials in response to single pulse stimulation of various intensities were simulated. The unknown parameter values were fitted by comparison to the amplitude and temporal characteristics (rising and falling times) of the experimentally measured receptor potential at various pheromone doses. The model obtained captures the main features of the dose–response curves: the wide dynamic range of six decades with the same amplitudes as the experimental data, the short rising time, and the long falling time. It also reproduces the second messenger kinetics. It suggests that the two main types of depolarizing ionic channels play different roles at low and high pheromone concentrations; the DAG-gated cationic channel plays the major role for depolarization at low concentrations, and the Ca2+-gated Cl− channel plays the major role for depolarization at middle and high concentrations. Several testable predictions are proposed, and future developments are discussed.
  Olfactory receptor neurons (ORNs) are essential for the recognition of odor molecules. In vertebrates this recognition is performed by several hundreds olfactory receptor proteins (ORs) borne by the ORN plasma membrane, each ORN expressing a single type of receptor [1]. In insects a smaller number of ORs have been identified [2]–[4]. In male moths, ORNs housed in antennal sensilla trichodea (Figure 1) can detect female-released sexual pheromone with exquisite sensitivity, specificity and efficiency [5]. These ORNs have been the subject of intensive studies during the last fifty years using molecular, radiochemical, pharmacological, electrophysiological, calcium imaging, behavioral and modeling techniques (reviewed in [6]–[8]). The latter contribution has been significant and ORNs have experienced a rich history of modeling, since reports that a male moth can find a pheromone releasing female from several miles away [9], [10] and that a single pheromone molecule is sufficient to elicit an action potential in the moth sensory neurons [11]. The system has been modeled at the level of behavior [12], [13], at the level of antenna as biomechanical filter for odor molecules [14]–[16], at the level of electrical circuits that give rise to action potentials (e. g. [17], [18]), and at the level of biochemical processes that lead to neuronal activation [19]–[22]. The most detailed model yet published is that of Kaissling [23] which attempted to account for the production of the “receptor potential” through the interactions of a process generally referred to as “perireceptor events”. In fact this process consists of a biochemical network of the carrier proteins (pheromone binding proteins, PBPs), ORs and odor degrading enzymes [8], [24] which occupy a common space surrounding the outer dendritic receptive membrane of ORNs. A reasonably complete picture of the transduction processes emerges from these studies, although some of the molecular and ionic channel mechanisms underlying the transduction process still remain elusive. After perireceptor processes, the pheromone bound ORs are believed to interact with a G-protein which in turn activates the effector enzyme phospholipase C-β (PLCβ) [25]. This enzyme catalyzes the production of second messenger molecules, inositol 1,4, 5-triphosphate (IP3) and diacylglycerol (DAG), which trigger the opening of a cascade of various ionic channels. The resulting ionic currents generate the receptor potential (RP) which passively propagates to the ORN soma and axon where it generates action potentials. Recently, this classical metabotropic mechanism has been challenged in insect ORNs and a direct coupling of the OR to a cationic channel has been proposed in parallel or in replacement [26]–[28]. These new developments are important from molecular, physiological and evolutionary points of view. The full description of such a complex signaling network, involving both feedforward and feedback processes, is a daunting task. Modeling can contribute to this description by integrating various effects and displaying quantitatively what results from the interplay of all molecular actors. The knowledge accumulated on the pheromonal ORN is sufficient to start building a model of its transduction cascade, and to test whether it can effectively link together some of the known facts and suggest new experiments. Thus, the first aim of our investigation was to develop a qualitative model of the pheromone transduction cascade integrating the known molecular and ionic mechanisms. The second aim was to translate these mechanisms, wherever possible, into a set of differential equations and to determine the quantitative values of their parameters. We made a systematic search of known values and determined the unknown values by fitting the model output to the properties of the experimentally measured RP. These properties were systematically determined in response to “square” pulses of pheromone of constant duration at several intensities [23], [29], [30]. They offer the most precise data on the transduction cascade available so far. These responses are characterized by a rapid rising phase, a slow falling phase, especially at high concentrations, and an extremely wide dynamic range of about 6 decades from threshold to saturation. In insects, most modeling efforts have been dedicated to the perireceptor and receptor processes in moth pheromone sensilla [23], [31]–[35]. Although interactions of ORs, G-proteins and effectors have been recently studied [36], no model has been proposed yet for post-effector processes in insects. The model we present here focuses on these processes in male moth pheromone ORNs and takes advantage of the modeling studies available on olfactory transduction in vertebrates [37]–[42]. Beyond fitting adequately the experimental dose-response curves we addressed the following related questions. What are the functional roles of the various currents? In particular, what could be the respective roles of the direct (ionotropic) and indirect (metabotropic) gating mechanisms of the initial cationic current? What are the mechanisms behind the characteristics of the concentration-response curves (broad dynamic range, short rising time and long falling time)? What are the processes that contribute most to the amplifying function of the cascade?
  Based on experimental results obtained in moth ORNs, complemented when necessary with data coming from other animal species and some reasonable assumptions, we developed a global qualitative model of pheromone transduction. A schematic diagram of the model is shown in Figures 2 and 3. This model is summarized in this section. Some of the experimental results and the main assumptions (denoted A to F) on which it rests, are briefly mentioned and listed in Table 1. Complementary justifications, references and comments are provided in the Discussion section. The qualitative description above, although indispensable, is not sufficient to gain a proper understanding of pheromone transduction. We must now turn to a formal description of the various steps involved. Note that abbreviations in roman (e. g. G*, E*, IP3 etc.) denote chemical species, whereas the corresponding symbols in italics (e. g. G*, E*, IP3 etc.) denote concentrations. The post-effector model described above includes 65 values: the initial values of the concentrations of modulators and the membrane potential (10 values given in Table 2) and 55 parameter values. These parameters fall in 3 different categories: (1) the rate constants of the reactions involving the modulators (10 values); (2) the characteristics of the 5 main currents with 4 to 7 parameters per current (28 values); (3) the 17 parameters describing the dendritic morphology (surface and volume), the electrical properties of the dendritic membrane and the conversion factors from currents to ionic fluxes. However, from a practical point of view, the most important distinction is between parameters with a known value (27) and those which were unknown (38). We considered as known, and used without change, any parameter value determined in a moth ORN, especially in Antheraea polyphemus. If a parameter value was known in a non-moth species, especially in the frog ([40] and references therein), or as a ubiquitous component presents in any cell, we used it as a starting value. The fixed parameters based on experimental determinations or calculated from experimental data are given in Table 3. The 38 fitted parameters are given in Table 4 for modulators and Table 5 for ionic currents. Their final values were obtained as explained below and in the Methods section by comparison with experimentally known response characteristics. Given the initial concentrations (Table 2) and the fixed parameter values (Table 3), computer simulations of the model were carried out. We searched for values of the unknown parameters, listed in Tables 4 and 5, yielding responses in accordance with experimental observations. Three sources of information were used. First, parameter values must remain in their physiological range. Second, the kinetic features of the second messengers and ionic currents must reproduce qualitatively the experimental observations. Third, the time evolution of the SP at various pheromone concentrations must agree quantitatively with the in vivo measurements of the SP performed in A. polyphemus [23], [30]. Following a 2-s square pheromone pulse, the SP grows to a maximum then returns progressively to zero (Figure 7H). This simple kinetics can be summarized with three numbers (Figure 8D), its height, its rising time, measured by the time it takes to reach half maximum, and its falling time, measured by the time to fall from the end of the stimulation to half-maximum. These three quantities depend on the dose of pheromone delivered to the system, measured either as a concentration in air in micromoles per liter or better as an uptake in µM per second. Zack [30] systematically determined the amplitude, rising time and falling time of the SP at various uptakes, from threshold to saturation. On this range, the amplitude increases, the rising time decreases 10 times and the falling time increases 10 times. These three dose-response functions (Figure 8, dotted lines) were our main criteria for the fine tuning of the parameter values because they are the only ORN responses experimentally measured with precision in vivo. We modified the unknown constants in the model to fit these experimental curves by using the same stimulation conditions as used by Zack [30]. We found a set of parameter values in agreement with known facts and giving good fits (see Methods), provided the mechanism of Ca2+ extrusion depends on membrane potential. With the pump mechanism (PMCA), the falling time could be fitted only on a restricted range; for example if correct at low uptakes, it was much too large at higher uptakes. Therefore, all following results are given for the potential-dependent extrusion only (NCX). The values of the fitted parameters are given in Tables 4 and 5. The corresponding simulated dose-response curves are illustrated in Figure 8 (solid red lines) for the SP and in Figure 9 for the other variables (chemical species and currents). With the parameter values at hand the main properties of the model can be described. Some aspects require specific attention: the kinetics of currents and chemical species, their relative importance, their relative responsiveness and the explanation of the dose-response curves. The sensitivity of the system to the parameters controlling each biochemical and electrical step was analyzed as explained in the Methods section. The effects produced on the SP responses by a change in the value of a single parameter at a time were examined. The main results of this analysis can be summarized as follows (see Text S1). (i) The sensitivity of SP to the parameters depends on the characteristic (height, rising or falling time) and on the dose. (ii) Each parameter has its greatest influence on one of the characteristics (Tables 4 and 5, rightmost column). The most influenced characteristic is usually the falling time (63%). (iv) The 12 most influential parameters are Kis, GMx, Kmcat, Kk, Kicat, Kpd, ncat, nCl, GMCa, nx, GMcat and KiCl. They are thus the best determined parameters. The 7 least sensitive parameters are sM, kcc2, niCa, iMCa, Ak, KmCl and iMcat. Tables in Text S1 list the parameters which most and least influence each characteristic. Finally, we determined the importance of the feedback controls on second-messenger production, main ionic channels and calcium extrusion, by removing them one at a time (see Text S1). We compared the action of PKC* on PLC for different types of activation and found that it is inhibitory when activated by Ca2+ only, but not when activated by DAG only.
  In the model the DAG-gated cationic current is the first ionic current in the transduction cascade (we attribute a different function to the parallel IP3-gated current, see below). A wide-spread assumption (our assumption A) is that this channel is indirectly coupled to the ORs via metabotropic pathway involving G-proteins and second messengers. This is based on the fact that like vertebrate ORs, insect ORs belong to the G-protein coupled receptor (GPCR) superfamily characterized by the canonical 7 transmembrane topology of its members [84]. However, recent bioinformatics and experimental investigations have revealed that the membrane topology of at least some Drosophila ORs differs from other GPCRs with an intracellular N-terminus and an extracellular C-terminus [58], [85]. The structural distinction between insect and mammalian ORs, with different membrane topology, put into question the coupling of insect ORs to G proteins [86], [87]. Recently, two studies indicated that, in expression systems, odorants can activate insect ORs and generate sensory currents independently of known G protein-coupled second messenger pathways, through a so-called ionotropic pathway [28], which may involve the cationic channel OR83b [27]. Proteins ortholog to OR83b are also expressed in moth pheromonal ORNs [56], which indicates that the ionotropic pathway may also be present in this neuron type. However, several experimental observations made both in vivo and in vitro provide strong support for the involvement of Gq proteins and PLC in the moth pheromone transduction cascade. First, Gq proteins are present in moth antennae [88], [89] and these proteins were localized in the outer dendrite of ORNs [54]. Second, the G protein activator, NaF, activates the firing activity of ORNs both in vivo [89] and in vitro (Lucas, unpublished results), demonstrating that Gq proteins are functional in moth pheromone-responding ORNs. Third, Xenopus oocytes transfected with B. mori conventional pheromone receptors and without OR83b do not respond to pheromone stimuli unless they are co-transfected with Gq proteins [90]. Fourth, pheromone stimulation activates the PLC activity as indicated by IP3 production [25], [49], [50], [52]. Two other arguments, obtained from non-pheromonal ORNs, strengthen the observations above in pheromonal ORNs. First, in addition to the fast and transient ionotropic pathway, at low concentration odorants also activate G proteins and the production of second messengers in expression systems [27]. This metabotropic current develops after a longer latency and with a slower kinetics but is more sensitive to odorants than the ionotropic pathway. Second, strong genetic evidence supports a role for a Gq- and PLCβ-mediated signaling cascade during olfactory transduction in Drosophila [53]. The results at hand suggest the coexistence of two signaling pathways in both pheromonal and non-pheromonal ORNs, one ionotropic, the other metabotropic. Remarkably, both pathways gate a cationic channel, although it is not known whether the cationic channels of both pathways are the same (OR83b) or not, and if different whether their conductances are the same. The ionotropic signal, not amplified, is rapid and transient, and the metabotropic, amplified, is sensitive and prolonged, the former being more visible at high odorant concentration and the latter at low concentration. So, the relative intensities and timings of the two currents might be significantly different. If these properties are confirmed in future studies, they would suggest that the intensity of the cationic current found in the present model should not be entirely attributed to the DAG-gated channel because part of it comes from the receptor-gated channel. An advantage of the ionotropic pathway, especially at high uptake when the number of activated receptor is the largest and the ionotropic current is expected to become significant, is that its energetic cost for the neuron is lower. According to this view one might expect that the G-protein pathway is more inhibited at high uptake than found in the present model. The suppressed current would be compensated by the energy-saving direct coupling mechanism and the global effect would be the same. At the present time, the relative contribution of the two channels cannot be specified, if only because the reaction rate of R* with OR83b is not known. A second advantage of the OR83b cationic current is its time course. The receptor-gated current is expected to appear first and therefore to trigger an action potential faster than the DAG-gated current. The ionotropic pathway could contribute to explain the high speed of response of ORNs, especially at high pheromone concentration, and the very fast behavioral response of moths, down to 150 ms, both to contact and loss of pheromone filaments during their oriented flight to calling females [91]. The latency of the initial response was not considered in the present study and deserves more attention in the future. It remains uncertain whether the role of the Cl− current is depolarizing or repolarizing, because the intracellular and sensillar concentrations of Cl− are not known. We assumed that this current is depolarizing (assumption C) for three reasons. First, in vertebrates, the Cl− equilibrium potential is more positive than the resting potential [92] and a similar Ca2+-activated Cl− current amplifies ORN depolarization ([93]–[96], see review [97]). Second, insect PBPs present in the mM range provide organic anions to the sensillum lymph [8] which decreases the sensillar Cl− concentration due to the principle of charge neutrality. Third, pheromone responses (SP and action potential firing) recorded in vivo were significantly higher when the sensillar Cl− concentration was lowered from 215 to 18 mM (Lucas, unpublished results). One of the main uncertainties on channels (and other proteins) concerns their exact spatial location, which entails uncertainty on the mechanisms of Ca2+ entry. Because the present model depends only on time, any diffusion or translocation is expressed in temporal (not spatial) terms and reflected in the reaction constants. This simplification has generally no incidence, except for two channel types. The first one relates to IP3-gated channels which have been located in the outer dendritic membrane, based on electrophysiology [45] and immunocytochemistry [54]. This location implies an inward flow of Ca2+ from the sensillum lymph. However, besides or in replacement of this dendritic membrane channel, channels located in the endoplasmic reticulum may be considered. In this case Ca2+ would come from intracellular stores. However this hypothesis is weakened by the fact that no intracellular Ca2+ stores have been found in the outer dendrite. In fact the present model is compatible with both possibilities and cannot discriminate them. A third possibility is discussed in the next paragraph. The second channel for which the spatial localization is important is the repolarizing K+ channel. A priori it can be located either in the outer dendrite or in the inner dendrite and soma region (as we did, our assumption D). The main objection against its outer dendritic location is that the equilibrium potential there is close to 0 mV, so that K+ cannot have a repolarizing function. Against the inner dendritic location one can mention their modulation by Ca2+ [72], [74], [75] because Ca2+ would have to diffuse from the outer to the inner membrane to trigger them which is unlikely due to the poor diffusive ability of this ion. A possible solution to this problem is that IP3-gated channels are in the inner dendrite. The IP3 synthesized in the outer dendrite would have to diffuse to the inner dendrite and soma, which is compatible with the diffusive property of this second messenger, and, there, it would gate the Ca2+ channel, possibly from Ca2+ stores. Ca2+ and voltage would then trigger the repolarizing K+ current with a delay due to IP3 diffusion. This distinguishes the functions of these two different channel types in the model. IP3-gated channels let Ca2+ ions flow into the cytoplasm, whereas DAG-gated channels also depolarize the ORN at low uptakes (see below). In fact it can be shown in the model that the IP3 pathway is unnecessary for the depolarization. On the basis of available experimental data we have included a feedback regulation of IP3 channels (via CaCaM, our assumption B), but not of Cl− channels (via PKC*). However, in the latter case, we studied the hypothesis of a feedback inhibition via PKC* [63]. We found that both regulations are not essential (Figure 5B). For reducing the computation time we ignored the known feedback regulations on the pre-effector steps [66], [98], [99]. They are likely important for repetitive pheromone pulses that occur in natural conditions. In single-pulse condition, as in this study, this simplification presents no inconvenience provided the inhibitory regulation of, say, the receptors is not much faster than that of the effector enzymes. All actors involved in the model have been shown to exist in the ORN membrane. The only exception is the Ca2+ extrusion pathway which has not yet been described in moth ORNs, although it plays an essential role in the falling phase of SP. Because of the poor intracellular diffusion of Ca2+ ions, it is likely that these mechanisms are located in the outer dendrite. We have shown that only a voltage-dependent extrusion is compatible with the experimental data, which strengthens the hypothesis of a Na+-Ca2+ exchange (NCX), without ruling out the presence of an ATPase pump (PMCA). We found that the NCX pump needs no negative feedback control and that its reversal potential is Ex≈−17 mV. Knowing the reversal potential of Ca2+ (ECa≈140 mV) and the relationship between Ex, ECa and ENa, see eq. (46) in Methods, this value of Ex implies ENa≈88 mV, which is a reasonable value. Further experimental and theoretical investigations are needed to clarify the Ca2+ extrusion mechanisms. For simplicity we have not included the Ca2+-gated Ca2+-permeable cationic channels described in M. sexta ORNs [100] because with Ca2+ entering the ORN and at the same time gating the channel, the control of the falling phase of the SP is made extremely difficult. Moreover, Ca2+-activated cationic channel could not be found in S. littoralis ORNs [61]. Perhaps qualitative differences exist in ion channel expression in ORNs across moth species providing different voltage- and time-dependent down-regulation mechanisms. Finally, when stimulated repetitively or for a longer time the ORN adapts and its response characteristics are different from those analyzed here [30]. Adaptation is beyond the scope of the present work as it may involve reactions other than those built in the present model. With the parameter values given in Tables 4 and 5 the model obtained accounts for several experimentally known properties of pheromone transduction in moths: The predicted SP reaches a maximum value and follows a time course, both in its rising and falling phases, that quantitatively agrees with the measured characteristics (depolarization, rising time and falling time) of the SP as a function of pheromone uptake (Figure 8). The model also accounts quantitatively for the transient course of IP3 production, as described in stop-flow experiments [25], with a very rapid increase followed by a quick decline at middle and high uptakes (Figure 7C). It follows that the kinetics of DAG and the currents gated by both IP3 and DAG must also be transient in this range of uptakes, which is the case in the model (Figure 7E). The transient course of second messengers at middle and high uptakes results from a strong feedback inhibition. At low uptake the inhibition is much weaker and thus the course is not transient. On the contrary, the intracellular Ca2+ concentration (Figure 7D) and the Ca2+-gated Cl− current (Figure 7F) become sustained in the range of uptakes where the second messengers and their gated currents become transient: they increase at a slow rate then decrease gradually. These features are in agreement with experimental findings. Moreover, this Cl− current (ICl) is the major component of the depolarizing currents at middle and high uptakes, which qualitatively agrees with experimental data in the frog [101]. The phase portraits E*-SP (Figure 13) show that at most uptakes except the highest, the maximum of E* and SP are reached at the same time and the trajectories of the rising and falling phases are close. This is equivalent to the fact, described by [23] that the concentration R1* giving a certain value of SP during the rising phase and the concentration R2* giving the same value of SP during the falling phase are equal, except at high uptake. The experimental variability of SP is the highest at high uptake [30]. This can be interpreted, in the framework of our model, because we have shown that relatively small changes in some parameter values in different ORNs can lead to relatively large SP changes in this range of uptakes (Table 5). However, several limitations affect the determination of the parameter values. Besides qualitative limitation regarding the completeness of the model, two other kinds of limitations must be taken into account. First, the model is based on ordinary differential equations depending only on time which entails two limitations: space is neglected, as discussed above, and very small concentrations are not adequately described because all chemical species must be in sufficiently large number to be considered as continuous variables. The total number of activated receptors per ORN is 30 when the uptake is 10−2. 5 µM/s [36]. The bottleneck of the whole cascade being the receptors, all uptakes greater than this can be considered as adequately described by ordinary differential equations. Presently, for uptakes less than 10−2. 5 µM/s, only mean values are obtained. A complete description in this range will require a stochastic approach, at least at the receptor level. Second, supposing the model qualitatively valid for the stimulus used, the problem of the precision of the parameter estimation arises. The sensitivity analysis we performed permitted a classification in two categories (see tables in Text S1): the parameters whose modification changes significantly the SP response (right columns), which were therefore estimated with good precision, and those which do not influence much the SP (left columns), which are less well estimated. For example the equilibrium potentials need not be precisely known. This analysis one parameter at a time gives only a partial view because some parameters are linked. For example, the concentration E* and the maximum synthetic rate of sM of activated effector appear as a product, see equations (29) and (30) in Methods, so they cannot be known independently. The value given for sM is valid under the assumption that E0* = 0. 136 µM [36]. However, it must be realized that the problem of parameter estimation is extremely constrained because the range of acceptable values of most parameters is restricted and because of the many feedback reactions. As a consequence the important parameters are not the same at all uptakes, especially for falling time. For this reason, finding a solution that works at all uptakes proved very difficult and suggests that a significantly different set of parameters in agreement with the experimental data available will not be easy to find. The present model helps to interpret global properties of the ORN, especially its performance at efficiently encoding the stimulus. The response characteristics (amplitude, rising and falling times) of the SP in the pheromonal ORN present three remarkable properties: a wide dynamic range, a short rising time which decreases with pheromone concentration, and a long falling time which increases with concentration. The proposed model explains all these features. They can be analyzed from two different points of view: the relative contribution of the pre- and post-effector steps and the mechanisms by which the post-effector steps contribute to the observed features. (1) First of all, the post-effector cascade contributes considerably to the large dynamic range which extends over about 6 decades from 10−4. 75 to 101. 5 µM/s. Indeed, the dynamic range of the effector response is 3. 25 decades only. The post-effector extension from 3. 25 to 6 decades results from the collaboration of the two main currents, the DAG-gated cationic current and the Ca2+-gated Cl− current, which present a large difference in responsiveness. The cationic current has an (uninhibited) efficient concentration EC50 of 0. 01 µM of DAG, which is reached at a pheromone uptake of 10−4. 25 µM/s (Figure 5A), whereas the Cl− current has an EC50 of 81 µM of Ca2+, which would be reached at an uptake of 50 µM/s (Figure 5B), a value not actually reachable because the perireception system saturates at ≈30 µM/s. This means that the cationic current is most active at low uptakes whereas the Cl− current is most active at high uptakes. They complement one another and contribute, by the separation of their EC50s, to widen the dynamic range of the ORN. (2) The fast rising time of the SP results from the cationic current because at all uptakes it is faster than the Cl− current (compare the dashed and solid curves of insets in Figure 10). (3) The long falling phase of the SP is also explained by the two main currents. It reflects primarily the time course of the cationic current at low uptakes and the time course of the Cl− current (which closely follows the time variation of intracellular Ca2+) at high uptakes. The long persistence of Ca2+ at high uptakes suggests that Ca2+ extrusion is not fast enough. However, as shown by the direct stimulation of receptors in the model, the bottleneck which limits the speed of rise and fall of SP is in the extracellular processes. Although involved and with many steps, the intracellular processes, as modeled here, are very fast, which confirms Kaissling' s analyses [23]. An important function of the cascade is to transform a weak initial signal (pheromone binding to OR) into a strong local signal (RP and its corresponding SP). As already shown in a previous work [36] the amplification provided by the pre-effector stage, as quantified by the ratio E*r/R*r, is relatively small at any uptake (always less than 7. 5). Consequently most of the amplification is provided by the post-effector stage, as quantified by SPr/E*r (Table 6). The main amplification is in the electrical stages (Icat and ICl), since the amplifications of all chemical stages are relatively small. Cationic and Cl− channels amplify the signal in different ways and they dominate the depolarization in response to different uptakes. Cationic channels amplify the signal at low uptakes with a short rising time. Cl− channels amplify the signal by a larger maximal conductance and a longer duration of depolarization. The high amplification factor at low uptakes involves several mechanisms acting inversely on the activation and inactivation processes. Moreover these mechanisms are not the same at different uptakes. At extremely low stimulation uptakes, inactivation processes have weaker effects. This can be partly explained by the value of Hill coefficients which is ≤1 for nCa, ncat, nx, and nicat but >1 for nis and niCa. This means that the inhibition of the production of second messenger (IP3 and DAG) and of the IP3-gated conductance develops on a relatively narrow range of concentration of their modulator, i. e. changes relatively abruptly in a threshold-like manner. On the contrary, the activation of the IP3- and DAG-dependent conductances and Ca2+ exchangers develops on a wider range of concentrations, i. e. more smoothly. The field of insect olfactory transduction has generated an ever increasing amount of data but dispersed and fragmentary. The complication and sometimes confusion that result from these circumstances justify an attempt to unite the parts in a comprehensive view and formal synthesis of how things might work. However, because it is clearly incomplete and involves several assumptions, the model presented here should not be considered uncritically as a faithful description of reality. Like any model, it is intended primarily as a starting point for addressing new questions, designing new experiments, and offering a tentative framework for their interpretation. This model calls for three kinds of experiments. First, experiments on specific components. For example, are the contributions of the ionotropic and metabotropic pathways to olfactory transduction in agreement with our tentative proposal? Is the membrane repolarized by a K+ channel depending on an IP3-gated Ca2+ channel, both located in the inner dendrite as suggested here? What is the exact mechanism of Ca2+ extrusion? Second, can the model be extended to account for adapted or periodically stimulated neurons? Can it be induced to oscillate like in mammals [41], [42], [102]. Finally, can qualitative differences in all these mechanisms be found between different ORN types in the same and different biological species? All these experiments have the potential to confirm or invalidate the views presented here and so to yield significant progress in our integrated view of the ORN functioning.
  The unknown parameters of the model were estimated by utilizing various search methods based on the following criteria. First, we imposed that each parameter be in a physiologically acceptable range of values compatible with the properties of our qualitative model of transduction: the order of magnitude of Hill coefficients is one; the IP3-gated channel is permeable to Ca2+ only while the DAG-gated channel is permeable to Ca2+ and other cations, so that fCa>fcat; the reversal potential of the Cl− channel ECl must be more positive than −97 mV to be depolarizing. Second, we considered a parameter set as acceptable if the predicted kinetics of the sensillar potential were close to the experimentally measured kinetics at all uptakes. For checking this condition, we minimized a cost function based on the three response characteristics, height (Hi), rising time (τrise, i) and falling time (τfall, i) at a series of uptakes i for which these characteristics were determined experimentally. The differences, ΔHi = |Hi−Ĥi|, between the values Ĥi predicted by the model for a given set θ of parameter values and the experimental values Hi, were determined at every uptake i. The differences Δτrise, i and Δτfall, i were determined in the same way. Because the three characteristics vary on different scales, the differences were weighted and summed to produce a single cost function (58) where n = 26 is the number of uptakes. Third, a solution was finally accepted only if it was in qualitative accordance with other available experimental facts: the transient feature of the kinetics of the second messengers and IP3-gated currents, the sustained property of the Cl− currents and K+ currents, and the condition that intracellular Ca2+ concentration must not exceed 200 µM. Two search methods were utilized in sequence to find the parameter values. First, for a global exploration of the parameter space, we relied on a trial-and-error method. We compared a few thousands parameter sets, drawn from sets θ obeying the first criterion above, at 3 uptakes (low, medium, high). Most sets led to unacceptable cost functions E (θ) and were rejected. The best sets were further selected on the third criterion then tested at more uptakes. Eight presumptive solutions tested at all uptakes with E (θ) in the range 2. 74–5. 38 were found fulfilling the three criteria. Second, the best presumptive solution for which E (θ) = 2. 74 was locally optimized utilizing the Matlab unconstrained minimizer fminsearch based on the Nelder-Mead simplex (direct search) method. The algorithm converged on the set of estimated parameters θ0 shown in Tables 4 and 5. With these parameter values E (θ0) = 2. 61. The sensitivity of a model response M to a single parameter bi can be expressed as a sensitivity function. This partial derivative was estimated as the central finite difference (59) using both the forward and backward differences (59) This equation is only valid for an infinitesimal variation (perturbation) of bi (). Practically, Δbi was implemented as the product Δbi = ξbi, where bi is the nominal parameter value as estimated above and ξ is a perturbation factor. We took ξ = 0. 01, large enough to avoid numerical inaccuracies and small enough to prevent the nonlinearity of the model to play a role in the sensitivity calculations. In our case, the model responses Mu are the height, half-rise time and half-fall time at different uptakes log10U (denoted here as subscript u), which have different units and take values of different orders of magnitudes. In order to compare their sensitivities Su (bi) we normalized them by the model response Mu (60) For each fitted parameter, the normalized sensitivities Sru of the 3 characteristics were calculated at 26 values of U from threshold to saturation. The characteristic and the uptake giving the largest absolute value Sr (bi) of the Sru (bi) were recorded (see Tables 4 and 5). It is conceivable that in the optimal parameter set θ0, the low sensitivity of specific parameters is a result of the local optimization procedure. We checked that this was not the case for each low-sensitivity parameter k by calculating the cost function E (θ′) where θ′ = θ0, except for k whose value was taken 10% smaller (and 10% larger) than its optimal value. We verified that in all cases E (θ′) ≈E (θ0).
summary: All sensory neurons transduce their natural stimulus, whether a molecule, a photon, or a mechanical force, in an electrical current flowing through their sensory membrane via similar molecular and ionic mechanisms. Olfactory receptor neurons (ORNs), whose stimuli are volatile molecules, are no exception, including one of the best known: the exquisitely sensitive ORNs of male moths that detect the sexual pheromone released by conspecific females. We provide a detailed computational model of the intracellular molecular mechanisms at work in this ORN type. We describe qualitatively and quantitatively how the initial event, the interaction of pheromone molecules with specialized receptors at the ORN surface, is amplified through a sequence of linked biochemical and electrical events into a whole cell response, the receptor potential. We detail the respective roles of the upward activating reactions involving a cascade of ionic channels permeable to cations, chloride and potassium, their control by feedback inactivating mechanisms, and the central regulatory role of calcium. This computational model contributes to an integrated understanding of this signalling pathway, provides testable hypotheses, and suggests new experimental approaches.
keywords: neuroscience/theoretical neuroscience
  neuroscience/sensory systems
  biophysics/theory and simulation
  computational biology/computational neuroscience
  cell biology/neuronal signaling mechanisms
  neuroscience/neuronal signaling mechanisms
  biophysics/membrane proteins and energy transduction
  computational biology/signaling networks
  biophysics/cell signaling and trafficking structures
  computational biology/systems biology
title: Computational Model of the Insect Pheromone Transduction Cascade