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* Institute of Applied Mathematics, University of Heidelberg, and WIN-Research Group of Olfactory Dynamics, Heidelberg Academy of Science and Humanities, Heidelberg, Germany; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany; and Fritz-Haber-Institute of the Max Planck Society, Berlin, Germany
Correspondence: Address reprint requests to Jens Starke, Tel.: 49-6221-548980; E-mail: starke{at}iwr.uni-heidelberg.de.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONSIGNAL TRANSDUCTION IN OLFACTORY...SOURCE OF INSTABILITY AND...KINETIC MODEL AND NUMERICAL...DISCUSSIONAPPENDIX: ALTERNATIVE PARAMETER...ACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONSIGNAL TRANSDUCTION IN OLFACTORY...SOURCE OF INSTABILITY AND...KINETIC MODEL AND NUMERICAL...DISCUSSIONAPPENDIX: ALTERNATIVE PARAMETER...ACKNOWLEDGEMENTSREFERENCES |
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). They typically arise from calcium-induced calcium release (CICR) mechanisms, in which an increase in cytoplasmic calcium level leads to increased Ca2+ influx from intracellular Ca2+ stores (2). Such positive feedback is implemented, e.g., through the regulation by Ca2+ of the Ca2+-permeable IP3 and ryanodine receptor channels of the endoplasmic reticulum (1).
Calcium oscillations arising from negative feedback have received much less attention. A notable exception is the case of coupled calcium and cyclic AMP oscillations. A mathematical analysis of conditions under which cAMP-induced Ca2+ influx coupled to Ca2+-modulated cAMP production leads to oscillations was given in Rapp and Berridge (3). A specific mechanism of this type was proposed in Cooper et al. (4), in which Ca2+ enters the cell through cyclic-nucleotide-gated (CNG) channels and Ca2+ is assumed to cooperatively suppress the activity of adenylyl cyclase. Oscillations in cAMP that arise from coupling to Ca2+ oscillations were recently reported in embryonic spinal neurons (5); in this case, however, the Ca2+ oscillations are due to a CICR mechanism and do not require varying cAMP. To the best of our knowledge, strictly mutually dependent oscillations of Ca2+ and cAMP, in which they are both essential (6,7), have not been conclusively observed.
The coupling of cyclic nucleotide dynamics to Ca2+ influx through CNG channels forms the basis of sensory signal transduction in vertebrate olfactory sensory neurons (8). Pronounced oscillations of the Ca2+ level in the cilia of olfactory sensory neurons were recently reported by Reisert and Matthews (9). In addition, oscillations in receptor current presumed to be indicative of Ca2+ oscillations were observed in Frings and Lindemann (10), Reisert and Matthews (11,12), and Suzuki et al. (13). It was argued in Reisert and Matthews (11) and Suzuki et al. (13) that the observed oscillations are probably due to the mechanism proposed in Cooper et al. (4). The corresponding cAMP oscillations have not, however, so far been confirmed experimentally. In addition, Ca2+ regulation of adenylyl cyclase in olfactory sensory neurons occurs on a timescale of minutes (14), whereas the period of Ca2+ oscillations is of the order of seconds.
In this article, we propose a mechanism for Ca2+ oscillations in olfactory sensory neurons that depends neither on cAMP variations nor on calcium-induced calcium release. The mechanism is based on direct negative regulation of CNG channels by calcium/calmodulin. This form of negative feedback is well established in olfactory sensory neurons (15,16), is known to play the key role in fast olfactory adaptation (17,18), and has been analyzed in detail biochemically (16,19,20). Using parameter values inferred from Bradley et al. (19), Munger et al. (20), Schild and Restrepo (21), and Reisert et al. (22), we construct a mathematical model of the coupled kinetics of calcium, calmodulin, and CNG channels. We then show that our model predicts persistent oscillations of Ca2+ for rate constants and concentrations in the physiological range and correctly captures fast adaptation (
1 s).
This article is organized as follows: in "Signal Transduction in Olfactory Cilia", we review some relevant facts about olfactory response. "Source of Instability and Hopf Bifurcation" suggests a biochemical mechanism and presents an analytical proof of the existence of oscillations using the formalism of stoichiometric network analysis. In the following section, we develop and numerically analyze a detailed kinetic model. We show that the model is in good agreement with available data and make predictions for the ranges of odorant concentration, calmodulin concentration, and system size in which Ca2+ oscillations should be observable. The relation of our model to other mechanisms of biochemical oscillations, as well as the limits of the validity of our model, are detailed in "Discussion".
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SIGNAL TRANSDUCTION IN OLFACTORY CILIA |
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)), where the concentration of odorant molecules outside of the cilium is transduced into a change of the level of the second messenger cAMP inside the cilium, leading to the opening of CNG-gated ion channels and the influx of Ca2+ (see Fig. 1 and its caption). Subsequently, the opening of Ca2+-gated chloride channels results in depolarization of the cilium that is sufficient to trigger the generation of action potentials in the soma of the sensory neuron.
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)). Several Ca2+-dependent negative feedback mechanisms have been identified: increase in the closing rate of the CNG channel; increase in PDE activity; and activation of CaMKII leading to decrease in adenylyl cyclase activity (see Fig. 1). These feedback mechanisms are expected to act simultaneously following the exposure to a strong odorant signal, and their relative importance is currently under debate. In the following we concentrate on the CaM4-mediated closing of channels (direct feedback), before addressing its possible interaction with other feedback loops in the discussion.
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SOURCE OF INSTABILITY AND HOPF BIFURCATION |
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) to check analytically whether the system of Fig. 1 can give rise to oscillations (for details of the procedure, see (31,32); for an accessible introduction to SNA, see (25,26); for a computational package, see (27,28); for recent developments in SNA, see (29,30)). The decisive advantage of the SNA formalism is that the complete set of steady states can be explicitly written down and their stability analyzed, therefore revealing the presence (or absence) of local bifurcations anywhere in the parameter space. In SNA the rate constants of the reactions and the concentrations at stationary points do not appear explicitly, i.e., can be left unspecified. The procedure of SNA allows for a substantial simplification of the bifurcation analysis and hence for fast checks of the qualitative dynamical behavior inherent in the reaction schemes. Nevertheless, it should be noted that the following SNA gives only existence results and for quantitative results, further knowledge of the rate constants is necessary.
In SNA, a reaction mechanism of n species and r reactions is completely specified by two n x r matrices: the stoichiometric matrix 
, which contains the (positive or negative) stoichiometric change of each species in each reaction; and the kinetic matrix 
, which contains the respective kinetic exponents, assuming that the kinetics of the jth reaction is given (or can be approximated) by power laws
where xi denotes the concentration of the ith species. For ease of visualization in network diagrams, is scaled so that it consists of small integers. All the rate constants kj are assumed to be able to take on any nonnegative value (complete parameter set (33)).
The elements of and can be written down explicitly or encoded in a network diagram (see Fig. 2). Thus, Fig. 2 A represents the part of Fig. 1 that contains the direct feedback (shown there with thick arrows) and is equivalent to the reaction scheme analyzed numerically in "Kinetic Model and Numerical Analysis" (i.e., encodes the stoichiometry and assumptions about kinetic exponents). Inspection of Fig. 2 A reveals that there are no autocatalytic loops (24,25,31,34) that could give rise to an instability; instead, the negative three-loop of CNGo—causing the influx of Ca2+ producing CaM4, which in turn blocks CNGo—is the only possible source of instability (35). We therefore examine a radically reduced network where only these three species and their six reactions are retained (Fig. 2 B). The reduced network no longer contains CNGi and CNGc. Denoting the concentration of open channels by X, of Ca2+ by Y, and of CaM4 by Z, the kinetic equations are
 | (1) |
). The term –k6Y
with effective exponent corresponds to extrusion of Ca2+ from the cilium by pumps and exchangers. In matrix form, Eq. 1 can be expressed as
 | (2) |
is the vector of concentrations of the three species,
 | (3) |
is the vector of velocities of the six reactions. The reaction vector can be written as
 | (4) |
 | (5) |
and . For the model specified by Eqs. 3 and 5 we first give the representation of its set of stationary states in SNA and then carry out its bifurcation analysis.
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with nonnegative coefficients ji of a certain number of undecomposable subnetworks 
(also referred to as extreme currents, basic feasible solutions, stationary flows, stoichiometric generators) (33). For the network of Fig. 2 B (with six reactions and three species), there are three such irreducible subnetworks
(
), which are shown in Fig. 2 C. These are the smallest subnetworks that still fulfill the stationary state condition that the production rates equal the degradation rates for all species of this subnetwork. In the language of the network diagrams, this means that for all vertices (species) the inflows are equal to the outflows, i.e., the number of (incoming) barbs and (outgoing) feathers are equal for every species.
For the first subnetwork
the Ca2+ inflow 
2 = k2X through open CNG channels is balanced by the outflow through the pumps 6 = k6Y, and all other reaction velocities are set to zero. This gives a stationary state with velocity vector 
where j1 
2 = 6 and 
Likewise, in the second subnetwork 
when 
with 
the production and degradation rates compensate each other for both species, Ca2+ and CaM4. The remaining reactions r1 and r5 generate the last subnetwork 
From linear algebra it follows that the complete set of subnetworks forms a basis of the right null space of the stoichiometric matrix , i.e., every velocity vector 
fulfilling the condition of stationarity 
is a linear combination of the extreme currents: 
The coefficients jk are nonnegative (i.e., forms a convex cone) because only the intersection with the nonnegative orthant are physically meaningful stationary states.
The Jacobian of the dynamical system (Eqs. 2 and
4) can be calculated as
where
\mathrm|<|diag|>|(
)
is defined as the diagonal matrix with the components of
as diagonal entries and 
is the vector of stationary concentrations. Since the multiplication with the positive stationary concentrations 
does not change the sign pattern of the Jacobian (24), it is often (as in our case) sufficient to analyze the stability properties of the reduced Jacobian: 
In this manner, the bifurcation analysis of the investigated model with six undetermined rate constants is reduced to depend on the three flux coefficients j1, j2, and j3. As scaling these parameters by a common positive factor does not change stability, we can choose j2 = 1, and are left with only two free parameters.
Occurrence of Hopf bifurcation
A necessary and sufficient condition for the occurrence of a Hopf bifurcation (based on a modified Routh scheme; see Clarke (25)) has been given in Eiswirth et al. (32). Essentially it arranges the sums ai over all principal minors 
i of a given order i of the Jacobian J or 
(i.e., the coefficients ai of the respective characteristic polynomial) in two rows and computes the elements of the further rows in a way analogous to 2 x 2 determinants.
For a system with three species, the characteristic polynomial is 
and the scheme becomes as shown in Table 1; see below.
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would have to change the sign.
Using this procedure, we now examine the stability of the steady state with velocity vector 
treating the as-yet-unspecified exponent (see Fig. 2) as an additional parameter. Evaluation of the sums over the minors results in a1 = 9 + j1 + j3; a2 = 9j3 + j1 + j1j3; and a3 = 2j1j3 + j1j3.
Since a3 is always positive, the only possibility of local loss of stability is via a Hopf bifurcation. In addition, no saddle-node bifurcation can occur since a3 cannot change sign; actually, this conclusion already follows from the absence of autocatalytic loops. To linear order in , the system becomes oscillatory for 
This condition can readily be fulfilled for 
i.e., when 
carries considerably higher weight than the other subnetworks (meaning the rates of the reactions in must be large), and the kinetic exponent is sufficiently small. The inequality 
is always satisfied: in terms of the rate constants of Table 2 we get 
Ca2+ is pumped out of the cell; near pump saturation, such a process is of very low order in [Ca2+]. Actually, no Hopf bifurcation could be numerically found for above 
0.05 (the exact value depending on the other parameters). Therefore, Ca2+ has to reach concentrations at or near pump saturation so that oscillations can occur. In contrast, the kinetic order of Ca2+ in the formation of CaM4 was uncritical, i.e., the existence of oscillations was robust using kinetic exponents of 1 or even 4.
Such analytical considerations not only showed what bifurcations to look for, but also proved very helpful in determining the range of parameters for subsequent numerical studies.
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KINETIC MODEL AND NUMERICAL ANALYSIS |
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in Eq. 6 increases with cAMP, which in turn depends on odorant concentration); Ca2+ binding to calmodulin in Eq. 7; irreversible binding of Ca2+/calmodulin (CaM4) to open channels in Eq. 8; and reversible binding of CaM4 to closed channels in Eq. 9,
 | (6) |
 | (7) |
 | (8) |
 | (9) |
The system contains two conservation constraints, so that only four of the six species are independent:
 | (10) |
 | (11) |
Square brackets with a subscript s denote surface, others volume concentrations. When volume species are produced by surface species or vice versa, the concentrations have to be converted at the respective location by division or multiplication by 
, which represents the thickness of the cytoplasmic layer in cilia. More generally we define as the ratio of cytoplasmic volume to membrane area; for a cylinder of diameter d, we get = d/4. Taking into account Eqs. 10 and 11, the reaction Eqs. 6–9 lead to four coupled ordinary differential equations for the concentrations of open channels, internal calcium, calcium-loaded calmodulin, and channels that bind calmodulin:
 | (12) |
 | (13) |
 | (14) |
 | (15) |
), therefore it is currently not possible to construct a mechanistically justified expression for Ca extrusion.) The coupled differential equations were solved numerically with a stiff solver using the parameter values given in Table 2. An alternative parameter set that leads to similar numerical results is presented in Appendix A.
We first verified that our model correctly captures the adaptation of olfactory response as seen in the experiments of Leinders-Zufall et al. (37). The odorant pulse is represented in the simulation by increasing from an assumed resting value of 1.6 x 10–5
to 5.5 for a period of 100 ms. The first pulse is presented at 0 s, the second after a time interval of 2 s, 4 s, 6 s, 8 s, 10 s, or 12 s, respectively. Shown in Fig. 3 A are the time courses of Ca2+ concentrations superimposed for these six different runs. For a better understanding of the underlying mechanism, the simulated time evolution of the concentrations of all species are shown in Fig. 3 B. The stimulus presentation is followed by the opening of CNG channels and rapid accumulation of Ca2+. The 100-µM concentration of Ca2+ reached in our simulation is consistent with levels up to 500 µM predicted by a detailed biophysical model of the steady state of a fully activated cilium in Lindemann (38). Within 400 ms, CaM4 (dotted curve in Fig. 3 B) reaches levels sufficient to fully inhibit the channels (thin solid curve) and therefore to reduce the open channel fraction (thick solid curve) to zero level. The saturation of the Ca2+ pump rate is reflected in the subsequent slow (almost linear) decay of Ca2+ concentration, completed within 1 s of the stimulus. The slowest process is the disinhibition of CNG channels (10). Most channels are still inhibited (thin solid curve) when the second pulse is presented at t = 4 s, resulting in a reduced response.
The time for recovery from adaptation is consistent with the experimental data of Fig. 8 A in Leinders-Zufall et al. (37), which is reprinted in Fig. 3 A. Our model also reproduces the results of double pulse experiments in Fig. 2, D–F, of Munger et al. (20). The shift in the onset of the Ca2+ peaks in the experiment compared to the simulation (Fig. 3) corresponds to the time delay between the application of odorant and the increase in cAMP concentration. The corresponding part of the signaling pathway is not included in our model.
We now model the experiments of Reisert and Matthews (9) in which the odorant was presented for a prolonged period and the intracellular Ca2+ concentration was measured directly. We keep all parameter values (except ) the same as above for the double-pulse experiments. To choose the appropriate value of we assume a linear relation between the concentrations of odorant and cAMP, and a quadratic relation between that of cAMP and (39).
As the concentration of odorant (cineole) used in this experiment was 1/10 of the concentration in the double-pulse experiment of Leinders-Zufall et al. (37), we choose 
Our model then produces calcium oscillations with a period of 2.5 s (Fig. 4 A), in good agreement with the data in Fig. 5 A of Reisert and Matthews (9), which is reprinted in Fig. 4 A. During these oscillations, the concentrations of open channels, internal Ca2+, CaM4, and inhibited channels reach their maximal values (Fig. 4 B) in the same order as in the double-pulse experiments. Throughout the oscillations, however, inhibition is never complete, and the open fraction remains above 0.02. The horizontal dashed line in Fig. 4 B indicates the open fraction kCa/(iCa[CNGtot]) = 0.039 at which the pumps are no longer able to compensate the influx through channels, leading to rapid accumulation of Ca2+. It is notable that the oscillations in the inhibited fraction (thin solid curve in Fig. 4 B) and in Ca2+ concentration (dashed curve) run almost in antiphase (maximum in inhibition coincides with minimum in Ca2+ concentration). Consequently, the oscillations are not damped out. Oscillations with increased period are obtained when the rate of Ca2+ extrusion kCa is reduced from the value given in Table 2. This is in qualitative agreement with the observation in Reisert and Matthews (9) that the period grew when the concentration of external Na+ and subsequently the pump rate of the Na/Ca exchanger was reduced.
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). Fig. 5 shows the range of values of 
, and [CaMtot], respectively, in which the oscillations exist (other parameter values remain as in Table 2). For = 0.05 µm, the range of stability spans three decades in which (under the assumptions given above), corresponds to a factor of 30 in odorant concentration. This is in agreement with the experiments of Reisert and Matthews (12), in which oscillations were observed for cineole concentrations of 30 µM and 100 µM, but not 10 µM or 300 µM. Fig. 5 B shows that our results are robust over a wide range of calmodulin concentration. In the shaded area, Ca2+ increases without bound. This is a consequence of the minimal assumptions of our model; in reality, the Ca2+ current through an open channel iCa will decay as the internal Ca2+ concentration increases. (To adequately describe the dependence of iCa on [Ca2+], it would be necessary to include the transmembrane voltage V as a dynamical variable of our model. This in turn would require the inclusion of the dynamics of Ca2+-gated chloride channels, making the model significantly more complex and increasing the number of unknown parameters.)
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DISCUSSION |
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,35,41–43), but are rarely found in reality. Circadian rhythms (44,45) and the synthetic system of Elowitz and Liebler (46) provide examples of oscillations of this type in genetic networks. To our knowledge, no chemical oscillator of this type without enzymes has been identified yet. Typically, nonautocatalytic oscillators differ from autocatalytic ones (i.e., which contain positive feedback) in that they do not give rise to bistability nor excitability.
It is well established that three Ca2+/CaM-mediated feedback loops act in olfactory cilia (see Fig. 1). All three loops can, in principle, lead to adaptation and to oscillations by themselves. In addition to modulating the CNG channel directly, CaM4 accelerates the degradation of cAMP through an enhancement of PDE activity, and reduces the production of cAMP through CaMKII by inhibition of adenylyl cyclase activity. The latter process is known to be quite slow (14) compared to the direct pathway, believed to be responsible for fast adaptation (23). The oscillations (with a period of the order of 1 s) seen in experiment are too fast to be attributable to the cyclase pathway. The timescale on which the PDE pathway operates is expected to be slower than the direct way, since its last step to close the loop involves the detachment of cAMP from the channels and its subsequent degradation. In addition, according to Kurahashi and Menini (17), the effect of PDE on fast adaptation is weak. We conclude that the direct CaM4 feedback is the primary cause for both fast adaptation and oscillation. In principle the interaction of, e.g., the PDE and the direct pathway may give rise to complex oscillations. Also, interaction with CICR is conceivable. Extension of the model to include these effects may become of interest in the future.
The described oscillatory mechanism is expected to manifest itself also in systems other than olfactory sensory neurons. Besides the presence of calmodulin, it requires calcium channels that are negatively regulated by CaM4, Ca2+ pumps or exchangers near saturation (which does not exclude that the dominant extrusion mechanisms at Ca2+ concentrations outside the oscillatory range are higher-order), as well as confinement to small volumes so that high concentrations of internal Ca2+ can be achieved and maintained.
Besides the CNG channels in olfactory cilia, other Ca2+-permeable channels that are strongly inhibited by CaM4 include CNG channels in photoreceptor rods and cones (47), NR1-type NMDA receptors (48), as well as L-type and P/Q-type voltage-dependent channels (49). Our assumption of a well-mixed system will remain reasonably well fulfilled for volume/area ratios up to 1 µm (e.g., for cilia and dendrites). For the parameter values used in our model, oscillations persist up to beyond this value (up to 5 µm in Fig. 5). Thus, any system sufficiently small to fulfill the homogeneity condition will lie well within the oscillatory range. Even for larger systems (e.g., a typical cell), at least transient oscillations may still be observable as the buildup and decay of the concentration gradient near the membrane may drive the system through the oscillatory regime. Including diffusion to describe time-dependent concentration gradients will be one of future extensions of our model.
In cellular compartments with internal Ca2+ stores, oscillations can arise based on the autocatalytic calcium-induced calcium release. A distinguishing feature of the nonautocatalytic mechanism (i.e., a mechanism without positive feedback) is that in contrast to CICR, it does not give rise to excitability and does not support corresponding Ca2+ waves. Excitability (i.e., a system makes a large excursion when subjected to a stimulus above a certain threshold before returning to its initial state) requires an autocatalysis to support the self-enhancement of the deviation from steady state. This mechanistic requirement is not fulfilled in a system based solely on negative feedback. Typically, excitable systems exhibit a refractory period (i.e., a certain time has to elapse before the next large excursion can be triggered). The present model shows signs of refractoriness (adaptation) without having the other characteristics of excitability.
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APPENDIX: ALTERNATIVE PARAMETER SET |
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,50–52) vary by up to two orders of magnitude. The numerical results of "Kinetic Model and Numerical Analysis" were obtained using 
and 
in the effective reaction Eq. 7. With this choice, the calcium-calmodulin binding curve and kinetics resemble those of the model of Bhalla (51) and Sivakumaran et al. (52). Cohen and Klee (36) and Delville et al. (50), however, imply significantly higher values of 
We therefore examined our model also for 
increased to 
To obtain adaptation and oscillation dynamics consistent with the experimental data in Figs. 3 and 4, it was then necessary to adjust the values of several other parameters as follows: 
; 
; 
; 
; 
; 
; 
(oscillations); and 
(adaptation). Other parameters remain as in Table 2. With the new parameter set, the time courses of [Ca2+] predicted by the model are very similar to the model curves in Figs. 3 and 4; however, the maximum [Ca2+] values are now 8 µM in the adaptation case, and 1 µM in the oscillations case. In addition, the oscillatory regions in the state diagrams are now significantly smaller than those with the original parameter set (Fig. 5, A and B). Both parameter sets are consistent with known physiological ranges of rate constants (see Table 2). However, we consider the Ca2+ concentrations predicted with the second parameter set to be too low in the case of oscillations, as 1 µM of [Ca2+] is not sufficient to fully activate chloride channels (21). |
ACKNOWLEDGEMENTS |
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We thank F. Zufall and H.R. Matthews for giving permission to reprint their experimental data, and R. Friedrich, T. Kuner, A. Schäfer, and H. Spors for discussions on olfaction.
J.R. and J.S. thank the Heidelberg Academy of Science and Humanities and the State Baden-Württemberg for financial support.
Submitted on January 18, 2005; accepted for publication October 6, 2005.
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Black indicates the occurrence of oscillations, white a stable fixed point. In the shaded area in panel B, no fixed point was reached; instead, there was unlimited increase of [Ca2+] (see the main text).